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abstract: 'Among the unitary reflection groups, the one on the title is singled out by its importance in, for example, coding theory and number theory. In this paper we start with describing the irreducible representations of this group and then examine the semi-simple structure of the centralizer algebra in the tensor representation.'
author:
- Masashi Kosuda and Manabu Oura
title: Centralizer algebras of the primitive unitary reflection group of order $96$
---
Introduction
============
The group, which we denote by $H_1$, on the title of this paper consists of $96$ matrices of size $2$ by $2$. It is the unitary group generated by reflections (u.g.g.r.), numbered as No.$8$ in Shephard-Todd [@ST]. This group, as well as No.$9$ in the same list, has long been recognized. The purpose of the present paper is to give a contribution to $H_1$ by decomposing the centralizer algebra of $H_1$ in the tensor representation into irreducible components.
We shall give an outline of the first statement in Abstract. The group $H_1$ naturally acts on the polynomial ring ${\mathbb C}[x,y]$ of $2$ variables over the complex number field ${\mathbb C}$, i.e. $$Af(x,y)=f(ax+by,cx+dy), \ \
A=\begin{pmatrix}a & b \\ c & d \end{pmatrix}\in H_1$$ for $f\in {\mathbb C}[x,y]$. We consider the invariant ring $${\mathbb C}[x,y]^{H_1}
=\{f\in {\mathbb C}[x,y]:\
\ Af=f, \ \ \forall A\in H_1\}$$ of $H_1$. This ring has a rather simple structure. It is generated by two algebraically independent homogeneous polynomials of degrees $8$ and $12$, and conversely this nature characterizes the u.g.g.r. Broué-Enguehard [@BE] found a map connecting this invariant ring with number theory. Take a homogeneous polynomial $f(x,y)$ of degree $n$ from the invariant ring. Introducing theta constants $$\theta_{ab}(\tau)
=\sum_{ m\in {{\mathbb Z}}}
\exp 2\pi i
\left[
\frac{1}{2} \tau
\left( m+\frac{a}{2} \right)^2
+\left( m+\frac{a}{2} \right)
\frac{b}{2}
\right],$$ we get a modular form $f(\theta_{00}(2\tau),\theta_{10}(2\tau))$ of weight $n/2$ for $SL(2,{{\mathbb Z}})$. Moreover this map is an isomorphism from the invariant ring of $H_1$ onto the ring of modular forms for $SL(2,{{\mathbb Z}})$.
Next we proceed to coding theory. Let ${{\mathbb F}}_2=\{0,1\}$ be the field of two elements and ${{\mathbb F}}_2^n$ the vector space of dimension $n$ over ${{\mathbb F}}_2$ equipped with the usual inner product $(u,v)=u_1v_1+\cdots + u_nv_n$. The weight of a vector $u$ is the number of non-zero coordinates of $u$. A code of length $n$ is by definition a linear subspace of ${{\mathbb F}}_2^n$. We impose two conditions on codes. The first one is the self-duality which says that a code $C$ coincides with its dual code $C^{\perp}$, that is, $C=C^{\perp}$ in which $$C^{\perp}=\{u \in {{\mathbb F}}_2^n:\
(u,v)=0,\ \ \forall v\in C\}.$$ The second one is the doubly-evenness which means $$wt(u)\equiv 0\pmod{4},\ \ \forall u\in C.$$ These two notions give rise to the relation with invariant theory via the weight enumerator $$W_C(x,y)=\sum_{v\in C}
x^{n-wt(v)}y^{wt(v)}$$ of a code $C$. In fact, if $C$ is self-dual, we have $$W_C((x-y)/\sqrt{2},(x+y)/\sqrt{2})=W_C(x,y)$$ and if $C$ is doubly even, we have $$W_C(x,iy)=W_C(x,y).$$ We mention that a self-dual and doubly even code of length $n$ exists if and only if $n$ is a multiple of $8$.
Now we can state the connections among all what we have mentioned. Take a positive integer $n\equiv 0\pmod{8}$. The weight enumerator of a self-dual doubly even code of length $n$ is an invariant of $H_1$ and $$W_C(\theta_{00}(2\tau),\theta_{10}(2\tau))$$ is a modular form of weight $n/2$ for $SL(2,{{\mathbb Z}})$. Gleason [@Gl] showed that the invariants of degree $n$ can be spanned by the weight enumerators of self-dual doubly even codes of length $n$. Finally any modular form of weight $n/2$ can be obtained from the weight enumerator of self-dual doubly even codes of length $k$. The whole theory with more general results could be found in [@Ru1], [@Ru2] from which our notation $H_1$ comes.
Besides the importance of $H_1$, the motivation of this paper could be found in Brauer [@Br], Weyl [@We]. One of the main ingredients there is the [*commutator algebra*]{} where invariant theory comes into play. We follow Weyl. Given any group of linear transformations in an $n$-dimensional space. Take covariant vectors $y^{(1)},\ldots,y^{(f)}$ and contravariant vectors $\xi^{(1)},\ldots,\xi^{(f)}$. A linear transformation acts on covariant vectors [*cogrediently*]{} and on contravariant vectors [*contragradiently*]{}. Then the matrices $\| b(i_1\cdots i_f;\ k_1\cdots k_f)\|$ in the tensor space obtained from the invariants $$\sum_{i;k}
b(i_1\cdots i_f;\ k_1\cdots k_f)\xi_{i_1}^{(1)}
\cdots \xi_{i_f}^{(f)}y_{k_1}^{(1)}\cdots y_{k_f}^{(f)}$$ form the commutator algebra of $H_1$ in the tensor representation. The problem here is to decompose this algebra into simple parts. It is quite natural to apply this philosophy to our group $H_1$ as we will in this paper ([*cf*]{}. [@Ba]).
Irreducible representations of $H_1$
====================================
In this section we determine the irreducible representations of $H_1$ which yields the character table. At the end of this section we discuss invariant theory of $H_1$ under the irreducible representations.
The unitary reflection group $H_1$ is a finite group in $U_2$ generated by the following matrices $T$ and $D$: $$T = \frac{1+i}{2}
\begin{pmatrix}
1 & 1\\
1 & -1
\end{pmatrix}
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon & \epsilon\\
\epsilon & \epsilon^5
\end{pmatrix}, \quad
D =
\begin{pmatrix}
1 & 0\\
0 & i
\end{pmatrix}
.$$ Here $\epsilon = \exp(2\pi i/8)$. It is known that the group size of $H_1$ is 96 and it has 16 conjugacy classes $\mathfrak{C}_{1}, \ldots, \mathfrak{C}_{16}$. Each conjugacy class has the following representative: $$\begin{aligned}
&\mathfrak{C}_{1} \ni 1
=\begin{pmatrix}1&0\\0&1\end{pmatrix},\ \mathfrak{C}_{2} \ni T
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon & \epsilon\\
\epsilon & \epsilon^5
\end{pmatrix},\ \mathfrak{C}_{3} \ni T^2
=\begin{pmatrix}i&0\\0&i\end{pmatrix},
&\\
&\mathfrak{C}_{4} \ni T^3
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon^3 & \epsilon^3\\
\epsilon^3 & \epsilon^7
\end{pmatrix},\ \mathfrak{C}_{5} \ni T^4
=\begin{pmatrix}-1&0\\0&-1\end{pmatrix},\ \mathfrak{C}_{6} \ni T^6
=\begin{pmatrix}-i&0\\0&-i\end{pmatrix},
&\\
&\mathfrak{C}_{7} \ni D
=\begin{pmatrix}1&0\\0&i\end{pmatrix},\ \mathfrak{C}_{8} \ni DT
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon & \epsilon\\
\epsilon^3 & \epsilon^7
\end{pmatrix},\ \mathfrak{C}_{9} \ni DT^2
=\begin{pmatrix}i&0\\0&-1\end{pmatrix},
&\\
&\mathfrak{C}_{10} \ni DT^3
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon^3 & \epsilon^3\\
\epsilon^5 & \epsilon
\end{pmatrix},\ \mathfrak{C}_{11} \ni DT^4
=\begin{pmatrix}-1&0\\0&-i\end{pmatrix},
&\\
&\mathfrak{C}_{12} \ni DT^5
=\frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon^5 & \epsilon^5\\
\epsilon^7 & \epsilon^3
\end{pmatrix},\ \mathfrak{C}_{13} \ni DT^6
=\begin{pmatrix}-i&0\\0&1\end{pmatrix},
&\\
&\mathfrak{C}_{14} \ni DT^7
= \frac{1}{\sqrt{2}}
\begin{pmatrix}
\epsilon^7 & \epsilon^7\\
\epsilon & \epsilon^5
\end{pmatrix},\ \mathfrak{C}_{15} \ni D^2
=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\
\mathfrak{C}_{16} \ni D^2T^2
=\begin{pmatrix}i&0\\0&-i\end{pmatrix}.
&\end{aligned}$$
Since the number of conjugacy classes and that of the non-isomorphic irreducible representations coincide, there exist 16 classes of the irreducible representations of $H_1$. In the following, we construct all of them one by one.
First we note that any group has the trivial representation which maps each element of the group to 1. We denote that of $H_1$ by $(\rho_1, V_1)$. The determinant which maps $T$ and $D$ to $-i$ and $i$ respectively also gives a one-dimensional irreducible representation. We call it $(\rho_3, V_3)$. The tensor product $\rho_3^{\otimes 2}$ also gives a one-dimensional irreducible representation, which maps both $T$ and $D$ to $-1$. We name it $(\rho_2, V_2)$. Also $\rho_2\otimes\rho_3$ defines a one-dimensional representation. We name it $(\rho_4, V_4)$.
Next we consider two-dimensional representations. The natural representation $(\rho_{10}, V_{10})$ which maps $T$ and $D$ to the defining matrices above is irreducible, since neither of one-dimensional $D$-invariant subspaces are $T$-invariant. Taking tensor products with the one-dimensional representations above and the natural representation, we have further 3 two-dimensional irreducible representations, $\rho_7 = \rho_3\otimes\rho_{10}$, $\rho_8 = \rho_2\otimes\rho_{10}$ and $\rho_9 = \rho_4\otimes\rho_{10}$. There are 2 more two-dimensional irreducible representations which we will deal with later.
As a subrepresentation of $\rho_{10}\otimes\rho_{10}$, we have a three-dimensional irreducible representation. Let $\langle {\mbox{\boldmath $e$}}_1, {\mbox{\boldmath $e$}}_2 \rangle$ be a basis of $V_{10}$ which gives the natural representation. Then $\langle {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}_1,
{\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}_2,
{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}_1,
{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}_2\rangle$ gives a basis for the tensor representation $\rho_{10}^{\otimes 2}$. With respect to this basis, the representation matrices of $T$ and $D$ are $$\rho_{10}^{\otimes 2}(T)
= \frac{i}{2}
\begin{pmatrix}
1& 1& 1& 1\\
1&-1& 1&-1\\
1& 1&-1&-1\\
1&-1&-1& 1
\end{pmatrix}\ \mbox{ and }\
\rho_{10}^{\otimes 2}(D)
= \mbox{diag}(1, i, i, -1).$$ If we put ${\mbox{\boldmath $e$}}'_1 = {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}_1$, ${\mbox{\boldmath $e$}}'_2 = {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}_2+{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}_1$ and ${\mbox{\boldmath $e$}}'_3 = {\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}_2$, then $\langle {\mbox{\boldmath $e$}}'_1, {\mbox{\boldmath $e$}}'_2, {\mbox{\boldmath $e$}}'_3\rangle$ is obviously a $D$-invariant subspace. It is easy to check that it is also $T$-invariant. Hence it gives a three-dimensional representation. We name it ($\rho_{13}$, $V_{13}$). The representation matrices with respect to this basis are $$\rho_{13}(T)
= \frac{i}{2}
\begin{pmatrix}
1& 2& 1\\
1& 0&-1\\
1&-2& 1
\end{pmatrix}\ \mbox{and}\
\rho_{13}(D)
= \mbox{diag}(1, i, -1).$$ Since each one-dimensional $D$-invariant subspace of $V_{13}$ is not $T$-invariant, the representation ($\rho_{13}$, $V_{13}$) is irreducible. Similarly to the previous case, we have further 3 three-dimensional irreducible representations, $\rho_{11} = \rho_3\otimes\rho_{13}$, $\rho_{12} = \rho_4\otimes\rho_{13}$ and $\rho_{14} = \rho_2\otimes\rho_{13}$.
Next we look for a four-dimensional irreducible representation in $(\rho_{10}\otimes\rho_{13}, V_{10}\otimes V_{13})$. Let $\langle {\mbox{\boldmath $e$}}_i\otimes{\mbox{\boldmath $e$}}'_j\ |\ i=1,2,\ j=1,2,3\rangle$ be a basis of $V_{10}\otimes V_{13}$ (lexicographical order). Then we have the following representation matrices of $T$ and $D$: $$\begin{aligned}
\rho_{10}\otimes\rho_{13}(T)
&= \frac{-1+i}{4}
\begin{pmatrix}
1& 2& 1& 1& 2& 1\\
1& 0&-1& 1& 0&-1\\
1&-2& 1& 1&-2& 1\\
1& 2& 1&-1&-2&-1\\
1& 0&-1&-1& 0& 1\\
1&-2& 1&-1& 2&-1
\end{pmatrix},\\
\rho_{10}\otimes\rho_{13}(D)
&= \mbox{diag}(1, i, -1, i,-1, -i).\end{aligned}$$ If we put ${\mbox{\boldmath $e$}}^{{\prime\prime}}_1 = {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}'_1$, ${\mbox{\boldmath $e$}}^{{\prime\prime}}_2 = {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}'_2+{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}'_1$, ${\mbox{\boldmath $e$}}^{{\prime\prime}}_3 = {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}'_3+{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}'_2$, and ${\mbox{\boldmath $e$}}^{{\prime\prime}}_4 = {\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}'_3$, according to the eigen values of $\rho_{10}\otimes\rho_{13}(D)$, then $\langle {\mbox{\boldmath $e$}}^{{\prime\prime}}_k\ |\ k=1,2,3,4 \rangle$ is obviously a $D$-invariant subspace. It is also easy to check that it is $T$-invariant. Hence it gives a four-dimensional representation. We name it ($\rho_{15}$, $V_{15}$). The representation matrices with respect to this basis are $$\rho_{15}(T)
= \frac{-1+i}{4}
\begin{pmatrix}
1& 3& 3& 1\\
1& 1&-1&-1\\
1&-1&-1& 1\\
1&-3& 3&-1
\end{pmatrix}\ \mbox{and}\
\rho_{13}(D)
= \mbox{diag}(1, i, -1, -i).$$ As we saw in the previous case, none of one-dimensional $D$-invariant subspaces of $V_{15}$ is $T$-invariant. Now consider two-dimensional $D$-invariant subspaces. Since all eigen spaces of $\rho_{15}(D)$ are one-dimensional, we find that a two-dimensional $D$-invariant subspace is of the form $\langle {\mbox{\boldmath $e$}}^{{\prime\prime}}_i, {\mbox{\boldmath $e$}}^{{\prime\prime}}_j\rangle (i\neq j)$. Let $W$ be $\langle {\mbox{\boldmath $e$}}^{{\prime\prime}}_1, {\mbox{\boldmath $e$}}^{{\prime\prime}}_2\rangle$ and take a non-zero vector ${\mbox{\boldmath $v$}}= a{\mbox{\boldmath $e$}}^{{\prime\prime}}_1+b{\mbox{\boldmath $e$}}^{{\prime\prime}}_2$ from $W$. Then we have $$\rho_{15}(T){\mbox{\boldmath $v$}}= \frac{-1+i}{4}
\left[
(a+3b){\mbox{\boldmath $e$}}^{{\prime\prime}}_1
+ (a+b){\mbox{\boldmath $e$}}^{{\prime\prime}}_2
+ (a-b){\mbox{\boldmath $e$}}^{{\prime\prime}}_3
+ (a-3b){\mbox{\boldmath $e$}}^{{\prime\prime}}_4
\right].$$ In order that $\rho_{15}(T){\mbox{\boldmath $v$}}\in W$, it must hold that $a=b=0$. This contradicts the assumption that ${\mbox{\boldmath $v$}}$ is non-zero vector. Hence we find that $W$ is not $T$-invariant. Similar arguments hold for all two-dimensional $D$-invariant subspaces $\{\langle {\mbox{\boldmath $e$}}^{{\prime\prime}}_i,
{\mbox{\boldmath $e$}}^{{\prime\prime}}_j\rangle\}_{1\leq i < j\leq 4}$. This implies there is no two-dimensional subrepresentation in $V_{15}$. Hence we find that ($\rho_{15}$, $V_{15}$) is irreducible. Similarly to the previous case, we have further four-dimensional irreducible representations, $\rho_2\otimes\rho_{15}$, $\rho_3\otimes\rho_{15}$, $\rho_4\otimes\rho_{15}$. The first one, however, coincides with $\rho_{15}$ and the second one and the third one are equivalent. Hence we have 2 four-dimensional irreducible representations, $\rho_{15}$ and $\rho_{16} = \rho_3\otimes\rho_{15}$.
Finally we look for the remaining irreducible representations in $(\rho_{10}\otimes\rho_{15}, V_{10}\otimes V_{15})$. Let $\langle {\mbox{\boldmath $e$}}_i\otimes{\mbox{\boldmath $e$}}^{{\prime\prime}}_j\ |\ i=1,2, j=1,2,3,4\rangle$ be a basis of $V_{10}\otimes V_{15}$ (lexicographical order). Then we have the following representation matrices of $T$ and $D$: $$\begin{aligned}
\rho_{10}\otimes\rho_{15}(T)
&= \frac{-1}{4}
\begin{pmatrix}
1& 3& 3& 1& 1& 3& 3& 1\\
1& 1&-1&-1& 1& 1&-1&-1\\
1&-1&-1& 1& 1&-1&-1& 1\\
1&-3& 3&-1& 1&-3& 3&-1\\
1& 3& 3& 1&-1&-3&-3&-1\\
1& 1&-1&-1&-1&-1& 1& 1\\
1&-1&-1& 1&-1& 1& 1&-1\\
1&-3& 3&-1&-1& 3&-3& 1
\end{pmatrix},\\
\rho_{10}\otimes\rho_{15}(D)
&= \mbox{diag}(1, i, -1,-i, i, -1,-i, 1).\end{aligned}$$ If we put ${\mbox{\boldmath $e$}}^{{\prime{\prime\prime}}}_1
= {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}^{{\prime\prime}}_1+{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}^{{\prime\prime}}_4$, and ${\mbox{\boldmath $e$}}^{{\prime{\prime\prime}}}_2
= {\mbox{\boldmath $e$}}_1\otimes{\mbox{\boldmath $e$}}^{{\prime\prime}}_3+{\mbox{\boldmath $e$}}_2\otimes{\mbox{\boldmath $e$}}^{{\prime\prime}}_2$, then $\langle {\mbox{\boldmath $e$}}^{{\prime{\prime\prime}}}_1, {\mbox{\boldmath $e$}}^{{\prime{\prime\prime}}}_2\rangle$ is $T$- and $D$-invariant subspace. We name it ($\rho_{5}$, $V_{5}$). The representation matrices with respect to this basis are $$\rho_{5}(T)
= \frac{-1}{2}
\begin{pmatrix}
1& 1\\
3&-1
\end{pmatrix}\ \mbox{and}\
\rho_{5}(D)
= \mbox{diag}(1, -1).$$ Similarly to the previous ones, we can check that this representation is irreducible. Further $\rho_6 = \rho_3\otimes\rho_5$ also defines a two-dimensional irreducible representation.
So far, we have got 16 irreducible representations. Since $H_1$ has 16 conjugacy classes, $\{(\rho_i, V_i)\}_{i=1}^{16}$ are a complete representatives of all irreducible representations of $H_1$. Accordingly, the character table of $H_1$ is also derived.
------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
$H_1$ $\mathfrak{C}_1$ $\mathfrak{C}_2$ $\mathfrak{C}_3$ $\mathfrak{C}_4$ $\mathfrak{C}_5$ $\mathfrak{C}_6$ $\mathfrak{C}_7$ $\mathfrak{C}_8$ $\mathfrak{C}_9$ $\mathfrak{C}_{10}$ $\mathfrak{C}_{11}$ $\mathfrak{C}_{12}$ $\mathfrak{C}_{13}$ $\mathfrak{C}_{14}$ $\mathfrak{C}_{15}$ $\mathfrak{C}_{16}$
$1$ $T$ $T^2$ $T^3$ $T^4$ $T^6$ $D$ $DT$ $DT^2$ $DT^3$ $DT^4$ $DT^5$ $DT^6$ $DT^7$ $D^2$ $D^2T^2$
order 1 8 4 8 2 4 4 6 4 12 4 3 4 12 2 4
$\chi_1$ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
$\chi_2$ 1 $-1$ 1 $-1$ 1 1 $-1$ 1 $-1$ 1 $-1$ 1 $-1$ 1 1 1
$\chi_3$ 1 $-i$ $-1$ $i$ 1 $-1$ $i$ 1 $-i$ $-1$ $i$ 1 $-i$ $-1$ $-1$ 1
$\chi_4$ 1 $i$ $-1$ $-i$ 1 $-1$ $-i$ 1 $i$ $-1$ $-i$ 1 $i$ $-1$ $-1$ 1
$\chi_5$ 2 0 2 0 2 2 0 $-1$ 0 $-1$ 0 $-1$ 0 $-1$ 2 2
$\chi_6$ 2 0 $-2$ 0 2 $-2$ 0 $-1$ 0 1 0 $-1$ 0 1 $-2$ 2
$\chi_7$ 2 0 $-2i$ 0 $-2$ $2i$ $-1+i$ 1 $1+i$ $-i$ $1-i$ $-1$ $-1-i$ $i$ 0 0
$\chi_8$ 2 0 $2i$ 0 $-2$ $-2i$ $-1-i$ 1 $1-i$ $i$ $1+i$ $-1$ $-1+i$ $-i$ 0 0
$\chi_9$ 2 0 $-2i$ 0 $-2$ $2i$ $1-i$ 1 $-1-i$ $-i$ $-1+i$ $-1$ $1+i$ $i$ 0 0
$\chi_{10}$ 2 0 $2i$ 0 $-2$ $-2i$ $1+i$ 1 $-1+i$ $i$ $-1-i$ $-1$ $1-i$ $-i$ 0 0
$\chi_{11}$ 3 1 3 1 3 3 $-1$ 0 $-1$ 0 $-1$ 0 $-1$ 0 $-1$ $-1$
$\chi_{12}$ 3 $-1$ 3 $-1$ 3 3 1 0 1 0 1 0 1 0 $-1$ $-1$
$\chi_{13}$ 3 $i$ $-3$ $-i$ 3 $-3$ $i$ 0 $-i$ 0 $i$ 0 $-i$ 0 1 $-1$
$\chi_{14}$ 3 $-i$ $-3$ $i$ 3 $-3$ $-i$ 0 $i$ 0 $-i$ 0 $i$ 0 1 $-1$
$\chi_{15}$ 4 0 $-4i$ 0 $-4$ $4i$ 0 $-1$ 0 $i$ 0 1 0 $-i$ 0 0
$\chi_{16}$ 4 0 $4i$ 0 $-4$ $-4i$ 0 $-1$ 0 $-i$ 0 1 0 $i$ 0 0
------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
We conclude this section with adding a few words on invariant theory of $H_1$ under irreducible representations ([*cf*]{}. [@Di]). Let $\rho$ be one of the $d$-dimensional irreducible representation of $H_1$. Then $\rho (H_1)$ acts naturally on the polynomial ring of $d$ variables. We denote the invariant ring under this action by ${\mathbb C}[\rho]^{H_1}$. The orders of $\rho_i (H_1)$ are $$\underbrace{1,2,4,4}_{\dim 1},
\underbrace{6,12,96,96,96,96}_{\dim 2},
\underbrace{24,24,48,48}_{\dim 3},
\underbrace{96,96}_{\dim 4}.$$ The dimension $1$ case aside, the invariant rings $${\mathbb C}[\rho_5]^{H_1},
{\mathbb C}[\rho_i]^{H_1}\ (i=7,8,9,10),
{\mathbb C}[\rho_{12}]^{H_1}$$ are weighted polynomial rings. In the sense of [@ST], all $\rho_i(H_1)\ (i=7,8,9,10)$ are equivalent each other, and $\rho_{15}(H_1)$ to $\rho_{16}(H_1)$. We already know the ring ${\mathbb C}[\rho_7]^{H_1}$. The ring ${\mathbb C}[\rho_5]^{H_1}$ can be generated by the polynomials of degrees $2$ and $3$, and the ring ${\mathbb C}[ \rho_{15}]^{H_1}$ by those of degrees $2,3$ and $4$. If we look at the degrees, we can find that $\rho_5(H_1)$ is equivalent to $G(3,3,2)$ and $\rho_{12}(H_1)$ to $G(2,2,3)$. The other cases up to dimension $3$ are modules of rank $2$ over the polynomial rings. The $\rho_{15}$ case has a somewhat complicated structure. The ring ${\mathbb C}[\rho_{15}]^{H_1}$ is a module of rank $32$ over the polynomial ring. We note that calculations here were done with Magma [@Bo].
Decomposition of tensor representations
=======================================
In the previous section, we have found complete representatives of all irreducible representations. In this section, we see how tensor powers of $\rho_{10}$ are decomposed into irreducible ones.
We begin with the general theory (see for example Curtis-Reiner[@CR]). Let $\chi_1,\ldots, \chi_s$ be the set of all irreducible characters of a finite group $G$. For any (not necessarily irreducible) representation $(\rho, V)$ of $G$, let $\chi$ be its character. Then $\chi$ can be uniquely expressed a sum of irreducible characters: $$\chi = m_1\chi_1+\cdots + m_s\chi_s.$$ Now suppose that $\chi$ has its character values ($k_1, \ldots, k_s$) on the conjugacy classes ($\mathfrak{C}_1, \ldots, \mathfrak{C}_s$). Then we get $$\begin{aligned}
(k_1,\ldots, k_s)
&= (\chi(\mathfrak{C}_1), \ldots, \chi(\mathfrak{C}_s))\\
&= m_1(\chi_1(\mathfrak{C}_1), \ldots, \chi_s(\mathfrak{C}_s))
+\cdots
+ m_s(\chi_s(\mathfrak{C}_1), \ldots, \chi_s(\mathfrak{C}_s))\\
&= (m_1, \ldots, m_s)
\begin{pmatrix}
\chi_1(\mathfrak{C}_1)&\cdots&\chi_1(\mathfrak{C}_s)\\
\vdots &\ddots& \vdots\\
\chi_s(\mathfrak{C}_1)&\cdots&\chi_s(\mathfrak{C}_s)
\end{pmatrix}.\end{aligned}$$ If we let ${\mbox{\boldmath $X$}}$ denote the matrix of the character table, then the above relation is simply written as $$(k_1, \ldots, k_s) = (m_1, \ldots, m_s){\mbox{\boldmath $X$}}.$$ By the linear independence of the irreducible characters, ${\mbox{\boldmath $X$}}$ is non-singular. Hence we have $$\label{eq:MulIrr}
(m_1, \ldots, m_s) = (k_1, \ldots, k_s){\mbox{\boldmath $X$}}^{-1}.$$
In order to examine the structure of the centralizer algebra of the tensor representation, it is useful to investigate how the tensor product of the natural and an irreducible representation is decomposed into the irreducible ones. In the following, we go back to our case and decompose $\rho_{10}\otimes\rho_i$ ($i = 1, 2, \ldots, 16$) one by one.
By the argument and/or the character table in the previous section, we already have the following: $$\begin{aligned}
\chi_{10}\cdot\chi_1 &= \chi_{10},\\
\chi_{10}\cdot\chi_2 &= \chi_{8},\\
\chi_{10}\cdot\chi_3 &= \chi_{7},\\
\chi_{10}\cdot\chi_4 &= \chi_{9}.\end{aligned}$$
Further, we can directly read the following from the character table: $$\begin{aligned}
\chi_{10}\cdot\chi_5 &= \chi_{16},\\
\chi_{10}\cdot\chi_6 &= \chi_{15}.\end{aligned}$$
Next, consider $\chi_{10}\cdot\chi_7(\mathfrak{C}_{1}, \ldots, \mathfrak{C}_{16})$. Again from the character table, we have $$\begin{aligned}
\lefteqn{
\chi_{10}\cdot\chi_7(\mathfrak{C}_{1}, \ldots, \mathfrak{C}_{16})
}\\
&= (4, 0, 4, 0, 4, 4,-2, 1,-2, 1, -2, 1, -2, 1, 0, 0).\end{aligned}$$ Using the identity , we have $$\begin{aligned}
\lefteqn{(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0)}\\
&= \chi_{10}\cdot
\chi_7(\mathfrak{C}_{1}, \ldots, \mathfrak{C}_{16}){\mbox{\boldmath $X$}}^{-1}.\end{aligned}$$ This means $$\chi_{10}\cdot\chi_7 = \chi_2 + \chi_{11}.$$ In a similar way we have $$\begin{aligned}
\chi\cdot\chi_8 &= \chi_4+\chi_{14}, \\
\chi\cdot\chi_9 &= \chi_1+\chi_{12}, \\
\chi\cdot\chi_{10} &= \chi_3 + \chi_{13}, \\
\chi\cdot\chi_{11} &= \chi_8 + \chi_{16}, \\
\chi\cdot\chi_{12} &= \chi_{10} + \chi_{16}, \\
\chi\cdot\chi_{13} &= \chi_7 + \chi_{15}, \\
\chi\cdot\chi_{14} &= \chi_9 + \chi_{15}, \\
\chi\cdot\chi_{15} &= \chi_5 + \chi_{11} + \chi_{12}, \\
\chi\cdot\chi_{16} &= \chi_6 + \chi_{13} + \chi_{14}. \\\end{aligned}$$ By the above calculation, we obtain the Hasse diagram of the decomposition of $\rho_{10}^{\otimes k}$ into irreducible ones. $$\begin{xy}
(20,0)="T1P10",
(20,-3)*{\rho_{10},1},
(80,-3)*{1^2 = 1}, (20,-5)="S1P10",
(10,-15)="T2P3", (30,-15)="T2P13",
(10,-18)*{\rho_{3},1},(30,-18)*{\rho_{13},1},
(80,-18)*{1^2+1^2 = 2}, (10,-20)="S2P3", (30,-20)="S2P13",
(20,-30)="T3P7",(30,-30)="T3P15",
(20,-33)*{\rho_7,2}, (30,-33)*{\rho_{15},1},
(80,-33)*{2^2+1^2 = 5}, (20,-35)="S3P7",(30,-35)="S3P15",
(10,-45)="T4P2",(20,-45)="T4P11",(30,-45)="T4P12",(40,-45)="T4P5",(10,-48)*{\rho_2,2},(20,-48)*{\rho_{11},3},(30,-48)*{\rho_{12},1},(40,-48)*{\rho_5,1},
(80,-48)*{2^2+3^2+1^2+1^2 = 15}, (10,-50)="S4P2",(20,-50)="S4P11",(30,-50)="S4P12",(40,-50)="S4P5",(10,-60)="T5P8",(20,-60)="T5P10",(30,-60)="T5P16",
(10,-63)*{\rho_8,5},(20,-63)*{\rho_{10},1},(30,-63)*{\rho_{16},5},
(80,-63)*{5^2+1^2+5^2 = 51}, (10,-65)="S5P8",(20,-65)="S5P10",(30,-65)="S5P16", (0,-75)="T6P4",(10,-75)="T6P3",(20,-75)="T6P14",(30,-75)="T6P13",(40,-75)="T6P6",
(0,-78)*{\rho_4,5},(10,-78)*{\rho_{3},1},(20,-78)*{\rho_{14},10},
(30,-78)*{\rho_{13},6},(40,-78)*{\rho_{6},5},
(80,-78)*{5^2+1^2+10^2+6^2+5^2 = 187}, (0,-80)="S6P4",(10,-80)="S6P3",(20,-80)="S6P14",(30,-80)="S6P13",(40,-80)="S6P6", (10,-90)="T7P9",(20,-90)="T7P7",(30,-90)="T7P15",
(10,-93)*{\rho_{9},15},(20,-93)*{\rho_{7},7},
(30,-93)*{\rho_{15},21},
(80,-93)*{15^2+7^2+21^2 = 715}, (10,-95)="S7P9",(20,-95)="S7P7",(30,-95)="S7P15", (0,-105)="T8P1",(10,-105)="T8P2",(20,-105)="T8P11",(30,-105)="T8P12",(40,-105)="T8P5",
(0,-108)*{\rho_{1},15},(10,-108)*{\rho_{2},7},
(20,-108)*{\rho_{11},28},(30,-108)*{\rho_{12},36},
(40,-108)*{\rho_{5},21},
(80,-108)*{15^2+7^2+28^2+36^2+21^2 = 2795}, (0,-110)="S8P1",(10,-110)="S8P2",(20,-110)="S8P11",(30,-110)="S8P12",(40,-110)="S8P5", (10,-120)="T9P8",(20,-120)="T9P10",(30,-120)="T9P16",
(10,-123)*{\rho_{8},35},(20,-123)*{\rho_{10},51},
(30,-123)*{\rho_{16},85},
(80,-123)*{35^2+51^2+85^2 = 11051}, {\ar @{{.}-{.}}"S1P10";"T2P3"},{\ar @{{.}-{.}}"S1P10";"T2P13"},
{\ar @{{.}-{.}}"S2P3";"T3P7"},{\ar @{{.}-{.}}"S2P13";"T3P7"}, {\ar @{{.}-{.}}"S2P13";"T3P15"},
{\ar @{{.}-{.}}"S3P7";"T4P2"},{\ar @{{.}-{.}}"S3P7";"T4P11"}, {\ar @{{.}-{.}}"S3P15";"T4P11"},{\ar @{{.}-{.}}"S3P15";"T4P12"},
{\ar @{{.}-{.}}"S3P15";"T4P5"},
{\ar @{{.}-{.}}"S4P2";"T5P8"}, {\ar @{{.}-{.}}"S4P11";"T5P8"},{\ar @{{.}-{.}}"S4P11";"T5P16"},
{\ar @{{.}-{.}}"S4P12";"T5P10"},{\ar @{{.}-{.}}"S4P12";"T5P16"},
{\ar @{{.}-{.}}"S4P5";"T5P16"},
{\ar @{{.}-{.}}"S5P8";"T6P4"},{\ar @{{.}-{.}}"S5P8";"T6P14"}, {\ar @{{.}-{.}}"S5P10";"T6P3"},{\ar @{{.}-{.}}"S5P10";"T6P13"},
{\ar @{{.}-{.}}"S5P16";"T6P14"},{\ar @{{.}-{.}}"S5P16";"T6P13"},
{\ar @{{.}-{.}}"S5P16";"T6P6"},
{\ar @{{.}-{.}}"S6P4";"T7P9"}, {\ar @{{.}-{.}}"S6P3";"T7P7"},
{\ar @{{.}-{.}}"S6P14";"T7P9"},{\ar @{{.}-{.}}"S6P14";"T7P15"},
{\ar @{{.}-{.}}"S6P13";"T7P7"},{\ar @{{.}-{.}}"S6P13";"T7P15"},
{\ar @{{.}-{.}}"S6P6";"T7P15"},
{\ar @{{.}-{.}}"S7P9";"T8P1"},{\ar @{{.}-{.}}"S7P9";"T8P12"}, {\ar @{{.}-{.}}"S7P7";"T8P2"},{\ar @{{.}-{.}}"S7P7";"T8P11"},
{\ar @{{.}-{.}}"S7P15";"T8P11"},{\ar @{{.}-{.}}"S7P15";"T8P12"},
{\ar @{{.}-{.}}"S7P15";"T8P5"},
{\ar @{{.}-{.}}"S8P1";"T9P10"}, {\ar @{{.}-{.}}"S8P2";"T9P8"},
{\ar @{{.}-{.}}"S8P11";"T9P8"},{\ar @{{.}-{.}}"S8P11";"T9P16"},
{\ar @{{.}-{.}}"S8P12";"T9P10"},{\ar @{{.}-{.}}"S8P12";"T9P16"},
{\ar @{{.}-{.}}"S8P5";"T9P16"},
\end{xy}$$ From this diagram we can read off the multiplicity of each irreducible representation $(\rho_i, V_i)$ in $V_{10}^{\otimes k}$ by counting the number of paths from the top vertex indexed by $\rho_{10}$ in the 1-st row to the corresponding vertex in the $k$-th row. We put the multiplicity on the right side of each irreducible representation. Further, we calculated the square sums of the multiplicities on the each row.
Let ${\cal A}_k = {\mbox{End}}_{H_1}(V_{10}^{\otimes k})$ be the centralizer algebra of $H_1$ in $V_{10}^{\otimes k}$, where $H_1$ acts on $V_{10}$ diagonally. By the Schur-Weyl reciprocity [@Sa; @We], this diagram is the Bratteli diagram of the algebra sequence $${\mathbb C}={\cal A}_1\subset{\cal A}_2\subset{\cal A}_3\subset \cdots .$$ (For the Bratteli diagram, see for example Goodman-de la Harpe-Jones [@GHJ], §2.3.) Accordingly, the square sum of the multiplicities on the $k$-th row is the dimension of $\mbox{End}_{H_1}(V_{10}^{\otimes k})$. We will examine it in detail in the next section.
Centralizer algebra
===================
In the previous section, we have seen that the dimensions of ${\cal A}_k = \mbox{End}_{H_1}(V_{10}^{\otimes k})$ ($k=1,2,\ldots$) are 1, 2, 5, 15, 51, 187, 715,…. According to “The On-Line Encyclopedia of Integer Sequences”[@Sl], these terms coincide with the fist few terms of the expression $(3\cdot 2^{k-2} + 2^{2k-3} + 1)/3$. This is indeed the case. In order to prove this, we calculate the size of each simple component of ${\cal A}_k$.
Let $d^{(i)}_j$ be the multiplicity of $\rho_{i}$ in the tensor representation $\rho_{10}^j$, which coincides with the size of the corresponding simple component of ${\cal A}_j$. By the Bratteli diagram of ${\cal A}_j$ given in the previous section, we have the recursive formulae as follows. First note that the irreducible representations $\rho_8$, $\rho_{10}$, $\rho_{16}$ of $H_1$ again appear in the bottom of the diagram, as well as the 5-th row of the diagram. This implies that the diagram periodically grows up as $k$ increases. The iteration is as follows: $$[\rho_8, \rho_{10}, \rho_{16}]\rightarrow
[\rho_4, \rho_3, \rho_{14}, \rho_{13}, \rho_6]\rightarrow
[\rho_9, \rho_7, \rho_{15}]\rightarrow
[\rho_1, \rho_2, \rho_{11}, \rho_{12}, \rho_5]\rightarrow\cdots.$$ Hence based on the Bratteli diagram of the 9-th row from the 5-th row, we can obtain the following recursive formulae: $$\begin{aligned}
&\left\{
\begin{array}{rcl}
d^{(4)}_{4\ell+2} &=& d^{(8)}_{4\ell+1},\\
d^{(3)}_{4\ell+2} &=& d^{(10)}_{4\ell+1},\\
d^{(14)}_{4\ell+2}
&=& d^{(8)}_{4\ell+1}+d^{(16)}_{4\ell+1},\\
d^{(13)}_{4\ell+2}
&=& d^{(10)}_{4\ell+1}+d^{(16)}_{4\ell+1},\\
d^{(6)}_{4\ell+2} &=& d^{(16)}_{4\ell+1},
\end{array}
\right.\\
&\left\{
\begin{array}{rcl}
d^{(9)}_{4\ell+3} &=& d^{(4)}_{4\ell+2}+d^{(14)}_{4\ell+2},\\
d^{(7)}_{4\ell+3} &=& d^{(3)}_{4\ell+2}+d^{(13)}_{4\ell+2},\\
d^{(15)}_{4\ell+3}
&=&
d^{(14)}_{4\ell+2}+d^{(13)}_{4\ell+2}+d^{(6)}_{4\ell+2},\\
\end{array}
\right.\\
&\left\{
\begin{array}{rcl}
d^{(1)}_{4(\ell+1)} &=& d^{(9)}_{4\ell+3},\\
d^{(2)}_{4(\ell+1)} &=& d^{(7)}_{4\ell+3},\\
d^{(11)}_{4(\ell+1)}
&=& d^{(7)}_{4\ell+3}+d^{(15)}_{4\ell+3},\\
d^{(12)}_{4(\ell+1)}
&=& d^{(9)}_{4\ell+3}+d^{(15)}_{4\ell+3},\\
d^{(5)}_{4(\ell+1)} &=& d^{(15)}_{4\ell+3},
\end{array}
\right.\\
&\left\{
\begin{array}{rcl}
d^{(8)}_{4(\ell+1)+1}
&=& d^{(2)}_{4(\ell+1)}+d^{(11)}_{4(\ell+1)},\\
d^{(10)}_{4(\ell+1)+1}
&=& d^{(1)}_{4(\ell+1)}+d^{(12)}_{4(\ell+1)},\\
d^{(16)}_{4(\ell+1)+1}
&=&
d^{(11)}_{4(\ell+1)}+d^{(12)}_{4(\ell+1)}+d^{(5)}_{4(\ell+1)}.
\end{array}
\right.\end{aligned}$$ Note also that if we allow the possibility $d_j^{(i)}=0$, the recursions above are still valid even for the 1-st to 4-th row. Hence we have the following: $$\begin{aligned}
\label{eq:rec8}\nonumber
d^{(8)}_{4\ell+1} &= d^{(2)}_{4\ell}+d^{(11)}_{4\ell}\\\nonumber
&= d^{(7)}_{4\ell-1}+(d^{(7)}_{4\ell-1}+d^{(15)}_{4\ell-1})\\\nonumber
&= 2d^{(7)}_{4\ell-1}+d^{(15)}_{4\ell-1}\\\nonumber
&= 2(d^{(3)}_{4\ell-2}+d^{(13)}_{4\ell-2})
+(d^{(14)}_{4\ell-2}+d^{(13)}_{4\ell-2}+d^{(6)}_{4\ell-2})\\\nonumber
&= 2d^{(3)}_{4\ell-2}+d^{(14)}_{4\ell-2}
+3d^{(13)}_{4\ell-2}+d^{(6)}_{4\ell-2}\\\nonumber
&= 2d^{(10)}_{4\ell-3}+(d^{(8)}_{4\ell-3}+d^{(16)}_{4\ell-3})
+3(d^{(10)}_{4\ell-3}+d^{(16)}_{4\ell-3})+d^{(16)}_{4\ell-3}\\
&= d^{(8)}_{4(\ell-1)+1}+5d^{(10)}_{4(\ell-1)+1}+5d^{(16)}_{4(\ell-1)+1}\
(\ell > 0).\end{aligned}$$ Similarly we have $$\label{eq:rec10}
d^{(10)}_{4\ell+1}
= 5d^{(8)}_{4(\ell-1)+1}+d^{(10)}_{4(\ell-1)+1}+5d^{(16)}_{4(\ell-1)+1}$$ and $$\label{eq:rec16}
d^{(16)}_{4\ell+1}
= 5d^{(8)}_{4(\ell-1)+1}+5d^{(10)}_{4(\ell-1)+1}+11d^{(16)}_{4(\ell-1)+1}$$ From the recursion , and , and the initial condition $(d^{(8)}_1, d^{(10)}_1, d^{(16)}_1) = (0,1,0)$, we obtain $$\begin{aligned}
d^{(8)}_{4\ell+1}
&=-\frac{(-4)^{\ell}}{2}+\frac{1}{3}+\frac{{16}^{\ell}}{6},\\
d^{(10)}_{4\ell+1}
&=\frac{(-4)^{\ell}}{2}+\frac{1}{3}+\frac{{16}^{\ell}}{6},\\
d^{(16)}_{4\ell+1}
&=-\frac{1}{3}+\frac{{16}^{\ell}}{3}.\end{aligned}$$ By the initial recursion formulae, we immediately obtain $$\begin{aligned}
d^{(4)}_{4\ell+2}
&=-\frac{(-4)^{\ell}}{2}+\frac{1}{3}+\frac{{16}^{\ell}}{6},\\
d^{(3)}_{4\ell+2}
&=\frac{(-4)^{\ell}}{2}+\frac{1}{3}+\frac{{16}^{\ell}}{6},\\
d^{(14)}_{4\ell+2}
&=-\frac{(-4)^{\ell}}{2}+\frac{{16}^{\ell}}{2},\\
d^{(13)}_{4\ell+2}
&=\frac{(-4)^{\ell}}{2}+\frac{{16}^{\ell}}{2},\\
d^{(6)}_{4\ell+2}
&=-\frac{1}{3}+\frac{{16}^{\ell}}{3},\end{aligned}$$ $$\begin{aligned}
d^{(9)}_{4\ell+3}
&=-(-4)^{\ell}+\frac{1}{3}+\frac{2\cdot{16}^{\ell}}{3},\\
d^{(7)}_{4\ell+3}
&=(-4)^{\ell}+\frac{1}{3}+\frac{2\cdot{16}^{\ell}}{3},\\
d^{(15)}_{4\ell+3}
&=-\frac{1}{3}+\frac{4\cdot{16}^{\ell}}{3}\end{aligned}$$ and $$\begin{aligned}
d^{(1)}_{4(\ell+1)}
&=-(-4)^{\ell}+\frac{1}{3}+\frac{2\cdot{16}^{\ell}}{3},\\
d^{(2)}_{4(\ell+1)}
&=(-4)^{\ell}+\frac{1}{3}+\frac{2\cdot{16}^{\ell}}{3},\\
d^{(11)}_{4(\ell+1)}
&=(-4)^{\ell}+2\cdot{16}^{\ell},\\
d^{(12)}_{4(\ell+1)}
&=-(-4)^{\ell}+2\cdot{16}^{\ell},\\
d^{(5)}_{4(\ell+1)}
&=-\frac{1}{3}+\frac{4\cdot{16}^{\ell}}{3}.\end{aligned}$$ Thus we have obtained the size of each simple component of ${\cal A}_k$. If we apply simple considerations to the order of simple components, the sizes are uniformly described as follows.
Let ${\cal A}_k = {\mbox{End}}_{H_1}(V_{10}^{\otimes k})$ be a centralizer algebra of $H_1$ in $V_{10}^{\otimes k}$, where $H_1$ acts on $V_{10}$ diagonally. Then ${\cal A}_k$ has the following multi-matrix structure. $${\cal A}_{k}
\cong
\left\{
\begin{array}{l}
M_{d_+(k)}({\mathbb C})
\oplus M_{d_-(k)}({\mathbb C})
\oplus M_{d_0(k)}({\mathbb C})
\\
\quad\mbox{if $k = 2m-1$},\\
M_{d_+(k)}({\mathbb C})
\oplus M_{d_-(k)}({\mathbb C})
\oplus M_{d_0(k)}({\mathbb C})
\oplus M_{e_+(k)}({\mathbb C})
\oplus M_{e_-(k)}({\mathbb C})
\\
\quad\mbox{if $k = 2m$},
\end{array}
\right.$$ where $$\begin{aligned}
d_{\pm}(k) &= \pm 2^{m-2}+\frac{1}{3}+\frac{2\cdot 4^{m-2}}{3},\\
d_0(k)& = -\frac{1}{3} + \frac{4^{m-1}}{3}\end{aligned}$$ and $$e_{\pm} =\pm 2^{m-2}+2\cdot 4^{m-2}.$$
Calculating the square sum of the dimensions of the simple components of ${\cal A}_k$ in cases $k=2m-1$ and $k=2m$, we finally obtain the following corollary as we expected.
$$\dim {\cal A}_k = 2^{k-2} + \frac{2^{2k-3}}{3} + \frac{1}{3}.$$
Again by the web cite [@Sl], this result suggests that the basis of the centralizer algebras of $H_1$ could be described in terms of the symmetric polynomials in 4 noncommuting variables [@Be] and/or the universal embedding of the symplectic dual polar space $DS_p(2k,2)$ [@Bl]. It would be interesting that these points become clear.
[**Acknowledgment**]{}. This work was supported by JSPS KAKENHI Grant Number 25400014.
[99]{}
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Broué, M. and Enguehard, M., Polynômes des poids de certains codes et fonctions thôta de certains réseaux, Ann. Sci. École Norm. Sup. (4) [**5**]{} (1972), 157-181.
Curtis, C. and Reiner, I, [*Methods of Representation Theory: With Applications to Finite Groups and Orders*]{}, Volume I, John Wiley & Sons, Inc., 1981.
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Gleason, A. M., Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, pp. 211-215. Gauthier-Villars, Paris, 1971.
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[Department of Mathematical Sciences, University of the Ryukyus, Okinawa, 903-0213, JAPAN]{}
[*E-mail address*]{}: kosuda@math.u-ryukyu.ac.jp
[Graduate School of Natural Science and Technology, Kanazawa University, Ishikawa, 920-1192 Japan]{}
[*E-mail address*]{}: oura@se.kanazawa-u.ac.jp
|
---
abstract: 'Supplemental material for paper: xxxx.'
title: |
[ Supplemental Material for\
Antideuteron production in [$\Upsilon(nS)$]{}decays and in [$\epem \to \qqbar$]{}]{}
---
Monmentum range () 0.35 – 0.55 0.70 0.80 0.90 1.00 1.15 1.30 1.55 2.25
---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Events in Signal Region 81.0 105.0 97.0 93.0 103.0 95.0 67.8 71.4 57.4
Branching Ratio ($\times 10^{-6} / (\gevc)$) 15.66 17.43 23.93 25.07 31.25 19.89 17.16 12.35 3.13
Statistical error (%) 10.49 9.21 9.62 9.58 8.92 9.29 18.51 10.30 40.16
Fit Biases 0.62 0.65 1.11 1.11 0.50 0.51 2.27 0.55 3.99
Background Model 0.30 0.32 0.16 0.52 0.65 0.90 0.27 0.23 7.72
Reconstruction Efficiency 2.64 2.52 3.02 4.10 4.86 6.67 2.84 10.47 6.47
Kinematic Acceptance 0.53 2.16 2.51 3.91 4.68 6.76 2.74 10.33 5.92
Material Interaction 2.79 2.98 3.26 4.30 5.04 7.01 2.79 10.48 6.65
Fake antideuterons -9.77 -2.09 -2.14 -1.66 -2.31 -3.30 -3.43 -1.73 -0.45
DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82
Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
Normalization 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20
Total systematic error (%) $^{ +7.49}_{ -10.85}$ $^{ +7.82}_{ -5.62}$ $^{ +8.24}_{ -6.21}$ $^{ +9.63}_{ -7.84}$ $^{ +10.59}_{ -9.15}$ $^{ +13.46}_{ -12.57}$ $^{ +8.32}_{ -6.86}$ $^{ +19.16}_{ -18.34}$ $^{ +15.40}_{ -14.27}$
Total error (%) $^{ +12.89}_{ -15.09}$ $^{ +12.08}_{ -10.79}$ $^{ +12.67}_{ -11.45}$ $^{ +13.58}_{ -12.38}$ $^{ +13.85}_{ -12.78}$ $^{ +16.35}_{ -15.63}$ $^{ +20.29}_{ -19.74}$ $^{ +21.76}_{ -21.03}$ $^{ +43.01}_{ -42.62}$
Bin range () 0.35 – 0.60 0.70 0.80 0.90 1.00 1.10 1.25 1.45 2.25
---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Events in Signal Region 89.2 93.0 92.0 61.5 91.0 70.2 75.5 71.2 118.5
Branching Ratio ($\times 10^{-6} / (\gevc)$) 12.63 19.18 20.42 13.80 24.14 19.39 13.68 10.34 5.78
Statistical error (%) 56.04 11.25 11.27 29.51 10.48 18.83 23.69 19.63 18.46
Fit Biases 2.78 0.10 0.70 9.57 0.64 3.41 6.55 5.31 4.89
Background Model 3.09 3.40 3.57 5.61 7.44 5.35 12.45 11.74 6.74
Reconstruction Efficiency 5.19 4.22 3.41 11.66 7.78 9.34 16.93 16.99 7.07
Kinematic Acceptance 4.42 3.62 3.59 10.82 7.34 9.38 16.34 16.39 4.85
Material Interaction 5.64 4.71 4.27 11.43 7.64 9.45 16.78 16.45 4.92
Fake antideuterons -2.90 -1.51 -1.07 -1.62 -1.99 -1.59 -2.94 -2.04 -0.97
DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82
Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
Normalization 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20 1.20
Total systematic error (%) $^{ +11.66}_{ -10.52}$ $^{ +10.26}_{ -8.58}$ $^{ +9.83}_{ -7.99}$ $^{ +23.40}_{ -22.72}$ $^{ +16.41}_{ -15.47}$ $^{ +18.58}_{ -17.72}$ $^{ +32.77}_{ -32.38}$ $^{ +32.17}_{ -31.70}$ $^{ +14.41}_{ -13.22}$
Total error (%) $^{ +57.24}_{ -57.02}$ $^{ +15.23}_{ -14.15}$ $^{ +14.95}_{ -13.81}$ $^{ +37.66}_{ -37.24}$ $^{ +19.47}_{ -18.69}$ $^{ +26.46}_{ -25.86}$ $^{ +40.44}_{ -40.12}$ $^{ +37.68}_{ -37.29}$ $^{ +23.42}_{ -22.71}$
Bin range () 0.35 – 0.65 0.85 1.00 1.20 2.25
---------------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Events in Signal Region 11.6 22.6 18.9 18.3 19.5
Branching Ratio ($\times 10^{-6} / (\gevc)$) 12.09 27.73 34.22 27.98 6.00
Statistical error (%) 54.14 28.75 34.42 30.55 40.43
Fit Biases 1.16 0.07 0.22 0.31 2.03
Background Model 7.58 0.86 2.22 1.65 3.83
Reconstruction Efficiency 7.08 1.68 3.57 1.28 3.23
Kinematic Acceptance 2.36 0.59 2.49 0.75 2.90
Material Interaction 7.27 2.47 2.84 1.95 2.97
Fake antideuterons -32.00 -1.90 -5.52 -3.46 -4.85
DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82
Event Selection 1.11 1.11 1.11 1.11 1.11
Normalization 0.24 0.24 0.24 0.24 0.24
Total systematic error (%) $^{ +14.22}_{ -34.53}$ $^{ +6.72}_{ -3.86}$ $^{ +8.20}_{ -7.99}$ $^{ +6.63}_{ -4.70}$ $^{ +9.04}_{ -8.44}$
Total error (%) $^{ +55.97}_{ -64.21}$ $^{ +29.52}_{ -29.01}$ $^{ +35.39}_{ -35.34}$ $^{ +31.26}_{ -30.91}$ $^{ +41.42}_{ -41.30}$
Bin range () 0.35 – 0.60 0.75 0.85 0.95 1.05 1.25 1.40 1.65 2.25
---------------------------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- -------------------------- --------------------------
Events in Signal Region -25.5 1.20 -10.1 -16.1 -22.5 -27.9 -11.5 27.7 39.7
Branching Ratio ($\times 10^{-6} / (\gevc)$) 1.17 -1.06 -2.42 -3.62 -3.79 -1.69 -2.05 0.39 1.72
Statistical error (%) 161.00 222.17 142.97 100.48 105.69 149.04 150.48 445.82 556.71
Fit Biases 4.64 16.06 10.40 4.72 4.53 6.56 2.38 32.11 0.35
Background Model 4.06 14.91 7.49 3.85 0.70 6.81 8.98 2.12 1.56
Reconstruction Efficiency 71.78 122.04 86.46 61.57 57.38 62.42 62.28 187.82 3.35
Kinematic Acceptance 68.12 120.62 84.95 62.45 56.45 61.11 60.97 187.15 3.25
Material Interaction 74.08 168.33 89.82 64.72 59.46 63.95 64.36 198.58 3.51
Fake antideuterons -1.02 -3.91 -1.96 -1.90 -1.99 -6.13 -5.60 -16.17 -0.61
DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82
Event Selection 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
Normalization 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
Total systematic error (%) $^{ +123.93}_{ -123.79}$ $^{ +241.45}_{ -241.41}$ $^{ +151.54}_{ -151.44}$ $^{ +109.34}_{ -109.21}$ $^{ +100.37}_{ -100.23}$ $^{ +108.85}_{ -108.87}$ $^{ +108.92}_{ -108.91}$ $^{ +332.88}_{ -333.22}$ $^{ +8.73}_{ -6.53}$
Total error (%) $^{ +203.17}_{ -203.09}$ $^{ +328.11}_{ -328.08}$ $^{ +208.34}_{ -208.27}$ $^{ +148.50}_{ -148.40}$ $^{ +145.76}_{ -145.66}$ $^{ +184.56}_{ -184.57}$ $^{ +185.77}_{ -185.76}$ $^{ +556.39}_{ -556.59}$ $^{ +556.78}_{ -556.75}$
Bin range () 0.35 – 0.60 0.75 0.85 0.95 1.05 1.25 1.40 1.65 2.25
--------------------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Events in Signal Region 157 130 108 102 117 176 128 178 308
Cross Section ($0.1 \rm{fb}/(\gevc)$) 66.46 54.13 72.22 71.70 79.84 64.24 66.38 47.28 18.97
Statistical error (%) 9.05 9.97 11.16 11.28 10.63 8.34 10.05 9.44 51.94
Fit Biases 0.05 0.05 0.18 0.07 0.02 0.02 0.06 0.00 0.19
Background Model 0.22 8.86 5.05 3.61 6.91 6.43 5.23 6.40 0.04
Reconstruction Efficiency 3.77 6.28 6.95 7.34 6.46 4.22 5.35 4.88 3.02
Kinematic Acceptance 1.47 6.17 8.32 8.40 7.69 5.93 4.08 1.37 4.17
Material Interaction 4.57 6.71 7.29 7.36 6.80 4.78 5.47 4.80 2.88
Fake antideuterons -0.56 -2.37 -2.04 -2.98 -2.93 -5.01 -5.37 -4.15 -1.73
DOCA Selection +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82 +5.82
Event Selection 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61
Normalization 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
Total systematic error (%) $^{ +9.64}_{ -7.70}$ $^{ +16.02}_{ -15.11}$ $^{ +15.86}_{ -14.90}$ $^{ +15.72}_{ -14.91}$ $^{ +15.83}_{ -15.00}$ $^{ +13.15}_{ -12.81}$ $^{ +12.57}_{ -12.37}$ $^{ +12.05}_{ -11.34}$ $^{ +9.50}_{ -7.71}$
Total error (%) $^{ +13.22}_{ -11.88}$ $^{ +18.87}_{ -18.10}$ $^{ +19.40}_{ -18.62}$ $^{ +19.35}_{ -18.69}$ $^{ +19.06}_{ -18.39}$ $^{ +15.57}_{ -15.28}$ $^{ +16.09}_{ -15.94}$ $^{ +15.31}_{ -14.75}$ $^{ +52.80}_{ -52.51}$
|
---
abstract: 'We study the transport properties of nonautonomous chaotic dynamical systems over a finite time duration. We are particularly interested in those regions that remain coherent and relatively non-dispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detects maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.'
author:
- |
Gary Froyland$^1$, Naratip Santitissadeekorn$^1$, and Adam Monahan$^2$\
$^1$School of Mathematics and Statistics\
University of New South Wales\
Sydney NSW 2052, Australia\
$^2$School of Earth and Ocean Sciences\
University of Victoria\
Victoria BC, Canada
bibliography:
- 'Bibchaos2010.bib'
title: 'Transport in time-dependent dynamical systems: Finite-time coherent sets'
---
Introduction
============
**Finite-time transport of time-dependent or nonautonomous chaotic dynamical systems has been the subject of intense study over the last decade. Existing techniques to analyse transport have evolved from classical geometric theory of invariant manifolds, where co-dimension 1 invariant manifolds are impenetrable transport barriers. In this work we take a very different approach, based on spectral information contained in a finite-time transfer (or Perron-Frobenius) operator. Our technique automatically identifies regions of state space that are maximally coherent or non-dispersive over a specific time interval, in the presence of an underlying chaotic system. These regions, called coherent sets, are robust to perturbation and are carried along by the chaotic flow with little transport between the coherent sets and the rest of state space. Thus, these coherent sets are ordered skeletons of the time-dependent dynamics around which the chaotic dynamics occurs relatively independently over the finite time considered. We develop the theory behind an optimization problem to determine these coherent sets and describe in detail a numerical implementation. Numerical results are given for a model system and real-world reanalysed data.**
Transport and mixing properties of dynamical systems have received considerable interest over the last two decades; see e.g. [@aref_02; @Meiss92; @wiggins92; @wiggins05] for discussions of transport phenomena. A variety of dynamical systems techniques have been introduced to explain transport mechanisms, to detect barriers to transport, and to quantify transport rates. These techniques typically fall into two classes: geometric methods, which exploit invariant manifolds and related objects as organizing structures, and probabilistic methods, which study the evolution of probability densities. Geometric methods include the study of invariant manifolds, the theory of lobe dynamics [@romkedar_etal90; @RW90; @wiggins92] in two- (and some three-) dimensions, and the notions of finite-time hyperbolic material lines [@haller00] and surfaces [@haller01]. The latter objects are often studied computationally via finite-time Lyapunov exponent (FTLE) fields [@haller00; @shadden05]. All of these geometric objects represent *transport barriers* and in this way influence (mitigate) global transport. Probabilistic approaches include a study of almost-invariant sets [@DJ97; @FD03; @froyland05; @froyland08] and very recently, coherent sets [@FLQ08; @FLS10]. For autonomous and time-dependent systems respectively, almost-invariant sets and coherent sets represent those regions in phase space which are *minimally dispersed* under the flow. Such regions provide an ordered skeleton often hidden in complicated flows. A recent comparison of the geometric and probabilistic approaches is given in [@FP09] for the time-independent and time-periodic settings.
The probabilistic methodologies provide important transport information that is often not well resolved by geometric techniques. Minimally dispersive regions need not be identified by geometric approaches. For example, recent work [@FP09] has shown that regions enclosed by FTLE ridges need not represent maximal transport barriers. Several authors [@MM10; @haller10] have noted other shortcomings of the FTLE-based approach: potential ambiguity in multiple FTLE “ridges”, ambiguity over flow duration for FTLE calculations, and a lack of correspondence between the strength of the ridge and the dispersal of mass across the ridge.
Probabilistic techniques have also been shown to be valuable analysis tools for geophysical systems. In such systems, physical quantities are often used to determine transport barriers. For example, lines of constant sea surface height (as proxies for streamlines under the assumption of geostrophy) are commonly used to determine locations of rotational trapping regions such as anticyclonic eddies and gyres [@CBT07], and maximum gradients of potential vorticity (PV) are used to determine “edges” of vortices in the stratosphere [@MP83; @Nash_etal96; @SPW97]. In both of these geophysical settings, the use of physical quantities has been shown to be non-optimal in determining the location of transport barriers [@FPET07; @SFM10].
Probabilistic and transfer operator approaches [@DJ97; @DJ99; @FD03; @SB07; @froyland05; @froyland08; @BS08; @FP09] have proven to be very effective for autonomous systems. Initial progress has been made in the development of these techniques for time-dependent systems [@FLQ08; @FLS10] over *infinite* time horizons. In the present work, we focus on transport analysis of time-dependent systems over a *finite* period of time, significantly expanding on concepts introduced in [@SFM10], and developing finite-time analogues of the time-asymptotic coherent set constructions in [@FLQ08; @FLS10]. We develop methodologies to identify those regions of phase space that are minimally dispersive, or maximally coherent, under the flow, for a specific finite time interval. We demonstrate the efficacy of our approach on two examples: an idealized stratospheric flow and a flow obtained from assimilated data sourced from the European Centre for Medium Range Weather forecasting. In the first example, we demonstrate that our new methodology easily detects an important dynamical separation of the domain; this separation is not clearly evident from an examination of the finite-time Lyapunov exponent (FTLE) field [@haller00; @haller01; @shadden05]. In the second example, we show that our new techniques can isolate the Antarctic polar vortex to high accuracy on a two-dimensional isentropic surface when compared to the commonly used potential vorticity criterion [@MP83]. We also illustrate our technique directly in three dimensions, going beyond the capabilities of existing techniques to image the vortex in three dimensions.
An outline of the paper is as follows. In section \[sect:to\] we describe our setting and outline our main computational tool, the transfer operator (or Perron-Frobenius operator), and our numerical approximation approach. In section \[sect:tech\] we motivate and detail our new computational approach. Section \[sect:comp\] describes the necessary computations and sections \[sect:eg1\] and \[sect:eg2\] illustrate our new methodology via two case studies.
Flows, coherent sets, and transfer operators {#sect:to}
============================================
Let $M\subset \mathbb{R}^d$ be a compact smooth manifold and consider a time-dependent vector field $f(z,t)$, $z\in M$, $t\in
\mathbb{R}$. Suppose that $f$ is smooth enough for the existence of a flow map $\Phi(z,t,\tau):M\times \mathbb{R}\times \mathbb{R}\to
M$, which describes the terminal location of an initial point $z$ at time $t$, flowing for $\tau$ time units.
Given a base time $t$ and a flow duration $\tau$, our motivation is to discover *coherent pairs* of subsets $A_t, A_{t+\tau}\subset M$ such that $\Phi(A_t,t,\tau)\approx A_{t+\tau}$. More precisely, we will call $A_t, A_{t+\tau}$ a *$(\rho_0,t,\tau)-$coherent pair* if $$\label{eq:invariant}
\rho_\mu(A_t,A_{t+\tau}):={\mu(A_t\cap \Phi(A_{t+\tau},t+\tau;
-\tau))}/{\mu(A_t)}\ge \rho_0,$$ and $\mu(A_t)=\mu(A_{t+\tau})$, where $\mu$ is a “reference” probability measure at time $t$. The measure $\mu$ describes the mass distribution of the quantity we wish to study the transport of over the interval $[t,t+\tau]$; $\mu$ need not be invariant under the flow $\Phi$.
We are only interested in coherent pairs that remain coherent under small diffusive perturbations of the flow; robust coherent pairs. Clearly, a $(1,t,\tau)$-coherent pair can be produced by choosing an arbitrary $A_t$ and setting $A_{t+\tau}=\Phi(A_t,t;\tau)$. However, such a pair may not be stable if some diffusion is added to the system. In a chaotic system, the set $A_{t+\tau}=\Phi(A_t,t;\tau)$ defined as above will experience stretching and folding, and for moderate to large $\tau$ will become very thin and geometrically irregular. A small amount of diffusion will then easily eject many particles from $A_{t+\tau}$, reducing the coherence ratio $\rho_\mu(A_t,A_{t+\tau})$. The requirement that coherent pairs be robust under diffusive perturbations favors coherent sets that are geometrically regular; these robust, regular sets are more likely to be more dynamically meaningful than non-robust, irregular sets.
Our basic tool for identifying sets satisfying is the transfer (or Perron-Frobenius) operator $\mathcal{P}_{t,\tau}:L^1(M,\ell)\circlearrowleft$ defined by $$\label{eq:pfeqn}
\mathcal{P}_{t,\tau}f(z):=f(\Phi(z,t+\tau;-\tau))\cdot|\det
D\Phi(z,t+\tau; -\tau)|$$ where $\ell$ is normalized Lebesgue measure on $M$. If $f(z)$ is a density of passive tracers at time $t$, $\mathcal{P}_{t,\tau}f(z)$ is the tracer density at time $t+\tau$ induced by the flow $\Phi$. In the autonomous setting, almost-invariant sets were determined [@DJ99; @FD03; @froyland05] by thresholding eigenfunctions $f$ of $\mathcal{P}_{t,\tau}$ ($=\mathcal{P}_\tau$ for all $t$) corresponding to positive eigenvalues $\lambda\approx
1$: $A=\{f<c\}$ or $\{f>c\}$.
The above calculations involved constructing a Perron-Frobenius operator for the action of $\Phi$ on the *entire domain* $M$. In the time-dependent setting, we wish to study transport from $X\subset M$ to a small neighborhood $Y$ of $\Phi(X,t;\tau)\subset M$. A global analysis would mean that $X=Y=M$ and a transfer operator would be constructed for all of $M$. However, often one is interested in the situation where the domain of interest $X$ is “open” and trajectories may leave $X$ in a finite time (our numerical examples in Sections \[sect:eg1\] and \[sect:eg2\] illustrate this). Moreover, the subset $X$ may be very small in comparison to $M$. In such instances, there are great computational savings if the analysis can be carried out using a non-global Perron-Frobenius operator defined on $X$ rather than $M$. *Our new methodology allows precisely this and is a significant theoretical and numerical advance over existing transfer operator numerics.*
We now describe a numerical approximation of the action of $\mathcal{P}_{t,\tau}$ from a space of functions supported on $X$ to a space of functions supported on $Y$. We subdivide the subsets $X$ and $Y$ into collections of sets $\{B_1,\ldots,B_m\}$ and $\{C_1,\ldots,C_n\}$ respectively. We construct a finite-dimensional numerical approximation of the transfer operator $\mathcal{P}_{t,\tau}$, using a modification of Ulam’s method [@ulam]: $$\label{eq:measureP}
\mathbf{P}^{(\tau)}(t)_{i,j}=\frac{\ell(B_i\cap\Phi(C_j,t+\tau;-\tau))}{\ell(B_i)},$$ where $\ell$ is a normalized volume measure. Clearly, the the matrix $\mathbf{P}^{(\tau)}(t)$ is row-stochastic by its construction. The value $\mathbf{P}^{(\tau)}(t)_{i,j}$ may be interpreted as the probability that a randomly chosen point in $B_i$ has its image in $C_j$. We numerically estimate $\mathbf{P}^{(\tau)}(t)_{i,j}$ by $$\label{eq:numericalulameqn}
\mathbf{P}^{(\tau)}(t)_{i,j}\approx{\#\{r:z_{i,r}\in B_i,
\Phi(z_{i,r},t;\tau)\in C_j\}}/{Q},$$ where $z_{i,r}$, $r=1,\ldots,Q$ are uniformly distributed test points in $B_i(t)$ and $\Phi(z_{i,r},t;\tau)$ is obtained via a numerical integration.
The numerical discretization has the useful side-benefit of producing a discretization-induced diffusion with magnitude the order of the image of box diameters (see Lemma 2.2 [@froyland95sbr]). Ultimately, in Section \[sect:comp\] we will construct coherent sets by thresholding vectors in $\sp\{\chi_{B_1},\ldots,\chi_{B_m}\}$ and $\sp\{\chi_{C_1},\ldots,\chi_{C_n}\}$. This discretization limits the irregularity of possible coherent sets, and in practice, high regularity is observed.
Coherent partitions {#sect:tech}
===================
For the remainder of the paper we set $P_{ij}=\mathbf{P}^{(\tau)}(t)_{i,j}$, fixing $t$ and $\tau$. We set $p_i=\mu(B_i), i=1,\ldots,m$ and assume that $p_i>0$ for all $i=1,\ldots,n$ (if some sets $B_i$ have zero reference measure, we remove them from our collection as there is no mass to be transported). Define $q=pP$ to be the image probability vector on $Y$; we assume $q>0$ (if not, we remove sets $C_j$ with $q_j=0$). The probability vector $q$ defines a probability measure $\nu$ on $Y$ via $\nu(Y')=\sum_{j=1}^n q_j\ell(Y'\cap C_j)$ for measurable $Y'\subset Y$. We may think of the probability measure $\nu$ as the discretized image of $\mu$.
Problem setup
-------------
To find the most coherent pair, we first try to partition $X$ and $Y$ as $X_1\cup X_2$ and $Y_1\cup Y_2$ in a particular way, where $X_1, X_2, Y_1, Y_2$ all have measure approximately 1/2. This restriction will be relaxed later. Let $I_1,I_2$ partition $\{1,\ldots,m\}$ and $J_1,J_2$ partition $\{1,\ldots,n\}$ and set $X_k=\cup_{i\in I_k} B_i$ and $Y_k=\cup_{j\in J_k} C_j$, $k=1,2$. We desire:
1. $\mu(X_k)=\sum_{i\in I_k}p_i\approx 1/2, \nu(Y_k)=\sum_{j\in J_k}q_i\approx 1/2, k=1,2$
(the sets $X_1, X_2$ and $Y_1, Y_2$ partition $X$ and $Y$ into two sets of roughly equal $\mu$-mass and $\nu$-mass respectively).
2. $\rho_\mu(X_k,Y_k)\approx 1, k=1,2$
(this is a measure-theoretic way of saying that $\Phi(X_k,t;\tau)\approx Y_k$, $k=1,2$).
Solution approach
-----------------
Introduce the inner products $\langle x_1,x_2\rangle_p=\sum_i x_{1,i}x_{2,i}p_i$ and $\langle y_1,y_2\rangle_q=\sum_j y_{1,j}y_{2,j}q_j$. We form a normalized matrix $L_{ij}=p_iP_{ij}/q_j$. The $L$ resulting from this normalization of $P$ ensures that $\mathbf{1}L=\mathbf{1}$. We think of $L$ as a transition matrix from the inner product space $(\mathbb{R}^m,\langle\cdot,\cdot\rangle_p)$ to the inner product space $(\mathbb{R}^n,\langle\cdot,\cdot\rangle_q)$, which takes a uniform density on $(\mathbb{R}^m,\langle\cdot,\cdot\rangle_p)$ (representing the measure $\mu$) to a uniform density on $(\mathbb{R}^n,\langle\cdot,\cdot\rangle_q)$ (representing the measure $\nu$). To describe 2-partitions of $X$ and $Y$ we consider vectors $x\in\{\pm 1\}^m, y\in \{\pm 1\}^n$ and define $X_1=\bigcup_{i:x_i=1}B_i, X_2=\bigcup_{i:x_i=-1}B_i, Y_1=\bigcup_{i:y_i=1}C_i, Y_2=\bigcup_{i:y_i=-1}C_i$. Thus, the partitions $I_1,I_2$ and $J_1,J_2$ are described by the parity of $x$ and $y$ respectively. We can write the condition $\mu(X_k)=\sum_{i\in I_k}p_i\approx 1/2, \nu(Y_k)=\sum_{j\in J_k}q_i\approx 1/2, k=1,2$ as $|\langle x,\mathbf{1}\rangle_p|, |\langle y,\mathbf{1}\rangle_q|<\epsilon$ for small $\epsilon$ (the $\epsilon$ is needed as it may be impossible to form finite collections of sets $B_i$ and $C_j$ with measure exactly 1/2).
Consider the problem: $$\label{combinatorial}
\max\{\langle xL,y\rangle_q: x\in\{\pm 1\}^m, y\in \{\pm 1\}^n, |\langle x,\mathbf{1}\rangle_p|, |\langle y,\mathbf{1}\rangle_q|<\epsilon\}$$ for some small $\epsilon$.
The objective $$\begin{aligned}
\label{eq:objective}
\langle xL,y\rangle_q&=&\left(\sum_{i\in I_1,j\in J_1} L_{ij}q_j+\sum_{i\in I_2,j\in J_2} L_{ij}q_j\right)-\left(\sum_{i\in I_1,j\in J_2} L_{ij}q_j+\sum_{i\in I_2,j\in J_1} L_{ij}q_j\right)\\
&=&\left(\sum_{i\in I_1,j\in J_1} p_iP_{ij}+\sum_{i\in I_2,j\in J_2} p_iP_{ij}\right)-\left(\sum_{i\in I_1,j\in J_2} p_iP_{ij}+\sum_{i\in I_2,j\in J_1} p_iP_{ij}\right)\\
&\approx&\left(\mu(X_1\cap \Phi(Y_1,t+\tau; -\tau))+\mu(X_2\cap \Phi(Y_2,t+\tau; -\tau))\right)\\
&\quad&-\left(\mu(X_1\cap \Phi(Y_2,t+\tau; -\tau))+\mu(X_2\cap \Phi(Y_1,t+\tau; -\tau))\right)\\
&=&\mu(X_1)\rho_\mu(X_1,Y_1)+\mu(X_2)\rho_\mu(X_2,Y_2)-\mu(X_1)\rho_\mu(X_1,Y_2)-\mu(X_2)\rho_\mu(X_2,Y_1).\end{aligned}$$ Thus, maximizing $\langle xL,y\rangle_q$ is a very natural way to achieve our aim of finding partitions so that $\rho_\mu(X_k,Y_k)\approx 1, k=1,2$. The approximation in the above reasoning occurs because $P_{ij}\approx \mu(B_i\cap\Phi(C_j,t+\tau;-\tau))/\mu(B_i)$[^1].
The problem (\[combinatorial\]) is a difficult combinatorial problem; as a heuristic means of finding a good solution we relax the binary restriction on $x$ and $y$ and allow them to take on continuous values. We will interpret the values of $x$ and $y$ as “fuzzy inclusions”; if $x_i$ is very positive, then $B_i$ is very likely to belong to $X_1$, and if $x_i$ is very negative, then $B_i$ is very likely to belong to $X_2$. Similarly for strong positivity or negativity of $y_i$ and inclusion of $B_i$ in $Y_1$ or $Y_2$ respectively. If the value of $x_i$ or $y_i$ is near to zero, the fuzzy inclusion is less certain and we use an optimization in Algorithm \[alg1\] (Section 3.3) to determine where $B_i$ belongs.
As $x$ and $y$ can now float freely, we can set $\epsilon=0$, and thus may insist that $\langle x,\mathbf{1}\rangle_p=\langle y,\mathbf{1}\rangle_q=0$. When restricting $x$ and $y$ to be elements of $\{\pm 1\}^m$ and $\{\pm 1\}^n$, we implicitly set the norms $\|x\|_p=\langle x,x\rangle_p^{1/2}$ and $\|y\|_q=\langle y,y\rangle_q^{1/2}$ to both be 1. Now that we let $x$ and $y$ freely float, we must include normalization terms in our objective. Thus, the relaxed problem is $$\label{relaxed}
\max_{x\in\mathbb{R}^m, y\in\mathbb{R}^n}\left\{\frac{\langle xL,y\rangle_q}{\|x\|_p\|y\|_q}: \langle x,\mathbf{1}\rangle_p=\langle y,\mathbf{1}\rangle_q=0\right\}$$ We will use the optimal $x$ and $y$ to create our partition $X_1, X_2$ and $Y_1, Y_2$ via $X_1=\bigcup_{i:x_i>b}B_i, X_2=\bigcup_{i:x_i<b}B_i, Y_1=\bigcup_{i:y_i>c}C_i, Y_2=\bigcup_{i:y_i<c}C_i$, where $b$ and $c$ are chosen so that $\sum_{i\in I_k}p_i\approx 1/2 \sum_{j\in J_k}q_i\approx 1/2, k=1,2$. As an extension to our heuristic, we may also relax the condition that the measures of $X_1, X_2$ and $Y_1, Y_2$ are all approximately 1/2, and only enforce $\mu(X_k)=\sum_{i\in I_k}p_i\approx \sum_{j\in J_k}q_i=\nu(Y_k), k=1,2$. This would mean that while there is some flexibility in the choice of $b$, the value $c$ is a function of $b$; see Algorithm \[alg1\].
We close this section with a lemma stating the solution to (\[relaxed\]). \[sect:comp\]
\[complemma\] Let $\Pi_p$ be an $m\times m$ diagonal matrix with $p$ on the diagonal and $\Pi_q$ be an $n\times n$ diagonal matrix with $q$ on the diagonal. Suppose that $PP^\top$ is an irreducible matrix[^2]. The value of (\[relaxed\]) is $\sigma_2$, the second largest singular value of $\Pi_p^{1/2}P\Pi_q^{-1/2}$, and the maximizing $x$ and $y$ in (\[relaxed\]) are given by $x=\hat{x}\Pi_p^{-1/2}$ and $y=\hat{y}\Pi_q^{-1/2}$, where $\hat{x}$ and $\hat{y}$ are the corresponding left and right singular vectors.
See appendix.
Extraction of coherent pairs
----------------------------
We now detail the procedure that extracts the coherent pairs $X_k, Y_k$ from the vectors $x$ and $y$ identified in Lemma \[complemma\]. We create sets that are unions of boxes with $x$ and $y$ values above certain thresholds. Define ${X}_1(b):=\bigcup_{i:x_i>b}B_i$ and ${Y}_{1}(c):=\bigcup_{j:y_j>c}C_j$, $b,c\in\mathbb{R}$. Define $$\label{eq:coherence}
\tilde{\rho}(\tilde{X}_1(b),\tilde{Y}_1(c))=\frac{\sum_{i:B_i\subset \tilde{X}_1(b),j:C_j\subset \tilde{Y}_1(c)}p_iP_{ij}}{\sum_{i:B_i\subset \tilde{X}_1(b)}p_i}.$$
The quantity $\tilde{\rho}$ measures the discretized coherence for the pair $\tilde{X}_1(b), \tilde{Y}_1(c)$. Our procedure to vary the thresholds $b$ and $c$ so as to select $\tilde{X}_1(b)$ and $\tilde{Y}_1(c)$ with largest $\tilde{\rho}$ value is summarized below:
\[alg1\]
1. Let $\eta(b)=\arg\min_{c'\in\mathbb{R}}\bigl|
\mu(\tilde{X}_1(b))-\nu(\tilde{Y}_1(c'))\bigr|$. This is to make $\nu(\tilde{Y}_1(c'))$ as close as possible to $\mu(\tilde{X}_1(b))$.
2. Set $b^*=\arg\max\tilde\rho(\tilde{X}_1(b),\tilde{Y}_1(\eta(b)))$. The value of $b^*$ is selected to maximize the coherence.
3. Define $A_t=X_1:=\tilde{X}_1(b^*)$ and $A_{t+\tau}=Y_1:=\tilde{Y}_1(\eta(b^*))$.
To obtain $X_2$ and $Y_2$, we define $X_2=\tilde{X}_2(b^*):=\bigcup_{i:x_i\le b^*}B_i$ and $Y_2=\tilde{Y}_{2}(\eta(b^*)):=\bigcup_{j:y_j\le \eta(b^*)}C_i$, the complements of $X_1$ and $Y_1$ in $X$ and $Y$, respectively. Thus, we select $X_1$ and $Y_1$ to be the most coherent pair and define $X_2$ and $Y_2$ as their respective complements. One now should repeat Algorithm \[alg1\] with $\tilde{X}_2(b):=\bigcup_{i:x_i\le b}B_i$ and $\tilde{Y}_{2}(c):=\bigcup_{j:y_j\le c}C_j$, $b,c\in\mathbb{R}$ in place of $\tilde{X}_1(b)$ and $\tilde{Y}_1(c)$ to search from “the negative end” of the vectors $x$ and $y$, possibly picking up a pair with higher coherence, and defining $X_1, Y_1$ as the complements of $X_2, Y_2$.
Example 1: Idealized Stratospheric flow {#sect:eg1}
=======================================
We consider the Hamiltonian system $\frac{dx}{dt}=-\frac{\partial\Phi}{\partial y},\frac{dy}{dt}=\frac{\partial\Phi}{\partial x}$ where $$\label{eq:idealizedflow}
\begin{split}
&\Phi(x,y,t)=c_3y-U_0L\tanh(y/L)+A_3U_0L{\mathrm{sech}}^2(y/L)\cos(k_1x)\\
&+A_2U_0L{\mathrm{sech}}^2(y/L)\cos(k_2x-\sigma_2t)+A_1U_0L{
\mathrm{sech}}^2(y/L)\cos(k_1x-\sigma_1t).\\
\end{split}$$ This quasi-periodic system represents an idealized stratospheric flow in the northern or southern hemisphere. Rypina *et al.* [@Rypina_etal07] show that there is a time-varying jet core oscillating in a band around $y=0$ and three Rossby waves in each of the regions above and below the jet core. The parameters studied in [@Rypina_etal07] are chosen so that the jet core forms a complete transport barrier between the two Rossby wave regimes above and below it. We modify some of the parameters to remove the jet core band and allow transport between the two Rossby wave regimes. We expect that the two Rossby wave regimes will form time-dependent coherent sets because transport between the two regimes is considerably less than the transport within regimes.We set the parameters as follows: $c_2/U_0=0.205$, $c_3/U_0=0.700$, $A_3=0.2, A_2=0.4$ and $A_1=0.075$, with the remaining parameters as stated in Rypina *et al.* [@Rypina_etal07].
Our initial time is $t=20$ days and our final time is $t+\tau=30$ days. At our initial time we set $X=S^1\times[-2.5, 2.5]$ Mm, where $S^1$ is a circle parameterised from 0 to $6.371\pi$ Mm, and subdivide $X$ into a grid of $m=28200$ identical boxes $X=\{B_1,\ldots,B_m\}$. This choice of $m$ is sufficiently large to represent the dynamics to a good resolution. We compute an approximation of $\Phi(X,20;30)$ by uniformly distributing $Q=400$ sample points in each grid box and numerically calculating $\Phi(z_{i,r},20;30)$ using the standard Runge-Kutta method. The choice of $Q$ is made so that over the flow duration, the image of boxes is well represented by the $Q$ sample points per box. These $Q\times m$ image points are then covered by a grid of $n=34332$ boxes $\{C_1,\ldots,C_n\}$ of the same size as the $B_i$, $i=1,\ldots,m$.
We set $Y=\bigcup_{j=1}^n C_j$, covering the approximate image of $X$. The transition matrix $P=\mathbf{P}^{(30)}_{20}$ is computed using (\[eq:numericalulameqn\]).
As the flow is area preserving, a natural reference measure $\mu$ is Lebesgue measure, which we normalize so that $\mu(X)=1$. Thus, $\mu(B_i)=p_i=1/m$, $i=1,\ldots,m$ and so $(\Pi_p)_{ii}=1/m$, $i=1,\ldots,m$. The vector $q$ is constructed as $q=pP$. We compute the second largest singular value of $\Pi_p^{1/2}P\Pi_q^{-1/2}$ and the corresponding left and right singular vectors and thus determine $x$ and $y$ from Lemma \[complemma\]. The top two singular values were computed to be $\sigma_1=1.0$ and $\sigma_2\approx0.996$. We expect $x$ to determine coherent sets at time $t=20$ days and $y$ to determine coherent sets at time $t+\tau=30$ days. Figure \[fig:subfig1\] and \[fig:subfig2\] illustrate the vectors $x$ and $y$, which provide clear separations into red (positive) and green (mostly negative) regions.
\[fig:compareSVSUFTLE\]
We apply the thresholding Algorithm \[alg1\] to the vectors $x$ and $y$ to obtain the pairs $(X_1,Y_1)$ and $(X_2,Y_2)$[^3] shown in Figures \[fig:subfig11\] and \[fig:subfig12\]. To demonstrate that $Y_1\approx \Phi(X_1,20;10)$, we plot the latter set in Figure \[fig:subfig13\]. When compared with Figure \[fig:subfig12\] we see that there is very little leakage from $Y_1$, just a few thin filaments. Similarly, Figures \[fig:subfig14\] and \[fig:subfig12\] compare $Y_2$ and $\Phi(X_2,20;10)$, again showing a small amount of leakage. This leakage is quantified by computing $\tilde{\rho}(X_1,Y_1)\approx\tilde{\rho}(X_2,Y_2)\approx0.98$.
\[fig:Interface\]
We compare our results with the attracting and repelling material lines computed via the finite-time Lyapunov exponent (FTLE) field [@haller00] with the flow time $\tau=10$. The ridges of the FTLE fields are commonly used to identify barriers to transport. Figures \[fig:subfig3\] and \[fig:subfig4\] present an overlay of forward- and backward-time FTLEs at $t=20$ and $t=30$, respectively. In this example, there are several FTLE ridges in the vicinity of the dominant transport barrier across the middle of the domain, and also several ridges far away from this barrier. The FTLE ridges do not crisply and unambiguously identify the dominant transport barrier shown in Figures \[fig:subfig1\] and \[fig:subfig2\].
Example 2: Stratospheric polar vortex as coherent sets {#sect:eg2}
======================================================
In our second example, we use velocity fields obtained from the ECMWF Interim data set (http://data.ecmwf.int/data/index.html). We focus on the stratosphere over the southern hemisphere south of 30 degrees latitude. In this region, there are strong persistent transport barriers to midlatitude mixing during the austral winter; these barriers give rise to the Antarctic polar vortex. We will apply our new methodology to the ECMWF vector fields in two and three dimensions to resolve the polar vortex as a coherent set.
Two dimensions
--------------
Our input data consists of two-dimensional velocity fields on a $121
\times 240$ element grid in the longitude and latitude directions, respectively. The ECMWF data provides updated velocity fields every 6 hours. The flow is initialised at September 1, 2008 on a 475K isentropic surface and we follow the flow until September 14. To a good approximation isentropic surfaces are close to invariant over a period about two weeks [@JL01].
We set $X= S^{1}\times[-90{\ensuremath{^\circ}}, -30{\ensuremath{^\circ}}]$, where $S^{1}$ is a circle parameterized from $0{\ensuremath{^\circ}}$ to $360{\ensuremath{^\circ}}$. The domain $X$ is initially subdivided into the grid boxes $B_i$, $i=1,\ldots,m$, where $m=13471$ in this example. Based on the hydrostatic balance and the ideal gas law, we set the reference measure $p_i=\mathrm{Pr}_i^{5/7}\mathrm{a}_i$ for all $i=1,\ldots,m$, where $\mathrm{Pr}_i$ is the pressure at the center point of $B_i$ and $\mathrm{a}_i$ is the area of box $B_i$.
Using $Q=100$ sample points $z_{i,r}$, $r=1,\ldots,Q$ uniformly distributed in each grid box $B_i$, $i=1,\ldots,m$ we calculate an approximate image $\Phi(X,t;\tau)$[^4] and cover this approximate image with $m=14395$ boxes $\{C_1,\ldots,C_n\}$ to produce the image domain $Y$. We construct $P=\mathbf{P}^{(t)}_{\tau}$ as described earlier using the same $Q\times m$ sample points.
We compute $x$ and $y$ as described in Lemma \[complemma\]; graphs of these vectors are shown in Figure \[fig:2D\] (Upper left and Upper right). Figure \[fig:2D\] (Lower left and Lower right) shows the result of Algorithm \[alg1\], extracting coherent sets $A_t$ and $A_{t+\tau}$ from the vectors $x$ and $y$. We calculate the coherent ratio $\rho_\mu(A_t,A_{t+\tau})\approx 0.991$, which means that 99.1% of the mass in $A_t$ (September 1, 2008) flows into $A_{t+\tau}$ (September 14, 2008), demonstrating a very high level of coherence.
To benchmark our new methodology, we will compare our result with a method commonly used in the atmospheric sciences to delimit the “edge” of the vortex. It has been recognized that during the winter a strong gradient of potential vorticity (PV) in the polar stratosphere is developed due to (1) strong mixing in the mid-latitudes (resulting from the breaking of Rossby waves emerging from the troposphere and breaking in the stratospheric “surf zone” [@MP83]) and (2) weak mixing in the vortex region. While potential vorticity depends only on the instantaneous vector field, potential vorticity is materially conserved for adiabatic, inviscid flow (both of which are good approximations in stratospheric flow over timescales of a week or two). Thus, PV may be viewed as a quantity derived from the Lagrangian specification of the flow and is therefore a meaningful comparator for these nonautonomous experiments who also use Lagrangian information. We used the method of Sobel et al. [@SPW97] to calculate a PV-based estimate of the vortex edge. The result is shown by the green curve in Figure \[fig:2D\] (Lower left and Lower right). Notating the area enclosed by the green curve at September 1, 2008 by $A^{PV}_t$ and at September 14, 2008 by $A^{PV}_{t+\tau}$, we compute $\rho_\mu(A^{PV}_t,A^{PV}_{t+\tau})\approx 0.984$; 98.4% of the mass in $A^{PV}_{t}$ flows into $A^{PV}_{t+\tau}$ over the 13 day period.
Our transfer operator methodology is clearly consistent with the accepted potential vorticity approach and in fact identifies a region that experiences slightly *greater* transport barriers across its boundary, indicated by the slightly larger coherence ratio: 99.1% versus 98.4%. In the next section we apply our methodology in three dimensions to estimate the three-dimensional structure of the vortex.
Three dimensions
----------------
Strong transport barriers to midlatitude mixing in the southern hemisphere are also known to exist even in the full 3D case, where strong descent occurs near the edges of polar vortex at each pressure altitude [@Plumb02; @Schoeberl_etal92]. In principle, PV-based methods could be extended to three-dimensions by (i) slicing the three-dimensional region of interest into several nearby isentropic surfaces, (ii) applying the PV methodology on each individual isentropic surface to obtain an estimate of the vortex boundary on that surface, and (iii) stitching together these curves to form a reasonable two-dimensional surface, with the hope that the surface represents an estimate of the boundary of the three-dimensional vortex. This stitching together of several curves is a nontrivial computational task and complicated geometries may be missed by this relatively simple construction. The PV approach is likely to be more susceptible to noise than our direct approach because the computation of PV relies on estimates of derivatives of the velocity field (vorticity is the curl of the velocity field). Finally, such an approach would not utilise the full three-dimensional vector field, but rather a series of vector fields on isentropic surfaces.
A key point of our new methodology is that it can easily applied in either two or three dimensions and works directly with the velocity fields to compute coherent regions with minimal external flux.
We set $X=S^{1}\times[-90{\ensuremath{^\circ}}, -30{\ensuremath{^\circ}}]\times[50,70]$, where the third (vertical) component of this direct product is in units of hPa. The ECMWF data is again provided on a $240\times 121$ grid in the longitude/latitude directions, and additionally at 7 pressure levels between 20 and 150 hPa. We use the full 3D velocity field from the ECMWF reanalysis data.
We subdivide $X$ into a grid of $m=4116\times8=32928$ (longitude-latitude$\times$pressure) boxes, where all boxes have the same area in the longitude-latitude directions and a “height” of $(70-50)/8=20/8$ hPa in the pressure direction. Following hydrostatic equilibrium considerations, we set the mass $p_i$ of box $B_i$ to be proportional to the base area of $B_i$ multiplied by the box “height” in hPa, and normalise so that $\sum_{i=1}^{32928}p_i=1$. We select $Q=250$ sample points in each grid box, uniformly distributed in the longitude-latitude direction and equally spaced in pressure direction. The $Q\times m$ images of these sample points are then covered by a grid of $n=51722$ boxes.
Repeating the approach of the two-dimensional study, the two largest singular values are computed to be $\sigma_1\approx1.0$ and $\sigma_2\approx0.9994$. A slice along the uppermost pressure level (50 hPa) of the optimal vectors $x$ and $y$ is shown in Figure \[fig:3Dslice\].
Applying Algorithm \[alg1\], we compute the coherent sets $A_t$ and $A_{t+\tau}$ shown in Figure \[fig:3dcoherents\] with $\rho_{\mu}(A_{1},A_{14})\approx 0.9890$. Figures \[fig:subfigz1\] and \[fig:subfigz2\] show that at 1 September 2008, a compact central domain with nearly vertical sides is extracted by Algorithm \[alg1\]. Figure \[fig:subfigz6\] shows that after 6 days of flow, this set is advected both upwards and downwards, and that this advection is not uniform over all latitudes. Figure \[fig:subfigz6\] and Figure \[fig:subfigz8\] (which gives a view from “below”), demonstrate that the upward flow occurs primarily near the centre of the vortex (high latitudes), while the downward flow is concentrated around the periphery (lower latitudes). A bowl-like shape is evident in Figure \[fig:subfigz8\] showing a thin layer at the core of the coherent set at 7 September, descending toward the troposphere near the edge of coherent set. This observation agrees with the motion of ozone masses in the lower stratosphere, where the mass in the mixing zone around the mid-latitude slowly moves downward and the mass in the vortex core moves within a thin stratospheric layer [@Plumb02; @Schoeberl_etal92].
Conclusions
===========
We introduced a methodology for identifying minimally dispersive regions (coherent sets) in time-dependent flows over a finite period of time. Our approach directly used the time-dependent velocity fields to construct an ensemble description of the finite-time dynamics; the Perron-Frobenius (or transfer operator). The transport of mass is explicitly calculated in terms of a reference measure considered to be most appropriate for the application by the practitioner. Singular vector computations of matrix approximations of the Perron-Frobenius operator directly yielded images of the coherent sets; the left singular vector described the coherent region at the initial time and the right singular vector at the final time. Our methodology is the first systematic transfer operator approach for handling time-dependent systems over finite time durations. A particular feature of our approach is that one can focus on small subdomains of interest, rather than study the entire domain; this leads to major computational savings.
In our first case study we used this new technique to show that an idealized stratospheric flow operates as two almost independent dynamical systems with a small amount of interaction across two Rossby wave regimes. Our second case study utilised reanalysed velocity data sourced from the European Centre for Medium Range Weather Forecasting (ECMWF) to estimate the location of the Southern polar vortex. Studying the dynamics on a two-dimensional isentropic surface, we found excellent agreement with traditional potential vorticity (PV) based approaches, and improved slightly over the PV methodology in terms of the coherence of the vortex. We also used the full three-dimensional velocity field to determine the vortex location in three dimensions, a computation not easily carried out with standard applications of the PV approach.
Proof of Lemma \[complemma\]
============================
We first show that the condition on $y$ in (\[relaxed\]) is unnecessary. $$\label{eqn1}\max_{x\in\mathbb{R}^m \atop y\in\mathbb{R}^n}\left\{\frac{\langle xL,y\rangle_q}{\|x\|_p\|y\|_q}: \langle x,\mathbf{1}\rangle_p=0\right\}=\max_{x\in\mathbb{R}^m \atop y\in\mathbb{R}^n}\left\{\frac{\langle xL,\frac{y}{\|y\|_q}\rangle_q}{\|x\|_p}: \langle x,\mathbf{1}\rangle_p=0\right\}=\max_{x\in\mathbb{R}^m}\left\{\frac{\|xL\|_q}{\|x\|_p}: \langle x,\mathbf{1}\rangle_p=0\right\},$$ with the maximizing $y$ being $y=xL$. Setting $y=xL$, we see that $\langle y,\mathbf{1}\rangle_q=\langle xL,\mathbf{1}\rangle_q=\langle x,\mathbf{1}L^*\rangle_p=\langle x,\mathbf{1}\rangle_p=0$ (it is straightforward to check $\mathbf{1}L^*=\textbf{1}$; $L^*=P^\top$). Thus, since the maximizing $y$ in (\[eqn1\]) satisfies $\langle y,\mathbf{1}\rangle_q=0$ when $\langle x,\mathbf{1}\rangle_p=0$, we see that the value of (\[relaxed\]) equals the value of the LHS of (\[eqn1\]), and both (\[relaxed\]) and (\[eqn1\]) have the same maximizing $x$ and $y$.
We now convert the RHS of (\[eqn1\]) to a maximization in the standard $\ell_2$ norm by noting that $\langle x_1,x_2\rangle_p=\langle x_1\Pi_p^{1/2},x_2\Pi_p^{1/2}\rangle_2$ and $\langle y_1,y_2\rangle_q=\langle y_1\Pi_q^{1/2},y_2\Pi_q^{1/2}\rangle_2$. $$\mbox{RHS of (\ref{eqn1})}=\max_{x\in\mathbb{R}^m}\left\{\frac{\|xL\Pi_q^{1/2}\|_2}{\|x\Pi_p^{1/2}\|_2}: \langle x\Pi^{1/2}_p,\mathbf{1}\Pi^{1/2}_p\rangle_2=0\right\}=\max_{\hat{x}\in\mathbb{R}^m}\left\{\frac{\|\hat{x}\Pi^{-1/2}_pL\Pi_q^{1/2}\|_2}{\|\hat{x}\|_2}: \langle \hat{x},p^{1/2}\rangle_2=0\right\},$$ where we have made the substitution $\hat{x}=x\Pi_p^{1/2}$. We claim that the leading singular value of $\Pi^{-1/2}_pL\Pi_q^{1/2}=\Pi^{1/2}_pP\Pi_q^{-1/2}$ is 1, with corresponding left singular vector $p^{1/2}$.
To prove this claim, we show that 1 is the leading singular value of $L$ with corresponding left singular vector $\mathbf{1}$ (where $L$ is always considered as a linear mapping from $\langle \cdot,\cdot\rangle_p$ to $\langle \cdot,\cdot\rangle_q$). Since $\mathbf{1}L=\mathbf{1}$ and $\mathbf{1}L^*=\mathbf{1}$, one has $\mathbf{1}LL^*=\mathbf{1}$; also $LL^*$ is irreducible iff $PP^\top$ is irreducible. By the Perron-Frobenius Theorem (eg. Thm 1.4 and 2.1 [@berman_plemmons]), 1 is the largest real eigenvalue of $LL^*$, and is simple; hence the largest singular value of $L$ is 1 and the left and right singular vectors are $\mathbf{1}\in\mathbb{R}^m$ and $\mathbf{1}\in\mathbb{R}^n$ respectively.
The result now follows from the Courant-Fischer theorem for symmetric matrices (see eg. Thm. 4.2.11 [@horn_johnson]), standard properties of singular vectors and the computation $y=xL=\hat{x}\Pi_p^{1/2}L=(\hat{x}\Pi_p^{-1/2}L\Pi_q^{1/2})\Pi_q^{-1/2}=\hat{y}\Pi_q^{-1/2}$ where $\hat{y}$ is the right singular vector of $\Pi_p^{-1/2}L\Pi_q^{1/2}$ corresponding to $\sigma_2$.
[^1]: If $\mu$ is absolutely continuous with a positive density that is Lipschitz on the interior of each $B_i$, then this error goes to zero with decreasing diameter of $B_i$ and $C_j$; see Lemma 3.6 [@froyland98].
[^2]: there exists a $k$ such that $(PP^\top)^k>0$.
[^3]: When determining $X_1$ and $Y_1$, Algorithm \[alg1\] produced values $b^*\approx0.0077$ and $\eta(b^*)\approx0.0005$.
[^4]: We use the standard Runge-Kutta method with step size of $3/4$ hours. Linear interpolation is used to evaluate the velocity vector of a tracer lying between the data grid points in the longitude-latitude coordinates. In the temporal direction the data is independently affinely interpolated.
|
---
abstract: 'A $(n +1)$-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological-type metrics, the equations of motion are reduced to a set of Lagrange equations. The effective Lagrangian contains two “minisuperspace” metrics on ${ {\mathbb R} }^{n}$. The first one is the well-known 2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric that is proportional to $n$-dimensional Berwald-Moor 4-metric. When a “synchronous-like” time gauge is considered the equations of motion are reduced to an autonomous system of first-order differential equations. For the case of the “pure” Gauss-Bonnet model, two exact solutions with power-law and exponential dependence of scale factors (with respect to “synchronous-like” variable) are obtained. (In the cosmological case the power-law solution was considered earlier in papers of N. Deruelle, A. Toporensky, P. Tretyakov and S. Pavluchenko.) A generalization of the effective Lagrangian to the Lowelock case is conjectured. This hypothesis implies existence of exact solutions with power-law and exponential dependence of scale factors for the “pure” Lowelock model of $m$-th order.'
---
**On cosmological-type solutions in multi-dimensional model with Gauss-Bonnet term**
**V. D. Ivashchuk\[1\]\[1\][e-mail: ivashchuk@mail.ru]{},**
*Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya ul., Moscow 119361, Russia*
*Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya ul., Moscow 117198, Russia*
Introduction
============
Here we deal with $D$-dimensional gravitational model with the Gauss-Bonnet term. The action reads $$S = \int_{M} d^{D}z \sqrt{|g|} \{ \alpha_1 R[g] +
\alpha_2 {\cal L}_2[g] \},
\label{1.1}$$ where $g = g_{MN} dz^{M} \otimes dz^{N}$ is the metric defined on the manifold $M$, ${\dim M} = D$, $|g| = |\det (g_{MN})|$ and
$${\cal L}_2 = R_{MNPQ} R^{MNPQ} - 4 R_{MN} R^{MN} +R^2
\label{1.2}$$
is the standard Gauss-Bonnet term. Here $\alpha_1$ and $\alpha_2$ are constants. The appearance of the renormalizable Gauss-Bonnet term as well as quadratic Riemann curvature terms in multidimensional gravity is motivated by string theory [@Zwiebach; @GBstrings1; @GBstrings2; @GBstrings3; @GBstrings4]. (For a review of fourth-order gravity in $D=4$, see [@Schmidt].)
At present, the so-called Einstein-Gauss-Bonnet (EGB) gravity and its modifications are intensively used in cosmology, see [@NojOd0; @CElOdZ] (for $D =4$), [@Ishihara; @ElMakObOsFil; @BambaGuoOhta; @TT; @KirMPTop; @PTop; @KirMak] and references therein, e.g. for explanation of accelerating expansion of the Universe following from supernovae (type Ia) observational data [@Kowalski]. Certain exact solutions in multidimesional EGB cosmology were obtained in [@Ishihara]-[@KirMak] and some other papers.
EGB gravity is also intensively investigated in a context of black-hole physics. The most important results here are related with the well-known Boulware-Deser-Wheeler solution (corresponding to the Schwarzschild-Tangherlini solution in general relativity) [@BoulDes; @Wheel] and its generalizations [@Wheel2; @Wilt; @Cai; @CvetNojOd], for a review and references, see [@GarGir; @Charm]. For certain applications of brane-world models with Gauss-Bonnet term, see review [@BrKonMel] and references therein.
Here we are interested in the cosmological (type) solutions with diagonal metrics (of Bianchi-I-like type) governed by scale factors depending upon one variable.
For $\alpha_2 = 0$, we have the Kasner type solution with the metric $$g= - d \tau \otimes d \tau +
\sum_{i=1}^{n} A_i^2 \tau^{2p^i} dy^i \otimes dy^i, \label{1.3}$$ where $A_i > 0$ are arbitrary constants, $D = n +1$ and parameters $p^i$ obey the relations $$\begin{aligned}
\sum_{i=1}^n p^i = 1, \label{1.4}\\
\sum_{i=1}^n (p^i)^2 = 1 \label{1.5}
\end{aligned}$$ and hence $$\sum_{ 1 \leq i < j \leq n} p^i p^j =
\frac{1}{2} ( \sum_{i=1}^n p^i )^2 -
\frac{1}{2} \sum_{i=1}^n (p^i)^2
= 0 . \label{1.5a}$$ For $D =4$, this is the well-known Kasner solution [@Kasner]. The set of eqs. (\[1.4\]), (\[1.5\]) is equivalent to the set of eqs. (\[1.4\]), (\[1.5a\]).
In [@Deruelle], a Einstein-Gauss-Bonnet (EGB) cosmological model was considered. For “pure” Gauss-Bonnet (GB) case $\alpha_1 = 0$ and $\alpha_2 \neq 0$, N. Deruelle has obtained a cosmological solution with the metric (\[1.3\]) for $n = 4, 5$ and parameters obeying the relations $$\begin{aligned}
\sum_{i=1}^n p^i = 3, \label{1.6}\\
\sum_{1 \leq i < j < k < l \leq n} p^i p^j p^k p^l = 0 \label{1.7}.
\end{aligned}$$ It was reported by A. Toporensky and P. Tretyakov in [@TT] that this solution was verified by them for $n = 6,7$. In recent paper by S. Pavluchenko [@Pavl] the power-law solution was verified for all $n$ (and also generalized to the Lowelock case [@Low]).
In this paper we give a derivation of the “power-law” (cosmological type) solution for arbitrary $n$. We also show that for $D \neq 4 $ this solution in “pure” GB cosmology is unique in a class of solutions with power-law dependence of scale factors: $a_i(\tau) = A_i \tau^{p^i}$, when the parameters $p^1,...,p^n$ contain more than two non-zero numbers. When $(n-2)$ parameters among $p^i$ are zero, say $p^3 =...= p^n =0$, than the metric (\[1.3\]) obeys the equations of motion (for $\alpha_1 =
0$) for arbitrary values of two Kasner-like parameters (say $p^1, p^2$).
The numerical analysis of cosmological solutions in EGB gravity for $D = 5, 6$ [@PTop] shows that the singular “power-law” solutions (\[1.3\]), (\[1.6\]), (\[1.7\]) (e.g. with a little generalization of scale factors $a_i(\tau) = A_i (\tau_0 \pm
\tau)^{p^i}$, where $\tau_0$ is constant) appear as asymptotical solutions for certain initial values as well as Kasner-type solutions (\[1.3\])-(\[1.5\]) do.
The paper is organized as follows. In Section 2 the equations of motion for $(n +1)$-dimensional EGB model are considered. For diagonal cosmological type metrics the equations of motion are reduced to a set of Lagrange equations corresponding to certain “effective” Lagrangian (in agreement with [@Deruelle; @Pavl] for cosmological case). Section 3 is devoted to the case of the “pure” Gauss-Bonnet model. Two exact solutions: with power-law and exponential dependence of scale factors (with respect to “synchronous-like” variable) are obtained. In Section 4 the equations of motion are reduced to an autonomous system of first order differential equations (when a “synchronous-like” time gauge is considered). For $\alpha_1 \neq 0$ and $\alpha_2 \neq 0$ it is shown that for any non-trivial solution with exponential dependence of scale factors $a_i(\tau) = A_i \exp( v^i
\tau)$, $i = 1,...,n$, there are no more than three different numbers among $v^1,...,v^n$. In Section 5 a generalization of the effective Lagrangian to the Lowelock case is conjectured and exact solutions with power-law and exponential dependence of scale factors for the “pure” Lowelock model of $m$-th order are presented. (See also [@Deruelle; @Pavl] for “power law” cosmological solutions.) Certain useful relations and proofs are collected in Appendix.
The cosmological type model and its effective Lagrangian
========================================================
The set-up
-----------
Here we consider the manifold $$M = { {\mathbb R} }_{*} \times M_{1} \times \ldots \times M_{n}, \label{2.1}$$ with the metric $$g= w e^{2{\gamma}(u)} du \otimes du +
\sum_{i=1}^{n} e^{2\beta^i(u)} { \varepsilon }_{i} dy^i \otimes dy^i, \label{2.2}$$ where $w = \pm 1$ and any $M_i$ is 1-dimensional manifold with the metric $g^i = { \varepsilon }_{i} dy^i \otimes dy^i$, $ { \varepsilon }_{i}= \pm 1$, $i = 1, \dots, n$. Here and in what follows ${ {\mathbb R} }_{*} = (u_{-},u_{+})$ is an open subset in ${ {\mathbb R} }$. (Here we identify $g^i$ with $\hat{g}^{i} = p_{i}^{*} g^{i}$ which is the pullback of the metric $g^{i}$ to the manifold $M$ by the canonical projection: $p_{i} : M \rightarrow M_{i}$, $i = 1,\ldots, n$.) The functions ${\gamma}(u)$ and $\beta^i(u)$, $i = 1,\ldots, n$, are smooth on ${ {\mathbb R} }_{*} = (u_{-},u_{+})$.
For $w = -1$, $ { \varepsilon }_{1}= ... = { \varepsilon }_{n} = 1$ the metric (\[2.2\]) is a cosmological one while for $w = 1$, $ { \varepsilon }_{1}= -1$, ${ \varepsilon }_{2} = ... ={ \varepsilon }_{n} = 1$ it describes static configurations.
According to Appendix A, the integrand in (\[1.1\]), when the metric (\[2.2\]) is substituted, reads as follows $$\sqrt{|g|} \{ \alpha_1 R[g] +
\alpha_2 {\cal L}_2[g] \} = L + \frac{df}{du},
\label{2.3}$$ where $$\begin{aligned}
L = \alpha_1 L_1 + \alpha_2 L_2,
\label{2.4}\\
L_1 = (-w) e^{-\gamma + \gamma_0} G_{ij} \dot{\beta}^i
\dot{\beta}^j,
\label{2.5} \\
L_2 = - \frac{1}{3} e^{- 3 \gamma + \gamma_0}
G_{ijkl} \dot{\beta}^i \dot{\beta}^j \dot{\beta}^k
\dot{\beta}^l,
\label{2.6}
\end{aligned}$$ $\gamma_0 = \sum_{i =1}^{n} \beta^i$ and $$\begin{aligned}
G_{ij} = \delta_{ij} -1,
\label{2.10} \\
G_{ijkl} = (\delta_{ij} -1)(\delta_{ik} -1)(\delta_{il} -1)
(\delta_{jk} -1)(\delta_{jl} -1)(\delta_{kl} -1)
\label{2.11}
\end{aligned}$$ are respectively the components of two “minisuperspace” metrics on ${ {\mathbb R} }^{n}$. (For cosmological case see also [@Deruelle; @Pavl; @Iv-09].) The first one is the well-known 2-metric of pseudo-Euclidean signature: $<v_1,v_2> = G_{ij}v^i_1 v^j_2$ and the second one is the Finslerian 4-metric: $<v_1,v_2,v_3,v_4> = G_{ijkl}v^i_1 v^j_2 v^k_3 v^l_4$, $v_s = (v^i_s) \in { {\mathbb R} }^n$, where $<.,.>$ and $<.,.,.,.>$ are respectively $2$- and $4$-linear symmetric forms on ${ {\mathbb R} }^n$. (Here we denote $\dot{A} = dA/du$ etc.)
In (\[2.3\]) the function $f = f(\gamma, \beta, \dot{\beta})$ has the following form: $$f = \alpha_1 f_1 + \alpha_2 f_2,
\label{2.7}$$ where $f_1$ and $f_2$ are defined in Appendix A (see (\[A.18f1\]) and (\[A.18f2\])).
The derivation of (\[2.4\])-(\[2.6\]) is based on the relations obtained in Appendix A (see (\[A.18L1\]), (\[A.18L2\])) and the following identities $$\begin{aligned}
G_{ij}v^i v^j = \sum_{i =1}^{n} (v^i)^2 -
(\sum_{i =1}^{n} v^i )^2,
\label{2.12} \\
G_{ijkl}v^i v^j v^k v^l = (\sum_{i =1}^{n} v^i )^4
- 6 (\sum_{i =1}^{n} v^i )^2
\sum_{j =1}^{n} (v^j)^2
\nonumber \\
+ 3 ( \sum_{i =1}^{n} (v^i)^2 )^2
+ 8 (\sum_{i =1}^{n} v^i )
\sum_{j =1}^{n} (v^j)^3 - 6 \sum_{i =1}^{n} (v^i)^4.
\label{2.13}
\end{aligned}$$ The first identity (\[2.12\]) is a trivial one. The second one (\[2.13\]) may be verified by straightforward calculations (see Appendix B).
It follows immediately from the definitions (\[2.10\]) and (\[2.11\]) that $$\begin{aligned}
G_{ij}v^i v^j = -2 \sum_{i < j} v^i v^j,
\label{2.14} \\
G_{ijkl}v^i v^j v^k v^l = 24 \sum_{i < j < k < l} v^i v^j
v^k v^l .
\label{2.15}
\end{aligned}$$
Due to (\[2.15\]), $G_{ijkl}v^i v^j v^k
v^l$ is zero for $n = 1, 2, 3$ ($D = 2, 3, 4$). For $n = 4$ ($D = 5$), $G_{ijkl}v^i v^j v^k v^l = 24 v^1 v^2 v^3
v^4$ and our 4-metric is proportional to the well-known Berwald-Moor 4-metric [@Berwald; @Moor] (see also [@Bogos; @GarPav] and references therein). We remind the reader that the 4-dimensional Berwald-Moor 4-metric obeys the relation: $<v,v,v,v>_{BM} =v^1 v^2 v^3v^4$. The Finslerian 4-metric with components (\[2.11\]) coincides up to a factor with the $n$-dimensional analogue of the Berwald-Moor 4-metric.
The equations of motion
------------------------
The equations of motion corresponding to the action (\[1.1\]) have the following form $${\cal E}_{MN} = \alpha_1 {\cal E}^{(1)}_{MN}
+ \alpha_2 {\cal E}^{(2)}_{MN} = 0,
\label{1.3e}$$ where $$\begin{aligned}
{\cal E}^{(1)}_{MN} = R_{MN} - \frac{1}{2} R g_{MN},
\label{1.3a} \\
{\cal E}^{(2)}_{MN} = 2(R_{MPQS}R_N^{\ \ PQS} - 2 R_{MP} R_N^{\ \ P}
\nonumber \\
-2 R_{MPNQ} R^{PQ} + R R_{MN}) - \frac{1}{2} {\cal L}_2 g_{MN}.
\label{1.3b}
\end{aligned}$$
The field equations (\[1.3e\]) for the metric (\[2.2\]) are equivalent to the Lagrange equations corresponding to the Lagrangian $L$ from (\[2.4\]). This follows from the relations $$\begin{aligned}
{\cal E}_{00}(-2w) \exp(\gamma_0 - \gamma) =
\frac{{\partial}L}{{\partial}\gamma},
\label{2.16a} \\
{\cal E}_{ii}(-2 { \varepsilon }_{i})
\exp(\gamma + \gamma_0 - 2\beta^i) =
\frac{{\partial}L}{{\partial}\beta^i} - \frac{d}{du} \frac{{\partial}L}{{\partial}\dot{\beta}^i}
\label{2.16b}, \\
{\cal E}_{0i} = 0,
\label{2.16c}
\end{aligned}$$ $i = 1,\ldots, n$.
Formulas (\[2.16a\])-(\[2.16c\]) may be verified just by straightforward calculations based on the relations for the Riemann tensor from Appendix A. But there exists a more “economic” way to prove these formulas using: [**(i)**]{} the diagonality of the matrix ${\cal E}_{MN}$ (in coordinates $(y^M) = (y^0 =u, y^i)$); [**(ii)**]{} the dependence of this matrix only on one variable $u$, i.e. ${\cal E}_{MN} = {\cal E}_{MN}(u)$; [**(iii)**]{} the relation (\[2.3\]). The proof of (\[2.16a\])-(\[2.16c\]) is given in Appendix C.
Thus, equations (\[1.3e\]) read as follows $$\begin{aligned}
w \alpha_1 G_{ij} \dot{\beta}^i \dot{\beta}^j
+ \alpha_2 e^{- 2 \gamma}
G_{ijkl} \dot{\beta}^i \dot{\beta}^j \dot{\beta}^k
\dot{\beta}^l = 0, \label{2.17} \\
\frac{d}{du}[ - 2w \alpha_1 G_{ij} e^{-\gamma +
\gamma_0}
\dot{\beta}^j \qquad \qquad \nonumber \\
- \frac{4}{3} \alpha_2 e^{- 3 \gamma + \gamma_0}
G_{ijkl} \dot{\beta}^j \dot{\beta}^k
\dot{\beta}^l] - L = 0, \label{2.18}
\end{aligned}$$ $i = 1,\ldots, n$. Due to (\[2.17\]) $$L= -w \frac{2}{3} e^{-\gamma + \gamma_0}
\alpha_1 G_{ij} \dot{\beta}^i \dot{\beta}^j.
\label{2.18a}$$
Exact solutions in Gauss-Bonnet model
=====================================
Now we put $\alpha_1 = 0$ and $\alpha_2 \neq 0$, i.e. we consider the cosmological type model governed by the action $$S_2 = \alpha_2 \int_{M} d^{D}z \sqrt{|g|} {\cal L}_2[g].
\label{3.1}$$
The equations of motion (\[1.3e\]) in this case read $${\cal E}^{(2)}_{MN} = {\cal R}^{(2)}_{MN} -
\frac{1}{2} {\cal L}_2 g_{MN} = 0,
\label{3.1a}$$ where $$\begin{aligned}
{\cal R}^{(2)}_{MN} =
2(R_{MPQS}R_N^{\ \ PQS} - 2 R_{MP} R_N^{\ \ P}
\nonumber \\
-2 R_{MPNQ} R^{PQ} + R R_{MN}).
\label{3.1b}
\end{aligned}$$
Due to identity $g^{MN} {\cal R}^{(2)}_{MN} = 2 {\cal L}_2$, the set of eqs. (\[3.1a\]) for $D \neq 4$ implies $${\cal L}_2 = 0.
\label{3.1d}$$ It is obvious that the set of eqs. (\[3.1a\]) is equivalent for $D \neq 4$ to the following set of equations $${\cal R}^{(2)}_{MN} = 0.
\label{3.1c}$$
Equations of motion (\[2.17\]) and (\[2.18\]) in this case read as follows $$\begin{aligned}
G_{ijkl} \dot{\beta}^i \dot{\beta}^j \dot{\beta}^k
\dot{\beta}^l = 0, \label{3.2} \\
\frac{d}{du} \left[ e^{- 3 \gamma + \gamma_0}
G_{ijkl} \dot{\beta}^j \dot{\beta}^k
\dot{\beta}^l \right] = 0, \label{3.3}
\end{aligned}$$ $i = 1,\ldots, n$. Here $L = 0$ due to (\[3.2\]).
Let us put $\ddot{\beta}^i = 0$ for all $i$ or, equivalently, $$\beta^i = c^i u + c^i_0,
\label{3.4}$$ where $c^i$ and $c^i_0$ are constants, $i = 1,\ldots, n$. We also put $$3 \gamma = \gamma_0 = \sum_{i =1}^{n} \beta^i,
\label{3.5}$$ i.e. a modified “harmonic” variable is used. Recall that in the case $\alpha_1 \neq 0$ and $\alpha_2 = 0$, the choice $\gamma = \gamma_0$ corresponds to the harmonic variable $u$ [@IM-top].
Then eqs. (\[3.3\]) are satisfied identically and eq. (\[3.2\]) gives us the following constraint $$G_{ijkl} c^i c^j c^k c^l = 24 \sum_{i < j < k < l} c^i c^j
c^k c^l = 0 \label{3.6}.$$
Thus, we have obtained a class of exact cosmological type solutions for the Gauss-Bonnet model (\[3.1\]) that is given by the metric (\[2.2\]) with the functions $\beta^i(u)$ and $\gamma(u)$ from (\[3.4\]) and (\[3.5\]), respectively, and integration constants $c^i$ obeying (\[3.6\]).
Solution with power-law dependence of scale factors
---------------------------------------------------
Let us consider the solutions with
$$\sum_{i=1}^n c^i \neq 0 \label{3.7}.$$
Introducing the synchronous-type variable
$$\tau = \frac{1}{c} \exp(c u + c_0) \label{3.8},$$
where $$c = \frac{1}{3} \sum_{i=1}^n c^i, \qquad
c_0 = \frac{1}{3} \sum_{i=1}^n c^i_0,
\label{3.9}$$
and defining new parameters $$p^i = c^i/c, \label{3.10}, \qquad
A_i = \exp[c^i_0 + p^i (\ln c - c_0)],$$ $i = 1,\ldots, n$, we get the “power-law” solution with the metric $$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} A_i^2 \tau^{2p^i} dy^i \otimes dy^i, \label{3.12}$$ where $w = \pm 1$, $ { \varepsilon }_i= \pm 1$; $A_i > 0$ are arbitrary constants, and parameters $p^i$ obey the relations $$\begin{aligned}
\sum_{i=1}^n p^i = 3, \label{3.13}\\
G_{ijkl} p^i p^j p^k p^l =
24 \sum_{i < j < k < l} p^i p^j p^k p^l = 0, \label{3.14}
\end{aligned}$$ following from (\[3.6\]), (\[3.7\]) and (\[3.10\]). This solution is a singular one for any set of parameters $p^i$, see Appendix D.
In the cosmological case when $w = -1$, ${ \varepsilon }_i= 1$ (for all $i$), this solution was obtained earlier in [@Deruelle] for $D = 5,6$ and verified recently in [@Pavl] for all $D > 4$.
[**Example 1.**]{} Let us consider the case $D = 6$ and $p_i \neq 0$, $i =1, \dots, 5$. Relations (\[3.13\]) and (\[3.14\]) read in this case as follows $$\begin{aligned}
p^1 + p^2 + p^3 + p^4 + p^5 = 3, \label{3.13e}\\
p^1 p^2 p^3 p^4 p^5
\left(\frac{1}{p^1} + \frac{1}{p^2} + \frac{1}{p^3}
+ \frac{1}{p^4} + \frac{1}{p^5} \right) = 0. \label{3.14e}
\end{aligned}$$ Let us put $p^1 = x > 0$, $p^2 = \frac{1}{x}$, $p^3 = z > 0$, $p^4 = y < 0$, $p^5 = \frac{1}{y}$. Then we get $$x + \frac{1}{x} + z + y + \frac{1}{y} = 3, \quad
x + \frac{1}{x} + \frac{1}{z} + y + \frac{1}{y} = 0. \label{3.14ee}$$ Subtracting the second relation in (\[3.14ee\]) from the first one we obtain $z - \frac{1}{z} = 3$ or $z = \frac{1}{2}(3 + \sqrt{13})$ ($z > 0$). For any $x > 0$ there are two solutions $y = y_{\pm}(x)
= \frac{1}{2}(- A \pm \sqrt{A^2 -4})$, where $A = x + \frac{1}{x} + \frac{1}{z} > 2$.
[**Proposition 1.**]{} [*For $D \neq 4$ the metric (\[3.12\]) is a solution to equations of motion (\[3.1a\]) if and only if the set of parameters $p = (p^1,...,p^n)$ either obeys the relations (\[3.13\]) and (\[3.14\]), or $p = (a,b,0,...,0), (a,0,b,0,...,0), \ldots
$, where $a$ and $b$ are arbitrary real numbers.*]{}
This proposition is proved in Appendix E. (For cosmological solutions in dimensions $D = 5,6$ see also [@Deruelle].)
For $D =4$ the metric (\[3.12\]) gives a solution to equations of motion (\[3.1a\]) for any set of parameters $p^i$.
Solution with exponential dependence of scale factors
-----------------------------------------------------
Now we consider the solution with
$$\sum_{i=1}^n c^i = 0 \label{3.15}.$$
Introducing the synchronous-type variable
$$\tau = u \exp(c_0) \label{3.16},$$
where $c_0$ is defined in (\[3.9\]) and defining new parameters $$v^i = c^i \exp(-c_0), \qquad
B_i = \exp(c^i_0), \label{3.17}$$ $i = 1,\ldots, n$, we are led to the cosmological-type solution with the metric $$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} B_i^2 e^{2v^i \tau} dy^i \otimes dy^i, \label{3.19}$$ where $w = \pm 1$, $ { \varepsilon }_i= \pm 1$; $B_i > 0$ are arbitrary constants, and parameters $v^i$ obey the relations $$\begin{aligned}
\sum_{i=1}^n v^i = 0, \label{3.20}\\
G_{ijkl} v^i v^j v^k v^l =
24 \sum_{i < j < k < l} v^i v^j v^k v^l = 0, \label{3.21}
\end{aligned}$$ following from (\[3.6\]), (\[3.15\]) and (\[3.17\]).
[**Example 2.**]{} Let $D = 6$ and $v_i \neq 0$, $i =1, \dots, 5$. Relations (\[3.20\]) and (\[3.21\]) read in this case as follows $$\begin{aligned}
v^1 + v^2 + v^3 + v^4 + v^5 = 0, \label{3.20e}\\
v^1 v^2 v^3 v^4 v^5
\left(\frac{1}{v^1} + \frac{1}{v^2} + \frac{1}{v^3}
+ \frac{1}{v^4} + \frac{1}{v^5} \right) = 0. \label{3.21e}
\end{aligned}$$ We put $v^1 = x > 0$, $v^2 = \frac{1}{x}$, $v^3 = 1$, $v^4 = y < 0$, $v^5 = \frac{1}{y}$. Then we get $$x + \frac{1}{x} + 1 + y + \frac{1}{y} = 0, \label{3.20ee}$$ For any $x > 0$ there are two solutions $y = y_{\pm}(x)
= \frac{1}{2}(- B \pm \sqrt{B^2 -4})$, where $B = x + \frac{1}{x} + 1 \geq 3$.
Some other solutions
--------------------
The solutions to equations of motion (\[3.2\]) and (\[3.3\]) are not exhausted by relations (\[3.4\])-(\[3.6\]). We give an example of another solution for $D > 4$: $$\begin{aligned}
e^{- 3 \gamma + \beta^1 + \beta^2 + \beta^3}
\dot{\beta}^1 \dot{\beta}^2 \dot{\beta}^3 = C,
\label{3.b1} \\
\beta^i(u) = \beta^i_0, \qquad i > 3,
\label{3.b2}
\end{aligned}$$ where $\beta^i_0$ ($i > 3$) and $C$ are arbitrary constants. In terms of “synchronous” variable $\tau$ (obeying $d \tau =
e^{\gamma(u)} du$) this solutions reads as follows $$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} a_i^2(\tau) dy^i \otimes dy^i, \label{3.a}$$ where $$\begin{aligned}
\left(\frac{d a_1}{d \tau}\right) \left(\frac{d a_2}{d\tau}\right)
\left(\frac{d a_3}{d\tau}\right) = C, \label{3.a1} \\
a_i(\tau) = a_i^0, \qquad i > 3, \label{3.a2}
\end{aligned}$$ where $a^i_0 > 0$ ($i > 3$) and $C$ are constants. This solution contains a special solution with $$a_1(\tau)= a_2(\tau)= a_3(\tau) = A \tau . \label{3.a3}$$
For $C = 0$ we get a special solution with arbitrary (smooth) functions $\gamma (u)$, $\beta^1(u)$, $\beta^2(u)$ and constant $\beta^i(u) = \beta^i_0$, for $i > 2$. In terms of synchronous variable this solution is described by the metric (\[3.a\]) with $$a_1(\tau), a_2(\tau) - {\rm arbitrary}, \qquad
a_i(\tau) = a_i^0 - {\rm constant}, \quad i > 2.
\label{3.c}$$
[**Remark 1.**]{} For $D = 4$, or $n= 3$, the equations of motion (\[3.2\]) and (\[3.3\]) are satisfied identically for arbitrary (smooth) functions $\beta^i(u)$ and $\gamma(u)$. This is in agreement with that fact that in dimension $D = 4$, the action (\[3.1\]) is a topological invariant and its variation is identically zero.
Reduction to an autonomous system of first order differential equations
=======================================================================
Now we put $\gamma = 0$, i.e. “the synchronous-like” time gauge is considered. We denote $u = \tau$. By introducing “Hubble-like” variables $h^i = \dot{\beta}^i$, we rewrite eqs. (\[2.17\]) and (\[2.18\]) in the following form
$$\begin{aligned}
w \alpha_1 G_{ij} h^i h^j
+ \alpha_2 G_{ijkl} h^i h^j h^k h^l = 0, \label{5.1} \\
\left[ -2 w \alpha_1 G_{ij} h^j
- \frac{4}{3} \alpha_2 G_{ijkl} h^j h^k h^l \right] \sum_{i=1}^nh^i
\qquad \nonumber \\
+ \frac{d}{d\tau} \left[ -2 w \alpha_1 G_{ij} h^j
- \frac{4}{3} \alpha_2 G_{ijkl} h^j h^k h^l \right]
- L = 0, \label{5.2}
\end{aligned}$$
$i = 1,\ldots, n$, where $$L = -w \alpha_1 G_{ij} h^i h^j
- \frac{1}{3} \alpha_2 G_{ijkl} h^i h^j h^k h^l.
\label{5.1a}$$
Due to (\[5.1\]), $$L = - \frac{2}{3} w \alpha_1 G_{ij} h^i h^j.
\label{5.1b}$$
Thus, we are led to the autonomous system of the first-order differential equations on $h^1(\tau), ..., h^n(\tau)$.
Here we may use the relations (\[2.12\]), (\[2.13\]) and the following formulas (with $v^i = h^i$) $$\begin{aligned}
G_{ij}v^j = v^i - S_1,
\label{5.3} \\
G_{ijkl} v^j v^k v^l
= S_1^3 + 2 S_3 -3 S_1 S_2 + 3 (S_2 - S_1^2) v^i
+ 6 S_1 (v^i)^2 - 6(v^i)^3,
\label{5.4}
\end{aligned}$$ $i = 1,\ldots, n$, where $S_k = S_k (v) = \sum_{i =1}^n
(v^i)^k$. Relation (\[5.4\]) is derived in Appendix B.
Let us consider the fixed point of the system (\[5.1\]) and (\[5.2\]): $h^i (\tau) = v^i$ with constant $v^i$ corresponding to the solutions $$\beta^i = v^i \tau +
\beta^i_0, \label{5.4a}$$ where $\beta^i_0$ are constants, $i = 1,\ldots, n$. In this case we obtain the metric (\[3.19\]) with exponential dependence of scale factors. For $\alpha_1 = 0$ we get the solution (\[3.19\])-(\[3.21\]).
Now we put $\alpha_1 \neq 0$ and $\alpha_2 \neq
0$. For the fixed point $v = (v^i)$ we have the set polynomial equations $$\begin{aligned}
G_{ij} v^i v^j
- \alpha_w G_{ijkl} v^i v^j v^k v^l = 0, \label{5.5} \\
\left[ 2 G_{ij} v^j
- \frac{4}{3} \alpha_w G_{ijkl} v^j v^k v^l \right] \sum_{i=1}^n v^i
- \frac{2}{3} G_{ij} v^i v^j = 0, \label{5.6}
\end{aligned}$$ $i = 1,\ldots, n$, where $\alpha_w = \alpha_2(-w)/\alpha_1$. For $n > 3$ this is a set of forth-order polynomial equations.
The trivial solution $v = (v^i) = (0, ..., 0)$ corresponds to a flat metric $g$.
For any non-trivial solution $v$ we have $\sum_{i=1}^n v^i \neq 0$ (otherwise one gets from (\[5.6\]) $G_{ij} v^i v^j = \sum_{i =1}^{n} (v^i)^2 -
(\sum_{i =1}^{n} v^i)^2 = 0$ and hence $v = (0, \dots, 0)$).
Let us consider the isotropic case $v^1 = ... = v^n = a$. The set of equations (\[5.5\]) and (\[5.6\]) is reduced to the equation $$n(n -1)a^2 + \alpha_w n(n -1)(n -2)(n -3) a^4 = 0.
\label{5.7}$$ For $n = 1$, $a$ is arbitrary and $a =0$ for $n = 2,3$. When $n > 3$, the non-zero solution to eq. (\[5.7\]) exists only if $\alpha_w < 0$ and in this case $$a = \pm \frac{1}{\sqrt{|\alpha_w| (n -2)(n -3)}}.
\label{5.8}$$ In cosmological case $w = -1$, this solution takes place when $\alpha_2/\alpha_1 < 0$.
Here the problem of classification of all solutions to eqs. (\[5.5\]), (\[5.6\]) for given $n$ arises. Some special solutions of the form $(a,...,a,b,...,b)$, e.g. in a context of cosmology with two factor spaces, for certain dimensions were considered in literature. See, for example, [@Ishihara; @ElMakObOsFil; @BambaGuoOhta; @KirMak].
Here we outline three properties of the solutions to the set of polynomial equations (\[5.5\]), (\[5.6\]).
[**Proposition 2.**]{}
*For any solution $v = (v^1,...,v^n)$ to polynomial eqs. (\[5.5\]) and (\[5.6\]):*
[**i)**]{} the vector $-v = (-v^1,...,-v^n)$ is also a solution;
[**ii)**]{} for any permutation $\sigma$ of the set of indices $\{1,..., n \}$ the vector $v = (v^{\sigma(1)},...,v^{\sigma(n)})$ is also a solution;
[**iii)**]{} there are no more than three different numbers among $v^1,...,v^n$, when $v = (v^1,...,v^n) \neq (0,...,0)$
.
[**Proof.**]{} The first item of the proposition is trivial. The second one follows just from relations (\[2.12\]), (\[2.13\]), (\[5.3\]) and (\[5.4\]).
Now we prove the item [**iii)**]{}. Let us suppose that there exists a non-trivial solution $v = (v^1,...,v^n)$ with more than three different numbers among $v^1,...,v^n$. Due to (\[5.4\]), (\[5.6\]) and $\sum_{i=1}^n v^i \neq 0$ any number $v^i$ obeys the cubic equation $C_0 + C_1 v^i + C_2 (v^i)^2 + C_3(v^i)^3 = 0$, with $C_3 \neq 0$, $i = 1,\ldots, n$, and hence at most three numbers among $v^i$ may be different. Thus, we are led to a contradiction. The proposition is proved.
This implies that in a future investigations of solutions to eqs. (\[5.5\]) and (\[5.6\]) for arbitrary $n$ we will need a consideration of three non-trivial cases when 1) $v = (a,...,a)$ (see (\[5.8\])); 2) $v = (a,...,a,b,...,b)$ ($a \neq b$); and 3) $v = (a,...,a,b,...,b,c,...,c)$ ($a \neq b$, $b \neq c$, $a \neq c$). One may put also $a > 0$ due to item [**i)**]{}.
The generalization to the Lowelock model
========================================
The action (\[1.1\]) is a special case of the Lowelock model [@Low] $$S = \int_{M} d^{D}z \sqrt{|g|}
\left\{ \sum_{k = 1}^{m} \alpha_k {\cal L}_k \right\},
\label{4.1}$$ where $\alpha_1, ...,\alpha_m$ are constants and ${\cal L}_k$ are defined as follows $${\cal L}_k = 2^{-k} \delta^{M_1...M_{2k}}_{N_1...N_{2k}}
R_{M_1 M_2}^{\ \ \ \ \ \ N_1 N_2} ...
R_{M_{2k -1} M_{2k}}^{\ \ \ \ \ \ \ \ N_{2k -1} N_{2k}} ,
\label{4.2}$$ $k = 1, \dots, m$. (Usually, $m$ is chosen as follows: $m = m(D) = [(D-1)/2]$; the terms with $k > m(D)$ will not give contributions into equations of motion.) Here $$\delta^{M_1...M_{2k}}_{N_1...N_{2k}} = \sum_{\sigma}
\varepsilon_{\sigma}
\delta^{M_1}_{N_{\sigma(1)}}... \delta^{M_{2k}}_{N_{\sigma(2k)}}
\label{4.2k}$$ is a generalized Kronecker tensor, totally antisymmetric in both groups of indices: $M_1,...,M_{2k}$ and $N_1,...,N_{2k}$. In (\[4.2k\]) a sum on all permutations of the set of indices $\{1,..., 2k \}$ is assumed. Here $\varepsilon_{\sigma} = \pm 1$ is the parity of the permutation $\sigma$.
It may be verified that ${\cal L}_1 = R[g]$ and ${\cal L}_2$ (from (\[4.2\])) is coinciding with the Gauss-Bonnet term (\[1.2\]).
The Lagrange approach
---------------------
Here we suggest the following conjecture: the equations of motion for the Lowelock action (\[4.1\]) when the metric (\[2.2\]) is substituted are equivalent to the Lagrange equations corresponding to the Lagrangian (for cosmological case see also [@Deruelle; @Pavl])
$$L = \sum_{k = 1}^m \alpha_k L_k, \label{4.3}$$
where $$L_k = \mu_k \exp[- (2k - 1) \gamma + \gamma_0]
G_{i_1 ... i_{2k}}^{(2k)} \dot{\beta}^{i_1}
\ldots \dot{\beta}^{i_{2k}},
\label{4.4}$$ $\gamma_0 = \sum_{i =1}^{n} \beta^i$, $\mu_k$ are rational numbers ($\mu_1 = -w, \mu_2 = - 1/3$) and $$G_{i_1 ... i_{2k}}^{(2k)} =
\prod_{1 \leq r < s \leq 2k }
(\delta_{i_r i_s} -1)
\label{4.5}$$ are the components of Finslerian $2k$-metric: $<v_1,...,v_{2k}>_{2k} = G_{i_1 ... i_{2k}}^{(2k)}
v^{i_1}_1 ... v^{i_{2k}}_{2k}$, $v_s = (v^i_s) \in { {\mathbb R} }^n$, where $<.,...,.>_{2k}$ is a $2k$-linear symmetric form on ${ {\mathbb R} }^n$, $k = 1, ..., m$. Here $G_{i_1 i_2}^{(2)} = G_{i_1 i_2}$ and $G_{i_1 i_2 i_3 i_4}^{(4)} = G_{i_1 i_2 i_3 i_4}$, see (\[2.10\]) and (\[2.11\]).
Cosmological type solutions for “pure” $m$-th Lowelock model
------------------------------------------------------------
Now we put $\alpha_1 = ... = \alpha_{m-1} = 0$ and $\alpha_{m} \neq 0$, i.e. we consider the cosmological type model governed by the “pure” $m$-th Lowelock action $$S_m = \alpha_m \int_{M} d^{D}z \sqrt{|g|} {\cal L}_m [g],
\label{4.6}$$ $m= 1, 2, 3, ...$.
It may be verified along a line as it was done in the Section 3 that our conjecture implies the existence of cosmological type solutions with the metrics (\[3.12\]) and (\[3.19\]).
For the “power-law” solution with the metric
$$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} A_i^2 \tau^{2p^i} dy^i \otimes dy^i$$
the parameters $p^i$ obey the following relations $$\begin{aligned}
\sum_{i=1}^n p^i = 2m - 1, \label{4.7}\\
G_{i_1 ... i_{2m}}^{(2m)} p^{i_1} ... p^{i_{2m}} =
(2m)! \sum_{i_1 <... <i_{2m}} p^{i_1} ... p^{i_{2m}} = 0. \label{4.8}
\end{aligned}$$ instead of (\[3.13\]) and (\[3.14\]). (For cosmological solutions see also [@Deruelle; @Pavl].)
For the “exponential” solution with the metric
$$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} B_i^2 e^{2v^i \tau} dy^i \otimes dy^i$$
the parameters $v^i$ should obey the relations (\[3.20\]): $\sum_{i=1}^n v^i = 0$ and $$G_{i_1 ... i_{2m}}^{(2m)} v^{i_1} ... v^{i_{2m}} =
(2m)! \sum_{i_1 <...< i_{2m}} v^{i_1} ... v^{i_{2m}} = 0. \label{4.10}$$ instead of (\[3.21\]).
The existence of these solutions corresponding to the “pure” Lowelock action (\[4.6\]) may be considered as test for the validity of the conjecture suggested above.
Conclusions
===========
Here we have considered the $(n +1)$-dimensional Einstein-Gauss-Bonnet model. For diagonal cosmological type metrics we have reduced the equation of motion to a set of Lagrange equations with the Lagrangian governed by two “minisuperspace” metrics on ${ {\mathbb R} }^{n}$: (i) the pseudo-Euclidean 2-metric (corresponding to the scalar curvature term) and (ii) the Finslerian 4-metric (corresponding to the Gauss-Bonnet term). The Finslerian 4-metric is proportional to $n$-dimensional Berwald-Moor 4-metric. Thus, we have found a rather natural and “legitime” application of $n$-dimensional Berwald-Moor metric in multidimensional gravity with the Gauss-Bonnet term. mth For the case of the “pure” Gauss-Bonnet model we have obtained two exact solutions: with power-law and exponential dependence of scale factors (w.r.t. “synchronous-like” variable). In the cosmological case (with $w = -1$, $ { \varepsilon }_{1}= ... = { \varepsilon }_{n} = 1$) the first (power-law) solution was obtained earlier by N. Deruelle for $n = 4, 5$ [@Deruelle] and verified by A. Toporensky and P. Tretyakov (for $n = 6,7$) [@TT] and by S. Pavluchenko (for all $n$) [@Pavl]. See also [@Iv-09].
When the “synchronous-like” time gauge was considered the equations of motion were reduced to an autonomous system of first order differential equations. It was shown that for any non-trivial solution with the exponential dependence of scale factors $a_i(\tau) = A_i \exp( v^i
\tau)$, $i = 1,...,n$, there are no more than three different numbers among $v^1,...,v^n$ (if $\alpha_1 \neq 0$ and $\alpha_2 \neq 0$.). This means that the solutions of such type have a “restricted” anisotropy. Such solutions may be used for constructing of new cosmological solutions, e.g. describing accelerated expansion of our 3-dimensional factor-space and small enough variation of the effective gravitational constant. For this approach, see [@BZhuk; @IKM-08] and references therein.
We have also proposed (without a proof) a generalization of the EGB effective\
(cosmological-type) Lagrangian to the Lowelock case (in agreement with [@Deruelle; @Pavl] for cosmological metrics). According to this conjecture a “pure” Lowelock term of $m$-th order in the action gives a contribution to the effective Lagrangian that contains a Finslerian $2m$-metric. This hypothesis implies the existence of cosmological solutions with power-law (see also [@Deruelle; @Pavl] for cosmological case) and exponential dependence of scale factors for the case of the “pure” Lowelock model of $m$-th order. A proof of the conjecture mentioned above may be the subject of a separate publication. Another generalization of the approach suggested in this paper will be connected with inclusion of a scalar field.
Here an open problem arises: do the generalized solutions (for arbitrary $n$) with “jumping” parameters $p^i, A_i$ appear as asymptotical solutions in EGB model when approaching a singular point? Recall that Kasner-type solutions with “jumping” parameters $p^i, A_i$ describe an approaching to a singular point in certain gravitational models, e.g. with matter sources, see [@BLK; @DHSp; @IKM-bil1; @IKM-bil2; @IM-bil1; @IM-bil2; @DamH1; @DHN; @IM-bil-rev] and references therein. This problem may be a subject of separate investigations. (Here it is worth to mention the paper of T. Damour and H. Nicolai [@DamNic], which includes a study of the effect of the 4th order in curvature gravity terms, including the Euler-Lovelock term octic in velocities, and its compatibility with the Kac-Moody algebra $E_{10}$.)
[**Acknowledgments**]{}
This work was supported in part by the Russian Foundation for Basic Research grants Nr. 09-02-00677-a. The author is also grateful to A.V. Toporensky and D.G. Pavlov for lectures at seminars of VNIIMS-RUDN, which stimulated the writing of this paper. The main results of this work were reported at Vth International Conference “Finsler Extensions of Relativity Theory” (27 September – 3 October 2009, Moscow - Fryazino, Russia). The author thanks the participants of this conference for fruitful discussions and numerous comments.
Appendix
========
Useful relations for $(1+n)$-splitting
--------------------------------------
Let us consider the metric defined on ${ {\mathbb R} }_{*} \times { {\mathbb R} }^{n}$ (${ {\mathbb R} }_{*} = (u_{-},u_{+})$ is an open subset in ${ {\mathbb R} }$) $$g= w e^{2{\gamma}(u)} du \otimes du +
\sum_{i,j =1}^{n} h_{ij}(u) dy^i \otimes dy^j. \label{A.1}$$
Here $(h_{ij}(u))$ is a symmetric non-degenerate matrix for any $u \in { {\mathbb R} }_{*}$, smoothly dependent upon $u$. The function ${\gamma}(u)$ is smooth.
The calculations give the following non-zero (identically) components of the Riemann tensor $$\begin{aligned}
R_{0i0j} = - R_{i00j} = - R_{0ij0} = R_{i0j0} =
\frac{1}{4} [-2 \ddot{h}_{ij} +
2 \dot{\gamma} \dot{h}_{ij} +
\dot{h}_{ik} h^{kl} \dot{h}_{lj}] , \label{A.2}\\
R_{ijkl} = \frac{1}{4} (-w) e^{- 2 \gamma}
(\dot{h}_{ik} \dot{h}_{jl} -
\dot{h}_{il} \dot{h}_{jk}), \label{A.3}
\end{aligned}$$ $i,j,k,l = 1, \dots, n$, where here and in what follows $h^{-1} = (h^{ij})$ is the matrix inverse to the matrix $h = (h_{ij})$. Here we denote $\dot{A} = dA/du$ etc.
For non-zero (identically) components of the Ricci tensor we get $$\begin{aligned}
R_{00} =
\frac{1}{2} [- h^{il} \ddot{h}_{li} +
\frac{1}{2} h^{ij} \dot{h}_{jk} h^{kl} \dot{h}_{li}+
h^{ik} \dot{h}_{ki} \dot{\gamma}], \label{A.4}\\
R_{ij} = \frac{1}{4} (-w) e^{- 2 \gamma}
[ 2 \ddot{h}_{ij} + \dot{h}_{ij}(h^{kl} \dot{h}_{lk} - 2
\dot{\gamma}) -2 \dot{h}_{ik} h^{kl} \dot{h}_{lj} ],
\label{A.5}
\end{aligned}$$ $i,j = 1, \dots, n$.
The scalar curvature reads $$R = \frac{1}{4} (-w) e^{- 2 \gamma} [ 4 {\rm tr}(\ddot{h}h^{-1})
+ {\rm tr}(\dot{h}h^{-1}) ({\rm tr}(\dot{h}h^{-1}) - 4 \dot{\gamma})
- 3 {\rm tr}(\dot{h} h^{-1}\dot{h}h^{-1})]. \label{A.6}$$
Let us denote $$M = \dot{h}h^{-1}, \label{A.8}$$ ($h = (h_{ij})$), then $$\dot{M} + M^2 = \ddot{h}h^{-1}. \label{A.10}$$
We obtain $$R \sqrt{|g|}= L_1 + \frac{df_1}{du}, \label{A.7}$$ where $$L_1 = \frac{1}{4} (-w) e^{- \gamma} \sqrt{|h|}
[ {\rm tr}M^2 - ({\rm tr}M)^2 ], \label{A.7L1}$$ $|h| = |{\rm det}(h_{ij})|$ and $$f_1 = (-w) e^{- \gamma} \sqrt{|h|} {\rm tr} M . \label{A.7f1}$$
In derivation of (\[A.7\]) the following relations were used: $$\frac{d\sqrt{|h|}}{du} = \frac{1}{2}\sqrt{|h|} {\rm tr}(\dot{h}h^{-1}),
\qquad
\sqrt{|g|} = e^{\gamma} \sqrt{|h|}. \label{A.12}$$
The calculations give us the following relations for quadratic invariants $$\begin{aligned}
R_{MNPQ} R^{MNPQ} = \frac{1}{8} e^{- 4 \gamma}
\{ ({\rm tr}M^2)^2 - {\rm tr}M^4 +
2 {\rm tr}(2 \dot M + M^2 - 2 \dot{\gamma} M)^2 \},
\label{A.13}\\
R_{MN} R^{MN} = \frac{1}{16} e^{- 4 \gamma}
\{ [ - 2 {\rm tr}\dot{M} - {\rm tr} M^2 +
2 \dot{\gamma} {\rm tr} M ]^2 +
{\rm tr}[2 \dot{M} + ({\rm tr} M - 2 \dot{\gamma})
M]^2 \}. \label{A.14}
\end{aligned}$$
Relations (\[A.6\]), (\[A.13\]) and (\[A.14\]) imply the following formula for the Gauss-Bonnet term (\[1.2\]) $$\begin{aligned}
{\cal L}_2 = \frac{1}{16} e^{- 4 \gamma}
\{ 2 ({\rm tr}M^2)^2 - 2 {\rm tr}M^4 +
[({\rm tr}M)^2 - {\rm tr}M^2] [8 {\rm tr} \dot{M}
+ 3 {\rm tr}M^2 \nonumber \\
+ ({\rm tr}M)^2 - 8 \dot{\gamma} {\rm tr} M ]
+ 4 {\rm tr}[( M^2 - ({\rm tr}M) M ) (4 \dot{M}
-4 \dot{\gamma} M + M^2 + ({\rm tr}M)M)] \}.
\label{A.15}
\end{aligned}$$
Relation (\[A.15\]) implies another important formula $${\cal L}_2 \sqrt{|g|} = L_2 + \frac{d}{du} f_2, \label{A.16}$$ where $$\begin{aligned}
L_2 = \frac{1}{48} e^{- 3 \gamma} \sqrt{|h|}
\{ 6 {\rm tr}M^4 - 3 ({\rm tr}M^2)^2
\nonumber \\
+ 6 {\rm tr}M^2 ({\rm tr}M)^2 - 8 ({\rm tr}M) {\rm tr}M^3
- {\rm tr}M^4 \}
\label{A.16L2}
\end{aligned}$$ and $$f_2 = \frac{1}{6} e^{- 3 \gamma} \sqrt{|h|}
\{ 2 {\rm tr}M^3 - 3 ({\rm tr}M) {\rm tr}M^2
+ ({\rm tr}M)^3 \}.
\label{A.16f2}$$
[**Diagonal metrics.**]{}
Now we consider the diagonal metric $$h_{ij}(u) = e^{2\beta^i(u)} { \varepsilon }_i \delta_{ij}, \label{A.17h}$$ ${ \varepsilon }_i = \pm 1$, $i = 1, \ldots, n$. Then, $M_{ij} = 2 \dot{\beta}^i \delta_{ij}$ and we get the following relations for “Lagrangians” $$\begin{aligned}
L_1 = (-w) e^{-\gamma + \gamma_0}
\left[\sum_{i =1}^{n} (\dot{\beta}^i)^2 -
(\sum_{i =1}^{n} \dot{\beta}^i)^2 \right]
\label{A.18L1} \\
L_2 = - \frac{1}{3} e^{- 3 \gamma + \gamma_0}
\left\{ (\sum_{i =1}^{n} \dot{\beta}^i)^4
- 6 (\sum_{i =1}^{n} \dot{\beta}^i)^2 \sum_{j =1}^{n}
(\dot{\beta}^j)^2 \right.
\nonumber \\
\left. + 3 (\sum_{i =1}^{n} (\dot{\beta}^i)^2)^2
+ 8 (\sum_{i =1}^{n} \dot{\beta}^i) \sum_{j =1}^{n} (\dot{\beta}^j)^3
- 6 \sum_{i =1}^{n} (\dot{\beta}^i)^4 \right\},
\label{A.18L2}
\end{aligned}$$ where $\gamma_0 = \sum_{i =1}^{n} \beta^i$.
The “f-functions” (\[A.7f1\]) and (\[A.16f2\]) read as follows $$\begin{aligned}
f_1 = 2 (-w) e^{-\gamma + \gamma_0}
\sum_{i =1}^{n} \dot{\beta}^i,
\label{A.18f1} \\
f_2 = \frac{4}{3} e^{- 3 \gamma + \gamma_0}
\left[ 2 \sum_{i =1}^{n} (\dot{\beta}^i)^3
- 3 (\sum_{i =1}^{n} \dot{\beta}^i) \sum_{j =1}^{n} (\dot{\beta}^j)^2
+ ( \sum_{i =1}^{n} \dot{\beta}^i)^3 \right].
\label{A.18f2}
\end{aligned}$$
Useful relations for Finslerian 4-metric
----------------------------------------
Here we consider a proof of identity (\[2.13\]). We decompose the product of $6$ terms in the definition of the 4-metric (\[2.10\]) into the sum (of “powers of $\delta$-s”) $$G_{ijkl} = \sum_{a= 0}^{6} G_{ijkl}^{a} \label{B.1}$$ where $$G_{ijkl}^{0} = 1, \quad
G_{ijkl}^{1} = - \delta_{ij} - \delta_{ik} - \delta_{il}
- \delta_{jk} - \delta_{jl} - \delta_{kl},
...,
G_{ijkl}^{6} = \delta_{ij} \delta_{ik} \delta_{il}
\delta_{jk} \delta_{jl} \delta_{kl}.$$
Then we get $$T = G_{ijkl}v^i v^j v^k v^l = \sum_{a= 0}^{6} T^a, \label{B.2}$$ where $T^a = G_{ijkl}^{a} v^i v^j v^k v^l$.
The calculations of $T^a$ give us the following results: $$\begin{aligned}
T^0 = S_1^4, \quad T^1 = - 6 S_1^2 S_2, \quad T^2 = 3 S_2^2 + 12 S_1 S_3,
\nonumber \\
T^3 = - 4 S_1 S_3 - 16 S_4, \quad T^4 = 15 S_4, \quad T^5 = - 6 S_4,
\quad T^6 = S_4, \label{B.3}
\end{aligned}$$ where $$S_k = S_k (v) = \sum_{i =1}^n (v^i)^k, \label{B.4}$$ $k = 1,2,3,4$.
The summation of all $T^a$ in (\[B.3\]) leads us to the relation $$T = G_{ijkl}v^i v^j v^k v^l =
S_1^4 - 6 S_1^2 S_2 + 3 S_2^2 + 8 S_1 S_3 - 6 S_4
\label{B.17}$$ coinciding with (\[2.13\]).
Now we prove relation (\[5.4\]). We get $$P_i = G_{ijkl} v^j v^k v^l = \sum_{a= 0}^{6}P^a_i, \label{B.18}$$ where $P^a_i = G_{ijkl}^{a} v^j v^k v^l$, $i = 1, \dots, n$.
The calculations of $P^a_i$ give us the following formulas $$\begin{aligned}
P^0_i = S_1^3, \quad P^1_i = - 3 S_1^2 v^i - 3 S_1 S_2,
\quad P^2_i = 3S_3 + 3S_2 v^i + 9 S_1 (v^i)^2,
\nonumber \\
P^3_i = - S_3 - 3S_1 (v^i)^2 - 16 (v^i)^3 , \quad P^4_i = 15 (v^i)^3,
\quad P^5_i = - 6 (v^i)^3,
\quad P^6_i = (v^i)^3, \label{B.19}
\end{aligned}$$ $i = 1, \dots, n$.
The summation of all $P^a_i$ in (\[B.19\]) leads us to the relation $$P_i = S_1^3 + 2 S_3 - 3 S_1 S_2 + 3(S_2 - S_1^2) v^i
+ 6 S_1 (v^i)^2 - 6(v^i)^3,
\label{B.20}$$ $i = 1, \dots, n$, coinciding with (\[5.4\]). This relation implies $P_i v^i = T$ in agreement with the definitions (\[B.2\]) and (\[B.18\]).
Lagrange equations
------------------
Here we prove the relations (\[2.16a\])-(\[2.16c\]) for the cosmological type metric (\[2.2\]) defined on manifold $M$ from (\[2.1\]). The tensor ${\cal E}_{MN}$ is obtained from the variation of the action $$S = \int_{M} d^{D}z \sqrt{|g|} {\cal L}[g],
\label{C.1}$$ with ${\cal L}[g] = \alpha_1 R[g]
+ \alpha_2 {\cal L}_2[g]$, i.e. $$\delta S = \int_{M} d^{D}z \sqrt{|g|} {\cal E}_{MN} \delta g^{MN},
\label{C.2}$$ and $\sqrt{|g|} {\cal E}_{MN} = \delta S/ \delta g^{MN}$.
Without loss of generality any 1-dimensional submanifold $M_i$ is chosen to be compact and coinciding with the circle of unit length: $M_i = S^1_r$ ($r = 1/2 \pi$ is the radius of the circle) and all coordinates $y^i$ (see (\[2.1\])) obey $0 < y^i <1$, $i = 1, \dots,n$.
Here we will use the following relations for the components ${\cal E}_{MN}$ in coordinates $(y^M) = (y^0 =u, y^i$) and ${\cal L}$ calculated for the metric (\[2.2\]): $$\begin{aligned}
{\cal E}_{MN} = \delta_{MN} {\cal E}_{NN}, \label{C.3} \\
{\cal E}_{MN} = {\cal E}_{MN}(u), \label{C.4} \\
\sqrt{|g|} {\cal L} = L + \frac{df}{du},
\label{C.5}
\end{aligned}$$ where $L = L(\gamma, \beta, \dot{\beta})$ and $f = f(\gamma, \beta, \dot{\beta})$ are defined in relations (\[2.4\]) and (\[2.7\]), respectively.
The first relation (\[C.3\]) may be readily verified using (\[1.3e\])-(\[1.3b\]) and formulas for the Riemann tensor (\[A.2\]) and (\[A.3\]). The second relation (\[C.4\]) is an obvious one and the third one (\[C.5\]) is coinciding with (\[2.3\]).
The substitution of the metric (\[2.2\]) into the functional (\[C.1\]) gives us (due to (\[C.5\]) and $0 < y^i <1$) $$S = \int_{u_{-}}^{u_{+}} du \left (L + \frac{df}{du} \right)
\label{C.6}$$ and hence $$\delta S = \int_{u_{-}}^{u_{+}} du \left\{
\frac{{\partial}L}{{\partial}\gamma} \delta \gamma
+ \sum_{i=1}^n \left( \frac{{\partial}L}{{\partial}\beta^i} -
\frac{d}{du} \frac{{\partial}L}{{\partial}\dot{\beta}^i}\right)
\delta \beta^i \right\}, \label{C.7}$$ where $\delta \gamma(u)$ and $\delta \beta^i(u)$ are smooth functions with compact support in $(u_{-}, u_{+})$ ($\delta \gamma(u_{\pm}) = \delta \beta^i(u_{\pm}) = 0)$, $i = 1, \dots,n$. On the other hand, using (\[C.2\])-(\[C.4\]), the relation $$(\delta g^{MN}) = {\rm diag} (-2w e^{- 2 \gamma} \delta \gamma,
- 2 { \varepsilon }_1 e^{- 2 \beta^1}\delta \beta^1, \dots, - 2 { \varepsilon }_n e^{- 2
\beta^n}\delta \beta^n)$$ and $0 < y^i <1$, we get $$\delta S = \int_{u_{-}}^{u_{+}} du
\{ {\cal E}_{00}(-2w) e^{\gamma_0 - \gamma} \delta \gamma +
\sum_{i=1}^n {\cal E}_{ii}(-2 { \varepsilon }_{i})
e^{\gamma + \gamma_0 - 2\beta^i} \delta \beta^i \}. \label{C.8}$$
Comparing (\[C.7\]) and (\[C.8\]) we get relations (\[2.16a\]) and (\[2.16b\]). Relations (\[2.16c\]) just follow from (\[C.3\]).
Riemann tensor squared
----------------------
Here we consider the Riemann tensor squared (Kretchmnann scalar) for the metric (\[3.12\])
$$g= w d \tau \otimes d \tau +
\sum_{i=1}^{n} { \varepsilon }_{i} A_i^2 \tau^{2p^i} dy^i \otimes dy^i.$$
From (\[A.13\]) we get $$R_{MNPQ} R^{MNPQ} = K \tau^{-4}, \label{D.2}$$ where $$K = 2 S_4 + 2 S_2^2 - 8 S_3 + 4 S_2 \label{D.3}$$ and $S_k = S_k (p) = \sum_{i =1}^n (p^i)^k$, $k = 1,2,3,4$.
Using the identities $$K = 4 \sum_{i = 1}^n (p^i - 1)^2 (p^i)^2 + 2 (S_2^2 - S_4). \label{D.4}$$ and $$S_2^2 - S_4 = 2 \sum_{i < j} (p^i)^2 (p^j)^2 \label{D.5}$$ we obtain that $K \geq 0$ and $K = 0$ if and only if the set of parameters $p = (p^1,...,p^n)$ is either trivial: $p = (0,...,0)$, or belongs to the Milne set: $$p = (1,0,...,0), \ldots, (0,...,0,1). \label{D.6}$$
For other sets $p$ we have $K > 0$ and the Riemann tensor squared diverges when $\tau \to + 0$.
The proof of Proposition 1
--------------------------
The equations of motion (\[5.1\]) and (\[5.2\]) corresponding to the metric (\[3.12\]) with $h^i = p^i/\tau$ (here $\alpha_1 =0$ and $\alpha_2 \neq 0$) read as follows $$\begin{aligned}
{\cal A} \equiv G_{ijkl}p^i p^j p^k p^l = 0,
\label{E.1} \\
{\cal D}_i \equiv G_{ijkl} p^j p^k p^l = 0,
\label{E.2}
\end{aligned}$$ $i = 1,\dots, n$.
Let $D = n+1 \neq 4$ and $$\begin{aligned}
{\cal B} \equiv \frac{1}{(n - 3)} \sum_{i =1}^{n} {\cal D}_i,
\label{E.6} \\
{\cal C}_i \equiv \frac{1}{3} ({\cal B} - {\cal D}_i) \label{E.7},
\end{aligned}$$ $i = 1,\dots, n$.
For $D \neq 4$ the set of equations (\[E.1\]) and (\[E.2\]) is equivalent to the following set of equations $$\begin{aligned}
{\cal A} = S_1^4 - 6 S_1^2 S_2 + 3 S_2^2 + 8 S_1 S_3 - 6 S_4 =
24 \sum_{i < j < k < l} p^i p^j p^k p^l = 0,
\label{E.10} \\
{\cal B} = (S_1 - 3)(S_1^3 - 3 S_1 S_2 + 2 S_3) =
6(S_1 - 3)\sum_{i < j < k } p^i p^j p^k = 0,
\label{E.11} \\
{\cal C}_i = (S_1 - 3) p^i [2 (p^i)^2 - 2 S_1 p^i + S_1^2 - S_2 ] = 0,
\label{E.12}
\end{aligned}$$ $i = 1,\dots, n$. Here $S_k = S_k (p) = \sum_{i =1}^n (p^i)^k$ and we used the identities (\[2.13\]), (\[5.4\]) and the following identity $$S_1^3 - 3 S_1 S_2 + 2 S_3 = G_{ijk}p^i p^j p^k = 6 \sum_{i < j < k } p^i p^j p^k,
\label{E.8}$$ where $$G_{ijk} = (\delta_{ij} -1)(\delta_{ik} -1)(\delta_{jk} -1)
\label{E.9}$$ are components of a Finslerian 3-metric. The identity (\[E.8\]) could be readily verified along a line as it was done in Appendix B for the Finslerian 4-metric. (We note that relation (\[E.11\]) may be also obtained using the formula (\[A.16\]).)
For $S_1 = 3$ we obtain the main solution governed by relations (\[3.13\]) and (\[3.14\]).
Now we consider another case $S_1 \neq 3$. Let $k$ be the number of all nonzero numbers among $p^1,...,p^n$. For $k = 0$ we get a trivial solution $(0,...,0)$. Let $k \geq 1$. We suppose without loss of generality that $p^1,...,p^k$ are nonzero. For $k = 1, 2$ all relations (\[E.10\])-(\[E.12\]) are satisfied identically. In all three cases $k = 0, 1, 2$ the solutions have the form $(a,b,0..,0)$ (plus permutations for general setup).
Now we consider $k \geq 3$. From (\[E.12\]) and $S_1 \neq 3$ we obtain $$2 (p^i)^2 - 2 S_1 p^i + S_1^2 - S_2 = 0,
\label{E.13}$$ $i = 1,\dots, k$. Summing on $i$ gives us $(2 -k) (S_2 - S_1^2) =
0$, or $S_2 = S_1^2$. Then we obtain from (\[E.11\]) $S_3 = S_1^3$ and from (\[E.10\]): $S_4 = S_1^4$. Thus, we get $S_4 = S_2^2$ implying $\Sigma = \sum_{1 \leq i < j \leq k } (p^i)^2 (p^j)^2 =
0$. But $\Sigma \geq (p^1)^2 (p^2)^2 > 0$. Hence, we are led to a contradiction. That means that for $S_1 \neq 3$, we have only solutions with $k \leq 2$ of the form $(a,b,0..,0)$ (plus permutations for general setup). The Proposition 1 is proved.
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---
abstract: 'Taking into account the drivers’state is a major challenge for designing new advanced driver assistance systems. In this paper we present a driver assistance system strongly coupled to the user. <span style="font-variant:small-caps;">Daaria</span> [^1] for Driver Assistance by Augmented Reality for Intelligent Automotive is an augmented reality interface informed by a several sensors. The detection has two focus: one is the obstacles position and the quantification of their dangerousness. The other is the driver behavior. Via a suitable visualization metaphor the driver can at any time perceive the location of relevant hazards while keeping his eyes ont the road. First results show that our method could be applied to vehicle but also to aerospace, fluvial or sea navigation.'
author:
- 'Paul George, Indira Thouvenin, Vincent Frémont and Véronique Cherfaoui [^2]'
title: '**DAARIA: Driver Assistance by Augmented Reality for Intelligent Automotive** '
---
Introduction
============
Driving relies on key concerns such as security and mediated perception for the driver: driving accidents are the 10th leading cause of death worldwide, killing every year 1,2 million people. The automotive industry has turned to support driving systems such as ABS (Anti-lock braking) or ESP (Electronic-Stability Program). These systems attempt to correct driving errors when they are happening. An original method is proposed in this paper : a Mobile Augmented Reality and Interactive Driving Assistance which provides the user the possibility to avoid obstacles and to anticipate driving difficulties. We designed a display module with Augmented Reality (AR) linking a module of obstacles perception and a module of observation of the driver’s behavior. We present in part 2 a state of the art of visualization metaphors for driving. Then in part 3 we describe our approach, and part 4 provides the <span style="font-variant:small-caps;">daaria</span> system design and realization for obstacle detection. The first results are given in part 5 and a conclusion is proposed in part 6.
Review of existing metaphors for the driver
===========================================
The visualization metaphor is defined by Averbukh [@c1] as “*a map establishing the correspondence between concepts and objects of the application domain under modeling and a system of some similarities and analogies. This map generates a set of views and a set of methods for communication with visual objects.*” These metaphors are used to inform the user by reusing standard interface concepts. It is essential to avoid cognitive overload when learning to use the new system. In this section we will extract from literature some existing visualization metaphors for car driving application.
Metaphor for navigation and planning assistance
-----------------------------------------------
The navigation assistance helps the driver to choose a direction while the planning assistance allows the user to reach its destination without prior knowledge on the road topology or on the followed path. We present in this section different metaphors related to the navigation and planning aid systems.
Narzt et al. [@c5] [@c6] describe two metaphors types in the field of car assistance system. The metaphor of the augmented road (Fig. \[fig1a\]) provides the user a highlighted way to follow when looking directly at the road. For example, it allows to notice that a back exit has been missed or to find the right exit on a roundabout without having to check. This metaphor is for planning assistance even if the information presented to the user is a middle term one (usefull till the next intersection). Another metaphor present a virtual car to follow as it accelerates, brakes or activates turn signal. In this case, the information is a short term one (about one second) (cf Fig.\[fig1b\]).
The metaphor of unrolled map [@c4] is an improvement of the metaphor of the augmented road. The aim is to give the user global knowledge of the environment. A map is held as and when the vehicle is moving. The part which is on the ground indicates the path to be taken immediately. The other part in the sky shows the structure of the surrounding road network. A curved area allows a continuous transition between the two modes of representation (Fig. \[fig1c\]). It provides a knowledge of the road network in the medium term, using the technique of LOD (Level Of Detail). The results of tests in simulation (i.e. virtual reality) show a reduction in navigation errors and problems caused by divided attention. This metaphor has been developed by CMU on the new HUD system “full windshield” of General Motors in order to verify these results in real situations.
Metaphor for driving safety
---------------------------
We define the driving safety assistance as all resources used to provide the user with information needed to drive as safely as possible. The aim is to overcome inattention and to compensate the ability to detect hazards.
The metaphor of the highlighter shown in Fig. \[fig1d\] allows to highlight some of the details existing in the driver’s field of view in order to draw his attention. It is possible to highlight other vehicles, pedestrians, and the lane limits[@c5]. We can notice that the highlight of the road is similar to the augmented road metaphor. However, the context and the goals are not the same. Indeed, in the first case, the aim is to guide the driver along a route. In the second case,the aim is to help the user to locate the path when the lane is not clear or when the visibility is poor.
The metaphor of the radar [@c8] looks like a top view of the vehicle in two dimensions. Imminent hazards are marked with an arrow indicating their direction. The user can see the dangers that are on front of the vehicle (Fig. \[fig1e\]). This metaphor is exocentric (i.e. external to), also the user must perform mental transformations to bring the information in the egocentric reference. The exocentric metaphors are not suitable for driving assistance.
The metaphor of the vane [@c8] is presented as a three-dimensional arrow pointing to imminent danger (Fig. \[fig1f\]). The arrow is attached to a virtual pole in front of the car, and helps preventing cognitive changes (otherwise some subjects perform a mental translation of the arrow to the location of their head). Unlike the metaphor of the radar, the metaphor of the wind vane keeps an egocentric view while providing the driver with information on items that are not in his visual field. The advantage of staying in egocentric visualizations is that it frees the driver to perform mental transformations before processing information.
Metaphors to discover points of interest
----------------------------------------
The discovery of points of interest is a mean to provide the user with additional knowledge without distraction. Narzt [@c5] offers the annotation metaphor with contextual information. He suggests to take into account the vehicle’s state in order to provide information suitable to the driver’s needs such as: location of a gas station and the fuel price when missing (Fig.\[fig2a\]).
A variant of the annotation metaphor is the use of a haptic touch pad representing the road and points of interest. The user can touch the points of interest to select them. He has both visual feedback via augmented reality, and haptic feedback through the pavement. He can finally click on a point of interest (always via the keypad), and get more information. Adding increased interaction in this context leads to interact with the interface safely.
Conclusion on metaphors uses for driving assistance
---------------------------------------------------
Many metaphors are available in the context of car driving. Displayed all together they would lead to an overload of the driver’s visual space. Our first objective will be to design a relevant metaphor. In addition, with existing metaphors, authors often assume that a future technology will be able to display on the whole windshield. We assume we have not this possibility. Our second objective will be to develop a prototype in order to conduct experiments for our solution and to realize an adapted interface.
Proposed Approach
==================
Our objective is to integrate the visualization metaphor in the design process of driving assistance system. In order to propose a visualization metaphor adapted to the driving situation, it is necessary to know what is the current situation and what the driver looks at. So we propose the coupling of an obstacle detection module with a system monitoring the driver in order to develop a display module for an application of Augmented Reality (AR). This section is organized as follows : we describe first the obstacle detection and the eye tracking system used for this study; then we describe our metaphor.
Obstacle detection and eye tracking system.
-------------------------------------------
To achieve obstacles detection, we will use the ADAS (Advanced Driver Assistance Systems and Driver Assistance) Mobileye$^\copyright$. A camera is mounted on the windshield of the vehicle (Fig.\[fig3a\]) and a computer (chipset) retrieves the video stream and processes it to extract more useful information for driver assistance applications: positions of pedestrians and vehicles ( Cars, trucks, bicycles and motorcycles), vehicle speed, road markings etc... The public version of Mobileye offers an interface for driver assistance notifying the presence of vehicles, pedestrians and lane (Fig.\[fig3b\]). The system is also equipped with loudspeakers to broadcast warnings. The professional version provides access to information calculated by the system and allows to exploit them differently from the original system. The data produced are broadcasted on the CAN bus of the vehicle. Video signal is not an output of the system.
To adapt the display to the metaphor, we have chosen to integrate the driver’s behavior to the system, especially his gaze. The idea is to inform the driver only when he needs help. FaceLab$^\copyright$ is a system able to capture head and eyes position and orientation. Two infrared emitters illuminate the user’s face, two cameras are located in the configuration of the stereo vision (cf Fig. \[fig4a\]). A computer processes the images to extract the following information : the position and orientation of the head, the position and orientation of the eyes, opening of the eyelids, pupil size, the frequency of eye blinking as shown in fig. \[fig4b\]).\
FaceLab is a commercial product and outputs are saved in a file or sent in the network.
The visualization metaphor
--------------------------
In order to help the driver with obstacle detection, we have chosen the metaphor of the weathervane. This metaphor has two main advantages : it is egocentric and it can be used to indicate several dangers at once. The metaphor proposed for the restitution to the driver is defined as follow.
### Indication of the type of danger
Rather than using a color code to indicate the nature of the dangers, we will refer to symbols which are familiar to the driver: those of the highway code (Fig. \[fig5a\]).
### Indication of dangerousness
To highlight the most critical dangers, we have adopted an intuitive color-code : the gradient green / yellow / orange / red. Indeed, it is generally accepted that red and green are associated respectively danger and safety (Fig. \[fig5b\]).
### Indication of criticality
In order to solve the problem of overlapping arrows and prioritize the display of arrows indicating the most critical dangers, we decided to convey information on the criticality of the danger through the height of the arrows. They are sorted so that the more dangerous, the more visible, thus allowing the driver to process them in an optimal way (Fig. \[fig5c\]).
### Animation of metaphor
To make the metaphor more pleasing to the eye, and give it a more credible behavior, we decided to animate it. In practice, the arrows motions are physics-driven, hence smoothing variations of metaphor states. Fig. \[fig5d\] is a diagram showing the design of our metaphor building on the original weathervane metaphor.
<span style="font-variant:small-caps;">Daaria</span> system
===========================================================
System Design
-------------
The <span style="font-variant:small-caps;">Daaria</span> system (Driver Assistance by Augmented Reality for Intelligent Automotive) is an augmented reality interface informed by two detection module. One is the obstacles position and the quantification of their dangerousness. The other is the user behavior. The <span style="font-variant:small-caps;">Daaria</span> architecture is described in figure \[fig17\].
![<span style="font-variant:small-caps;">Daaria</span> Architecture[]{data-label="fig17"}](ArchiDAARIA){width="0.9\columnwidth"}
The experimental platform <span style="font-variant:small-caps;">carmen</span> has been used to develop <span style="font-variant:small-caps;">Daaria</span>. <span style="font-variant:small-caps;">Carmen</span> is an experimental vehicle dedicated to research on intelligent vehicles. The Figure \[fig18\] shows the cameras of Mobileye and FaceLab systems. Another camera (scene) has been added in order to have a video output of the scene and to monitor the system. In the same way, the display allows to check that all components are working correctly when experimenting.
![Integration on board CARMEN[]{data-label="fig18"}](Capture5.jpg){width="0.8\columnwidth"}
Calibration method
------------------
The data to combine in <span style="font-variant:small-caps;">Daaria</span> come from heterogenous sources : Mobileye, FaceLAB and a wide angle camera. Since the geometrical informations are expressed in different frames (see Fig. \[fig19a\]) , it is crucial to perform an extrinsic calibration procedure between the systems [@c10]. For each camera system, we use the Caltech Calibration toolbox [@c12] to estimate the intrinsic parameters of each camera. If the cameras have enough overlapping, the calibration toolbox can be used to estimate the extrinsic parameters i.e. rotation and translation between the cameras frames. In our case, the Mobileye and the FaceLAB cameras are looking at opposite direction, therefore classical calibration procedure is impossible. Moreover, it is impossible to access to the video images in the FaceLAB system. The only available information comes from the gaze tracker which gives the eyes direction in the FaceLAB frame.
Therefore we developed a new idea to solve the calibration problem (see Fig. \[fig19b\] ):
- We take pictures of the calibration target with the Mobileye System.
- During the pictures acquisition, someone seats on the driver side, looking at some reference point on the target and we save the gaze direction vector.
- Simultaneously, we place a Laser pointer on the driver’s forehead pointing on the calibration target, and we save the measured distance.
- If one repeats the procedure several times, one obtains 3D points in the Mobileye frame and in the FaceLAB frame. Then we use a classical registration approach such as ICP [@c11] to estimate the transformation between the two frames.
![Frames[]{data-label="fig19a"}](reperes){width="0.8\columnwidth"}
![Calibration System[]{data-label="fig19b"}](calib){width="0.8\columnwidth"}
Objects and driver gaze restitution in scene image
--------------------------------------------------
This module is developed in order to have the possibility to check and monitor the metaphor computation. It calculates the projection of the driver’s gaze direction and detected objects on two views : a bird view (Fig \[fig20a\]) and a scene view given by the scene camera (Fig. \[fig20b\]). This is done thanks to calibration data. The blue boxes represent pedestrians, the green boxes represent the cars and red line or circle gives the direction of the driver gaze.
Metaphor computation
--------------------
The metaphor computation module calculates the configuration to be taken by the metaphor of the weathervane. To achieve this, we combine information from the viewing direction (FaceLAB) and the obstacles location (Mobileye) and can get obstacles information looked at. Determining obstacles dangerousness relies on several criteria:
- Object dynamics : the stationary objects are not considered for the metaphor.
- Driver’s visual activity: it avoids to point out a danger checked by the driver. The current implementation is very simple : it carries out tests of intersection between the eye direction and obstacles.
- Time-to-Collision (TTC) : using the status of ego-vehicle and information about the detected objet, we can approximate a TTC. The lower, the more dangerous. The arrows in the weathervane metaphor are ordered by level of danger.
Display device
--------------
The most suitable device for an AR interface compatible with the the driving task would be the HUD (Head Up Display). Its main advantage over the HDD (Head Down Display) is the driver time saving. Indeed, the eyes back and forth between the road and the dashboard deprive the driver of valuable time, as the need to constantly refocus his eyes to distance changes. HUD solves both problems: it is close to the road, which limits the round-trip of gaze and in addition the image formed is located behind the windshield, which reduces the visual rehabilitation. We proposed a low-cost solution that uses a tablet-PC as a HUD. The solution is to put it under the windshield of the car and to display our metaphor as shown in Fig. \[fig21\].
Preliminary results and discussions
====================================
The first real road tests allowed us to validate the choice of components as well as the feasibility of our approach. <span style="font-variant:small-caps;">Carmen</span> is able to collect data that we can replay for analysis (Figure \[fig22\]).
{width="75.00000%"}
Characterization of the consideration of an obstacle
----------------------------------------------------
To integrate the driver’s behavior in our system, we chose a very simple rule based on the intersection of vector of the gaze of the driver and detected obstacles. This simple rule has several basic flaws: no consideration of gaze tracking, the field of view is not a vector, the imprecise or incorrect detection can lead to wrong conclusions. We could propose several improvements concerning the visual attention model, the finer characterization of the obstacle consideration and the cognitive map managing of the driver taking into account the information life cycle.
Characterization of dangerous obstacles
---------------------------------------
To integrate the situation analysis to DAARIA, we chose a simple rule: the closer the obstacle, the more dangerous. More information could be taken into account: the obstacle nature (pedestrian or vehicle), its trajectory and its speed, its behavior (a pedestrian about to cross the street), the consideration of traffic (for example, priority of the drivers on the right).
Conclusion and outlooks
=======================
A prototype of driver assistance system in augmented reality has been presented. It is based on the weathervane metaphor adapted for a use in real conditions. The metaphor is computed according to the driving situation and the driver’s attention. The driver can perceive at any time the location of the dangers without leaving the eyes of the road. The system adapts its behavior to provide him the most relevant information. A lot of problem of integration (calibration, synchronization - not described here- and data exchange) have been resolved to conduct first experiments and prove the feasibility.
Future works concern three major directions. The first one is the uncertainties consideration propagated in this complex system and especially the uncertainties of detected objects [@c2] and of the driver’s attention. The second axe for future research is focused on the metaphor improvement. The third axe is the analysis of metaphor performances in virtual environnement [@c3] in order to validate it and test it with critical scenarios.
ACKNOWLEDGMENTS
===============
The authors gratefully acknowledge the contribution of Gérald Dherbomez.
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http://www.vision.caltech.edu/bouguetj/
[^1]: The authors are with *University of Technology of Compiègne(UTC)*, *CNRS Heudiasyc UMR* 7253*,* [firstname.familyname@utc.fr]{}
[^2]: The authors are with *University of Technology of Compiègne (UTC)*,*CNRS Heudiasyc UMR* 7253*,* *Contact* [firstname.familyname@utc.fr]{}
|
---
abstract: 'In this work, we present a simple and efficient generator of polymeric linear chains, based on a random self-avoiding walk process. The chains are generated using a discrete process of growth, in cubic networks and in a finite time, without border limits and without exploring all the configurational space. First, we thoroughly describe the chains morphology exploring the statistics of two characteristic distances, the radius of gyration and the end-to-end distance. Moreover, we examine the dependence of mean characteristic distances with the number of steps ($N$). Despite the simplicity of our procedure, we obtain universal critical exponents, which are in very good agreement with previous values reported in the literature. Moreover, studying the balance between the monomer-monomer interaction and the bending energy, we find that initially, the chains develop by multiple doubling, forming a cluster and increasing its energy. After reaching a given number of steps, the chains stretch and flee from the cluster, which results in a reduction of its interaction energy. However, the behaviour of the bending energy reveals that the chains follow the same folding pathway in both regimes. Additionally, we also characterize the energy of the obtained chains, combining the local interaction energy with its corresponding bending energy but in a discrete version. This analysis is relevant because it allows differentiating between chains of equal interaction energy but with different structures.'
address:
- 'Departamento de Estatística e Informática, Universidade Federal Rural de Pernambuco, Recife, Pernambuco, CEP 52171-900, Brazil'
- 'Departamento de Física, Universidade Federal Rural de Pernambuco, Recife, Pernambuco, CEP 52171-900, Brazil.'
author:
- 'David R. Avellaneda B.'
- 'Ramón E. R. González'
bibliography:
- 'bibliograp.bib'
title: 'A simple self-avoiding walking process as a reasonable non-conventional generator of polymeric linear chains.'
---
Self-avoiding random walk ,polymer chains ,critical exponents ,interaction and bending energy ,radius of gyration ,end-to-end distance.
INTRODUCTION {#Intro}
============
Although the conformational properties of polymer chains in a good solvent have been the subject of intensive experimental, theoretical and numerical studies their full understanding is still an open challenge. Besides its simplicity, the natural self-avoiding random walk (SAW), proposed over half a century ago, is widely accepted as the principal model for dilute polymers [@book_flory; @medio_seculo1; @cn2d]. SAW describes well a large spectrum of real systems with diverse details (such as bond angles and monomer-monomer potential). Moreover, the equivalence of SAW with the $n\rightarrow 0$ limit of the $n$-vector model [@nvector] has provided an important connection with the theory of phase transitions and critical phenomena [@degennes].\
One of the first (but still widely used) theoretical approaches to this subject is the Flory theory [@book_flory], which with simple mean field arguments involving the concept of excluded volume brought about the understanding of underlying power laws and the role of dimensionality. The subsequent analytical approaches span rigorous methods that have achieved only limited success [@SAW1993], approximate methods such as perturbation theory and self-consistent field theory, which break down for long chains [@yamakawa1971], and renormalization group (RG) [@WILSON1974] which has yielded reasonably accurate estimates for critical exponents and some universal amplitude ratios.\
In parallel with theoretical developments, numerical methods have shown to be a fundamental tool in establishing properties of long SAWs [@sokal1995]. Exact enumeration methods have been used to find the number of all possible SAWs of finite length $N$, from which universal properties are estimated using techniques such as the ratio method, Pade approximants or differential approximants. Thus, results up to $N=71$ steps for the square lattice [@guttmann2001; @jensen2004] and up to $N=21$ steps for the cubic lattice [@SAW1993] have been reported. A large number of Monte Carlo sampling techniques have been proposed ever since the 1950s (see, e.g., [@sokal1995] for a comprehensive overview), where the pivot algorithm [@medio_seculo1] has been shown to perform in time $O(N)$, and studies with SAWs of length up to $N=80000$ have been reported [@Li1995].\
Typically, an ensemble of self-repelling chains considers all possible configurations of a given length. Moreover, the growth process is directed by probability values, which are linked to the Boltzmann factor in terms of the potential energy of interaction and are proportional to the number of interactions [@trueSAW_Peliti].\
Rather than attempting to improve the performance previous algorithms, in the current work we focus on different goals. Instead, in this paper we study relatively small SAW chain ensembles, which might be relevant describing polymer solutions where the polymer growth process has been going on for a limited time. We present a method to generate such ensembles using very simple and efficient numerical algorithm and, surprisingly, we obtain universal critical exponents, which are in very good agreement with the previous reported in the literature. In addition, we carefully examine the balance between the monomer-monomer interaction and the bending energy, addressing its relation with the structure of the chains. This analysis is relevant because it allows differentiating between chains of equal interaction energy but with different structures.\
The paper is organized as follows. In the next section we briefly review the SAW and the principal conformation measures, the algorithm, and the energy measures. The subsequent section is devoted to the results of our simulations, and finally the conclusions are drawn.
METODOLOGY {#Metodologia}
==========
Our aim is to efficiently generates an ensemble of linear homopolymer chains in a good solvent. For simplicity, we use the approximation that the solvent molecules are considered the same size as the monomers.\
The chains are generated using the pathway of a particle that moves randomly in a cubic network with an unlimited boundary conditions. Following that approach, the particle is not allowed to occupy the sites it has visited before (a particle with property self-avoiding). We use the idea of a model known as “true” self-avoiding random walk [@trueSAW_Peliti] but in 3d. Thus, each steps given by the walker can be interpreted as a monomer (or a set of monomers of the same type), and the steps not visited by the walker (empty sites of the network) can be considered as molecules of the solvent, thus, the trajectory described by this particle defines a homopolymeric chain in a good solvent [@trueSAW_Peliti]. The probability of the walker taking a step in the direction $i$ in this type of chain depends on the number of times $n_j$ that the next time the site to be occupied is already “visited”, and is given by: $$p_i = \frac{e^{-gn_{i}}}{\displaystyle\sum_{j=1}^{3d}e^{-gn_{j}}}= \frac{1}{\displaystyle\sum_{j=1}^{3d}e^{-g(n_{j}-n_{i})}}.
\label{eq:trueSAW}$$ In general, the sum runs through all possible $3d$ paths from the position occupied by the walker at each instant of time, including the address $i$, and $g$ is a positive parameter which measures the intensity with which the walk avoids it self. For the sake of simplicity, in this work we implement the limiting case $g=\infty$, which corresponds in the $3d$ case to a discrete domain of probabilities $p_i(g=\infty)=[1/6, 1/5, 1/4, 1/3, 1/2, 1]$.
Numerical Algorithm {#Algoritmo}
-------------------
For a $d-$dimensional network with free boundaries the algorithm to generate a one chain is as follows:
1. Choose the number of attempts $N'$.
2. Choose the origin of the polymer, which in our case is the origin of the coordinate system.
3. Generate the first step randomly or choose it arbitrarily from a point in the cubic network.
4. Choose the following step randomly from one of the $2\times d$ possible steps.
5. If the given step leads to self-intersection, go to item 4. and try again with another step. This step is most important to ensure the SAW.
6. If the step leads to an available location, add the step to the walk.
7. If the number of attempts is reached or if the number of possible steps is zero (the walker gets stuck), the simulation is accepted and saved.
Thus, the random chain is formed by $N$ steps generated from $N'$ attempts, being that, for the three-dimensional case $N'>N$, this is because the chains can get trapped before reaching the total number of attempts.\
After generating the random chain, we store the positions of each of the monomers that constitute it and proceed to compute the characteristic measurements of its configuration. Starting by calculating the displacement ${\bf{r}}_i$ of each monomer in order to obtain the mass center ${\bf{r}}_c$ given in Eq. (\[eq:centro\_de\_massa\]) and with which we calculate the radius of gyration, ${\bf{R}}_g$, given in Eq. (\[eq:raio\_de\_giro\]). Next, we compute the end-to-end vector module, ${\bf{R}}_{ee}$, which can be easily derived from the distance of the $N$th monomer to the origin of the chain as described in Eq. (\[eq:Ree\]).\
Characteristic distances
------------------------
As is proven below, this simple algorithm efficiently generates an ensemble of linear homopolymer chains in a good solvent; each of them is formed by $N$ monomers in positions $\{r_{0},r_{1},...,r_{N}\}$ in the space of dimension $d$. The separation distance between a monomer and its nearest neighbor is $b = r_{i} - r_{i-1}$, for $i=1,2,...,N$, which would be equivalant to a Kuhn segment [@book_flory; @Rubinstein].\
Moreover, we thoroughly describe the chains morphology exploring the behaviour of two characteristic distances, the end-to-end distances ${\bf{R}}_{ee}$ and the radius of gyration ${\bf{R}}_g$. The end-to-end distance is defined as the mean squared variance of the displacement and is reads as, $${\bf{R}}_{ee}^{2} = \langle({\bf{r}}_N-{\bf{r}}_0)^2\rangle,
\label{eq:Ree}$$ the variables ${\bf{r}}_N$ and ${\bf{r}}_0$ are the positions of the ends of the chain. The radius of gyration ${\bf{R}}_g$, whose square is the second moment around the center of mass ${\bf{r}}_c$ given by: $${\bf{r}}_c = \frac{1}{N+1} \sum_{i=0}^N {\bf{r}}_i,
\label{eq:centro_de_massa}$$ so that, the radius of gyration takes the form [@polimer_textbook; @Rubinstein]: $${\bf{R}}_g^2 = \frac{1}{N+1} \sum_{i=0}^N \langle({\bf{r}}_i-{\bf{r}}_c)^2\rangle.
\label{eq:raio_de_giro}$$ For real (non-Gaussian, with excluded volume) chains, a relationship between these distances is [@polimer_textbook]: $$\frac{6{\bf{R}}_g^2}{{\bf{R}}_{ee}^{2}} = 0.952.
\label{eq:relationship}$$ For this type of chain, the radius of gyration depends on $N$, size of the Kuhn segment $b$ and exponent $\nu$: $${\bf{R}}_g = bN^{\nu}.
\label{eq:exp_flory}$$ The exponent $\nu$ is called the size exponent or the Flory exponent [@book_flory; @Rubinstein; @flory_theor_polym], which comes from Flory’s theory, which had remarkable success in explaining the experimental evidence in the swelling of real polymers.\
Energy of linear chains {#Energia}
-----------------------
In our analysis, we use a definition of the interaction energy, which is based on the compactness of the chains [@pedro; @entropia_energia; @energia_chao_tang]. Complementary, we also examine the bending energy of the chains, which in our case is a discrete variable. This magnitude characterises the flexibility of the chains, as well as, its tangential correlations. Furthermore, the total energy of the chain is the sum of the bending energy and the interaction energy.
### Interaction Energy Term {#energia_interacao}
Interaction energy accounts for the energy of the chain due to its compactness and it quantifies the short-range interactions (Von Neumann neighborhood) for non-continuous monomers [@pedro; @entropia_energia; @energia_chao_tang; @REM]. It reads as, $$E=\sum_{i<j}e_{\upsilon_{i}\upsilon_{j}}\Delta({\bf{r}}_i-{\bf{r}}_j),
\label{eq:energia_interacao}$$ where $\Delta({\bf{r}}_i-{\bf{r}}_j)=1$ if ${\bf{r}}_i$ and ${\bf{r}}_j$ are attached to the network, but $i$ and $j$ are not adjacent positions along the chain sequence and $\Delta({\bf{r}}_i-{\bf{r}}_j)=0$ for otherwise (see Fig. \[figura1\]). The value of the factor, $e_{\upsilon_{i}\upsilon_{j}}$, depends on the type of contact between the monomers and represents the potential energy of interaction between the monomers located in the position ${\bf{r}}_i$ and ${\bf{r}}_j$ respectively. In our case, it is $e_{\upsilon_{i}\upsilon_{j}}= -1$ because we consider an attractive monomer-monomer interaction in a homopolymeric linear chain.
![14-step 2d-chain that shows the values that the interaction takes $\Delta({\bf{r}}_i-{\bf{r}}_j)$, where it adopts the values: $1$, for the attachments but not adjacent sites (blue color) and $0$ for the otherwise.[]{data-label="figura1"}](imag1){width="6.5cm"}
### Bending Energy {#energia_bending}
One of the basic characteristics of all macromolecules is their flexibility [@Grosberg_solvent]. The polymer chains, in the pure state or in a dissolution, may adopt different conformations depending on their flexibility. When the flexibility is high, the chain may have large changes of direction within a few links. On the contrary, if the flexibility is low, the chain will be more rigid and will tend, in the limit, to behave as a hard stick. The flexibility of the polymer chain is related to the persistence length $l_p$. This can be defined as the average value of the maximum linear length of the chain configuration (it is also related to the Kuhn segment as $b=2l_{p}$ [@Rubinstein]). At distances greater than $l_p$, fluctuations in relation to itself or to the surroundings destroys the memory bound to the direction of the chain. Thus, the polymers are not completely flexible, some energy is required to fold them, which can happen up to at most the $l_p$.\
The correlation (in turn related to the flexibility) between $\bf{u}$ and $\bf{u'}$, two unit vectors that join three points of the chain (monomers in the Fig. \[fig:tikz1\]) and that are separated by a distance $l$ is given by [@polimer_textbook]: $$\langle {\bf{u}} \cdot {\bf{u}}' \rangle = [(1-(b/l_{p}))^{1/b}]^{l} \sim \exp{(-l/l_{p})}.
\label{eq:correla_orient_exp}$$
At the limit, the conformation of the chain is a smooth curve as described in Fig. \[fig:tikz2\]. Using Eq. (\[eq:correla\_orient\_exp\]) can obtain the correlation function between ${\bf{u}}(s)$ and ${\bf{u}}(s')$ of two segments of the chain, $s$ and $s'$, as a function of the persistence length $l_{p}$, given by [@polimer_textbook]: $$c(s,s')=\langle {\bf{u}}(s) \cdot {\bf{u}}(s')\rangle = \exp{(-|s-s'|/l_{p})},
\label{eq:correlation}$$ this shows that the directional correlation of two segments of a macromolecule, decreases exponentially with the growth of the chain length [@Grosberg_solvent; @flexibility].\
For bending energy we propose the following: we consider our polymer chain taking into account the interactions with other monomers of the same chain and the interactions with its environment. These interactions are described by an effective potential that represent the energy cost for its formation and whose stability is determined by two forces, one elastic with negative signal, that leads the chain to a collapse, and another repulsive of positive signal, that makes the chain is stretched. This energy cost is reflected in the chain in the form of free energy, for example, the number of conformations decreases with the increase of the vector end-to-end but increases the free energy of the same due to the high correlation that exists in the chain.\
Chains with high flexibility experience changes of direction at a distance of few links tending to turn to itself, while low flexibility chains tend to become rigid, because the two-segment correlation function in the chain decreases exponentially with the distance between them as shown in Eq. (\[eq:correlation\]). This correlated behavior occurs in the same way in a system of continuous mechanics, for the flexion model due to a force acting on a thin rod with stiffness constant $k$ [@Landau; @energia_corr]. Flexing generates a differentiable curve on the rod, where, at a point ${\bf{r}}(s)$ of the curve there is a tangent vector ${\bf{u}}(s)$ generating a behavior similar to that described in Fig. \[fig:tikz2\] for polymers. The Hamiltonian describing the internal energy of the rod of length $l_{c}$ is given by, $$\mathscr{H}=\frac{k}{2}\int_{0}^{l_{c}}\left(\frac{\partial{\bf{u}}(s)}{\partial s}\right)^{2} ds.
\label{eq:hamiltoniano_vara}$$ By virtue of this, we propose a discretization of bending energy in order to be adapted to our chains, in this way, part of the internal energy of the chain which is described in terms of its configuration and which is equivalent to Eq. (\[eq:hamiltoniano\_vara\]) is given by the following relation: $$\mathscr{H} \simeq H =\frac{k}{b}\sum_{i,j=1}^{N}\varepsilon_{ij}p_{i},
\label{eq:energia_desvio}$$ where the weight function $\varepsilon_{ij}$ can take values of $(1)$ or $(-1)$ depending on whether or not the direction of the $i-$th step changes as compared to the previous step (see Fig. \[figura3\]) and $p_{i}$ represents the probability that each step will find any of its accessible microstates (Eq. (\[eq:trueSAW\])), $k$ is a constant of units of energy times distance and finally, our model adopts the $b=1$ as the length of Kuhn.
![4-step 2d-chain that shows the value of the weight function $\varepsilon_{i}$ used to compute the bending energy. Left: linear chain without deviation with its weight per step equal to $(1)$. Right: chain with mixed deviations, when direction changes $\varepsilon_{i}$ adopts a weight equal to $(-1)$, as is the case with steps $2$ and $4$.[]{data-label="figura3"}](imag31 "fig:"){width="7.5cm"} ![4-step 2d-chain that shows the value of the weight function $\varepsilon_{i}$ used to compute the bending energy. Left: linear chain without deviation with its weight per step equal to $(1)$. Right: chain with mixed deviations, when direction changes $\varepsilon_{i}$ adopts a weight equal to $(-1)$, as is the case with steps $2$ and $4$.[]{data-label="figura3"}](imag32 "fig:"){width="5cm"}
The bending energy, in its discrete version $H$, describes the behavior of the polymer chain from its tangencial correlations, for example, for a highly correlated polymer chain (Fig. \[figura3\], left) this energy will be purely positive (high free energy) what is expected from the Hamiltonian described by the rod and the correlation given in Eq. (\[eq:correlation\]).\
In our simulations the calculation of bending energy takes into account the term we call function weight as well as the relative probability of each step of the chain, which has the form of Eq. (\[eq:trueSAW\]). This energy can be positive or negative depending on the winding of the chain.
RESULTS AND DISCUSSION {#Resultados}
======================
In order to obtain the results, eleven thousand three-dimensional chains was generated from $2400$ attempts ($N'=2400$).
Chain Length, End-to-end Distance and Radius of Gyration {#comprimento}
--------------------------------------------------------
The behavior of the distribution of the number of steps for three-dimensional chains is shown in Fig. \[figura4\]. The maximum of the distribution, is obtained for large chains (compared to $N'$). The maximum corresponds to $N = 1775$ steps for the chains generated from $N'=2400$. This way, chains with approximately this number of steps appear with greater probability under the conditions established in the simulation.\
![Normalized histogram of the number of steps $N$ with mean $\mu=1775.42$ and standard deviation $\sigma=33.10$. Behavior obtained for random chains in $3d$ generated from $N'=2400$. The inset illustrates the histogram in a wide interval in which a Gaussian behavior occurs.[]{data-label="figura4"}](Histo_N){width="8.7cm"}
In Fig. \[figura5\] and Fig. \[figura6\], the distribution of the characteristic distances (${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$) of the chains is shown. The graphs in Fig. \[figura5\] represent adjustments using Lhuillier’s proposal [@Lhuillier_Daniel]: $$P({\bf{R}}) \sim \exp{(-{\bf{R}}^{-\alpha d}- {\bf{R}}^{\delta})}.
\label{eq:Lhuillier_Daniel}$$ The distribution behavior can be described separated in two regions that follow different exponential laws. For small values of characteristic distances, the distribution behavior is described by: $\sim\exp(-{\bf{R}}^{-\alpha d})$, where $\alpha=(\nu d-1)^{-1}$ and $d$ is a spatial dimension. For large values of the characteristic distances, the expresion for the distribution is: $~\exp(-{\bf{R}}^{\delta})$, $\delta=(1-\nu)^{-1}$ is the Fisher exponent [@Victor_Lhuillier].
![Normalized histograms of the radius of gyration ${\bf{R}}_g$ and end-to-end discance ${\bf{R}}_{ee}$. Behavior obtained for the ensemble of random chains generated from $N'=2400$ in $3d$. It is observed that a distributions agrees very well with the expression derived by Lhuillier [@Lhuillier_Daniel; @Victor_Lhuillier], both, the ${\bf{R}}_g$ and ${\bf{R}}_{ee}$ distribution.[]{data-label="figura5"}](Histo_R_LR){width="8.7cm"}
The Fig. \[figura6\] shows the graphs of the ${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$ distribution, adjusted according to the Fisher-McKenzie-Moore-des Cloiseaux law [@Rubinstein; @McKenzie_1971; @Cloizeaux_1974; @cloizeaux1990polymers]. This law is commonly used to describe the distribution of extreme-extreme distance and is of the following type:
$$P({\bf{R}}) \sim {\bf{R}^{\theta} \exp({-\bf{R}}^{\delta})}.
\label{eq:Victor_Lhuillier}$$
The exponent $\theta = (\gamma - 1)/\nu$ characterizes the shorts-distance intra-chain correlations between two segments of a long polymer in a good solvent. The total number of chain conformations is indirectly determined by the exponent $\gamma$. As opposed to ideal chains, where $\gamma = 1$, for real chains, $\gamma > 1$, i.e. there is a reduction of the probabilities in the distribution of ${\bf{R}}$ for short chains.\
![Normalized histograms of the radius of gyration ${\bf{R}}_g$ and end-to-end discance ${\bf{R}}_{ee}$. Behavior obtained for the ensemble of random chains generated from $N'=2400$ in $3d$. It is observed that a distributions agrees very well with the function proposed by McKenzie and Moore [@McKenzie_1971] and des Cloizeaux [@Cloizeaux_1974; @cloizeaux1990polymers], both, the ${\bf{R}}_g$ and ${\bf{R}}_{ee}$ distribution.[]{data-label="figura6"}](Histo_R_Cloizeaux){width="8.7cm"}
It is important to note that while Fisher-McKenzie-Moore-des Cloizeaux theoretical distribution parameters are contructed by fixing the number of steps of the chains and study the distance end-to-end of the different configurations generated, in our approach we generate a random ensemble of chains without fixing a priory the number of steps. However we have been demonstrated that the chains obtained follow the same distribution [@Rubinstein; @Cloizeaux_1974].\
The value of Flory exponent $\nu$, which describes the size of the polymer chain, was calculated by computing the mean value of the radius of gyration. The values of $\nu$ obtained from the simulation are incorporated in Fig. \[figura7\] and Fig. \[figura8\]. These results closely approximate the expected theoretical value for the Flory exponent that is $\nu \approx 0.59$ for the three-dimensional case. The calculation of $\nu$ from the behavior of ${\bf{R}}_g$ results in a value closer to the expected theoretical value than that calculated from the ${\bf{R}}_{ee}$.\
The Flory exponent was also calculated indirectly from the delta ($\delta$) exponent that results from the distribution of characteristic distances, shown in Fig. \[figura5\] and Fig. \[figura6\]. The values of the alpha ($\alpha$) and theta ($\theta$) exponents, the first indirectly and the second directly, were also obtained from these distributions.\
![Log-log scale representation of the end-to-end distance (${\bf{R}}_{ee}$) as a function of the number of steps ($N$) with its respective value of $\nu$.[]{data-label="figura7"}](Ree_Vs_N){width="8.7cm"}
![Log-log scale representation of the radius of gyration (${\bf{R}}_g$) as a function of the number of steps ($N$) with its respective value of $\nu$.[]{data-label="figura8"}](Rg_Vs_N){width="8.7cm"}
The Table \[tabla1\] shows a comparison between the main critical exponents reported by various authors, both analytically and numerically, and the values reported by our simulations. The above results lead us to conclude that although the two behaviors, ${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$, as a function of $N$ fit a power law, the value of the characteristic exponent of ${\bf{R}}_{g}$ is closer to the theoretical value of Flory ($\nu=0.60$ for 3d chains) and to the values reported in [@Rubinstein; @Ree_dis_polymers; @Fluctuating_SoftSphere]($\nu=0.588$).\
Although both propossed functions fit well the distributions (${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$), the values of the critical exponents obtained for each function, specifically the delta exponent, are different and correspond to different laws. The ${\bf{R}}_{g}$ distribution responds to the law proposed by Lluillier and the distribution of ${\bf{R}}_{ee}$, to the Fisher-McKenzie-Moore-des Cloiseaux law, as already reported in the literature.\
$\nu$ $\delta$ $\alpha$ $\theta$
------------------------------------------- --------------------- ------------------- ---------- -------------------
Rubinstein [@Rubinstein] $0.588$ $2.43$ $1.31$ $0.28$
Caracciolo et al. [@Ree_dis_polymers] $0.58758\pm0.00007$ $2.4247\pm0.0004$ $1.311$ $0.2680\pm0.0011$
Vectorel et al. [@Fluctuating_SoftSphere] $0.588$ $2.38$ $1.31$ —
${\bf{R}}_{g}$ Vs $N$ — —
${\bf{R}}_{ee}$ Vs $N$ $0.631\pm0.011$ — $1.12$ —
${\bf{R}}_{g}$ (Lluillier) —
${\bf{R}}_{ee}$ (Lluillier) $0.724$ $3.63$ $0.853$ —
${\bf{R}}_{g}$ (DC) $0.225$ $1.29$ — $2.54$
${\bf{R}}_{ee}$ (DC) $0.91$
: Main critical exponents calculated directly and indirectly from our simulations, compared with values reported by other authors (the results in blue, are values closer to those reported in the literature [@Rubinstein; @Ree_dis_polymers; @Fluctuating_SoftSphere]). Our exponents appear from the fourth row. The fourth and fifth rows show the exponents calculated from the behavior of the characteristic distances as a function of $N$. The exponents calculated using the distribution of ${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$ appear in the next four rows. The exponents of the sixth and seventh rows were obtained using the Lhuillier distribution for both distributions. For the calculation of the exponents that appear in the last two rows, the Fisher-McKenzie-Moore-des-Cloiseuax distribution was used.
\[tabla1\]
Energy of the Polymeric Chains {#energy}
------------------------------
To study the energy of the random polymer chains generated by our simulation, we analyzed the energy given by the interactions between monomer in each chain, as well as the energy associated with the bending of the chains. As generated chains are homopolymeric, they are expected to show a uniform behavior or have a low amount of metastable states [@entropia_energia; @energias].\
In the contact potential described in Eq. (\[eq:energia\_interacao\]), the value of the constant $e_{\upsilon_{i}\upsilon_{j}}$ has the information about of interaction energy between the non-continuous and adjacent monomers. This constant adopts the value of $-1$, for each contact, thus generating a “folding” force in the chain, known in proteins as a hydrophobic force [@energia_chao_tang]. The sum of all the interactions in chain defines the energy of the system, called *interaction energy* ($E$).\
To take into account the flexibility of the chain and, consequently, its tangent correlations, we proposed adding to the interaction term $E$, the *bending energy* ($H$) in its discrete version, proposed in Eq. (\[eq:energia\_desvio\]).\
In the Fig. \[figura9\] we can see the characteristic histograms of each energy for three-dimensional chains generated using our algorithm. The shape of the distribution is similar to the results obtained previously for the distribution of number of steps.\
![Interaction energy $E$ (left side of figure) and bending energy $H$ (right side) histograms, for three-dimensional chains. Gaussian distributions with standard deviations, $\sigma_{E}=50.66$ and $\sigma_{H}=30.52$, and mean values $\mu_{E}=-701.60$, and $\mu_{H}=-791.77$ respectively.[]{data-label="figura9"}](Histo_E "fig:"){width="7.7cm"} ![Interaction energy $E$ (left side of figure) and bending energy $H$ (right side) histograms, for three-dimensional chains. Gaussian distributions with standard deviations, $\sigma_{E}=50.66$ and $\sigma_{H}=30.52$, and mean values $\mu_{E}=-701.60$, and $\mu_{H}=-791.77$ respectively.[]{data-label="figura9"}](Histo_H "fig:"){width="7.7cm"}
One of the most important results obtained here is that, even when considering both attractive and repulsive behavior of the chains, the resulting chains have negative energy, because in the SAW models for polymers in a good solvent, the attractions prevail over the repulsions [@Rubinstein; @polimer_textbook].\
The total energy of the chain is described by the sum of interaction and bending energies, which determine the structural configuration. In the Fig. \[figura10\], the total energy distribution is shown. The most probable energy values are between $-1650$ and $-1350$ energy units, corresponding to long chains. The drop in probability density at the tail of the distribution at energy values approaching zero correspond to “small” chains. These small chains can also be identified in the in the probality density function of $N$, which shows the same drop on the left tail of the distribution. This may be due to the fact that the total number of configurations, for real chains, is smaller than for ideal chains because of the reduction of probabilities in the distributions of the characteristic distances, mentioned above, for ${\bf{R}}_{ee}\lesssim40$ and ${\bf{R}}_{g}\lesssim10$ (see Fig. \[figura5\]), which corresponds to $N\lesssim500$.\
Following the same procedure used to study the Flory exponent, the mean values of interaction energy and bending energy were computed for eleven thousand three-dimensional chains. The average energy was plotted as a function of N (Fig. \[figura11\]). The plot shows a linear and uniform behavior for the two energy (which is expected [@entropia_energia; @energias]). When $N$ is large, the behavior of the interaction energy changes from decreasing to increasing, having less negative values, which can be interpreted as representative of more stretched chains, with fewer contacts but with small lengths of persistence still prevailing.\
![Total Energy histogram ($E + H$) for three-dimensional chains with mean with mean $\mu=-1499.14$ and standard deviation $\sigma=53.24$.[]{data-label="figura10"}](Histo_H+E){width="8.7cm"}
![Interaction energy $E$ and bending energy $H$ as a function of the number of steps $N$. The horizontal dotted lines represent the values between which the energies are distributed (see Fig. \[figura10\]). On the other hand, the vertical ones, represent the values between which distribute all sizes of the three-dimensional chains generated.[]{data-label="figura11"}](E_Vs_N){width="8.7cm"}
The great majority of the three-dimensional chains generated are large ($N>1700$). These chains, with energies between $-1350$ and $-1650$, belong to the second regime of the graph of the total energy and follow a Gaussian distribution (see Fig. \[figura10\]). The rest of the chains, with energy between $0$ and $-1350$, belong to the first regime of the energy graph and follow a uniform distribution.\
In the “discrete” bending energy approach, the second regime does not appear (see Fig. \[figura11\]). This term has the same linear behavior, independent of the presence of neighbors interacting with other monomers. The folding of the chains in the large $N$ regime adds negative energy to the system. The graph of Fig. \[figura11\] shows a lower density of points for energies between $-450 and 0$ (corresponding to $0 \lesssim N \lesssim 500$) confirming the decrease in the number of configurations for small chains, as shown in the dot density of Fig. \[figura7\] and Fig. \[figura8\] for $N\lesssim 500$.\
The Fig. \[figura12\] a typical linear polymeric chain of $N=1772$ generated by the simulation is illustrated. The generated chain forms two clusters separated by a tail. The interaction energy decreases because the chain “escapes” and does not have neighbors that contribute to this energy. On the other hand, the bending energy remains constant, since the flexibility of the chain is maintained and causes the chain to continue folding even in the “bridge” that separates the clusters.\
![Typical structure of $N=1772$ generated by the simulation which shows the formation of two clusters. The spheres represent the monomers of a homopolymeric chain.[]{data-label="figura12"}](simul2){width="7.5cm"}
In the total energy graph of Fig. \[figura11\], two regimes can also be seen due to the contribution of the interaction term. For chains of size below $N = 1775$ (mean value of $N$), the energy decreases with $N$ at a rate of $0.86$, practically twice the rate of decrease of each term separately, because the two energy contributes in this regime in the same proportion. For larger chains, the energy increases with $N$ and the characteristic rate of this increase is $0.29$. Here the different contributions of the two types of energy make the increase less significant than in the case of the interaction energy.\
CONCLUSIONS
===========
In this work, an algorithm based on natural self-avoiding random walk in a cubic network with no boundary was used to generate linear chains. The implementation of this algorithm is simpler and its computational cost is significantly lower than other approaches generally used to generate self-repelling chains. Although our method does not consider all possible configurations for $N$ size of the chain we obtain values of the characteristic critical exponent’s of the biopolymers, similar to those reported in literature. The distribution of the characteristic distances for the linear chains obtained from our simulation showed a reasonable correspondence with that reported in the literature. Specifically, we prove that although it is possible to fit the two distributions (${\bf{R}}_{g}$ and ${\bf{R}}_{ee}$) using both Lhuillier’s theory and the law proposed by Fisher-McKenzie-Moore-des Cloiseaux, the obtained values of critical exponents such as $\delta$, $\alpha$, $\nu$ show us that the distribution of ${\bf{R}}_{g}$ responds better to the theory of Lhuillier, while in the case of the distribution of ${\bf{R}}_{ee}$, the best fit is obtained with the Fisher-McKenzie-Moore-des Cloizeaux function. These results are in full agreement with the literature and reinforce the validity of the algorithm used when characterizing this type of system.\
Both ${\bf{R}}_{ee}$ and ${\bf{R}}_{g}$ resulted in power functions of $N$ and the values of the Flory exponent, in both cases, are quite close to the theoretical value, especially the value corresponding to ${\bf{R}}_{g}$, showing that this characteristic distance is more appropriate when characterizing structurally this type of chains. By comparing the behavior with $N$ of these two distances, we can validate the theoretical linear dependence between these two parameters in the case of real chains.\
Our study revealed interesting behaviors related to the flexibility of the chains. Small chains are more correlated and consequently, less flexible. Medium-sized chains with N values below $1650$ behave uniformly, being more flexible as $N$ increases.\
Comparing the behavior of the energy $E$, related to the monomer-monomer interactions of the chains, with the energy $H$, related to its flexibility, we found that chains start with clusters and doubles over distances of the order of ${\bf{R}}_{g}$, increasing its energy. When the chains reach a large number of steps, the interaction energy revealed that the chain stretches and escapes from the cluster, which results in a loss of interaction energy. The behavior of the bending energy reveals that in this escape regime, the chains keep the folding behavior they showed before the escape. This process lasts until the clusters reach a size comparable with the mean radius of gyration.\
The analysis presented in this work, including both the bending and interaction energies is important because it allows differentiating between chains of equal interaction energy but with different structures and hence different bending energies.\
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has received financial support from CAPES and CNPq (Brazilian Federal Grant Agencies) and from FACEPE (Pernambuco State Grant Agency). We thank Pedro Hugo Figueirêdo, Raúl Cruz Hidalgo and Juan Miguel Parra Robles for useful comments of this work.\
References {#references .unnumbered}
==========
|
---
abstract: 'In this article we extend on work which establishes an analogy between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater than one which do all act as good resources for one-way quantum computation, and sets of fractal lattices with dimension greater than one all of which do not. The difference is put down to other topological factors such as ramification and connectivity. This work adds confidence to the analogy and highlights new features to what we require for universal resources for one-way quantum computation.'
author:
- Damian Markham
- Janet Anders
- Michal Hajdušek
- Vlatko Vedral
bibliography:
- 'C:/DAMO/work/GenBib.bib'
title: Measurement Based Quantum Computation on Fractal Lattices
---
Introduction
============
Drawing analogies is often a very powerful tool in science. It can allow not only deepened understanding through new perspectives opened up, but it can also allow technical tools from one discipline to be applied to another, often very fruitfully. In [@Anders07] an analogy between measurement based quantum computation and thermodynamics is made, viewing the computation itself as a phase transition. This is in spirit reverse to the thought provoking analogy made by Toffoli [@Toffoli98] where physics is viewed as a computation, rather [@Anders07] tries to understand computation itself as a physical process. In doing so key features of useful resources for one-way quantum computers were identified, in direct analogy to the identification of critical systems in thermodynamics. In particular rather elegant and simple methods first developed by Peierls [@Peierls36] to show that one dimensional spin chains are not critical, where as two dimensional chains are, were translated into arguments of the dependence of dimension on universal resources for one-way quantum computation. In this work we extend this theme to look for other important features of universal resources, following the work on fractal lattices by Gefen et. al. [@Gefen80]. There it is shown that critical behaviour in spin systems relies not just on the dimension of the lattice, but also other features such as order of ramification and connectivity. We again see an exact mirroring of results, highlighting these also as features crucial for universal resources for one-way quantum computation. As examples we will see that there exists a set of fractal lattices (Sierpinski carpets) for which any dimension greater than one guarantees it can act as a universal resource. On the other hand we will also see examples of dimension greater than one which are not universal, highlighting the importance of the other topological features (ramification, connectivity and lacunarity).
The analogy: Phase Transition and Measurement Based Quantum Computation
=======================================================================
We start by reviewing the problems addressed in this analogy. In the case of thermodynamics and many-body physics, the problem which is of interest is the existence, or not, of some critical phenomena or phase transition. Simply put, a phase transition is when a small change in some parameters of a given system gives rise to a large macroscopic change of state, or phase. For example, at just below zero degrees water becomes ice, and just above it becomes water again. These two phases of matter are clearly very different. In spin systems the macroscopic property of interest is whether the system is magnetised or not. This happens when sufficiently many spins point in the same direction - we call this the ‘ordered state’. The effect is witnessed by the amount of magnetisation $M$ present - which is called an ‘order parameter’. In the Ising model the ground state (corresponding to zero temperature) is ordered, and for high temperatures, the orientation of the spin becomes random and it is not ordered - its magnetisation is zero. The question is then whether or not there is a finite, non-zero, temperature $T_{crit}$ below which the system is ordered. If this is the case we say there can be a phase transition from non-magnetised to magnetised at temperature $T_{crit}$. It is known that for one-dimensional spin chains with nearest neighbour interactions only, there is no phase transitions, where as for two dimensional lattices there are. This will be explained via the Peierls argument below.
In the case of measurement based quantum computation, the problem of interest is the ability, or not, to perform universal quantum computation. In one-way quantum computation (1-way QC, in this work we take it to be synonymous with measurement based quantum computation) [@Raussendorf01], a computation is carried out, first by preparing a highly entangled multipartite quantum state (which we call a ‘resource state’, and which is independant of the actual computation to be performed) and then performing local measurements and local corrections on individual sites. The choice of measurements, and how they depend on each other determines the computation which is performed. During the process of measurement entanglement is destroyed, and in this sense consumed by the computation. At the end of the computation, the classical information $I$ of the solution is obtained as the measurement outcomes of the last few measurements. Since its invention a large amount of effort has gone into finding out what constitutes a good initial resource state (see e.g. [@VandenNest07; @Gross09]). Given a particular set of states, the question is, whether or not it can act as a universal resource for quantum computation. The analogy we will now draw goes towards answering this question. Note that we will always consider resource states as graph states in this work.
A list of the analogous quantities between thermodynamics on the one hand, and one-way quantum computation on the other is given in Fig. \[fig:comparison\].
![\[fig:comparison\] (Borrowed from [@Anders07]) On the left, a set of quantities from thermodynamics and on the right their analogous counterparts for one-way quantum computation.](table){width="35.00000%"}
The second law of thermodynamics states that systems interacting with a thermal bath will always conspire to minimise the free energy $F$, given by $$\begin{aligned}
\label{eqn: 2nd Law}
F=U-TS.\end{aligned}$$ Intuitively we can understand the second law as saying that by the process of thermalisation, nature insists that at a given temperature $T$, the energy $U$ is spread out as much as possible, by maximising the entropy $S$.
When making our analogy the quality we are insisting upon for our 1-way QC computation is that it should be universal. To this end, we postulate a kind of ‘law of 1-way QC’ whereby we insist (as ‘mother nature’ of the 1-way QC - it is after all us who designs and controls it) that the quantum computer be as universal at each step as possible. That is, we insist that at each time the computation is carried out in such a way as to maximise the potential. In terms of quantities, we say that for a given amount of entanglement $E$, we insist that any computation at any time $t$ maximises the number of ways it can be used - which we call the computational capacity $C$. We thus phrase our ‘second law of universal 1-way QC’ as that at each time $t$, the potential $P$, given by $$\begin{aligned}
\label{eqn: 2nd Law of QC}
P=E-1/tC\end{aligned}$$ must be minimised (i.e. the potential should be consumed as fully as possible).
Following an argument first put forward by Peierls [@Peierls36], and cleaned up by Griffiths [@Griffiths64], which shows that one dimensional spin chains are not critical, but two dimensional spin chains are, an intuitive argument as to why a one dimensional cluster state is not a universal resource for 1-way QC, where as a two dimensional cluster state is a universal resource was given [@Anders07]. Peierls’ argument goes as follows. If we want to test whether an ‘ordered state’ (i.e. one with a large number of spins pointing in the same direction such that there is overall a positive magnetisation) is possible for some nonzero temperature $T$, we simply check wether small pertubations to this state will raise or lower the free energy of that state (the very physicsy ‘shake it and see’ approach). Any such pertubation will change the free energy $$\begin{aligned}
\label{eqn: Pertbn of F}
\Delta F = \Delta U - T \Delta S,\end{aligned}$$ If by perturbing it we can [*reduce*]{} the free energy, the state clearly is not a valid thermal state, by the 2nd law. In terms of equation (\[eqn: Pertbn of F\]) this is then a question of balance between the change in energy and the change in entropy. If perturbing the system increases the entropy more than the energy, the state before pertubation was not a valid thermal state. In the case of a one dimensional spin chain, the cost of any pertubation in terms of entropy is much greater (it scales with the number of spins $n$) than the cost in energy (which is fixed). In the case of a two dimensional spin, they scale with $n$ in the same way, hence a balance can be found. By finding the fixed point of the free energy (the point where the pertubation makes no change - by setting (\[eqn: Pertbn of F\]) to zero), a critical temperature can be found above which the system is not ordered. Remarkably, given the simplicity of this approach, this is very close to the actual critical temperature, below which it can be shown also that the system is ordered.
When testing if a system can be used for 1-way QC, the ordered state is the ‘solution state’ (the state after all measurements have been made in the 1-way QC) and the test is, if it is possible for some finite time $t$. Again, we test this by perturbing it and seeing if it violates our 2nd law of 1-way QC. If it does, it is definitely not a valid state according to our second law - that is, no computation satisfying our ‘law of 1-way QC’ can find such a state at a time $t$. Any partubation results in a change in computational potential $$\begin{aligned}
\label{eqn: Pertbn of P}
\Delta P = \Delta E - 1/t \Delta C.\end{aligned}$$ This then bares out as a balance between the entanglement $E$ and the number of ways of using it $C$, for a given $t$. As above, in the case of a one dimensional cluster state, this is balance can not be met - the number of ways of using the entanglement is larger than the entanglement available, hence some choice must be made about its use, sacrificing universality. Alternatively it says that there is no finite time length at which it could be achieved, so if it were possible, it would take an infinite amount of time. On the other hand as for the spin case, a two dimensional lattice these quantities do balance. Again as above it is possible to approximate a critical time $t_{crit}$ below which the computation cannot be completed in a universal fashion, by setting by setting (\[eqn: Pertbn of P\]) to zero. In fact by seeing how both of these factors scale with dimension $D$, it is possible to arrive at the following formula $$\begin{aligned}
t_{crit} \propto \frac{ln{D}}{D},\end{aligned}$$ which agrees with both our intuition and examples that higher dimensional states can allow for greater speed in computation.
Computing on Fractal Lattices
=============================
We can now extend this analogy to cover another interesting set of examples from many-body physics, where it is shown that not only does dimension play a role in spin criticality, but also other topological features. In [@Gefen80] similar techniques to those of Peierls and Griffiths described above are used to test the criticality of spin systems, this time based over several self-similar fractal lattices. Again the arguments are testing the ability of a lattice to balance the change in energy and entropy for small pertubations. Examples are presented which both do and do not allow criticality for all (fractal) dimensions greater than one. The additional features which capture the existence of criticality are shown to be topological including ramification, connectivity and lacunarity.
We follow the same analogy as before to show exact mirrors of these results in 1-way QC. We see that graph states of the fractal lattices of Koch curves and Sierpinski gaskets are not universal resources for 1-way QC, where as Sierpinski carpets are, independent of dimension. As in [@Gefen80] we can interpret this as the role of other topological factors including the ramification.
The Koch curve is illustrated in Fig. \[FIG: KochCurve\]. For our purposes this behaves exactly as in the 1D case in the previous section, where we argue it is indeed not a good universal resource [@Anders07]. Proofs to this effect are also known in the literature ([@VandenNest07]). The Sierpinski gasket is shown in Fig. \[FIG: SierpinskiGasket\]. Again, the same Peierls like arguments show it is not a valid possibility for a universal resource follow as those made above. That is the balance between the entanglement present and the number of ways to use the entanglement can not be found for some finite $t$. In analogy to the spin case [@Gefen80], a significant pertubation of the solution state by adding entanglement can be done in many more ways than the amount of entanglement that is added, causing a negative change in P (equation (\[eqn: Pertbn of P\])). This is unfortunately not a rigorous proof of non-universality, since our analogy (and in particular our ‘law of 1-way QC’) is not proven, rather it is justified. We can however prove that the Sierpinski gasket is not a universal resource by methods introduced in [@VandenNest06]. There it is shown that if the entanglement does not scale with a family of resource states (such as our lattices), then it cannot be a universal resource for 1-way QC [@VandenNest06; @VandenNest07]. The entanglement measure they use is the *entanglement width* $E_{wd}$ defined as $$\begin{aligned}
E_{wd}(|\psi\rangle):= \min_T \max_e E^{bi}_{T,e}(|\psi\rangle),\end{aligned}$$ where $E^{bi}_{T,e}(|\psi\rangle)$ is the bipartite entropy of entanglement across the bipartite cut defined by $T$ and $e$. $T$ is a subcubic graph with $n$ leaves (edges not leading to a vertex at one end) and $e$ is an edge of $T$. Each leaf corresponds to a qubit. The bipartite cut is defined by removing edge $e$ to give two separate trees. The leaves of one tree correspond to one side of the cut, and the other tree the other side of the cut. It can easily be seen that for the Sierpinski gasket $|\psi_{SG}\rangle$ a tree can be defined with the same self similar properties, such that the best cut $e$ also has self similar properties and gives entanglement $E^{bi}_{T,e}(|\psi_{SG}\rangle)=3$ which does not grow. Hence the entanglement width is bounded $E_{wd}(|\psi_{SG}\rangle)\leq3$ for any lattice size. Since it does not scale with the size of the lattice, it cannot be a universal resource.
On the other hand Sierpinski carpets (see Fig. \[FIG: Sierpinski\]) are universal for all dimensions greater than one. The arguments to show it is a valid possibility for a universal resource follow along the same lines as the Peierls like argument made above. That is the balance of entanglement present and the number of ways to use the entanglement can always be found for some finite $t$. This is of course not a proof that it is a universal resource, since, even if we assume our ‘law of 1-way QC’, it only shows that it is a possible resource, i.e. that it doesn’t violate the law of 1-way QC. We can however construct exact proofs for all cases. To show explicitly that these are universal, we adopt a similar technique to that used in [@Browne08], which is to actively construct a standard 2D lattice by taking out vertices using local $Z$ and local $X$ measurements, which in turn is known to be a universal resource [@Raussendorf01]. The idea is that, given an arbitrary lattice (which may even be irregular, as in the case of [@Browne08]), if we can draw a standard 2D grid over this lattice, we can measure away the extra qubits to leave only the ideal 2D lattice. This is possible because of the way $X$ and $Z$ measurements convert one graph state to another (see e.g. [@Hein06]). It is easy to see by looking at the Sierpinksi carpet Fig. \[FIG: Sierpinski\], it is always possible to draw a 2D grid which grows with the size of the carpet. Thus we always have a way to get a known universal resource for any dimension greater than one.
We thus see that the ability of a lattice to act as a universal resource for 1-way QC does not just depend on dimension. In particular, for any dimension between $1$ and $2$ we can find a Sierpinski carpet which is a universal resource, where as we have seen two examples with dimension in this range which are not universal. As in [@Gefen80] we can then infer that it is down to other topological properties. One such property that resonates in the case of 1-way QC is *ramification*. The ramification $R$ is the minimal number of edges that must be removed to separate a part of the lattice of arbitrary size. It tells us something about how globally connected the lattice is or how easily the lattice can be separated into chunks. The lower the ramification the easier it is to separate parts of the lattice off and the less globally connected it is. From our examples, those lattices with finite ramification are not universal resources, where as those with infinite ramification are (the 2D lattice also has infinite ramification). This is very similar in flavour to the idea behind the entanglement width introduced in [@VandenNest06] to check for universality of resource states (and used above). In a sense this also looks for some global connectedness, by the nature of the min max definition above. Here too an infinite scaling is required for universality. We may imagine there could be a connection between the two. We may also wonder whether an alternative entanglement measure can be defined with respect to ramification (or indeed other topological properties), which could be used to see their usefulness as resources for 1-way QC. This is beyond the scope of the current manuscript, but poses interesting possibilities.
Conclusions
===========
We have seen that the analogy developed in [@Anders07] can be used to argue that fractal lattices can also act as universal resources for 1-way QC, and that not only is dimensionality important, but also other topological features such as ramification. By providing further examples where the analogy succeeds we have strengthened its validity. It also highlights new features that we can expect good resources should posses for 1-way QC. We can also ask how this corresponds to known conditions for good resources, such as the entanglement conditions in [@VandenNest07]. In this context perhaps it is possible that the important topological features could also correspond to particular entanglement features. Another possible connection to existing conditions would be to the existence of Flow on these lattices. Flow (and gFlow) are known sufficient conditions for a lattice, or graph to allow 1-way QC [@Danos06; @Browne07]. The fact that for example in the Sierpinski carpets we can always reduce them to a 2D lattice implies that we can always extract a flow in some sense. Perhaps the topological features presented here are also important for the existence of flow. We also note that the techniques, and indeed the lattices used are very similar to those in [@Browne08], which arise in the context of 2D lattices with noise. In a sense this is no surprise since it is similar situations which may give rise to fractal lattices in many-body physics also. But it may also indicate that the analogy used could be useful in treating noise over fixed lattices. On foundational level, this analogy, and its reenforcement by this work, opens up many interesting questions and possibilities. For example, can these analogies be made more solid by a kind of path integral approach to 1-way QC? How deep can we take these analogies beyond 1-way QC, can it work for example for other models of computation? We hope this work will stimulate further research in these areas.
*Acknowledgements* DM acknowledges support from ANR Projet FREQUENCY (ANR-09-BLAN-0410-03). JA thanks the Royal Society for support in form of a Dorothy Hodgkin Research Fellowship. VV acknowledges financial support from the National Research Foundation and Ministry of Education in Singapore. VV is a fellow of Wolfson College Oxford.
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---
abstract: 'The electronic structure of benzene on graphite (0001) is computed using the GW approximation for the electron self-energy. The benzene quasiparticle energy gap is predicted to be 7.2 eV on graphite, substantially reduced from its calculated gas-phase value of 10.5 eV. This decrease is caused by a change in electronic correlation energy, an effect completely absent from the corresponding Kohn-Sham gap. For weakly-coupled molecules, this correlation energy change is seen to be well described by a surface polarization effect. A classical image potential model illustrates trends for other conjugated molecules on graphite.'
author:
- 'J. B. Neaton'
- 'Mark S. Hybertsen'
- 'Steven G. Louie'
date: 'June 18, 2006'
title: 'Renormalization of Molecular Electronic Levels at Metal-Molecule Interfaces'
---
There is renewed interest in using organic molecules as components in nanoscale electronic and optoelectronic devices [@ref1; @ref2], and thus a critical need has emerged for improved knowledge and control of charge transport phenomena in organic molecular assemblies [@ref3]. Understanding transport across the interface between the active organic layer and the metallic electrode has proved particularly challenging, especially in the single-molecule limit. Fundamentally, charge transport is controlled in such systems by the electronic coupling of frontier molecular orbitals to extended states in the electrode, and the energetic position of these orbitals relative to the contact Fermi level. Several recent measurements of organic thin films, self-assembled monolayers (SAMs), and single-molecule junctions have emphasized the important role of Coulomb interactions between the added hole or electron in the frontier orbitals and the metal substrate [@ref4; @ref5; @ref6; @ref7; @ref8; @ref9; @ref10; @ref30]. However, most theoretical calculations of transport through organic molecules have continued to rely on some implementation of density functional theory (DFT) or semiempirical one-particle Hamiltonians [@ref3]. The limitations of DFT for describing excited-state energies are well known [@ref11], and implications for a DFT-based theory for nanoscale conductance have been recently discussed [@ref12].
When a molecule is brought in contact with a metal, several physical effects will influence its ionization level (highest occupied molecular orbital, HOMO) and affinity level (lowest unoccupied molecular orbital, LUMO). First, the self-consistent interaction between molecule and surface will rearrange the electron density and modify the alignment of frontier orbital energies. Second, electronic coupling to extended states in the metal will further shift orbital energies and broaden discrete molecular levels into resonances. Finally, the Coulomb interaction between the added hole or electron associated with the ionization or affinity level will result in a polarization of the metal substrate. This additional correlation energy further stabilizes the added hole or electron, reducing the gap between affinity and ionization levels as illustrated in Fig. 1. An accurate DFT-based approach should correctly capture the first effect [@ref12a], although the use of DFT to calculate the width of resonances is under debate [@ref12]. Importantly however, the surface polarization response, as we show here, is completely absent from frontier orbital energies computed in DFT.
{width="6.5cm"}
In this Letter, we compute the electronic excited states for a clean, weakly-coupled system consisting of an aromatic molecule (benzene) physisorbed on the graphite (0001) surface. Electronic correlations are included directly within a first-principles many-electron Green function approach [@ref13]. The electron self-energy is calculated from first principles within the GW approximation (GWA) [@ref14] using a methodology [@ref15] that has proved accurate for a wide range of systems [@ref16]. While more generally including dynamical electronic correlation, the GWA is well known to include static, long-range image potential effects for an electron near an interface [@ref17]. Using this theoretical approach, we predict a strong renormalization of the electronic gap of the benzene system (relative to its molecular gas-phase value) when it is physisorbed on a graphite (0001) surface. The change in the electron correlation energy on adsorption can be understood as a polarization effect in this case. An image potential model is used to illustrate trends for other aromatic molecules.
Equilibrium geometries of molecular benzene in the gas-phase, condensed in a bulk crystalline phase, and physisorbed on graphite (0001) are determined using DFT within the local density approximation (LDA). Norm-conserving pseudopotentials [@ref18] are used with a plane-wave basis for the electron wavefunctions (80 Ry cutoff) for structural relaxations. The surface is modeled with a 3$\times$3 supercell containing 4 layers of graphite, a single benzene molecule, and the equivalent of 7 layers of vacuum. The theoretical in-plane bulk lattice parameter is used ($a$=2.45 Å, $c$=6.62 Å). In the most stable site for adsorption, benzene rests flat on the surface centered on a three-fold site 3.25 Å above a substrate carbon atom, in agreement with a previous study [@ref19]. For comparison, benzene is also considered in an upright position, centered above a hollow site with its closest hydrogen atom 2.21 Å from the surface. Solid crystalline benzene has an orthorhombic unit cell containing four molecules (Pbca); the atomic positions within the unit cell are optimized keeping the lattice parameters $a$, $b$ and $c$ fixed to their experimental values of of 7.44, 9.55 and 6.92 Å respectively [@ref20]. The gas phase is modeled using a cubic supercell ($a$=13.22 Å). For each system, matrix elements of the self-energy operator are evaluated using a 50 Ry energy cutoff for the electronic wavefunctions, a 6 Ry cutoff for the momentum-space dielectric matrix, and a 2.9 Ry cutoff for the sum on the virtual states. This choice of parameters results in quasiparticle energy gaps converged within $\sim$ 0.2 eV.
The electron addition and removal energies of a benzene molecule in the gas-phase, calculated in the present GW approach, result in a HOMO-LUMO (quasiparticle) gap of 10.51 eV. This value agrees well with an independent GW calculation [@ref21], total energy difference calculations based on DFT [@ref22; @ref23], and experiment [@ref24]. By contrast, the Kohn-Sham gap (within LDA) is 5.1 eV, substantially smaller.
{width="6.5cm"}
The electronic structure of benzene on the graphite (0001) surface along the $\Gamma$-K’ direction is shown in Fig. 2. Comparing the surface-projected band structures (shaded regions) in Fig. 2(a) and 2(b), the quasiparticle bandwidth increases by about 15% relative to LDA, in agreement with previous works [@ref25; @ref26]. The bold horizontal lines interpolate between the benzene HOMO and LUMO states computed at $\Gamma$ and K’; the filled circles at these high-symmetry points indicate states with significant weight on the molecule. For physisorbed benzene, the Kohn-Sham (LDA) gap is 5.05 eV throughout the zone, unchanged from the corresponding LDA gas-phase value. Relative to the LDA value, the quasiparticle gap of the molecule flat on the graphite surface is much [*larger*]{}, 7.35 eV. However, the predicted quasiparticle gap is substantially [*smaller*]{} than the gas-phase value of 10.51 eV.
Table 1 summarizes the calculated HOMO-LUMO gaps of benzene in the four environments considered in this study. Interestingly, the LDA gaps are identical for all environments. In contrast, the GW self-energy corrections exhibit noticeable variation. To understand this variation, we analyze the self energy change relative to the gas-phase. The change $\Delta\Sigma$ for each frontier level is decomposed into Coulomb-hole ($\Delta\Sigma_{\rm CH}$), screened-exchange ($\Delta\Sigma_{\rm SX}$), and bare exchange or Fock ($\Delta\Sigma_{\rm X}$) contributions. We find that $\Delta\Sigma_{\rm CH}$ is nearly equal for the occupied and empty frontier states, and that $\Delta\Sigma_{\rm X}$ is quite small (0.1-0.2 eV). Interestingly, the screened exchange term that is responsible for most of the difference: for the HOMO, we observe $\Delta\Sigma_{\rm SX}$ $\sim$ -2$\Delta\Sigma_{\rm CH}$, while for the LUMO $\Delta\Sigma_{\rm SX}$ $\sim$ 0. Put together, the change in correlation energy ($\Delta\Sigma_{\rm Corr}=\Delta\Sigma_{\rm CH}+\Delta\Sigma_{\rm SX}$) reported in Table 1 turns out to be nearly symmetric between the ionization and affinity levels for the benzene in each environment studied. This result is qualitatively the same as that obtained from the derivation of the image potential effect for an electron near a metal surface [@ref17].
----------------------------------- ------- ------------- ------------- -------------
Gas Flat Perp Crystal
phase graphite graphite phase
$\Delta E_{\rm gap}$ (LDA) 5.16 5.05 5.11 5.07
$\Delta E_{\rm gap}$ (GW) 10.51 7.35 8.10 7.91
$\Delta\Sigma_{\rm Corr}$ 1.45, -1.51 1.18, -1.17 1.16, -1.15
$\Delta\Sigma_{\rm Corr}$ (Model) 1.50, -1.43 0.97, -0.96
----------------------------------- ------- ------------- ------------- -------------
: \[displacements\] Benzene HOMO-LUMO gaps in the gas phase, crystal phase, and adsorbed on the graphite surface (flat and perpendicular). First and second lines are Kohn-Sham (LDA) and quasiparticle (GW) gaps. (For the crystal, we average over the $\pi$ and $\pi^*$ manifolds.) Third and fourth lines are calculated changes in correlation energy for the HOMO and LUMO, relative to the gas phase, determined from the full GW calculations and from an image potential model. Energies are in eV.
To develop a more detailed model of our self energy results, we recognize that the benzene frontier orbitals are only weakly coupled to the environment. When the orbitals have negligible overlap with the metal, the correction to the molecular self-energy operator upon adsorption depends only on the change in the screened Coulomb interaction W, i.e. $$\Delta \Sigma_{SX}({\bf r,r'}; E) = \sum_{j}^{occ} \phi_j({\bf r})\phi^*_j({\bf r'}) \Delta W ({\bf r,r'};E-E_j),$$ where $\phi_{\rm j}$ are molecular wavefunctions and $E_j$ their eigenvalues. A corresponding expression exists for the Coulomb-hole term. For sufficiently large metal-molecule separations, $\Delta$W is smooth and slowly-varying over the spatial extent of the molecular orbitals, and only the self-term contributes to single-particle matrix elements of Eq. (1). Then the change in correlation energy from the surface can be reduced to $$\begin{split}
\Delta E_{\rm HOMO} &= \langle \phi_{\rm HOMO}|\Delta\Sigma_{\rm SX}+\Delta\Sigma_{\rm CH}|\phi_{\rm HOMO}\rangle \\
&\approxeq 2P_{\rm HOMO} - P_{\rm HOMO} = P_{\rm HOMO} \\
\end{split}$$ and $$\begin{split}
\Delta E_{\rm LUMO} &= \langle \phi_{\rm LUMO}|\Delta\Sigma_{\rm SX}+\Delta\Sigma_{\rm CH}|\phi_{\rm LUMO}\rangle \\
&\approxeq 0 - P_{\rm LUMO} = -P_{\rm LUMO}, \\
\end{split}$$ where $P$ is the static polarization integral $$P_{j} = - {1\over 2} \int\int d{\bf r}d{\bf r'} \phi_j({\bf r})\phi^*_j({\bf r'}) \Delta W ({\bf r,r'}) \phi_j({\bf r'})\phi^*_j({\bf r}).$$ For benzene on graphite, the full GW calculations indicate that dynamical effects make a negligible contribution to $\Delta\Sigma_{\rm Corr}$, and that the self-term accounts for more than 90% of $\Delta\Sigma_{\rm Corr}$, supporting the simplified picture of Eqs. (2-4).
{width="6cm"}
Significant further simplification is achieved if an image potential model is sufficient for $\Delta W({\bf r},{\bf r})$. In Fig. 3, the screening potential through the molecular adsorbate is illustrated by considering $\Delta W({\bf r},{\bf r})$ = $\Delta V_{\rm scr}({\bf r},{\bf r})$, where $V_{\rm scr}({\bf r},{\bf r})$ results from the screening response to the added electron (or hole) [@ref15]. The difference between the adsorbed molecule and isolated molecule, $\Delta V_{\rm scr}({\bf r},{\bf r})$, is compared with the image potential model, $\Delta V_{\rm scr}({\bf r},{\bf r})$$\rightarrow$$1/4|z-z_0|$. The image plane position z$_0$ is explicitly determined, in a separate calculation [@ref27], to be 1 Å beyond the last surface plane for our graphite slab. From Fig. 3 it can be seen that, over the spatial range of the molecular orbital, an image potential captures the main effect. Using the value of the image plane position computed above and the frontier orbitals, $P_{\rm HOMO}$ and $P_{\rm LUMO}$ for benzene flat and perpendicular on the graphite surface are calculated using the image potential model. As shown in Table 1, the image potential model is quite accurate for the flat case and captures most of the effect for the perpendicular case (within 0.2 eV). The simple image potential model neglects the internal screening response of the molecule to the polarization of the metal surface. While small for a flat molecule oriented parallel to a surface, a significant molecular polarizability perpendicular to the metal surface leads to an additional energy gain, increasing the $P_{\rm j}$.
------------- --------- --------- --------------- ---------------- ---------------- ---------------
Molecule Expt IP Expt EA Gas-phase gap $P_{\rm HOMO}$ $P_{\rm LUMO}$ Adsorbate gap
Naphthalene 8.14 -0.20 8.34 1.41 1.39 5.54
Anthracene 7.44 0.53 6.91 1.32 1.30 4.29
Tetracene 6.97 0.88 6.09 1.24 1.23 3.62
Pentacene 6.63 1.39 5.24 1.18 1.18 2.88
Coronene 7.29 0.47 6.82 1.19 1.17 4.46
------------- --------- --------- --------------- ---------------- ---------------- ---------------
Provided that molecular resonances are well separated from the metal Fermi energy, the polarization model for $\Delta\Sigma$ should be broadly applicable. To illustrate the impact of the change in correlation energy, we use the image model to predict the renormalized gaps for members of the acene series and coronene adsorbed flat on graphite (Table 2). For the larger molecules in the series, the change in gap becomes dramatic, e.g. the pentacene gap is predicted to diminish by nearly a factor of two on a graphite surface.
The role of geometry and morphology on changes in polarization energy in organic systems can be subtle [@ref28], but the impact has been directly measured for organic films on metal substrates using photoemission and inverse photoemission [@ref6]. Adsorbate frontier orbital energies can be probed directly by STM, provided the HOMO-LUMO gap is small enough for the resonant tunneling regime to be experimentally accessible [@ref6; @ref29]. From Table 2, tetracene and pentacene are within typical measurement range ($\pm$2.5 V), while anthracene and coronene are marginal. In a recent study of pentacene adsorbed on ultrathin NaCl on Cu(111) [@ref30], gaps of 3.3, 4.1 and 4.4 eV are observed for NaCl thicknesses of one, two and three monolayers, respectively. Our modeled value of 2.9 eV for direct adsorption on the graphite surface fits well with this progression.
In conclusion, we find that the correlation contribution to the frontier molecular orbital energies depends sensitively on environment. In the examples studied here, the change in correlation energy is dominated by a polarization effect. The impact of electrode surface polarization on spectroscopic measurements must be carefully assessed for each metal-molecule system. For organic films or SAMs on a metal, the polarization contribution from neighboring molecules can also be quite significant. For molecular systems where the frontier orbitals have stronger electronic coupling to the metal and the resultant resonances overlap with the Fermi energy, the role of dynamical charge transfer is expected to be considerable. Future investigations must address the nature of additional contributions to the self energy in the event of stronger coupling.
We thank Prof. G. W. Flynn and Dr. C. D. Spataru for useful discussions. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, the National Science Foundation under Award No. DMR04-39768, the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award Number CHE-0117752, and the New York State Office of Science, Technology, and Academic Research (NYSTAR). Computational resources were provided by NERSC and NPACI.
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---
abstract: 'In the task of quantum state exclusion we consider a quantum system, prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures can not have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper we formulate the general problem as a Semidefinite Program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty and give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result of Pusey, Barrett and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analogue of Tsirelson’s bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst case error.'
author:
- Somshubhro Bandyopadhyay
- Rahul Jain
- Jonathan Oppenheim
- Christopher Perry
title: Conclusive exclusion of quantum states
---
Introduction
============
Suppose we are given a single-shot device, guaranteed to prepare a system in a quantum state chosen at random from a finite set of $k$ known states. In the quantum state discrimination problem, we would attempt to identify the state that has been prepared. It is a well known result [@Nielsen2010] that this can be done with certainty if and only if all of the states in the set of preparations are orthogonal to one another. By allowing inconclusive measurement outcomes [@Ivanovic1987; @Dieks1988; @Peres1988] or accepting some error probability [@Helstrom1976; @Holevo1973; @Yuen1975], strategies can be devised to tackle the problem of discriminating between non-orthogonal states. For a recent review of quantum state discrimination, see [@Barnett2009]. What however, can we deduce about the prepared state with certainty?
Through state discrimination we effectively attempt to increase our knowledge of the system so that we progress from knowing it is one of $k$ possibilities to knowing it is one particular state. We reduce the size of the set of possible preparations that could have occurred from $k$ to $1$. A related, and less ambitious task, would be to exclude $m$ preparations from the set, reducing the size of the set of potential states from $k$ to $k-m$. If we rule out the $m$ states with certainty we say that they have been conclusively excluded. Conclusive exclusion of a single state is not only interesting from the point of view of the theory of measurement, but it is becoming increasingly important in the foundations of quantum theory. It has previously been considered with respect to quantum state compatibility criteria between three parties [@Caves2002] where Caves et al. derive necessary and sufficient conditions for conclusive exclusion of a single state from a set of three pure states to be possible. More recently it has found use in investigating the plausibility of $\psi$-epistemic theories describing quantum mechanics [@Pusey2012].
As recognized in [@Pusey2012] for the case of single state exclusion, the problem of conclusive exclusion can be formulated in the framework of Semidefinite Programs (SDPs). As well as being efficiently numerically solvable, SDPs also offer a structure that can be exploited to derive statements about the underlying problem they describe [@Vandenberghe1996; @Watrous2011]. This has already been applied to the problem of state discrimination [@Jezek2002; @Eldar2003b; @Eldar2003]. Given that minimum error state discrimination forms a subclass ($m=k-1$) of the general exclusion framework, it is reasonable to expect that a similar approach will pay dividends here.
For minimum error state discrimination, SDPs provide a route to produce necessary and sufficient conditions for a measurement to be optimal. Similarly, the SDP formalism can be applied to obtain such conditions for the task of minimum error state exclusion and we derive these in this paper. By applying these requirements to exclusion problems, we have a method for proving whether a given measurement is optimal for a given ensemble of states.
From the SDP formalism it is also possible to derive necessary conditions for $m$-state conclusive exclusion to be possible for a given set of states and lower bounds on the probability of error when it is not. A special case of this result is the fact that state discrimination can not be achieved when the set of states under consideration are non-orthogonal. By regarding perfect state discrimination as $(k-1)$-state conclusive exclusion, we re-derive this result.
As an application of our SDP and its properties we consider a game, motivated by the argument, due to PBR [@Pusey2012], against a class of hidden variable theories. Assume that we have a physical theory, not necessarily that of quantum mechanics, such that, when we prepare a system, we describe it by a state, $\chi$. If our theory were quantum mechanics, then $\chi$ would be identified with ${|\psi\rangle}$, the usual quantum state. Furthermore, suppose that $\chi$ does not give a complete description of the system. We assume that such a description exists, although it may always be unknown to us, and denote it $\lambda$. As $\chi$ is an incomplete description of the system, it will be compatible with many different complete states. We denote these states $\lambda\in\Lambda_{\chi}$. PBR investigate whether for distinct quantum descriptions, ${|\psi_0\rangle}$ and ${|\psi_1\rangle}$, it is possible that $\Lambda_{{|\psi_0\rangle}}\cap\Lambda_{{|\psi_1\rangle}}\neq\emptyset$. Models that satisfy this criteria are called $\psi$-epistemic, see [@Harrigan2010] for a full description.
Consider now the following scenario. Alice gives Bob a system prepared according to one of two descriptions, $\chi_1$ or $\chi_2$, and Bob’s task is to identify which preparation he has been given. Bob observes the system and will identify the wrong preparation with probability $q$. Note that $0\leq q\leq 1/2$, as Bob will always have the option of randomly guessing the description without performing an observation. If $\Lambda_{\chi_1}\cap\Lambda_{\chi_2}\neq\emptyset$ then, even if Bob has access to the complete description of the system, $\lambda$, $q>0$ as there will exist $\lambda$ compatible with both $\chi_1$ and $\chi_2$.
Now suppose Bob is given $n$ such systems prepared independently, and we represent the preparation as a string in $\{0,1\}^n$. Bob’s task is to output such an $n$-bit string and he wins if his is not identical to the string corresponding to Alice’s preparation, i.e., he attempts to exclude one of the $2^n$ preparations. We refer to this as the ‘PBR game’ and we will consider two scenarios for playing it. Under the first scenario, Bob can only perform measurements on each system individually. We refer to this as the separable version of the game. In the second scenario we allow Bob to perform global measurements on the $n$ systems he receives. We refer to this as the global version, and we are interested in how well quantum theory performs in this case. We shall make a key assumption of PBR: that the global complete state of $n$ independent systems, $\Omega$, is given by the tensor product of the individual systems’ complete states. This second, quantum, task is related to the problem of ‘Hedging bets with correlated quantum strategies’ as introduced in [@Molina2012] and expanded upon in [@Arunachalam2013].
By calculating Bob’s probability of success in the PBR game under each of these schemes we gain a measure of how the predictions of quantum mechanics compare with the predictions of theories in which both $\Lambda_{\chi_1}\cap\Lambda_{\chi_2}\neq\emptyset$ and $\Omega=\otimes_{i=1}^{n}\lambda_i$ hold. As such, the result can be seen as similar in spirit to Tsirelson’s bound [@Tsirelson1980] in describing how well quantum mechanical strategies can perform at the CHSH game.
This paper is organized as follows. First, in Section \[Section: The State Exclusion SDP\], we formulate the quantum state exclusion problem as an SDP, developing the structure we will need to analyze the task. Next, in Section \[Section: The optimal exclusion measurement\], we derive sufficient and necessary conditions for a measurement to be optimal in performing conclusive exclusion. It is these conditions that will assist us in investigating the entangled version of the PBR game. In Section \[Section: Necessary condition\] we derive a necessary condition on the set of possible states for single state exclusion to be possible and in Section \[Section: Lower Bound\] we give a lower bound on the probability of error when it is not. We apply the SDP formalism to the PBR game in Section \[Section: The PBR game\] and use it to quantify the discrepancy between the predictions of a class of hidden variable theories and those of quantum mechanics. Finally, in Section \[Section: Alternative measures of exclusion\], we present alternative formulations of state exclusion and construct the relevant SDPs.
The State Exclusion SDP {#Section: The State Exclusion SDP}
=======================
More formally, what does it mean to be able to perform conclusive exclusion? We first consider the case of single state exclusion and then show how it generalizes to $m$-state exclusion. Let the set of possible preparations on a $d$ dimensional quantum system be $\mathcal{P}=\left\{\rho_{i}\right\}^{k}_{i=1}$ and let each preparation occur with probability $p_i$. For brevity of notation we define $\tilde{\rho}_i=p_i \rho_i$. Call the prepared state $\sigma$. The aim is to perform a measurement on $\sigma$ so that, from the outcome, we can state $j\in\{1,\ldots,k\}$ such that $\sigma\neq\rho_j$.
Such a measurement will consist of $k$ measurement operators, one for attempting to exclude each element of $\mathcal{P}$. We want a measurement, described by $\mathcal{M}=\{M_i\}_{i=1}^{k}$, that never leads us to guess $j$ when $\sigma=\rho_j$. We need: $${\textnormal{Tr}}\left[\rho_i M_i\right]=0, \quad \forall i, \label{single exclusion}$$ or equivalently, since $\rho_i$ and $M_i$ are positive semidefinite matrices and $p_i$ is a positive number: $$\alpha=\sum_{i=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right]=0. \label{excl cond}$$ There will be some instances of $\mathcal{P}$ for which a $\mathcal{M}$ can not be found to satisfy Eq. (\[excl cond\]). In these cases our goal is to minimize $\alpha$ which corresponds to the probability of failure of the strategy, ‘If outcome $j$ occurs say $\sigma\neq\rho_j$’.
Therefore, to obtain the optimal strategy for single state exclusion, our goal is to minimize $\alpha$ over all possible $\mathcal{M}$ subject to $\mathcal{M}$ forming a valid measurement. Such an optimization problem can be formulated as an SDP: $$\begin{aligned}
\begin{split}
\underset{\mathcal{M}}{\textrm{Minimize: }}&\alpha=\sum_{i=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right]. \\
\textrm{Subject to: }&\sum_{i=1}^{k}M_i=\mathbb{I},\\
& M_i\geq0, \quad \forall i. \label{SDP Primal}
\end{split}\end{aligned}$$ Here $\mathbb{I}$ is the $d$ by $d$ identity matrix and $A\geq 0$ implies that $A$ is a positive semidefinite matrix. The constraint $\sum_{i=1}^{k}M_i=\mathbb{I}$ corresponds to the fact that the $M_i$ form a complete measurement and we don’t allow inconclusive results.
Part of the power of the SDP formalism lies in constructing a ‘dual’ problem to this ‘primal’ problem given in Eq. (\[SDP Primal\]). Details on the formation of the dual problem to the exclusion SDP can be found in Appendix \[State Exclusion SDP Formulation\] and we state it here:
$$\begin{aligned}
\begin{split}
\underset{N}{\textrm{Maximize: }}&\beta={\textnormal{Tr}}\left[N\right].\\
\textrm{Subject to: }&N\leq \tilde{\rho}_i, \quad\forall i,\\
&N\in \textrm{Herm}. \label{SDP Dual}
\end{split}\end{aligned}$$
For single state exclusion, the problem is essentially to maximize the trace of a Hermitian matrix $N$ subject to $\tilde{\rho}_i-N$ being a positive semidefinite matrix, $\forall$ $i$.
What of $m$-state conclusive exclusion? Define $Y_{(k,m)}$ to be the set of all subsets of the integers $\{1,\ldots,k\}$ of size $m$. The aim is to perform a measurement on $\sigma$ such that from the outcome we can state a set, $Y\in Y_{(k,m)}$, such that $\sigma\notin\{\rho_y\}_{y\in Y}$. Such a measurement, denoted $\mathcal{M}_m$, will consist of ${k \choose m}$ measurement operators and we require that, for each set $Y$: $${\textnormal{Tr}}\left[\tilde{\rho}_y M_Y\right]=0,\quad \forall y\in Y.$$ If we define: $$\hat{\rho}_Y = \sum_{y\in Y} \tilde{\rho}_y,$$ then this can be reformulated as requiring: $${\textnormal{Tr}}\left[\hat{\rho}_Y M_Y\right]=0, \quad \forall Y\in Y_{(k,m)}. \label{m exclusion}$$ Eq. (\[m exclusion\]) is identical in form to Eq. (\[single exclusion\]). Hence we can view $m$-state exclusion as single state exclusion on the set $\mathcal{P}_m=\{\hat{\rho}_Y\}_{Y\in Y_{(k,m)}}$. Furthermore, we can generalize this approach to an arbitrary collection of subsets that are not necessarily of the same size. With this in mind we restrict ourselves to considering single state exclusion in all that follows.
The tasks of state exclusion and state discrimination share many similarities. Indeed, if we instead maximize $\alpha$ in Eq. (\[SDP Primal\]) and minimize $\beta$ in Eq. (\[SDP Dual\]) together with inverting the inequality constraint to read $N\geq\tilde{\rho}_i$, we obtain the SDP associated with minimum error state discrimination. It is also possible to recast each problem as an instance of the other. Firstly, state discrimination can be put in the form of an exclusion problem by taking $m=k-1$ because if we exclude $k-1$ of the possible states, then we can identify $\sigma$ as the remaining state.
Following the observation of [@Nakahira2012] regarding minimum Bayes cost problems, state exclusion can be converted into a discrimination task. To see this, from $\mathcal{P}$ define: $$\mathcal{R}=\left\{\vartheta_i=\frac{1}{k-1}\sum_{j\neq i}\tilde{\rho}_j\right\}_{i=1}^{k}.$$ Writing $P_{\textit{error}}^{\textit{dis}}$ and $P_{\textit{error}}^{\textit{exc}}$ to distinguish between the probability of error in discrimination and exclusion, in state discrimination on $\mathcal{R}$ we would attempt to minimize: $$\begin{aligned}
P_{\textit{error}}^{\textit{dis}}\left(\mathcal{R}\right)&=1-\sum_{i=1}^{k}{\textnormal{Tr}}\left[\vartheta_i M_i\right]
\intertext{which can be rearranged to give (see Appendix \ref{exc-disc derivation}):}
P_{\textit{error}}^{\textit{dis}}\left(\mathcal{R}\right)&=\frac{k-2}{k-1}+\frac{1}{k-1}P_{\textit{error}}^{\textit{exc}}\left(\mathcal{P}\right). \label{exc-disc}\end{aligned}$$ Hence, minimizing the error probability in discrimination on $\mathcal{R}$ is equivalent to minimizing the probability of error in state exclusion on $\mathcal{P}$ and the optimal measurement is the same for both. This interplay between the two tasks enables us to apply bounds on the error probability of state discrimination (see for example [@Qiu2010]) to the task of state exclusion.
Returning to the SDP, let us define the optimum solution to the primal problem to be $\alpha^{*}$ and the solution to the corresponding dual to be $\beta^{*}$. It is a property of all SDPs, known as weak duality, that $\beta\leq\alpha$. Furthermore, for SDPs satisfying certain conditions, $\alpha^{*}=\beta^{*}$ and this is known as strong duality. The exclusion SDP does fulfill these criteria, as shown in Appendix \[Slater’s Theorem applied to Exclusion SDP\]. Using weak and strong duality allows us to derive properties of the optimal measurement for the problem, a necessary condition on $\mathcal{P}$ for conclusive exclusion to be possible and a bound on the probability of error in performing the task.
The optimal exclusion measurement {#Section: The optimal exclusion measurement}
=================================
Strong duality gives us a method for proving whether a feasible solution, satisfying the constraints of the primal problem, is an optimal solution. If $\mathcal{M}^{*}$ is an optimal measurement for the conclusive exclusion SDP, then, by strong duality, there must exist a Hermitian matrix $N^{*}$, satisfying the constraints of the dual problem, such that: $$\sum_{i=1}^{k} {\textnormal{Tr}}\left[\tilde{\rho}_i M_{i}^{*}\right]={\textnormal{Tr}}\left[N^{*}\right].$$ Furthermore, the following is true:
\[SD theorem\] Suppose a state $\sigma$ is prepared at random using a preparation from the set $\mathcal{P}$ according to some probability distribution $\{p_i\}_{i=1}^{k}$. Applying the measurement $\mathcal{M}$ to $\sigma$ is optimal for attempting to exclude a single element from the set of possible preparations if and only if: $$N=\sum_{i=1}^{k} \left[\tilde{\rho}_i M_i\right], \label{N construct}$$ is Hermitian and satisfies $N\leq \tilde{\rho}_{i}$, $\forall i$.
The proof of Theorem \[SD theorem\] is given in Appendix \[Necessary and sufficient conditions for a measurement to be optimal\] and revolves around the application of strong duality together with a property called complementary slackness. It is similar in construction to Yuen et al.’s [@Yuen1975] derivation of necessary and sufficient conditions for showing a quantum measurement is optimal for minimizing a given Bayesian cost function. This result provides us with a method for proving a measurement is optimal; we construct $N$ according to Eq. (\[N construct\]) and show that it satisfies the constraints of the dual problem. It is this technique which will allow us to analyze the PBR game in the quantum setting.
Necessary condition for single state conclusive exclusion {#Section: Necessary condition}
=========================================================
Through the application of weak duality we can also gain insight into the SDP. As the optimal solution to the dual problem provides a lower bound on the solution of the primal problem, any feasible solution to the dual does too although it may not necessarily be tight. This relation can be summarized as: $${\textnormal{Tr}}\left[N^{feas}\right]\leq{\textnormal{Tr}}\left[N^{*}\right]=\beta^{*}=\alpha^{*}.$$ In particular if, for a given $\mathcal{P}$, we can construct a feasible $N$ with ${\textnormal{Tr}}\left[N\right]>0$, then we have $\alpha^{*}>0$ and hence conclusive exclusion is not possible.
Constructing such an $N$ gives rise to the following necessary condition on the set $\mathcal{P}$ for conclusive exclusion to be possible:
\[Necc Cond Theorem\] Suppose a system is prepared in the state $\sigma$ using a preparation chosen at random from the set $\mathcal{P}=\{\rho_i\}_{i=1}^{k}$. Single state conclusive exclusion is possible only if: $$\sum_{j\neq l=1}^{k} F(\rho_j,\rho_l)\leq k(k-2), \label{Necc Con}$$ where $F(\rho_j,\rho_l)$ is the fidelity between states $\rho_j$ and $\rho_l$.
The full proof of this theorem is given in Appendix \[Necessary Condition for Conclusive Exclusion\]. but we sketch it here. Define $N$ as follows: $$N=-p\sum_{r=1}^{k}\rho_r +\frac{1-\epsilon}{k-2}p\sum_{1\leq j<l\leq k}\left(\sqrt{\rho_j}U_{jl}\sqrt{\rho_l}+\sqrt{\rho_l}U_{jl}^{*}\sqrt{\rho_j}\right),$$ where the $U_{jl}$ are unitary matrices chosen such that: $${\textnormal{Tr}}\left[N\right]=-kp+\frac{1-\epsilon}{k-2} p\sum_{j\neq l=1}^{k} F(\rho_j,\rho_l).$$ $N$ is Hermitian and for suitable $p$ and $\epsilon$ it can be shown that $\rho_i-N\geq 0$, $\forall i$. Eq. (\[Necc Con\]) follows by determining when ${\textnormal{Tr}}[N]>0$ and letting $\epsilon \rightarrow 0$. Note that the probability with which states are prepared, $\{p_i\}_{i=1}^{k}$, does not impact on whether conclusive exclusion is possible or not.
This is only a necessary condition for single state conclusive exclusion and there exist sets of states that satisfy Eq. (\[Necc Con\]) for which it is not possible to perform conclusive exclusion. Nevertheless, there exist sets of states on the cusp of satisfying Eq. (\[Necc Con\]) for which conclusive exclusion is possible. For example, the set of states of the form: $${|{\psi}_i\rangle}=\sum_{j\neq i}^{k}\frac{1}{\sqrt{k-1}}{|j\rangle},$$ for $i=1$ to $k$, can be conclusively excluded by the measurement in the orthonormal basis $\{{|i\rangle}\}_{i=1}^{k}$ and yet: $$\begin{aligned}
\begin{split}
\sum_{j\neq l=1}^{k} F\left({|\psi_j\rangle\langle \psi_j|},{|\psi_l\rangle\langle \psi_l|}\right)&=\sum_{j\neq l=1}^{k}|{\langle \psi_j|\psi_l\rangle}|\\
&= k(k-2).
\end{split}\end{aligned}$$
It can be shown that the necessary condition for conclusive state discrimination can be obtained from Theorem \[Necc Cond Theorem\] and the interested reader can find this derivation in Appendix \[Necessary condition for conclusive state discrimination\].
Lower bound on the probability of error {#Section: Lower Bound}
=======================================
Weak duality can also be used to obtain the following lower bound on $\alpha^{*}$:
\[lower bound\] For two Hermitian operators, $A$ and $B$, define $\min\left(A,B\right)$ to be: $$\min\left(A,B\right)=\frac{1}{2}\left[A+B-|A-B|\right].$$ Given a set of states $\mathcal{P}=\{\rho_i\}_{i=1}^{k}$ prepared according to some probability distribution $\{p_i\}_{i=1}^{k}$ and a permutation $\varepsilon$, acting on $k$ objects, taken from the permutation group $S_k$, consider: $$N_{\varepsilon}=\min\left(\tilde{\rho}_{\varepsilon(k)},\min\left(\tilde{\rho}_{\varepsilon(k-1)},\min\left(\ldots,\min\left(\tilde{\rho}_{\varepsilon(2)},\tilde{\rho}_{\varepsilon(1)}\right)\right)\right)\right). \label{Nepsilon}$$ Then: $$\alpha^{*}\geq\underset{\varepsilon\in S_k}{\textnormal{Maximum: }}{\textnormal{Tr}}\left[N_{\varepsilon}\right].$$
The proof of this result is given in Appendix \[lower bound proof\] and relies upon showing that $\min(A,B)\leq A \textrm{ and }B$, together with the iterative nature of the construction of $N_{\varepsilon}$. Note that by considering a suitably defined $\max$ function, analogous to the $\min$ used in Theorem \[lower bound\], it is possible to derive a similar style of bound for the task of minimum error state discrimination. We omit it here however, as it is beyond the scope of this paper.
The PBR game {#Section: The PBR game}
============
We now turn our attention to the PBR game. Suppose Alice gives Bob $n$ systems whose preparations are encoded by the string $\vec{x}\in\{0,1\}^n$. The state of system $i$ is $\chi_{x_i}$. Bob’s goal is to produce a string $\vec{y}\in\{0,1\}^n$ such that $\vec{x}\neq\vec{y}$.
Separable version
-----------------
In the first scenario, where Bob can only observe each system individually and we consider a general theory, we can represent his knowledge of the global system by: $$\Gamma=\gamma_1\otimes\ldots\otimes\gamma_n,$$ with $\gamma_i\in\{\Gamma_0,\Gamma_1,\Gamma_?\}$, representing his three possible observation outcomes. If $\gamma_i\in\Gamma_0$ he is certain the system preparation is described by $\chi_0$, if $\gamma_i\in\Gamma_1$ he is certain the system preparation is described by $\chi_1$ and if $\gamma_i\in\Gamma_?$ he remains uncertain whether the system was prepared in state $\chi_0$ or $\chi_1$ and he may make an error in assigning a preparation to the system. We denote the probability that Bob, after performing his observation, assigns the wrong preparation description to the system as $q$. Provided that $\Gamma_?\neq\emptyset$, then $q>0$.
Bob will win the game if for at least one individual system he assigns the correct preparation description. His strategy is to attempt to identify each value of $x_i$ and choose $y_i$ such that $y_i\neq x_i$. Bob’s probability of outputting a winning string is hence: $$P_{\textit{win}}^{S}=1-q^n. \label{P win sep}$$
Global version
--------------
Now consider the second scenario and when the theory is quantum and global (i.e., entangled) measurements on the global system are allowed. We can write the global state that Alice gives Bob, labeled by $\vec{x}$, as: $${|\Psi_{\vec{x}}\rangle}=\bigotimes^{n}_{i=1}{|\psi_{x_i}\rangle}.$$ Bob’s task can now be regarded as attempting to perform single state conclusive exclusion on the set of states $\mathcal{P}=\{{|\Psi_{\vec{x}}\rangle}\}_{\vec{x}\in\{0,1\}^n}$; he outputs the string associated to the state he has excluded to have the best possible chance of winning the game.
To calculate his probability of winning $P_{\textit{win}}^{G}$ we need to construct and solve the associated SDP. Without loss of generality, we can take the states ${|\psi_0\rangle}$ and ${|\psi_1\rangle}$ to be defined as: $$\begin{aligned}
\begin{split}
{|\psi_0\rangle}&=\cos\left(\frac{\theta}{2}\right){|0\rangle}+\sin\left(\frac{\theta}{2}\right){|1\rangle}, \\
{|\psi_1\rangle}&=\cos\left(\frac{\theta}{2}\right){|0\rangle}-\sin\left(\frac{\theta}{2}\right){|1\rangle},
\end{split}\end{aligned}$$ where $0\leq\theta\leq\pi/2$. The global states ${|\Psi_{\vec{x}}\rangle}$ are then given by: $${|\Psi_{\vec{x}}\rangle}=\sum_{\vec{r}}\left(-1\right)^{\vec{x}\cdot\vec{r}}\left[\cos\left(\frac{\theta}{2}\right)\right]^{n-|\vec{r}|}\left[\sin\left(\frac{\theta}{2}\right)\right]^{|\vec{r}|}{|\vec{r}\rangle},$$ where $\vec{r}\in\{0,1\}^n$ and $|\vec{r}|=\sum^{n}_{i=1}r_i$.
From [@Pusey2012], we know that single state conclusive exclusion can be performed on this set of states provided $\theta$ and $n$ satisfy the condition: $$2^{1/n}-1\leq \tan\left(\frac{\theta}{2}\right). \label{PBR Criterion}$$ When this relation holds, $P_{\textit{win}}^{G}=1$. What however, happens outside of this range? Whilst strong numerical evidence is given in [@Pusey2012] that it will be the case that $P_{\textit{win}}^{G}<1$, can it be shown analytically?
Through analyzing numerical solutions to the SDP (performed using [@Lofberg2004], [@Sturm1999]), there is evidence to suggest that the optimum measurement to perform when Eq. (\[PBR Criterion\]) is not satisfied is given by the projectors: $$\begin{aligned}
{|\zeta_{\vec{x}}\rangle}=\frac{1}{\sqrt{2^n}}\left({|\vec{0}\rangle}-\sum_{\vec{r}\neq\vec{0}}\left(-1\right)^{\vec{x}\cdot\vec{r}}{|\vec{r}\rangle}\right), \label{PBR Projector}\end{aligned}$$ which are independent of $\theta$. That the set $\{{|\zeta_{\vec{x}}\rangle}\}_{\vec{x}\in\{0,1\}^n}$ is the optimal measurement for attempting to perform conclusive exclusion is shown in Appendix \[PBR Game\]. If we construct $N$ as per Eq. (\[N construct\]) and consider the trace, we can determine how successfully single state exclusion can be performed. This is done in Appendix \[PBR Game\] and we find: $${\textnormal{Tr}}\left[N\right]=\frac{1}{2^n}\left[\cos\left(\frac{\theta}{2}\right)\right]^{2n}\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right)^2.$$ This is strictly positive and hence we have shown that Eq. (\[PBR Criterion\]) is a necessary condition for conclusive exclusion to be possible on the set $\mathcal{P}$.
In summary, we have: $$\begin{aligned}
\begin{split}
&\textrm{If: } 2^{1/n}-1\leq \tan\left(\frac{\theta}{2}\right), \\
&\quad P_{\textit{win}}^{G}=1. \\
&\textrm{Else:} \\
&\quad P_{\textit{win}}^{G}=1-\frac{1}{2^n}\left[\cos\left(\frac{\theta}{2}\right)\right]^{2n}\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right)^2
\end{split}\end{aligned}$$ which characterizes the success probability of the quantum strategy.
Comparison
----------
What is the relation between $P_{\textit{win}}^{S}$ and $P_{\textit{win}}^{G}$? If, in the separable scenario, we take the physical theory as being quantum mechanics and Bob’s error probability as arising from the fact that it is impossible to distinguish between non-orthogonal quantum states, we can write [@Helstrom1976]: $$\begin{aligned}
\begin{split}
q&=\left(\frac{1}{2}\right)\left(1-\sqrt{1-\left|{\langle \psi_0|\psi_1\rangle}\right|^2}\right)\\
&=\left(\frac{1}{2}\right)\left(1-\sin\left(\theta\right)\right).
\end{split}\end{aligned}$$ With this substitution we find that $P_{\textit{win}}^{S}\leq P_{\textit{win}}^{G}$, $\forall n$. This is unsurprising as the first scenario is essentially the second but with a restricted set of allowable measurements.
Of more interest however, is if we view $q$ as arising from some hidden variable completion of quantum mechanics. If $\Lambda_{{|\psi_0\rangle}}\cap\Lambda_{{|\psi_1\rangle}}=\emptyset$, then if an observation of each ${|\psi_{x_i}\rangle}$ were to allow us to deduce $\lambda_{x_i}$ then $q=0$ and $P_{\textit{win}}^{S}=1\geq P_{\textit{win}}^{G}$. However, if $\Lambda_{{|\psi_0\rangle}}\cap\Lambda_{{|\psi_1\rangle}}\neq\emptyset$, then we have $q>0$ and $P_{\textit{win}}^{S}$ will have the property that Bob wins with certainty only as $n\rightarrow\infty$. On the other hand, $P_{\textit{win}}^{G}=1$ if and only if Eq. (\[PBR Criterion\]) is satisfied and we have analytically proven the necessity of the bound obtained by PBR. Furthermore, we have defined a game that allows the quantification of the difference between the predictions of general physical theories, including those that attempt to provide a more complete description of quantum mechanics, and those of quantum mechanics.
Alternative measures of exclusion {#Section: Alternative measures of exclusion}
=================================
There exist multiple strategies and figures of merit when undertaking state discrimination. In addition to considering minimum error discrimination or unambiguous discrimination, further variants may try to minimize the maximum error probability [@Kosut2004] or allow only a certain probability of obtaining an inconclusive measurement result [@Fiuravsek2003]. Similarly, alternative methods to minimum error can be defined for state exclusion and in this section unambiguous exclusion and worst case error exclusion are defined and the related SDPs given.
Unambiguous State Exclusion
---------------------------
In unambiguous state exclusion on the set of preparations $\mathcal{P}=\{\tilde{\rho}_i\}_{i=1}^{k}$ we consider a measurement given by $\mathcal{M}=\{M_1, \ldots, M_k, M_?\}$. If we obtain measurement outcome $i$ $(1\leq i\leq k)$, then we can exclude with certainty the state $\rho_i$. However, if we obtain the outcome labeled $?$, we can not infer which state to exclude. We wish to minimize the probability of obtaining this inconclusive measurement: $$\alpha=\sum_{i=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_i M_?\right],$$ which can be rewritten as: $$\alpha={\textnormal{Tr}}\left[\sum_{j=1}^{k}\tilde{\rho}_j\left(\mathbb{I}-\sum_{i=1}^{k}M_i\right)\right].$$
Defining $\tilde{\alpha}=1-\alpha$, the primal SDP associated with this task is given by: $$\begin{aligned}
\begin{split}
\underset{\mathcal{M}}{\textrm{Maximize: }}&\tilde{\alpha}={\textnormal{Tr}}\left[\sum_{j=1}^{k}\tilde{\rho}_j\sum_{i=1}^{k}M_i\right]. \\
\textrm{Subject to: }&\sum_{i=1}^{k}M_i\leq\mathbb{I},\\
& {\textnormal{Tr}}\left[\tilde{\rho_i}M_i\right]=0, \quad 1\leq i\leq k, \\
& M_i\geq0, \quad 1\leq i\leq k. \label{Unambig Prime}
\end{split}\end{aligned}$$ Here, the first and third constraints ensure that $\mathcal{M}$ is a valid measurement whilst the second, ${\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right]=0$, $1\leq i\leq k$, encapsulates the fact that when measurement outcome $i$ occurs we should be able to exclude state $\rho_i$ with certainty.
The dual problem can be shown to be (see Appendix \[Unambiguous State Exclusion SDP\]): $$\begin{aligned}
\begin{split}
\underset{N, \{a_i\}_{i=1}^{k}}{\textrm{Minimize: }}&\beta={\textnormal{Tr}}\left[N\right].\\
\textrm{Subject to: }&a_i\tilde{\rho}_i+N\geq \sum_{j=1}^{k}\tilde{\rho}_j, \quad 1\leq i\leq k,\\
&a_i\in\mathbb{R}, \quad \forall i,\\
&N\geq 0. \label{Unambig Dual}
\end{split}\end{aligned}$$
Unambiguous state exclusion has recently found use in implementations of quantum digital signatures [@Collins2013], enabling such schemes to be put into practice without the need for long term quantum memory.
Worst Case Error State Exclusion
--------------------------------
The goal of the SDP given in Eqs. (\[SDP Primal\]) and (\[SDP Dual\]) is to minimize the average probability of error, over all possible preparations, of the strategy, ‘If outcome $j$ occurs say $\sigma\neq\rho_j$’. An alternative goal would be to minimize the worst case probability of error that occurs: $$\alpha=\max_{i}{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right].$$
The primal SDP associated with this task is: $$\begin{aligned}
\begin{split}
\underset{\mathcal{M}}{\textrm{Minimize: }}&\alpha=\lambda. \\
\textrm{Subject to: }&\lambda\geq{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right], \quad \forall i,\\
&\sum_{i=1}^{k}M_i=\mathbb{I},\\
& \lambda\geq0\in\mathbb{R}, \\
& M_i\geq0, \quad 1\leq i\leq k. \label{Worst Case Prime}
\end{split}\end{aligned}$$ These constraints again encode that $\mathcal{M}$ forms a valid measurement and ensure that $\alpha$ picks out the worst case error probability across all possible preparations.
The associated dual problem is: $$\begin{aligned}
\begin{split}
\underset{N, \{a_i\}_{i=1}^{k}}{\textrm{Maximize: }}&\beta={\textnormal{Tr}}\left[N\right].\\
\textrm{Subject to: }&N\leq a_i\tilde{\rho}_i, \quad \forall i,\\
&\sum_{i=1}^{k} a_i\leq1,\\
&a_i\geq0\in\mathbb{R}, \quad \forall i,\\
&N\in \textrm{Herm}. \label{Worst Case Dual}
\end{split}\end{aligned}$$ The derivation of this is given in Appendix \[Worst Case Error State Exclusion SDP\].
Conclusion
==========
In this paper we have introduced the task of state exclusion and shown how it can be formulated as an SDP. Using this we have derived conditions for measurements to be optimal at minimum error state exclusion and a criteria for the task to be performed conclusively on a given set of states. We also gave a lower bound on the error probability. Furthermore, we have applied our SDP to a game which helps to quantify the differences between quantum mechanics and a class of hidden variable theories.
It is an open question, posed in [@Caves2002], whether a POVM ever outperforms a projective measurement in conclusive exclusion of a single pure state. Whilst it can be shown from the SDP formalism that this is not the case when the states are linearly independent and conclusive exclusion is not possible to the extent that ${\textnormal{Tr}}\left[M_i\rho_i\right]>0$, $\forall i$, further work is required to extend it and answer the above question. It would also be interesting to see whether it is possible to find further constraints and bounds, similar to Theorem \[Necc Cond Theorem\] and Theorem \[lower bound\], to characterize when conclusive exclusion is possible.
Finally, the main SDP, as given in Eq. (\[SDP Primal\]), is just one method for analyzing state exclusion in which we attempt to minimize the average probability of error. Alternative formulations were presented in Section \[Section: Alternative measures of exclusion\] and it would be interesting to study the relationships between them and that defined in Eq. (\[SDP Primal\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of this work was completed while S.B. and J.O. were visiting the Center for Quantum Technologies, Singapore and while R.J. was visiting the Bose Institute, Kolkata, India. S.B. thanks CQT for their support. The work of R.J. is supported by the Singapore Ministry of Education Tier 3 Grant and also the Core Grants of the Centre for Quantum Technologies, Singapore. J.O. is supported by the Royal Society and an EPSRC Established Career fellowship.
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This supplementary material contains five sections. In Appendix \[State Exclusion SDP Formulation\], we give the general definition of an SDP, derive the dual problem for the state exclusion SDP and show the relation to state discrimination. Next, in Appendix \[Strong Duality\], we show that the SDP exhibits strong duality and give the proof of Theorem \[SD theorem\] from the main text. Appendix \[Necessary Conditions\] derives the necessary condition for conclusion exclusion to be possible given in Theorem \[Necc Cond Theorem\] as well as the associated corollary. It also contains the proof of the bound on the error probability of state exclusion, Theorem \[lower bound\]. The PBR game is analyzed in Appendix \[PBR Game\]. Finally, in Appendix \[Alternative State Exclusion SDPs\], we state alternative state exclusion SDPs.
State Exclusion SDP Formulation {#State Exclusion SDP Formulation}
===============================
Contains:
- General definition of an SDP.
- Derivation of the state exclusion SDP dual.
- Recasting of state exclusion as a discrimination problem.
General SDPs {#General SDPs}
------------
In this section we state the general form of a Semidefinite Program as given in [@Watrous2011]. A semidefinite program is defined by three elements $\{A,B,\Phi\}$. $A$ and $B$ are Hermitian matrices, $A\in \textit{Herm}(\mathcal{X})$ and $B\in \textit{Herm}(\mathcal{Y})$, where $\mathcal{X}$ and $\mathcal{Y}$ are complex Euclidean spaces. $\Phi$ is a Hermicity preserving super-operator which takes elements in $\mathcal{X}$ to elements in $\mathcal{Y}$.
From these three elements, two optimization problems can be defined: $$\begin{aligned}
\begin{split} \label{Prime}
\textrm{Primal Problem}\\
\underset{X}{\textrm{Minimize}} :& \quad \alpha={\textnormal{Tr}}[AX].\\
\textrm{Subject to} :& \quad \Phi(X)= B,\\
& \quad X\geq0.
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split} \label{Dual}
\textrm{Dual Problem}\\
\underset{Y}{\textrm{Maximize}} :& \quad \beta={\textnormal{Tr}}[BY].\\
\textrm{Subject to} :& \quad \Phi^{*}(Y)\leq A,\\
& \quad Y\in \textit{Herm}(\mathcal{Y}).
\end{split}\end{aligned}$$ Here $\Phi^*$ is the dual map to $\Phi$ and is defined by: $$\begin{aligned}
{\textnormal{Tr}}[Y\Phi(X)]={\textnormal{Tr}}[X\Phi^{*}(Y)]. \label{Phi* equation}\end{aligned}$$ We define the optimal solutions to the primal and dual problems to be $\alpha^{*}=\textrm{inf}_X \alpha$ and $\beta^{*}=\textrm{sup}_Y \beta$ respectively.
State Exclusion SDP {#State Exclusion SDP}
-------------------
Looking at the state exclusion primal problem, Eq. (\[SDP Primal\]), we see that for the exclusion SDP:
- $A$ is a $kd$ by $kd$ block diagonal matrix with each $d$ by $d$ block, labeled by $i$, given by $\tilde{\rho}_i$: $$A=\left(\begin{array}{ccc}
\tilde{\rho}_1 & & \\
& \ddots & \\
& & \tilde{\rho}_k
\end{array}\right).$$
- $B$ is the $d$ by $d$ identity matrix.
- $X$, the variable matrix, is a $kd$ by $kd$ block diagonal matrix where we label each $d$ by $d$ block diagonal by $M_i$: $$X=\left(\begin{array}{ccc}
M_1 & & \\
& \ddots & \\
& & M_k
\end{array}\right).$$
- $Y$ is the $d$ by $d$ matrix we call $N$.
- The map $\Phi$ is given by $\Phi(X)=\sum_{i} M_{i}$.
Using Eq. (\[Phi\* equation\]) we see that $\Phi^*$ must satisfy: $$\begin{aligned}
{\textnormal{Tr}}\left[N\sum_{i=1}^{k} M_i\right]={\textnormal{Tr}}\left[\left(\begin{array}{ccc}
M_1 & & \\
& \ddots & \\
& & M_{k}
\end{array} \right)\Phi^{*}(N)\right],\end{aligned}$$ and hence $\Phi^{*}(N)$ produces a $kd$ by $kd$ block diagonal matrix with $N$ in each of the block diagonals: $$\Phi^{*}(N)=\left(\begin{array}{ccc}
N & & \\
& \ddots & \\
& & N\end{array}\right).$$
Substituting these elements into Eq. (\[Dual\]), we obtain the dual SDP for state exclusion as stated in Eq. (\[SDP Dual\]) .
The relation between state discrimination and state exclusion {#exc-disc derivation}
-------------------------------------------------------------
Here we give the derivation of Eq. (\[exc-disc\]).
Given $\mathcal{P}$ we define: $$\mathcal{R}=\left\{\vartheta_i=\frac{1}{k-1}\sum_{j\neq i}\tilde{\rho}_j\right\}_{i=1}^{k}.$$ Then, in state discrimination on $\mathcal{R}$ we would attempt to minimize: $$\begin{aligned}
P_{\textit{error}}^{\textit{dis}}\left(\mathcal{R}\right)&=1-\sum_{i=1}^{k}{\textnormal{Tr}}\left[\vartheta_i M_i\right],\\
&=1-\sum_{i=1}^{k}\sum_{j\neq i}\frac{1}{k-1}{\textnormal{Tr}}\left[\tilde{\rho}_j M_i\right],\\
&=1-\frac{1}{k-1}\sum_{i=1}^{k}\sum_{j=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_j M_i\right]+\frac{1}{k-1}\sum_{i=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right],\\
&=\frac{k-2}{k-1}+\frac{1}{k-1}P_{\textit{error}}^{\textit{exc}}\left(\mathcal{P}\right).\end{aligned}$$
Strong Duality {#Strong Duality}
==============
Contains:
- Statement of Slater’s Theorem.
- Proof that the exclusion SDP satisfies the conditions of Slater’s Theorem.
- Derivation of necessary and sufficient conditions for a measurement to be optimal for performing exclusion (proof of Theorem \[SD theorem\] ).
Slater’s Theorem {#Slater's Theorem}
----------------
Slater’s Theorem provides a means to test whether an SDP satisfies strong duality ($\alpha^{*}=\beta^{*}$).
[(Slater’s Theorem.)]{} The following implications hold for every SDP:
1. If there exists a feasible solution to the primal problem and a Hermitian operator $Y$ for which $\Phi^*(Y)<A$, then $\alpha^{*}=\beta^{*}$ and there exists a feasible $X^*$ for which ${\textnormal{Tr}}[AX^*]=\alpha^{*}$.
2. If there exists a feasible solution to the dual problem and a positive semidefinite operator $X$ for which $\Phi(X)=B$ and $X>0$, then $\alpha^{*}=\beta^{*}$ and there exists a feasible $Y^*$ for which ${\textnormal{Tr}}[BY^*]=\beta^{*}$.
Slater’s Theorem applied to Exclusion SDP {#Slater's Theorem applied to Exclusion SDP}
-----------------------------------------
To see that the exclusion SDP satisfies the conditions of Slater’s Theorem consider $X=\frac{1}{k} \mathbb{I}$ and $N=-\mathbb{I}$ (where the Identity matrices are taken to have the correct dimension). $X$ is strictly positive definite and so strictly satisfies the constraints of the primal problem. $N<0$ and hence $N<\tilde{\rho}_{i}$, $\forall i$, so $N$ strictly satisfies the constraints of the dual problem.
Necessary and sufficient conditions for a measurement to be optimal {#Necessary and sufficient conditions for a measurement to be optimal}
-------------------------------------------------------------------
To prove Theorem \[SD theorem\] we will need the following fact about SDPs:
[(Complementary Slackness.)]{} \[Slackness\] Suppose $X$ and $Y$, which are feasible for the primal and dual problems respectively, satisfy ${\textnormal{Tr}}[AX]={\textnormal{Tr}}[BY]$. Then it holds that: $$\Phi^*(Y)X=AX \textrm{ and }\Phi(X)Y=BY.$$
We now give the proof for Theorem \[SD theorem\] .
[(Proof of Theorem \[SD theorem\] .)]{}
Suppose we are given a valid measurement, $\mathcal{M}=\{M_i\}_{i=1}^{k}$, and that $N$, defined by: $$N=\sum_{i=1}^{k} \tilde{\rho}_i M_i,$$ satisfies the constraints of the dual problem. Then: $$\begin{aligned}
\beta&={\textnormal{Tr}}[N],\\
&={\textnormal{Tr}}\left[\sum_{i=1}^{k} \tilde{\rho}_i M_i\right],\\
&=\sum_{i=1}^{k}{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right],\\
&=\alpha.\end{aligned}$$ Hence, by strong duality, $\mathcal{M}$ is an optimal measurement.
Now suppose $\mathcal{M}$ is an optimal measurement. By Proposition \[Slackness\], an optimal $N$ satisfies: $$\begin{aligned}
\Phi^{*}(N)\left(\begin{array}{ccc}
M_1 & & \\
& \ddots & \\
& & M_{k}
\end{array} \right)
&=
\left(\begin{array}{ccc}
\tilde{\rho}_1 M_1 & & \\
& \ddots & \\
& & \tilde{\rho}_k M_{k}
\end{array} \right),\\
\Rightarrow\quad\left(\begin{array}{ccc}
N M_1 & & \\
& \ddots & \\
& & N M_{k}
\end{array} \right)
&=
\left(\begin{array}{ccc}
\tilde{\rho}_1 M_1 & & \\
& \ddots & \\
& & \tilde{\rho}_k M_{k}
\end{array} \right),\end{aligned}$$ which implies that: $$N M_i = \tilde{\rho}_i M_i, \quad \forall i.$$ Taking the sum over $i$ on both sides and using the fact that $\sum_i M_i=\mathbb{I}$, we obtain: $$N=\sum_{i=1}^{k} \tilde{\rho}_i M_i,$$ as required.
Necessary Conditions and Bounds {#Necessary Conditions}
===============================
Contains:
- Derivation of a necessary condition for conclusive exclusion to be possible (proof of Theorem \[Necc Cond Theorem\] ).
- Derivation of the necessary condition for conclusive state discrimination to be possible .
- Derivation of the lower bound on the error probability for the exclusion task (proof of Theorem \[lower bound\]).
Necessary Condition for Conclusive Exclusion {#Necessary Condition for Conclusive Exclusion}
--------------------------------------------
Here we derive the necessary condition for single state conclusive exclusion to be possible that was given in Theorem \[Necc Cond Theorem\] .
[(Proof of Theorem \[Necc Cond Theorem\] .)]{}
Suppose that $\mathcal{P}=\{\rho_i\}_{i=1}^{k}$. A feasible solution to the dual SDP, $N$, must be Hermitian and satisfy $N\leq\rho_i$, $\forall i$. Our goal is to construct such an $N$ with the property ${\textnormal{Tr}}[N]>0$. If this is possible, conclusive exclusion is not possible.
First we define $U_{jl}$ to be a unitary such that ${\textnormal{Tr}}\left[\sqrt{\rho_l}\sqrt{\rho_j}U_{jl}\right]=F(\rho_j,\rho_l)$ and note that $U_{lj}=U_{jl}^{*}$. We construct $N$ as follows (for $p,\epsilon \in (0,1) $): $$\begin{aligned}
N=-p\sum_{r=1}^{k}\rho_r +\frac{1-\epsilon}{k-2}p\sum_{1\leq j<l\leq k}\left(\sqrt{\rho_j}U_{jl}\sqrt{\rho_l}+\sqrt{\rho_l}U_{jl}^{*}\sqrt{\rho_j}\right),\end{aligned}$$ and note that $N$ is Hermitian. Now consider: $$\begin{aligned}
\rho_1-N&=(1+p)\rho_1+p\sum_{r=2}^{k}\rho_r-\frac{1-\epsilon}{k-2}p\sum_{1\leq j<l\leq k}\left(\sqrt{\rho_j}U_{jl}\sqrt{\rho_l}+\sqrt{\rho_l}U_{jl}^{*}\sqrt{\rho_j}\right),\\
&=\sum_{r=2}^{k}\left[\frac{1+p}{k-1}\rho_1+\epsilon p\rho_r -\frac{1-\epsilon}{k-2}p\left(\sqrt{\rho_1}U_{1r}\sqrt{\rho_r}+\sqrt{\rho_r}U_{1r}^{*}\sqrt{\rho_1}\right)\right]\\
&\quad +\frac{1-\epsilon}{k-2}p\sum_{2\leq j<l\leq k} \left[\rho_j+\rho_l -\sqrt{\rho_j}U_{jl}\sqrt{\rho_l}-\sqrt{\rho_l}U_{jl}^{*}\sqrt{\rho_j}\right],\\
&=\sum_{r=2}^{k}\left[\frac{1+p}{k-1}\rho_1+\epsilon p\rho_r -\frac{1-\epsilon}{k-2}p\left(\sqrt{\rho_1}U_{1r}\sqrt{\rho_r}+\sqrt{\rho_r}U_{1r}^{*}\sqrt{\rho_1}\right)\right]\\
&\quad + \frac{1-\epsilon}{k-2}p\sum_{2\leq j<l\leq k} \left(\sqrt{\rho_j}\sqrt{U_{jl}}-\sqrt{\rho_l}\sqrt{U_{jl}^{*}}\right)\left(\sqrt{U_{jl}^{*}}\sqrt{\rho_j}-\sqrt{U_{jl}}\sqrt{\rho_l}\right).\end{aligned}$$ The terms in the second summation on the last line are positive semidefinite. Consider, individually, the terms in the first summation: $$\begin{aligned}
&\frac{1+p}{k-1}\rho_1+\epsilon p\rho_r -\frac{1-\epsilon}{k-2}p\left(\sqrt{\rho_1}U_{1r}\sqrt{\rho_r}+\sqrt{\rho_r}U_{1r}^{*}\sqrt{\rho_1}\right),\\
=&\left[\frac{1+p}{k-1}-\left(\frac{(1-\epsilon)p}{k-2}\right)^2\frac{1}{\epsilon p}\right]\rho_1\\
&\quad+\left[\left(\frac{(1-\epsilon)p}{k-2}\right)^2\frac{1}{\epsilon p}\right]\rho_1 +\epsilon p\rho_r -\frac{1-\epsilon}{k-2}p\left(\sqrt{\rho_1}U_{1r}\sqrt{\rho_r}+\sqrt{\rho_r}U_{1r}^{*}\sqrt{\rho_1}\right),\\
=&\left[\frac{1+p}{k-1}-\left(\frac{(1-\epsilon)p}{k-2}\right)^2\frac{1}{\epsilon p}\right]\rho_1\\
&\quad+\left(\frac{(1-\epsilon)p}{(k-2)\sqrt{\epsilon p}}\sqrt{\rho_1}\sqrt{U_{1r}}-\sqrt{\epsilon p}\sqrt{\rho_r}\sqrt{U_{1r}^{*}}\right)\left(\frac{(1-\epsilon)p}{(k-2)\sqrt{\epsilon p}}\sqrt{U_{1r}^{*}}\sqrt{\rho_1}-\sqrt{\epsilon p}\sqrt{U_{1r}}\sqrt{\rho_r}\right).\end{aligned}$$ Hence, for $\rho_1-N$ to be positive semidefinite, we need the first term in the last line to be positive: $$\begin{aligned}
\left[\frac{1+p}{k-1}-\left(\frac{(1-\epsilon)p}{k-2}\right)^2\frac{1}{\epsilon p}\right]&\geq 0, \nonumber\\
\frac{\epsilon}{\frac{(k-1)(1-\epsilon)^2}{(k-2)^2}-\epsilon}&\geq p. \label{p epsilon condition}\end{aligned}$$ Therefore, provided $p$ and $\epsilon$ satisfy Eq. (\[p epsilon condition\]), $N\leq \rho_1$. Similarly, one can argue that $\rho_i\leq N$, $\forall i$ and hence $N$ is a feasible solution to the dual problem.
We now wish to know under what conditions we have ${\textnormal{Tr}}[N]>0$: $$\begin{aligned}
\begin{array}{crl}
&{\textnormal{Tr}}\left[N\right]>&0,\\
\Rightarrow&-kp+\frac{1-\epsilon}{k-2}p\sum_{1\leq j<l\leq k}{\textnormal{Tr}}\left[\sqrt{\rho_j}U_{jl}\sqrt{\rho_l}+\sqrt{\rho_l}U_{jl}^{*}\sqrt{\rho_j}\right]>&0,\\
\Rightarrow&\sum_{j\neq l=1}^{k} F(\rho_j,\rho_l)>&\frac{k(k-2)}{1-\epsilon}.
\end{array}\end{aligned}$$ Letting $\epsilon\rightarrow 0$ and using weak duality we obtain our result. Conclusive exclusion is not possible if $\sum_{j\neq l=1}^{k} F(\rho_j,\rho_l)>k(k-2)$.
Necessary condition for conclusive state discrimination {#Necessary condition for conclusive state discrimination}
-------------------------------------------------------
Here we show how the necessary condition for perfect state discrimination to be possible can be derived from our necessary condition on conclusive state exclusion, Theorem \[Necc Cond Theorem\] .
\[State Discrimination\] Conclusive state discrimination on the set $\mathcal{P}=\{\rho_i\}_{i=1}^{k}$ is possible only if $\mathcal{P}$ is an orthogonal set.
[(Proof of Corollary \[State Discrimination\] .)]{} For $\mathcal{P}=\{\rho_i\}_{i=1}^{k}$, define: $$\hat{\rho}_j=\frac{1}{k-1}\sum_{i\neq j} \rho_i.$$ Let $j\neq l$ and consider: $$A=\frac{1}{k-1}\sum_{r\neq j,l}\rho_r.$$ We first show that $F(\hat{\rho}_j,\hat{\rho}_l)\geq F(\hat{\rho}_j,A)$. Consider: $$\begin{aligned}
F(\hat{\rho}_j,A)&={\textnormal{Tr}}\left[\sqrt{\sqrt{\hat{\rho}_j}A\sqrt{\hat{\rho}_j }}\right],\\
&\leq{\textnormal{Tr}}\left[\sqrt{\sqrt{\hat{\rho}_j}\hat{\rho}_l\sqrt{\hat{\rho}_j }}\right],\\
&=F(\hat{\rho}_j,\hat{\rho}_l).\end{aligned}$$ The inequality follows from the following facts:
1. It can be easily seen from the definitions that $A\leq\hat{\rho}_l$.
2. If $B\geq C$ then $D^{*}BD\geq D^{*}CD$, $\forall D$. Hence: $$\begin{aligned}
\sqrt{\hat{\rho}_j}A\sqrt{\hat{\rho}_j}\leq\sqrt{\hat{\rho}_j}\hat{\rho}_l\sqrt{\hat{\rho}_j}.\end{aligned}$$
3. The square root function is operator monotone, so: $$\begin{aligned}
\sqrt{\sqrt{\hat{\rho}_j}A\sqrt{\hat{\rho}_j}}\leq\sqrt{\sqrt{\hat{\rho}_j}\hat{\rho}_l\sqrt{\hat{\rho}_j}}.\end{aligned}$$
4. The trace function is operator monotone and so finally: $$\begin{aligned}
{\textnormal{Tr}}\left[\sqrt{\sqrt{\hat{\rho}_j}A\sqrt{\hat{\rho}_j}}\right]\leq{\textnormal{Tr}}\left[\sqrt{\sqrt{\hat{\rho}_j}\hat{\rho}_l\sqrt{\hat{\rho}_j}}\right].\end{aligned}$$
Using a similar argument to the above, it is possible to show that: $$F(\hat{\rho}_j,A)\geq F(A,A) =\frac{k-2}{k-1}.$$
If ${\rho}_j$, ${\rho}_l$ and $A$ are pairwise orthogonal, then $\hat{\rho}_j$ and $\hat{\rho}_l$ commute and are simultaneously diagonalizable. This means that: $$\begin{aligned}
F(\hat{\rho}_j,\hat{\rho}_l)&=\left|\left|\sqrt{\hat{\rho}_j}\sqrt{\hat{\rho}_l}\right|\right|_{\textit{Tr}},\\
&=\left|\left|A\right|\right|_{\textit{Tr}},\\
&=F(A,A),\\
&=\frac{k-2}{k-1}.\end{aligned}$$
Now suppose that $\rho_j$ and $A$ are not orthogonal. We take $\{a_r\}$ to be the eigenvalues and $\{{|v_r\rangle}\}$ to be the eigenvectors of $\sqrt{A}$, so: $$\begin{aligned}
F(\hat{\rho}_l,A)&\geq{\textnormal{Tr}}\left[\sqrt{\hat{\rho_l}}\sqrt{A}\right],\\
&=\sum_{r}a_r{\langle v_r|}\sqrt{\hat{\rho_l}}{|v_r\rangle}.\end{aligned}$$ We know that $\sqrt{\hat{\rho}_l}\geq\sqrt{A}$ and hence: $${\langle v_r|}\sqrt{\hat{\rho}_l}{|v_r\rangle}\geq a_r,\quad \forall r.$$ As $\rho_j$ and $A$ are not orthogonal: $$\begin{aligned}
\sum_r{\langle v_r|}\sqrt{\hat{\rho}_l}{|v_r\rangle}>\sum_r a_r,\end{aligned}$$ and there must exist some $r$ such that: $${\langle v_r|}\sqrt{\hat{\rho}_l}{|v_r\rangle}> a_r.$$ Hence: $$\begin{aligned}
F(\hat{\rho}_l,A)&\geq\sum_{r}a_r{\langle v_r|}\sqrt{\hat{\rho_l}}{|v_r\rangle},\\
&>\sum_r a_{r}^{2},\\
&={\textnormal{Tr}}\left[A\right], \\
&=\frac{k-2}{k-1}.\end{aligned}$$ So $F(\hat{\rho}_j,\hat{\rho}_l)=(k-2)/(k-1)$, $\forall l\neq j$, if and only if $\mathcal{P}$ is an orthogonal set.
By Theorem \[Necc Cond Theorem\] , for conclusive $(m-1)$-state exclusion (and hence conclusive state discrimination) to be possible, we require that: $$\begin{aligned}
\sum_{j\neq l=1}^{k} F(\hat{\rho}_j,\hat{\rho}_l)=k(k-2),\end{aligned}$$ which implies that $\mathcal{P}$ must be an orthogonal set.
Bound on success probability {#lower bound proof}
----------------------------
In this section we give the proof of Theorem \[lower bound\].
[(Proof of Theorem \[lower bound\].)]{} The goal is to show that $N_{\varepsilon}\leq\tilde{\rho}_i$, $\forall i$, where $N_{\varepsilon}$ is defined in Eq. (\[Nepsilon\]). Recall that given two Hermitian operators, $A$ and $B$, $\min\left(A,B\right)$ is defined by: $$\min\left(A,B\right)=\frac{1}{2}\left[A+B-\left|A-B\right|\right].$$ Note that $\min(A,B)\leq A$ and $\min(A,B)\leq B$ as: $$\begin{aligned}
A-\min(A,B)&=\frac{1}{2}\left[A-B+\left|A-B\right|\right],\\
&=\frac{1}{2}\left[\sum_{i=1}^{d}\lambda_i{|u_i\rangle\langle u_i|}+\sum_{i=1}^{d}\left|\lambda_i\right|{|u_i\rangle\langle u_i|}\right],\\
&\geq 0,\end{aligned}$$ and similarly $B-\min(A,B)\geq 0$. Here $\sum_{i=1}^{d}\lambda_i{|u_i\rangle\langle u_i|}$ is the spectral decomposition of $A-B$.
The bound is obtained by constructing $N_{\varepsilon}$ iteratively as follows: $$\begin{aligned}
N_{\varepsilon}^{(2)}&=\min\left(\tilde{\rho}_{\varepsilon(2)},\tilde{\rho}_{\varepsilon(1)}\right),\\
N_{\varepsilon}^{(3)}&=\min\left(\tilde{\rho}_{\varepsilon(3)},N_{\varepsilon}^{(2)}\right),\\
\vdots&=\vdots\\
N_{\varepsilon}=N_{\varepsilon}^{(k)}&=\min\left(\tilde{\rho}_{\varepsilon(k)},N_{\varepsilon}^{(k-1)}\right).\end{aligned}$$ Using the fact that $\min(A,B)\leq A$ and $\min(A,B)\leq B$, by construction we have $N_\varepsilon\leq\tilde{\rho}_i$, $\forall i$.
PBR Game {#PBR Game}
========
Contains:
- Proof that the set of projectors $\mathcal{M}=\{{|\zeta_{\vec{x}}\rangle}\}_{\vec{x}\in\{0,1\}^n}$, as given in Eq. (\[PBR Projector\]) , forms a valid measurement.
- Derivation of the conditions under which $\mathcal{M}$ is the optimal measurement for performing exclusion in the PBR game.
- Derivation of how well $\mathcal{M}$ performs at the exclusion task.
Proof that $\mathcal{M}$ is a measurement {#Proof that M is a measurement}
-----------------------------------------
To see that $\mathcal{M}=\{{|\zeta_{\vec{x}}\rangle}\}_{\vec{x}\in\{0,1\}^n}$, where: $${|\zeta_{\vec{x}}\rangle}=\frac{1}{\sqrt{2^n}}\left({|\vec{0}\rangle}-\sum_{\vec{r}\neq\vec{0}}\left(-1\right)^{\vec{x}\cdot\vec{r}}{|\vec{r}\rangle}\right),$$ forms a valid measurement we shall show that it is a set of orthogonal vectors. Consider: $$\begin{aligned}
{\langle \zeta_{\vec{s}}|\zeta_{\vec{t}}\rangle}&=\frac{1}{2^n}\left({\langle \vec{0}|}-\sum_{\vec{r}\neq\vec{0}}\left(-1\right)^{\vec{s}\cdot\vec{r}}{\langle \vec{r}|}\right)\left({|\vec{0}\rangle}-\sum_{\vec{q}\neq\vec{0}}\left(-1\right)^{\vec{t}\cdot\vec{q}}{|\vec{q}\rangle}\right),\\
&=\frac{1}{2^n}\left(1+\sum_{\vec{r},\vec{q}\neq\vec{0}}\left(-1\right)^{\vec{s}\cdot\vec{r}}\left(-1\right)^{\vec{t}\cdot\vec{q}}{\langle \vec{r}|\vec{q}\rangle}\right),\\
&=\frac{1}{2^n}\sum_{\vec{r}}\left(-1\right)^{\left(\vec{s}+\vec{t}\right)\cdot\vec{r}},\\
&=\delta_{\vec{s}\vec{t}}.\end{aligned}$$ Hence $\mathcal{M}$ is a set of orthogonal vectors and therefore a valid measurement basis.
Derivation of conditions under which $\mathcal{M}$ is an optimal measurement {#Derivation of conditions under which M is an optimal measurement}
----------------------------------------------------------------------------
To show that this measurement, $\mathcal{M}$, is optimal for certain pairs of $n$ and $\theta$, we need to construct an $N$ as per Eq. (\[N construct\]) and show that it satisfies the constraints of the dual problem. Writing $\tilde{\rho}_{\vec{x}}=\frac{1}{2^n}{|\Psi_{\vec{x}}\rangle\langle \Psi_{\vec{x}}|}$ and $M_{\vec{x}}={|\zeta_{\vec{x}}\rangle\langle \zeta_{\vec{x}}|}$, we have: $$N=\frac{1}{2^n}\sum_{\vec{x}}{|\Psi_{\vec{x}}\rangle}{\langle \Psi_{\vec{x}}|\zeta_{\vec{x}}\rangle}{\langle \zeta_{\vec{x}}|}.$$ Note that: $$\begin{aligned}
\langle\Psi_{\vec{x}}|\zeta_{\vec{x}}\rangle&=\frac{1}{{\sqrt{2^n}}}\left(\left[\cos\left(\frac{\theta}{2}\right)\right]^n-\sum^{n}_{i=1}{n\choose{i}}\left[\cos\left(\frac{\theta}{2}\right)\right]^{n-i}\left[\sin\left(\frac{\theta}{2}\right)\right]^i \right),\\
&=\frac{1}{{\sqrt{2^n}}}\left[\cos\left(\frac{\theta}{2}\right)\right]^n\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right).\end{aligned}$$ So we have: $$\begin{aligned}
N=C\left(\theta\right)\left[{|\vec{0}\rangle\langle \vec{0}|}-\sum_{\vec{r}\neq\vec{0}}\left[\tan\left(\frac{\theta}{2}\right)\right]^{|\vec{r}|}{|\vec{r}\rangle\langle \vec{r}|}\right], \label{PBR N}\end{aligned}$$ where $C(\theta)$ is given by: $$C\left(\theta\right)=\frac{1}{2^n}\left[\cos\left(\frac{\theta}{2}\right)\right]^{2n}\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right). \label{C(theta)}$$ Note also that $N$ is a real, diagonal matrix and hence is Hermitian so it remains to determine under what conditions $\rho_i-N$ is a positive semidefinite matrix for all $i$.
Let us define the matrices $A_i$ by: $$A_i=-N+\rho_i.$$ The goal is to prove that none of the $A_i$ have a negative eigenvalue. Say $A_i$ has eigenvalues $\{a^{r}_{i}\}$ where $a^{1}_{i}\geq a^{2}_{i}\geq\ldots a^{2^n}_{i}$. The matrix $-N$ has eigenvalues $\{v^{r}\}$ where for $1\leq r\leq 2^n-1$: $$\begin{aligned}
v^{r}&=C\left(\theta\right)\left[\tan\left(\frac{\theta}{2}\right)\right]^{|\vec{r}|},\end{aligned}$$ and for $r=2^n$: $$\begin{aligned}
v^{2^n}&=-C\left(\theta\right).\end{aligned}$$ Each $\rho_i$ is a rank 1 density matrix and hence have eigenvalues $u^{1}_{i}=1$ and $u^{r}_{i}=0$ for $2\leq r\leq 2^n$.
By Weyl’s inequality: $$\begin{aligned}
v^{r}+u^{2^n}_{i}\leq a^{r}_{i}.\end{aligned}$$ So, provided $C(\theta)>0$, we have $a^{r}_{i}>0$ for $1\leq r\leq2^n-1$. Hence at most one eigenvalue of $A_i$ is non-positive. Investigating this non-positive eigenvalue further, consider $A_i$ acting on the state ${|\zeta_i\rangle}$: $$\begin{aligned}
A_i{|\zeta_i\rangle}&=\rho_i{|\zeta_i\rangle}-\sum_{j=1}^{2^n}\rho_j{|\zeta_j\rangle}{\langle \zeta_j|\zeta_i\rangle}, \\
&=0.\end{aligned}$$ Hence the non-positive eigenvalue of $A_i$ is 0 implying that $A_i\geq 0$, $\forall i$, which in turn implies that $N\leq\rho_i$, $\forall i$, provided $C(\theta)>0$. As $\left[\cos\left(\theta/2\right)\right]^{2n}\geq 0$, we have shown that $\{{|\zeta_{\vec{x}}\rangle}\}_{\vec{x}\in\{0,1\}^n}$ , as defined in Eq. (\[PBR Projector\]) , is the optimal measurement for exclusion provided: $$\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right)>0. \label{PBR Criterion Complement}$$ This region is the complement of that given in Eq. (\[PBR Criterion\]) so we know the optimal measurement to perform for all values of $n$ and $\theta$.
Derivation of how well $\mathcal{M}$ performs at the exclusion task {#Derivation of how well M performs at the exclusion task}
-------------------------------------------------------------------
Is conclusive exclusion possible in the region defined by Eq. (\[PBR Criterion Complement\])? To answer this we must consider the trace of the $N$ given in Eq. (\[PBR N\]): $${\textnormal{Tr}}[N]=\frac{1}{2^n}\left[\cos\left(\frac{\theta}{2}\right)\right]^{2n}\left(2-\left[1+\tan\left(\frac{\theta}{2}\right)\right]^n\right)^2.$$ This is strictly positive provided and hence conclusive exclusion is not possible. ${\textnormal{Tr}}[N]$ does however, tell us how accurately we can perform state exclusion when we can not do it conclusively.
Alternative State Exclusion SDPs {#Alternative State Exclusion SDPs}
================================
Contains:
- Derivation of unambiguous state exclusion SDP dual.
- Derivation of worst case error SDP dual.
Unambiguous State Exclusion SDP {#Unambiguous State Exclusion SDP}
-------------------------------
In this section the dual problem for the primal SDP for unambiguous state exclusion as given in Eq. (\[Unambig Prime\]) is derived.
Comparing Eq. (\[Unambig Prime\]) with Eq. (\[Prime\]), we see that here:
- $A$ is a $kd$ by $kd$ block diagonal matrix with each $d$ by $d$ block containing $\sum_{j=1}^{k}\tilde{\rho}_j$: $$A=\left(\begin{array}{ccc}
\sum_{j=1}^{k}\tilde{\rho}_j & & \\
& \ddots & \\
& & \sum_{j=1}^{k}\tilde{\rho}_j
\end{array}\right).$$
- $B$ is a $(d+k)$ by $(d+k)$ matrix with the top left $d$ by $d$ block being an identity matrix and all other elements being $0$: $$B=\left(\begin{array}{cc}
\mathbb{I} & 0\\
0 & 0
\end{array}\right).$$
- $X$, the variable matrix, is a $kd$ by $kd$ block diagonal matrix where we label each $d$ by $d$ block diagonal by $M_i$: $$X=\left(\begin{array}{ccc}
M_1 & & \\
& \ddots & \\
& & M_k
\end{array}\right).$$
- $Y$ is a $(d+k)$ by $(d+k)$ matrix whose top left $d$ by $d$ block we call $N$ and the remaining $k$ diagonal elements we label by $a_i$. $$Y=\left(\begin{array}{cccc}
N & & &\\
& a_1 & &\\
& & \ddots & \\
& & & a_k
\end{array}\right).$$
- The map $\Phi$ is given by: $$\Phi(X)=\left(\begin{array}{cccc}
\sum_{i=1}^{k}M_i & & &\\
& {\textnormal{Tr}}\left[\tilde{\rho}_1 M_1\right] & &\\
& & \ddots &\\
& & & {\textnormal{Tr}}\left[\tilde{\rho}_k M_k\right]
\end{array}\right).$$
Using Eq. (\[Phi\* equation\]) we see that $\Phi^*$ must satisfy: $$\begin{aligned}
{\textnormal{Tr}}\left[N\sum_{i=1}^{k} M_i\right]+\sum_{i=1}^{k}a_i{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right]={\textnormal{Tr}}\left[\left(\begin{array}{ccc}
M_1 & & \\
& \ddots & \\
& & M_{k}
\end{array} \right)
\Phi^{*}\left[\left(\begin{array}{cccc}
N & & &\\
& a_1 & &\\
& & \ddots & \\
& & & a_k
\end{array}\right)\right]\right],\end{aligned}$$ and hence $\Phi^{*}(Y)$ produces a $kd$ by $kd$ block diagonal matrix: $$\Phi^{*}(Y)=\left(\begin{array}{ccc}
N+a_1 \tilde{\rho}_1 & & \\
& \ddots & \\
& & N+a_k \tilde{\rho}_k\end{array}\right).$$
Substituting these elements into Eq. (\[Dual\]) and taking into account the fact that we are maximizing rather than minimizing in the primal problem, we obtain the dual SDP as stated in Eq. (\[Unambig Dual\]).
Worst Case Error State Exclusion SDP {#Worst Case Error State Exclusion SDP}
------------------------------------
In this section the dual problem for the primal SDP for worst case error state exclusion as given in Eq. (\[Worst Case Prime\]) is derived.
Comparing Eq. (\[Worst Case Prime\]) with Eq. (\[Prime\]), we see that here:
- $A$ is a $(kd+1)$ by $(kd+1)$ matrix with $A_{11}=1$ being the only non-zero element: $$A=\left(\begin{array}{cccc}
1 & & &\\
& 0 & & \\
& & \ddots &\\
& & & 0
\end{array}\right).$$
- $B$ is a $(d+k)$ by $(d+k)$ where the bottom right $d$ by $d$ block is the identity matrix. All other elements are zero: $$B=\left(\begin{array}{cc}
0 & 0\\
0 & \mathbb{I}
\end{array}\right).$$
- $X$, the variable matrix, is a $kd+1$ by $kd+1$ block diagonal matrix where $X_{11}=\lambda$ and we label each subsequent $d$ by $d$ block diagonal by $M_i$: $$X=\left(\begin{array}{cccc}
\lambda & & &\\
& M_1 & & \\
& & \ddots & \\
& & & M_k
\end{array}\right).$$
- $Y$ is a $(d+k)$ by $(d+k)$ matrix whose bottom right $d$ by $d$ block we call $N$ and the remaining $k$ diagonal elements we label by $a_i$. $$Y=\left(\begin{array}{cccc}
a_1 & & &\\
& \ddots & & \\
& & a_k &\\
& & & N
\end{array}\right).$$
- The map $\Phi$ is given by: $$\Phi(X)=\left(\begin{array}{cccc}
\lambda-{\textnormal{Tr}}\left[\tilde{\rho}_1 M_1\right] & &\\
& \ddots & &\\
& & \lambda-{\textnormal{Tr}}\left[\tilde{\rho}_k M_k\right] &\\
& & & \sum_{i=1}^{k}M_i
\end{array}\right).$$
Using Eq. (\[Phi\* equation\]) we see that $\Phi^*$ must satisfy: $$\begin{aligned}
\lambda\sum_{i=1}^{k} a_i - \sum_{i=1}^{k} a_i{\textnormal{Tr}}\left[\tilde{\rho}_i M_i\right]={\textnormal{Tr}}\left[\left(\begin{array}{cccc}
\lambda & & &\\
& M_1 & & \\
& & \ddots & \\
& & & M_k
\end{array}\right)
\Phi^{*}\left[\left(\begin{array}{cccc}
a_1 & & &\\
& \ddots & & \\
& & a_k &\\
& & & N
\end{array}\right)\right]\right],\end{aligned}$$ and hence $\Phi^{*}(Y)$ produces a $kd$ by $kd$ block diagonal matrix: $$\Phi^{*}(Y)=\left(\begin{array}{cccc}
\sum_{i=1}^{k} a_i & & &\\
& N-a_1 \tilde{\rho}_1 & & \\
& & \ddots & \\
& & & N-a_k \tilde{\rho}_k\end{array}\right).$$
Substituting these elements into Eq. (\[Dual\]), we obtain the dual SDP as stated in Eq. (\[Worst Case Dual\]).
|
---
abstract: 'We report on the fabrication and electrical characterization of an InAs double - nanowire (NW) device consisting of two closely placed parallel NWs coupled to a common superconducting electrode on one side and individual normal metal leads on the other. In this new type of device we detect Cooper-pair splitting (CPS) with a sizeable efficiency of correlated currents in both NWs. In contrast to earlier experiments, where CPS was realized in a single NW, demonstrating an intrawire electron pairing mediated by the superconductor (SC), our experiment demonstrates an *interwire* interaction mediated by the common SC. The latter is the key for the realization of zero-magnetic field Majorana bound states, or Parafermions; in NWs and therefore constitutes a milestone towards topological superconductivity. In addition, we observe transport resonances that occur only in the superconducting state, which we tentatively attribute to Andreev Bound states and/or Yu-Shiba resonances that form in the proximitized section of one NW.'
author:
- 'S. Baba'
- 'C. Jünger'
- 'S. Matsuo'
- 'A. Baumgartner'
- 'Y. Sato'
- 'H. Kamata'
- 'K. Li'
- 'S. Jeppesen'
- 'L. Samuelson'
- 'H.-Q. Xu'
- 'C. Sch[ö]{}nenberger'
- 'S. Tarucha'
bibliography:
- 'literature.bib'
title: 'Cooper-pair splitting in two parallel InAs nanowires'
---
[^1]
[^2]
Introduction
============
Topologically protected electronic states in nanostructures have recently attracted wide attention, as they may provide fundamental building blocks for quantum computation.[@Sarma2015; @Hoffman2016] Recent advances in material science and device fabrication resulted in considerable progress towards the generation and detection of topologically protected bound states in topologically non-trivial semiconducting nanowires (NWs), so called Majorana Fermions (MF).[@Mourik2012; @Deng2012; @Albrecht2016; @Deng2016; @Guel2018] Two MFs can combine together into a regular Fermion that is why MFs are also known as $Z_2$ Fermions. MFs are predicted to have non-Abelian braid statistics and may provide a platform for topological quantum computation.[@Alicea2010; @Oreg2010] However, one can not implement all required operations for universal quantum computation in quantum bits (qubits) based on MFs by using topologically protected braiding. In this respect, $Z_4$ Fermions, also known as Parafermions (PFs), are better as they allow for a larger set of operations.[@Pachos2012] Recently, it has theoretically been predicted that PFs can be generated in a system based on two NWs with different spin orbit interaction coupled to a common superconducting electrode.[@Keselman2013; @Klinovaja2014; @Gaidamauskas2014] The SC induces both a pairing interaction within each NW and between the two NWs due to Cooper-pair splitting (CPS). In order to realize PFs, the interwire coupling must dominate.[@Klinovaja2014] It has been shown that this is possible in systems with strong electron-electron interaction, such as nanoscaled semiconducting wires.[@Sato2012; @Bena2002; @Recher2002] Another advantage of the parallel two-wire approach, even if PFs are not formed, is the fact, that for large interwire pairing an external magnetic field is not required or only a small field is enough to reach the topological phase.[@Thakurathi2018] A large magnetic field is a limiting factor, because of the critical magnetic field of the SC. It is therefore crucial both for MF-based charge qubits and for the realization of PFs to demonstrate an appreciable magnitude of interwire pairing interaction mediated by a SC. This coupling is also known as crossed-Andreev reflection or CPS.[@Recher2001; @Sato2012; @Chevallier2012] In the last few years, several CPS experiments have been performed using different platforms, mainly [*single*]{} NWs,[@Hofstetter2009; @Hofstetter2011; @Das2012; @Fueloep2014; @Fueloep2015] Carbon Nanotubes [@Herrmann2010; @Schindele2012] and graphene.[@Tan2015; @Borzenets2016] Splitting efficiencies close to 100% have been reported,[@Schindele2012] demonstrating that intrawire pairing can exceed local Cooper pair tunneling. Up to now, all NW based CPS devices consisted of a single NW contacted by two normal metal electrodes and one superconducting contact in between. In order to assess the interwire pairing in double NW structures, it is essential to investigate CPS in such a system. In this work we demonstrate CPS in a parallel double NW device, which is an important first step towards topological quantum computation with PFs.
Scheme and sample
=================
We investigate a device shown schematically in figure \[fig:CPS-DeviceSetup\](a). Two InAs semiconducting NWs with large spin orbit interaction are placed in parallel ($\textrm{NW}_1$ green, $\textrm{NW}_2$ red) and electrically coupled by a common superconducting electrode $\textrm{S}$ (blue). Both NWs are contacted by individual normal metal leads $\textrm{N}_{1/2}$ (yellow). Sidegates $\textrm{SG}_{1/2}$ (yellow) are located on each side of the NWs, in order to separately tune the chemical potentials of the quantum dots (QDs), which form between $\textrm{N}_{1/2}$ and $\textrm{S}$. We note, that the exact location of the QDs is not known, since we do not use additional barrier gates to terminate the QDs.[@Fasth2005] However, it is clear that both $\textrm{N}_{1/2}$ and $\textrm{S}$ induce a potential step from which (partial) electron reflection is possible and QD bound states can form. We also point out already here that the electronic boundary conditions on the $\textrm{S}$ side may change if $\textrm{S}$ is in the normal or superconducting state, due to the proximity effect. Besides local Cooper pair tunneling from S to $\textrm{N}_{1/2}$,[@Gramich2015] Cooper pairs (white circle with red/black dot) can be split, resulting in a non-local current consisting of entangled single electrons. This process is expected to be large if both QDs, $\textrm{QD}_{1/2}$, are in resonance and the electrons can sequentially tunnel from the SC to the two normal metal leads.
![(a) Schematic depiction of Cooper pair splitting in a double NW setup. Two InAs NWs, $\textrm{NW}_{1/2}$, are located in parallel, coupled by a common superconducting lead S and individual normal metal leads, $\textrm{N}_{1/2}$. $\textrm{QD}_{1/2}$ are tuned separately by local sidegates $\textrm{SG}_{1/2}$. (b) Scanning electron microscope image with false color of the device consisting of $\textrm{NW}_{1/2}$ with a common Al contact (S) on the right and individual Au contacts ($\textrm{N}_{1/2}$) on the left and sidegates, $\textrm{SG}_{1/2}$. The measurement setup is also shown.[]{data-label="fig:CPS-DeviceSetup"}](DeviceSetup.pdf){width="0.9\columnwidth"}
Fabrication and Characterization
--------------------------------
The InAs NWs used in this study are grown by Chemical Beam Epitaxy along the $\langle$111$\rangle$ direction. They have a diameter of about and possess pure Wurtzite crystal structure. After transferring NWs from the growth chip to the substrate by standard dry transfer, we use scanning electron microscopy, to select NWs naturally lying next to each other. It is important to note that the NWs are electronically disconnected by their native oxide, which is about thick surrounding each NW. Next, we deposit the common superconducting lead $\textrm{S}$ made of Ti/Al (thickness: /) after removing the native oxide at the contact area using a solution of $({NH}_{4})_2S_x$.[@Suyatin2007] Afterwards, the individual normal metal contacts $\textrm{N}_{1/2}$ made of Ti/Au (/) are deposited at the same time as the local sidegates $\textrm{SG}_{1/2}$. A false color scanning electron microscopy image of the device is shown in figure \[fig:CPS-DeviceSetup\](b). The distance between the source Al contact and the Au drain contacts is about .\
\
All measurements were carried out in a dilution refrigerator with a base temperature of about . Differential conductance has been measured for the respective NWs simultaneously using synchronized lock-in techniques (see figure \[fig:CPS-DeviceSetup\](b)). Characterization measurements (see figure \[fig:D\_diamonds\] in appendix) indicate two individual QDs $\textrm{QD}_1$ and $\textrm{QD}_2$ in each of the NWs, similar to previous measurements.[@Baba2017] From Coulomb blockade measurements we extract the following parameters for the two QDs for the charging energy $U$, single particle level spacing $\epsilon$ and the life-time broadening of the QD eigenstates $\Gamma$ to the leads: $U_{1,2} = 0.5-0.7$meV, $\epsilon_1 = 0.3-0.5$meV, $\epsilon_2 = 0.1 - 0.3$meV, $\Gamma_1 = 0.1-0.2$meV and $\Gamma_2 = 0.2 - 0.3$meV for $\textrm{QD}_1$ and $\textrm{QD}_2$, respectively. Both quantum dots hold similar properties, implying that each QD is formed between the Al contact and individual Au contacts. In addition, we observe a slight suppression of conductance for some regions within the superconducting energy gap $\delta$ of about , which is similar to other experiments, see appendix.[@Das2012] We note here, that since $\Delta < \Gamma$, local pair tunneling should exceed CPS.[@Recher2012; @Schindele2012] We also emphasize that we cannot distinguish the individual tunnel coupling strengths of each QD to either $\textrm{S}$ or $\textrm{N}$. The respective tunnel-rate ratio has an important effect on the magnitude of CPS. In particular, CPS can appear to be suppressed in the experiment if tunneling out of the QD into the drain electrode is the rate-limiting step.[@Schindele2012; @Fueloep2015]
Cooper pair Splitting in double NW
==================================
![(a) Differential conductance $G_1$ of $\textrm{QD}_1$ as a function of $V_{SG1}$ and $V_{SG2}$ and (b) $G_2$ respectively for $\textrm{QD}_2$. (c) cross sections along dashed white lines of (a) and (b). (d) cross section along black dashed line of (a) and (b).[]{data-label="fig:CPS-gate"}](A_CPS.pdf){width="0.9\columnwidth"}
In figure \[fig:CPS-gate\](a) and \[fig:CPS-gate\](b) the simultaneously measured differential conductance $G_1$ through $\textrm{QD}_1$ and $G_2$ through $\textrm{QD}_2$ are shown, both as a function of the side-gate voltages $V_{SG1}$ and $V_{SG2}$. These measurement are done at zero bias and without an external magnetic field. Varying the sidegate voltage $V_{SG1}$ tunes $\textrm{QD}_1$ through several Coulomb blockade resonances, resulting in conductance peaks in $G_1$. Similarly $V_{SG2}$ tunes the resonances of $\textrm{QD}_2$ in $\textrm{NW}_2$. Each resonance signifies a change in charge state of the respective QD. The charge on one QD can be sensed by the other QD, due to the capacitive coupling between $\textrm{QD}_1$ and $\textrm{QD}_2$. In our experiment, $\textrm{QD}_2$ acts as a good sensor for the charge on $\textrm{QD}_1$, as the Coulomb blockade resonance lines of $\textrm{QD}_1$ shift whenever the charge on $\textrm{QD}_2$ changes by one electron, see Fig.\[fig:CPS-gate\](b). Due to capacitive crosstalk from $V_{SG1}$ on $\textrm{QD}_2$ ($V_{SG2}$ on $\textrm{QD}_1$ respectively) the resonance positions are slightly tilted in both graphs. At certain gate voltages, when both QDs are in resonance, an increase of conductance can be observed on both sides. This can be seen more clearly in the cross sections indicated by black and white dashed lines Fig.\[fig:CPS-gate\](a,b). We observe an enhancement of $\textrm{G}_1$ along the black dashed line in Fig.\[fig:CPS-gate\](a) at the peak positions of $\textrm{G}_2$ for the two resonances at $V_{SG2}\approx 1.87$V and $V_{SG2}\approx 1.91$V (see arrows), while other possible correlations are less clear and disappear in the background noise. A similar characteristics can be found in cross sections of $\textrm{G}_2$ along the white dashed line in Fig.\[fig:CPS-gate\](b) at peak positions of $\textrm{G}_1$. Here, the positive correlation is very clear at $V_{SG1}\approx 0.32$V (arrow), while there is only a weak quite broadened correlation visible for the other two resonances at $V_{SG1}\approx 0.4$V and $V_{SG1}\approx 0.49$V. Hence, we observe clear positive correlation between $\textrm{G}_1$ and $\textrm{G}_2$ on three resonances, which we assign to CPS from $\textrm{S}$ into $\textrm{QD}_1$ and $\textrm{QD}_2$. We repeat the same measurement in the absence of superconductivity by applying an out of plane magnetic field of , which is larger than the critical field of the Al contact. In this case the previously positive correlations between $\textrm{G}_1$ and $\textrm{G}_2$ disappear fully, proofing that the positive correlation in conductance originates from CPS (see figure \[fig:E\_CPS\_Bfield\] in appendix). We define the CPS efficiency as 2$\textrm{G}_{CPS}$/$\textrm{G}_{total}$ resulting in a maximum efficiency of $\approx$ 20%, similar to the largest reported values on single NW devices.
In the following we investigate the same type of measurement as discussed in figure \[fig:CPS-gate\], for a different gate voltage region. The data is shown in figure \[fig:B\_2nCPS\]. Here, we observe two sets of resonances for $\textrm{QD}_1$. Besides the QD levels, we already discussed in figure \[fig:CPS-gate\] (indicated with ${\rm I}$), we detect a second set of resonances (referred to ${\rm II}$). For $\textrm{QD}_2$ we observe only one set of QD resonances. The resonances denoted with ${\rm II}$ are significantly different from the one denoted with ${\rm I}$.
![(a) Differential conductance $G_1$ of $\textrm{QD}_1$ as a function of $V_{SG1}$ and $V_{SG2}$, second set of resonances are denoted with ${\rm II}$ (black arrows). (b) Differential Conductance $G_2$ respectively for $\textrm{QD}_2$. (c) cross sections along dashed white lines of (a) and (b). (d) cross section along black dashed line of (a) and (b).[]{data-label="fig:B_2nCPS"}](B_2nCPS.pdf){width="0.9\columnwidth"}
First, the amplitude of type ${\rm II}$ resonances is only half of the value of type ${\rm I}$. Furthermore, type ${\rm II}$ have a different slope than type ${\rm I}$, indicating different capacitive coupling ratio between $\textrm{SG}_1$ and $\textrm{SG}_2$. The broadening differs roughly by a factor of two: $\Gamma_{\rm I}$= whereas $\Gamma_{\rm II}$= . Most strikingly, the resonances of type ${\rm II}$ vanish completely when an external magnetic field is applied, i.e. these resonances are only present in the superconducting state (not shown). In addition, $\textrm{QD}_{2}$ only reacts on a change in charge state of $\textrm{QD}_{1}$ for resonances of type ${\rm I}$ (e.g. green arrow in \[fig:B\_2nCPS\](b)). For the resonances of type ${\rm II}$, there seems to be no change in the charge state, as $\textrm{QD}_{2}$ does not sense these resonances (e.g. yellow arrows in \[fig:B\_2nCPS\](b)). We therefore conclude that the resonance lines of type ${\rm II}$ are not Coulomb blockade resonances. They must have their origin within the superconducting phase, most likely near $\textrm{S}$. Though these features are not Coulomb blockade resonances, they can still be used to test CPS by conductance correlations. We observe a clear positive correlation between the conductances $\textrm{G}_1$ and $\textrm{G}_2$ in figure \[fig:B\_2nCPS\](d) as well as in \[fig:B\_2nCPS\](c) for both types of resonances. We estimate a splitting efficiency of about $\approx$13% for type ${\rm II}$, which is of similar magnitude as the one obtained for resonances of type ${\rm II}$ in the gate region of Fig.\[fig:CPS-gate\].\
The measurements of type ${\rm II}$ resonances suggest the existence of sub-gap states which are not located in $\textrm{QD}_{1}$, but rather in the lead connecting to $\textrm{S}$. Since these states are gate-tunable, they are not fully screened by $\textrm{S}$. We therefore propose that a proximitized region is formed in NW1 that extends to some distance out from $\textrm{S}$. $\textrm{QD}_{1}$ is coupled to this proximitized lead. Within the lead, bound states can form due to potential fluctuations and residual disorder. There are two kinds of bound states, Andreev bound states (ABS) [@Pillet2010; @Dirks2011; @Gramich2017] or Yu-Shiba Rusinov (YSR)[@Jellinggaard2016] states. These states do not usually occur at zero energy, but can be tuned electrically to zero energy, signaling a ground state transition between the proximitized lead region and the bulk of the SC. This gives in effect rise to a density-of-state peak in the gap of the SC, enhancing the subgap conductance which we measure. In this case the two electrons that are launched by CPS are transmitted in a different way to the respective drain electrodes. The electron that takes the path through $\textrm{QD}_{2}$ is transferred by the usual (resonant) sequential tunneling, while the one that takes the path through $\textrm{QD}_{1}$ is transferred by co-tunneling. One might expect that this suppresses CPS as the latter process corresponds to a low probability for out-tunneling into the drain contact. However, due to the sub-gap resonance in the proximitized lead, this process is enhanced and one can therefore reach almost similar CPS efficiencies.
Conclusion
==========
In summary, we demonstrate the fabrication of an electronic device, consisting of two closely placed parallel InAs NWs, contacted by a common superconducting lead and individual normal metal leads. By addressing individual sidegates we can separately tune the QDs formed in each NW. When both QDs are in resonance, we observe CPS with efficiencies up to 20%. For certain gate voltages, we detect a second set of QD resonances in one of the NWs, which only appears in the superconducting state. The second set of resonances do not correspond to Coulomb blockade resonances, hence, are not related to a change in the charge state of the QD. Since they only appear in the superconducting state, we tentatively assign the second set to subgap states in the lead that connects to the SC, which are most likely caused by superconducting bound states (Andreev and or Yu-Shiba Rusinov states). For the first time we provide a platform, suitable to implement the next milestone in topological quantum computation, namely Parafermions.
Acknowledgements
================
This work was partially supported by a Grant-in-Aid for Young Scientific Research (A) (Grant No. JP15H05407), Grant-in-Aid for Scientific Research (A) (Grant No. JP16H02204), Grant-in-Aid for Scientific Research (S) (Grant No. JP26220710), JSPS Research Fellowship for Young Scientists (Grant No. JP14J10600), and the JSPS Program for Leading Graduate Schools (MERIT) from JSPS, Grants-in-Aid for Scientific Research on Innovative Area “Nano Spin Conversion Science” (Grants No. JP15H01012 and No. JP17H05177) and a Grant-in-Aid for Scientific Research on Innovative Area “Topological Materials Science” (Grant No. JP16H00984) from MEXT, JST CREST (Grant No. JPMJCR15N2), Murata Science Foundation and the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). Part of this research was performed within the Nanometer Structure Consortium/NanoLund-environment, using the facilities of Lund Nano Lab, with support from the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF) and from Knut and Alice Wallenberg Foundation (KAW). It was also supported by the Swiss National Science Foundation, the Swiss Nanoscience Institute (SNI), the NCCR on Quantum Science and Technology and the H2020 project QuantERA.
Appendix {#appendix .unnumbered}
========
![(a) Differential conductance $G_1$ of $\textrm{QD}_1$ as a function of $\textrm{SG}_1$ and source drain bias $\textrm{V}_{SD}$. White dashed lines indicate a suppression of conductance due to the superconducting energy gap. Cross section along green dashed line shown in green. (b) Differential conductance $G_2$ of $\textrm{QD}_2$ as a function of $\textrm{SG}_2$ and $\textrm{V}_{SD}$ respectively. Cross section along green dashed line shown in green, representing an enhancement of Andreev Reflection due to the stronger coupling of $\textrm{QD}_2$ to the SC.[]{data-label="fig:D_diamonds"}](D_diamonds.pdf){width="0.95\columnwidth"}
![(a,b) Differential conductance $G_1$ and $G_2$ of $\textrm{QD}_1$ and $\textrm{QD}_2$ as a function of $V_{SG1}$ and $V_{SG2}$. During this measurements a constant out-of-plane external magnetic field of was applied. (c,d) Cross sections along the dashed white and black lines shown in (a) and (b). The positive correlation, which was evident in Fig.\[fig:CPS-gate\] in the main text, is now either absent or replaced by a negative correlation. The latter is expected for classical correlations that can be described by a simple resistor network.[@Hofstetter2009] In (d) a linear background was removed from $G_1$, therefore its denoted as $\Delta G_1$. []{data-label="fig:E_CPS_Bfield"}](E_CPS_Bfield.pdf){width="0.95\columnwidth"}
[^1]: These two authors contributed equally
[^2]: These two authors contributed equally
|
---
abstract: 'We investigate the reliability of mass estimators based on the observable velocity dispersion and half-light radius $R_\mathrm{h}$ for dispersion-supported galaxies. We show how to extend them to flattened systems and provide simple formulae for the mass within an ellipsoid under the assumption the dark matter density and the stellar density are stratified on the same self-similar ellipsoids. We demonstrate explicitly that the spherical mass estimators [@Walker2009; @Wolf2010] give accurate values for the mass within the half-light ellipsoid, provided $R_\mathrm{h}$ is replaced by its ‘circularized’ analogue $R_\mathrm{h}\sqrt{1-\epsilon}$. We provide a mathematical justification for this surprisingly simple and effective workaround. It means, for example, that the mass-to-light ratios are valid not just when the light and dark matter are spherically distributed, but also when they are flattened on ellipsoids of the same constant shape.'
author:
- 'Jason L. Sanders'
- 'N. Wyn Evans'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Mass estimators for flattened dispersion-supported galaxies'
---
\[firstpage\]
Introduction
============
Accurate estimates of the dark matter content of dwarf spheroidal galaxies (dSphs) are crucial for furthering our understanding of galaxy formation and structure. Calculating reliable mass estimates has historically been an awkward problem as with only line-of-sight (l.o.s.) velocity measurements the mass profile of a spherical galaxy can only be inferred by making an assumption about the degree of velocity anisotropy i.e. the ratio of radial to tangential motion.
Through comparisons to solutions of the Jeans equations, it has been shown that the mass contained near the half-light radius of a dispersion-supported galaxy is approximately independent of the velocity anisotropy and the radial profile of the dark and luminous matter and is simply related to the half-light radius $\Rh$ and the luminosity-averaged l.o.s. velocity dispersion $\sqrt{\slosT}$. There exist several different forms for these formulae in the literature [@Walker2009; @Wolf2010; @Amorisco2012; @Campbell2016] that may be summarised as $$M_\mathrm{sph}(<r_x) = \frac{C_x \slosT\Rh}{G}
\label{Eqn::WalkerWolf}$$ where $M_\mathrm{sph}(<r_x)$ is the mass contained within a sphere of radius $r_x$ and $G$ the familiar gravitational constant. $C_x$ is a constant that depends on the choice of radius $r_x$. [@Walker2009] proposed that if $r_x=\Rh$ then $C_x=2.5$ based on a simple example of the stellar distribution following a Plummer profile and the dark matter following a cored isothermal profile although this was validated through fuller testing. [@Wolf2010] demonstrated that for $r_x\approx\tfrac{4}{3}\Rh$ (approximately the 3D spherical half-light radius for a range of observationally-motivated profiles) that $C_x=4$ reproduced the results from full Jeans analyses and was also shown to be mathematically true under the assumption of a near-flat velocity dispersion profile.
Although spherical mass estimators have proved useful for understanding dSphs, they cannot give the full picture as they do not consider the fundamentally aspherical shape of these galaxies. Our aim in this Letter is to find mass estimators equivalent to equation applicable to flattened systems. We begin by inspecting the validity of the spherical mass estimators and go on to investigate the applicability of the estimator when considering flattened systems in which the dark and light matter are stratified on the same self-similar ellipsoids. We give formulae similar to equation that may be used when the 3D shape of the system is known. By marginalizing over prior assumptions on the intrinsic shape and alignment, we show how the mass can be estimated when the intrinsic shape and alignment are not known.
Spherical mass estimators
=========================
For a spherical stellar luminosity density $j_\star(r)$ with a constant mass-to-light ratio in a spherical mass density $\rho_\mathrm{DM}(r)$ with mass profile $M(r)$ sourcing potential $\Phi(r)$, the potential energy can be written in terms of the surface brightness $S(R)$ as $$W=\tfrac{1}{2}\int\drm V\,j_\star(r)\Phi(r)=4\pi G\int_0^\infty\drm r\,I(r)M'(r),$$ where $$I(r)=\int_r^\infty\drm r\,rj_\star(r)=-\frac{1}{\pi}\int_r^\infty\drm R\,(R^2-r^2)^{1/2}{\frac{\drm S}{\drm R}}$$ From the virial theorem, we know that the l.o.s. velocity dispersion is related to the total luminosity $L$ by $\slosT = -W/3L$ which gives the constant $\C$ as $$C_x=\frac{1}{R_\mathrm{h}}\Big[\int_0^{r_x}\drm r\,r^2\rho_\mathrm{DM}(r)\Big]\Big[\int_0^{\infty}\drm r\,r^2J(r)\rho_\mathrm{DM}(r)\Big]^{-1}
\label{Eqn::WynC}$$ where $J(r)=(4\pi/3L)I(r)$. The constant $\C$ depends only on the profile of the halo model $\rho_\mathrm{DM}$ and the surface brightness profile $J(r)$.
$$\includegraphics[width=0.9\columnwidth]{Figure1.pdf}$$
$$\includegraphics[width=\textwidth]{Figure2.pdf}$$
We use this to test the validity of the spherical mass estimator. In Fig. \[Fig::SphericalMassEstimator\], we show the result of equation computed numerically for two models with differing ratios of dark to stellar scale-lengths ($r_\mathrm{DM}/\Rh$). They are an NFW dark matter profile $\rho_\mathrm{DM}(r)\propto r^{-1}(1+r/r_\mathrm{DM})^{-2}$ and a cored isothermal profile of the form $$\rho_\mathrm{DM}(r)=\frac{v_0^2}{4\pi G}\frac{3r_\mathrm{DM}^2+r^2}{(r_\mathrm{DM}^2+r^2)^2}.$$ The stellar tracer profile follows a Plummer law $\rho_\star(r)\propto
(1+(r/r_\star)^2)^{-5/2}$ for which $R_\mathrm{h}=r_\star$. The constant $\C$ is computed at the two radii recommended by [@Walker2009] and [@Wolf2010]. The constants given by these two authors are shown with horizontal lines along with the uncertainty found by [@Campbell2016] from inspecting cosmological hydrodynamical simulations. The variation of $\C$ with respect to $r_\mathrm{DM}/\Rh$ is smallest for the NFW profile and is consistent with the bracket found by [@Campbell2016]. In the cored isothermal profile with $r_\mathrm{DM}/\Rh\approx1$, both estimators perform well. However, again as $r_\mathrm{DM}/\Rh$ is increased, $\C$ deviates significantly and so the estimators perform poorly for $r_\mathrm{DM}/\Rh>2$.
We now explore how the mass estimators perform as the parameters of a double power-law dark matter density profile are altered. We use a fixed Plummer profile for the stars with a $\mathrm{sech}$ truncation at $10R_\mathrm{h}$. In Fig. \[Fig::PlummerNFW\_rrat1\], we show the mass profiles of different dark matter profiles that all produce the same luminosity-averaged l.o.s. velocity dispersion. The default parameters are those of an NFW profile with $r_\mathrm{DM}/\Rh=1$ and a $\mathrm{sech}$ truncation at $10r_\mathrm{DM}$. We alter the outer slope $\beta$, inner slope $\gamma$ and the ratio $r_\mathrm{DM}/\Rh$. We find that when varying the inner and outer slopes the pinch point where the mass is the same for all profiles is around $\tfrac{4}{3}\Rh$ i.e. the radius recommended by [@Wolf2010]. Varying $r_\mathrm{DM}/\Rh$ produces a pinch point further out. This helps explain why mass estimators derived for use on realistic halos with $r_\mathrm{DM}>\Rh$ can constrain the mass at larger radii [e.g. @Amorisco2012; @Campbell2016].
Flattened mass estimators
=========================
We now turn to adapting the spherical mass estimators for application to flattened systems. We work with models with both the dark and stellar density stratified on the same concentric self-similar ellipsoids labelled with the coordinate $m$ such that $m^2=x^2/a^2+y^2/b^2+z^2/c^2$ with $a>b>c$. The axis ratios of the ellipsoids are $p=b/a$ and $q=c/a$. We view the model along the spherical polar unit vector defined by the angles $(\vartheta,\varphi)$, where $\vartheta$ is the co-latitudinal angle and $\varphi$ the azimuthal angle defined with respect to a Cartesian coordinate system aligned with the principal axes (see Fig. \[Fig::Diagram\]. When oblate and prolate spheroids are viewed ‘face-on’, they appear round. The spherical mass estimator underestimates (overestimates) the mass within a sphere for the oblate (prolate) case, as mass is added to (removed from) the sphere. Similarly, the formulae give (smaller) under- and overestimates for the mass within the corresponding ellipsoid. We seek an appropriate modification to equation that is applicable to flattened systems, namely $$M_\mathrm{ell}(<m_x)=\frac{C_xf_\sigma\slosT f_r\Rh}{G},
\label{Eqn::WalkerWolf_Flattened}$$ where $M_\mathrm{ell}(<m_x)$ is the mass within an ellipsoid (that is the same shape as the equidensity contours) with major axis length $r_x$. We imagine creating an ellipsoidal model by deforming a spherical model that obeys the spherical mass estimator formulae outlined in the previous section. The total mass is conserved if $abc=1$ and the mass within an ellipsoid of major-axis length $r_x$ is identical to the mass within a sphere of radius $m_x=r_x/a=r_x(pq)^{1/3}$. However, to estimate this parent spherical model mass from the spherical mass estimator formulae, we must relate the observed l.o.s velocity dispersion to the spherical velocity dispersion and the observed half-light major-axis length to the intrinsic major-axis length of the considered ellipsoid. Assuming the total velocity dispersion (the average of the dispersions along the principal axes) is conserved as we deform the model[^1], the factor $f_\sigma$ accounts for the relationship between the l.o.s velocity dispersion and the total dispersion of the ellipsoidal model. The factor $f_r$ accounts for the relationship between the observed major-axis length and the intrinsic major-axis length of the equivalent ellipsoid (and that of the parent spherical model).
Velocity scaling
----------------
For triaxial systems, the velocity scaling $f_\sigma=\stotT/\slosT$ is given by
$$f_\sigma=\frac{1}{3}\frac{1+r_{xz}+r_{yz}}{\cos^2\vartheta+r_{xz}\sin^2\vartheta\cos^2\varphi+r_{yz}\sin^2\vartheta\sin^2\varphi}$$
where $$r_{ij}=\langle\sigma_i^2\rangle/\langle \sigma_j^2\rangle=W_{ii}/W_{jj}\text{ (no sum)}.$$ For dSphs in which the stellar and dark-matter density profiles are stratified on the same self-similar ellipsoids, $r_{ij}$ depends only on the shape of the ellipsoids [@Roberts1962; @BinneyTremaine]. That is to say, it is independent of the ‘radial’ density profile of the light and dark matter. Therefore, $f_\sigma$ is a function of $p$, $q$ and the viewing angles: $f_\sigma=f_\sigma(\vartheta,\varphi,p,q)$. Expressions for $W_{ij}$ are given in Table 2.2 of [@BinneyTremaine].
Radial scaling
--------------
$$\includegraphics[width=\columnwidth]{Figure3.pdf}$$
We decompose the radial scaling into two components, $f_r=f_1
f_2$. $f_2$ describes the relationship between the ellipsoidal major-axis length and the parent spherical radius so (as described above) $f_2=(pq)^{1/3}$. The other factor $f_1$ gives the relationship between the observed major-axis length of the half-light ellipse $\Rh$ and the intrinsic major-axis length of the corresponding ellipsoid $r_\mathrm{maj}$. In the spherical case, these quantities are equal. In the ellipsoidal case, the relationship between these quantities depends on the viewing angles and the intrinsic shape $f_1=f_1(\vartheta,\varphi,p,q)$. We approximate $f_1$ by the relationship between the major-axis length of an ellipsoid and the major-axis length of its projected ellipse. This neglects any subtleties related to the extended nature of the true density distribution. However, if the 3D stellar light profile falls off sufficiently rapidly then our relationship is a good approximation.
To derive our approximation for $f_1$, we use a coordinate system $(x',y',z')$ related to the intrinsic coordinate system by (see Fig. \[Fig::Diagram\]) $$\begin{split}
x&=-x'\sin\varphi-y'\cos\vartheta\cos\varphi+z'\sin\vartheta\cos\varphi\\
y&=x'\cos\varphi-y'\cos\vartheta\sin\varphi+z'\sin\vartheta\sin\varphi\\
z&=y'\sin\vartheta+z'\cos\vartheta.
\end{split}$$ We consider the set of points where the ellipsoidal surface is tangential to $\hat{\boldsymbol{z}}'$ which results in a rotated ellipse in the $(x',y')$ plane. We diagonalize the resultant quadratic surface to find the major axis length $\Rh$ as $$f_1^{-2}=(\Rh/r_\mathrm{maj})^2= 2C/(A-\sqrt{B}),
\label{Eqn::Fsig_exp}$$ where $$\begin{aligned}
A&=&(1-q^2)\cos^2\vartheta+(1-p^2)\sin^2\vartheta\sin^2\varphi+p^2+q^2,\nonumber\\
B&=&[(1-q^2)\cos^2\vartheta-(1-p^2)\sin^2\vartheta\sin^2\varphi-p^2+q^2]^2\nonumber\\
&+& 4(1-p^2)(1-q^2)\sin^2\vartheta\cos^2\vartheta\sin^2\varphi,\nonumber\\
C&=& p^2\cos^2\vartheta+q^2\sin^2\vartheta(p^2\cos^2\varphi+\sin^2\varphi).\end{aligned}$$ As given in [@Weijmans2014], the observed ellipticity $\epsilon$ satisfies $(1-\epsilon)^2=(A-\sqrt{B})/(A+\sqrt{B})$.
For an oblate spheroid ($p=1$), eqn (\[Eqn::Fsig\_exp\]) simplifies to $\Rh=r_\mathrm{maj}$. For a prolate spheroid $p=q$, so we find $$f_1^{-2}=\cos^2\vartheta+\sin^2\vartheta(q^2\cos^2\varphi+\sin^2\varphi).$$ In Fig \[Fig::Diagram\], we show the major axis length for a prolate figure and a triaxial figure as a function of the viewing angle.
The ellipsoidal half-light radius $m_h$ is well approximated by $\tfrac{4}{3}f_r\Rh$ which should be compared to the radius of $\tfrac{4}{3}\Rh\sqrt{1-\epsilon}$ that is empirically used [e.g. @Koposov2015; @Sanders2016] as $\Rh\sqrt{1-\epsilon}$ approximately reproduces the circularly-averaged half-light radius of the dSph[^2].
Near-spherical limits {#Sect::NearSph}
---------------------
Using eqn (\[Eqn::Fsig\_exp\]), we can find the modification factor $f_\sigma f_r$ for the simple cases of viewing down the principal axes of a near-spherical triaxial ellipsoid and compare to the alternative factor $\sqrt{1-\epsilon}$. When viewing down the major axis ($\vartheta=\pi/2,\varphi=0$), we find $$f_\sigma f_r\approx1+\tfrac{2}{5}(1-p)-\tfrac{3}{5}(1-q).$$ The observed ellipticity $\epsilon=1-p/q$ so the *circularized* factor $\sqrt{1-\epsilon}\approx1+\tfrac{1}{2}(1-p)-\tfrac{1}{2}(1-q)$ which is a close approximation to our factor $f_\sigma f_r$. Similarly, for viewing down the intermediate axis, we find $$f_\sigma f_r\approx1+\tfrac{1}{5}(1-p)-\tfrac{3}{5}(1-q),$$ whilst $\sqrt{1-\epsilon}=\sqrt{q}\approx1-\tfrac{1}{2}(1-q)$. Finally, viewing down the minor axis, we find $$f_\sigma f_r\approx1-\tfrac{3}{5}(1-p)+\tfrac{1}{5}(1-q),$$ whilst $\sqrt{1-\epsilon}=\sqrt{p}\approx1-\tfrac{1}{2}(1-p)$. We note that the flattening in the line-of-sight direction (e.g. $p$ in the intermediate axis case) has a smaller contribution to the factor $f_\sigma f_r$. This demonstrates that the simple factor $\sqrt{1-\epsilon}$ goes a long way to account for the velocity and radial scalings we propose.
Results
-------
$$\includegraphics[width=0.9\columnwidth]{Figure4.pdf}$$
In Fig. \[Fig::PlummerNFW\_rrat1\], we show the mass estimates using our formulae for an oblate and prolate model viewed edge-on. The models have the same ellipsoidal mass profile as the spherical model shown. The factors we have introduced correctly deproject the observed quantities producing an unbiased mass estimate. Fig. \[Fig::CorrNoCorr\] shows the constant in the half-light ellipsoid mass estimator (eqn. ) for three models of flattened Plummer profiles embedded in equivalently flattened NFW halos with $m_\mathrm{DM}/m_\star=5$. We show an oblate, prolate and triaxial $p=\tfrac{1}{2}(1+q)$ model. Simply using the spherical mass estimator with $R_\mathrm{h}$ underestimates/overestimates the ellipsoidal mass for the oblate/prolate case viewed face-on (down the minor/major axis). Similarly, the edge-on case (major for oblate, minor for prolate) produces overestimates of the mass for both oblate and prolate models. For the triaxial model the spherical mass estimator produces an overestimate when viewing down the major axis and (for this particular case) is largely unbiased when viewing down the minor axis. The results using the correction factors $f_\sigma$ and $f_r$ are unbiased estimates of the mass within the ellipsoid $m=\tfrac{4}{3}m_\star$ and using the spherical mass estimator with the ‘circularized’ radius $R_\mathrm{h}\sqrt{1-\epsilon}$ produces very similar results to the corrected version. This echoes a result in [@Sanders2016] who demonstrated that the correction to the D-factor (important for interpreting dark-matter decay signals) is almost independent of the flattening for edge-on systems. The near-spherical expansions of § \[Sect::NearSph\] are also shown, which replicate the trends over the full $q$ range.
$$\includegraphics[width=\columnwidth]{Figure5.pdf}$$
Our proposed modifications correctly reproduce the mass within ellipsoids. However, this relies on knowing the intrinsic shape and alignment of the dSph. Such information is not accessible, but we can put priors on possible models which reproduce the observables. We choose to put priors on the triaxiality $T=(1-p^2)/(1-q^2)$, flattening $q$ and the viewing angles $(\vartheta,\varphi)$. We consider three priors:
1. Flat prior – $T\sim\mathcal{U}(0,1)$, $q\sim\mathcal{U}(0.05,1)$, $\cos\vartheta\sim\mathcal{U}(0,1)$, $\varphi\sim\mathcal{U}(0,\pi/2)$.
2. Major-axis prior – $T\sim\mathcal{U}(0,1)$, $q\sim\mathcal{U}(0.05,1)$, $\vartheta\sim\mathcal{N}(\pi/2,0.1\mathrm{\,rad})$, $\varphi\sim\mathcal{U}(0,0.1\mathrm{\,rad})$.
3. Fixed-shape prior – $T\sim\mathcal{N}(0.55,0.04)$, $q\sim\mathcal{N}(0.49,0.12)$, $\cos\vartheta\sim\mathcal{U}(0,1)$, $\varphi\sim\mathcal{U}(0,\pi/2)$.
where the final prior is taken from a fit to the shapes of the Local Group dSphs from [@SanchezJanssen2016]. The major-axis prior is inspired by the observation from simulations that the major-axes of subhaloes points towards the centre of the host halo [e.g. @Barber2015]. We sample from the priors folded with a normal distribution on the observed ellipticity with width $\sigma_\epsilon=0.05$ [using *emcee* @ForemanMackey2012] and for each sample compute the mass within the half-light ellipsoid from equation . The results for a range of observed ellipticities are shown in Fig. \[Fig::massaverage\]. We show the mass estimates over the spherical mass estimator using the ‘circularized’ radius. We see that using the spherical mass estimator in this way reproduces the mass within the half-light ellipsoid over the full range of ellipticities[^3]. The uncertainty in the estimator increases with increasing ellipticity but is only $\sim10-20\percent$ for $\epsilon\sim0.4$ (a typical dSph flattening). There is the tendency for the mass within the half-light ellipsoid to be overestimated for large $\epsilon$, but only by $\sim5\percent$. We also show the distribution of $f_r/\sqrt{1-\epsilon}$ for each prior assumption (i.e. the ratio of the size of the ellipsoid to the $\Rh\sqrt{1-\epsilon}$ approximation). For the uniform prior, this ratio is unity (within $\sim10-20\percent$) so the ‘size’ of the dSphs are well approximated by $\Rh\sqrt{1-\epsilon}$. For the other two priors, the ratio increases with ellipticity as the intrinsic ellipsoids are on average more elongated along the line-of-sight so larger than $\Rh\sqrt{1-\epsilon}$.
We have demonstrated that the mass within the half-light ellipsoid can be accurately estimated using the spherical mass estimator formulae. Although we do not know the shape or orientation of this half-light ellipsoid, we can say with confidence the mass within it. Therefore, we can accurately estimate the mass-to-light ratio using the mass within the half-light ellipsoid and half the total luminosity $L$. [*We conclude that using the spherical mass estimators [@Walker2009; @Wolf2010] with the ‘circularized’ half-light radius produces accurate estimates of the mass-to-light ratio of dSphs, irrespective of flattening, provided the light and dark matter are stratified on the same self-similar concentric ellipsoids*]{}.
Conclusions
===========
This [*Letter*]{} has answered the question: how should the mass of a flattened, dispersion-supported galaxy like a dwarf spheroidal be estimated? If the galaxy were spherical, then the answer is well-established. Accurate mass estimators depending on the observable half-light radius and the velocity dispersion of the stars have been devised by a number of investigators [@Walker2009; @Wolf2010; @Amorisco2012; @Campbell2016].
We have shown how to modify the spherical mass estimators so that they work for flattened systems in which the light and dark matter are stratified on the same concentric self-similar ellipsoids. This represents a limiting case as simulations indicate the dark matter distribution is in fact rounder than the light [@Ab10; @Ze12] due to baryonic feedback effects, particularly for the more massive dSphs. The modifications require knowledge of the intrinsic shape and alignment of the triaxial figure and reproduce the mass within ellipsoids by deprojecting the half-light radius and line-of-sight velocity dispersion. The resulting mass estimates are independent of details of the radial profile and are as accurate as the corresponding spherical formulae.
This would be of little use if we require knowledge of intrinsic properties. However, we have also shown that, when averaging over triaxial configurations that are consistent with the observed ellipticity $\epsilon$, major-axis half-light length $R_\mathrm{h}$ and line-of-sight velocity dispersion, the mass within the half-light ellipsoid is well approximated by the spherical mass estimate using the ‘circularized’ half-light radius of $R_\mathrm{h}\sqrt{1-\epsilon}$. The scatter in the estimate increases with ellipticity but is only $10-20\percent$ for $\epsilon\sim0.6$. In turn, this observation implies that mass-to-light ratios using spherical estimators, together with a luminosity of $L_{1/2}=L/2$, are accurate and insensitive to the flattening of the dSph. This therefore provides a surprisingly simple, flexible and effective way to account for the effects of flattening.
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\[lastpage\]
[^1]: To leading order in the flattening, the ratio of the total dispersion of the flattened model to the spherical model with the same mass is $\langle\sigma_\mathrm{tot}^2\rangle_\mathrm{flat}/
\langle\sigma_\mathrm{tot}^2\rangle_\mathrm{sph}\approx1-
\tfrac{4}{45}\Big[(1-p)^2-(1-p)(1-q)+(1-q)^2\Big]$ .
[^2]: For example, a flattened ($q=1-\epsilon$) Plummer surface profile produces a circularly-averaged half-light radius equal to $\Rh\sqrt{\tfrac{1}{6}(1+q^2+\sqrt{1+14q^2+q^4})}$ which for small flattenings is $\Rh(1-\tfrac{1}{2}\epsilon+\mathcal{O}(\epsilon^4))$ so well approximated by $\Rh\sqrt{1-\epsilon}$.
[^3]: We note that a similar observation was made by [@Laporte2013] who found that the variations of $\Rh$ and $\slosT$ with triaxiality compensated each other to give an unbiased mass estimate.
|
---
abstract: 'New concept of conditional differential invariant is discussed that would allow description of equations invariant with respect to an operator under a certain condition. Example of conditional invariants of the projective operator is presented.'
---
[**Differential Invariants and Construction\
of Conditionally Invariant Equations** ]{}
Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv 4, Ukraine\
E-mail: iyegorch@imath.kiev.ua
Introduction
============
Importance of investigation of symmetry properties of differential equations is well-established in mathematical physics. Classical methods for studying symmetry properties and their utilisation for finding solutions of partial differential equations were originated in the papers by S. Lie, and developed by modern authors (see e.g. [@yehorchenko:Ovs-eng; @yehorchenko:Olver1; @yehorchenko:Bluman; @Kumei; @book; @yehorchenko:FSS]).
We start our consideration from some symmetry properties and solutions of the nonlinear wave equation $$\label{yehorchenko:nl wave eq}
\Box u = F (u, u^*)$$ for the complex-valued function $u = u(x_0,x_1,\ldots,x_n)$, $x_0 =t$ is the time variable, $x_1,\ldots,x_n$ are $n$ space variables. $F$ is some function. $\Box u $ is the d’Alembert operator $$\Box u = - \frac{\partial^2 u}{\partial x^2_0} + \frac{\partial^2 u }{%
\partial x^2_1} +\cdots + {\frac{\partial^2 u }{\partial x^2_n}}.$$
It is well-known that the equation (\[yehorchenko:nl wave eq\]) may be reduced to a nonlinear Schrödinger equation with the number of space dimensions smaller by 1, when the nonlinearity $F$ has a special form $F=u
f(|u|)$, where $|u|=(uu^*)^{1/2}$, an asterisk designates complex conjugation.
Further we are trying to generalise this relation between the nonlinear wave equation and the nonlinear Schrödinger equation into a relation between differential invariants of the respective invariance algebrae, and introduce new concepts of the reduction of fundamental sets of differential invariants and of conditional differential invariants. Conditional differential invariants may be utilised to describe conditionally invariant equations under certain operators and with the certain conditions, in the same manner as absolute differential invariants of a Lie algebra may be utilised for description of all equations invariant under this algebra.
The concept of non-classical, or conditional symmetry, originated in its various facets in the papers [@yehorchenko:Bluman; @Cole; @yehorchenko:Olver; @Rosenau; @yehorchenko:F; @Tsyfra; @yehorchenko:F; @Zhdanov; @cond; @yehorchenko:Clarkson; @Kruskal; @yehorchenko:Levi; @Winternitz] and later by numerous authors was developed into the theory and a number of algorithms for studying symmetry properties of equations of mathematical physics and for construction of their exact solutions. Here we will use the following definition of the conditional symmetry:
**Definition 1** The equation $F(x,u,{\mathop {u}\limits_1},\ldots ,{%
\mathop
{u}\limits_l})=0$ where ${\mathop {u}\limits_k}$ is the set of all $k\,%
\mathrm{th}$-order partial derivatives of the function $u=(u^1,u^2,\ldots
,u^m)$, is called conditionally invariant under the operator $$Q=\xi ^i(x,u)\partial _{x_i}+\eta ^r(x,u)\partial _{u^r}$$ if there is an additional condition $$G(x,u,{\mathop {u}\limits_1},\ldots ,{\mathop {u}\limits_{l_1}})=0,
\label{yehorchenko:G=0}$$ such that the system of two equations $F=0$, $G=O$ is invariant under the operator $Q$.
If (\[yehorchenko:G=0\]) has the form $G=Qu$, then the equation $F=0$ is called $Q$-conditionally invariant under the operator $Q$.
Differential invariants and description of invariant equations
==============================================================
Differential invariants of Lie algebrae present a powerful tool for studying partial differential equations and construction of their solutions [@yehorchenko:Lie; @yehorchenko:Tresse; @yehorchenko:Olver2].
Now we will present some basic definitions that we will further generalise. For the purpose of these definitions we deal with Lie algebrae consisting of the infinitesimal operators $$\label{yehorchenko:X}
X = \xi^i (x, u) \partial_{x_i} + \eta^r (x, u) \partial_{u^r}.$$ Here $x=(x_1,x_2,\ldots,x_n)$, $u=(u^1,u^2,\ldots,u^m)$.
**Definition 2** The function $F=F(x,u,{\mathop {u}\limits_1},\ldots ,{%
\mathop {u}\limits_l}),$ is called a differential invariant for the Lie algebra $L$ with basis elements $X_i$ of the form (\[yehorchenko:X\]) $%
(L=\langle X_i\rangle )$ if it is an invariant of the $l\,\mathrm{th}$ prolongation of this algebra: $${\mathop {X}\limits_l}_sF(x,u,{\mathop {u}\limits_1},...,{\mathop {u}%
\limits_l})=\lambda _s(x,u,{\mathop {u}\limits_1}...,{\mathop {u}\limits_l}%
)F,$$
where the $\lambda _s$ are some functions; when $\lambda _i=0,F$ is called an absolute invariant; when $\lambda _i\ne 0$, it is a relative invariant.
Further when writing “differential invariant” we would imply “absolute differential invariant”.
**Definition 3** A maximal set of functionally independent invariants of order $r\leq l$ of the Lie algebra $L$ is called a functional basis of the $l\,\mathrm{th}$-order differential invariants for the algebra $L$.
While writing out lists of invariants we shall use the following designations $$\begin{gathered}
u_a \equiv \frac{\partial u}{\partial x_a}, \qquad u_{ab} \equiv \frac{%
\partial^2 u}{\partial x_a \partial x_b}, \qquad S_k (u_{ab}) \equiv
u_{a_1a_2} u_{a_2a_3} \cdots u_{a_{k-1} a_k} u_{a_k a_1}, \nonumber \\
S_{jk} (u_{ab}, v_{ab}) \equiv u_{a_1a_2} \cdots u_{a_{j-1} a_j} v_{a_j
a_{j+1}} \cdots v_{a_k a_1}, \nonumber \\
R_{k}(u_{a},u_{ab})\equiv u_{a_1}u_{a_k}u_{a_1 a_2}u_{a_2 a_3 }\cdots
u_{a_{k-1} a_k}. \label{yehorchenko:SR}\end{gathered}$$
In all the lists of invariants $j$ takes the values from 0 to $k$. We shall not discern the upper and lower indices with respect to summation: for all Latin indices $x_a x_a \equiv x_a x^a \equiv x^a x_a =x_1^2 +x_2^2 +\cdots +
x_n^2$.
Fundamental bases of differential invariants for the standard scalar representations of the Poincaré and Galilei algebra of the types (\[yehorchenko:poin\]), (\[yehorchenko:Galilei1\]) were found in [@yehorchenko:F; @Ye; @DifInvs]. Fundamental bases of differential invariants allow describing all equations invariant under the respective Lie algebrae.
Construction of conditional differential invariants would allow describing all equations, conditionally invariant with respect to certain operators under certain conditions.
**Definition 4** $F=F(x,u,{\mathop {u}\limits_1},\ldots ,{\mathop
{u}\limits_l})$ is called a conditional differential invariant for the operator with $X$ of the form (\[yehorchenko:X\]) if under the condition $$\begin{gathered}
G(x,u,{\mathop {u}\limits_1},\ldots ,{\mathop {u}\limits_{l_1}})=0,
\label{yehorchenko:Gcond} \\
{\mathop {X}\limits_{l_{\max }}}F(x,u,{\mathop {u}\limits_1},\ldots ,{%
\mathop {u}\limits_l})=0,\qquad {\mathop {X}\limits_{l_{\max }}}G(x,u,{%
\mathop {u}\limits_1},\ldots ,{\mathop {u}\limits_{l_1}})=0,
\label{yehorchenko:Xcond}\end{gathered}$$ ${\mathop {X}\limits_{l_{\max }}}$ being the $l_{\max }\,\mathrm{th}$ prolongation of the operator $X$. The order of the prolongation $l_{\max }={%
\max }(l,l_1)$.
Nonlinear wave equation, nonlinear Schrödinger equationand relation between their symmetries
============================================================================================
The Galilei algebra for $n-1$ space dimensions is a subalgebra of the Poincaré algebra for $n$ space dimensions (see e.g. [@yehorchenko:Gomis]) and references therein), and this fact allows reduction of the nonlinear wave equation (\[yehorchenko:nl wave eq\]) to the Schrödinger equation. We will consider the nonlinear wave equations for three space variables, and its symmetry properties in relation to the symmetry properties of the nonlinear Schrödinger equation for two space variables. However, all the results can be easily generalised for arbitrary number of space dimensions.
Reduction of the nonlinear wave equation (\[yehorchenko:nl wave eq\]) to the Schrödinger equation can be performed by means of the ansatz $$\label{yehorchenko:ansatz}
u = \exp((-im/2) (x_0+x_3)) \Phi (x_0-x_3,x_1,x_2).$$
Substitution of the expression (\[yehorchenko:ansatz\]) into (\[yehorchenko:nl wave eq\]) gives the equation $\exp(({-im/2}) (x_0+x_3))
(2im\Phi_{\tau} + \Phi_{11}+ \Phi_{22})= F(u,u^*).$ Here we adopted the following notations: $\tau = x_0+x_3$ is the new time variable, $%
\Phi_{\tau}= \frac{\partial \Phi}{\partial \tau}$, $\Phi_{a}= \frac{\partial
\Phi}{\partial x_a }$, $\Phi_{ab}=\frac{\partial^2 \Phi}{\partial x_a
\partial x_b }.$
Further on we adopt the convention that summation is implied over the repeated indices. If not stated otherwise, small Latin indices run from 1 to 2.
If the nonlinearity in the equation (\[yehorchenko:nl wave eq\]) has the form $F=uf(|u|)$, then it reduces to the Schrödinger equation $$\label{yehorchenko:Schr2}
2im\Phi_{\tau} + \Phi_{11}+ \Phi_{22}= \Phi f (|\Phi|).$$
Such reduction allowed construction of numerous new solutions for the nonlinear wave equation by means of the solutions of a nonlinear Schrödinger equation [@yehorchenko:Basarab2; @yehorchenko:Basarab1]. We show that this reduction allowed also to describe additional symmetry properties for the equation (\[yehorchenko:nl wave eq\]), related to the symmetry properties of the equation (\[yehorchenko:Schr2\]).
Lie symmetry of the equation (\[yehorchenko:Schr2\]) was described in [@yehorchenko:FMosk; @yehorchenko:W1]. With an arbitrary function $f$ it is invariant under the Galilei algebra with basis operators $$\begin{gathered}
\partial_\tau =\frac{\partial}{\partial \tau}, \qquad \partial_a = \frac{%
\partial}{\partial x_a}, \qquad J_{12} =x_1\partial_2 - x_2 \partial_1,
\nonumber \\
G_a =t\partial_a +ix_a (\Phi \partial_\Phi - \Phi^* \partial_{\Phi^*}) \quad
(a=1,2), \qquad J =(\Phi \partial_\Phi - \Phi^* \partial_{\Phi^*}).
\label{yehorchenko:Galilei1}\end{gathered}$$
When $f = \lambda |u|^2$, where $\lambda$ is an arbitrary constant, the equation (\[yehorchenko:Schr2\]) is invariant under the extended Galilei algebra that contains besides the operators (\[yehorchenko:Galilei1\]) also the dilation operator $$\label{yehorchenko:D}
D = 2\tau \partial_\tau + x_a \partial_a - I,$$ where $I = \Phi\partial_\Phi + \Phi^* \partial_{\Phi^*}$, and the projective operator $$\label{yehorchenko:A}
A= \tau^2 \partial_\tau + \tau x_a \partial_a +{\frac{im }{2}}x_a x_a J-
\tau I.$$ Lie reductions and families of exact solutions for multidimensional nonlinear Schrödinger equations were found at [@yehorchenko:FSerov; @Schr; @yehorchenko:W1; @yehorchenko:W2; @yehorchenko:W3; @yehorchenko:W4; @yehorchenko:W5]. Note that the ansatz (\[yehorchenko:ansatz\]) is the general solution of the equation $$\label{yehorchenko:condition}
u_0 + u_3 + imu=0.$$
We can regard the equation (\[yehorchenko:condition\]) as the additional condition imposed on the nonlinear wave equation with the nonlinearity $%
F=\lambda u|u|^2$. Solution of the resulting system $$\label{yehorchenko:nl wave2}
\Box u = \lambda u|u|^2,$$ with the equation (\[yehorchenko:condition\]) would allow to reduce number of independent variables by one, and obtain the same reduced equation, invariant under the extended Galilei algebra with the projective operator. This allows establishing conditional invariance of the nonlinear wave equation (\[yehorchenko:nl wave2\]) under the projective operator. It is well-known that it is not invariant under this operator in the Lie sense.
The maximal invariance algebra of the equation (\[yehorchenko:nl wave eq\]) that may be found according to the Lie algorithm (see e.g. [@yehorchenko:Ovs-eng; @yehorchenko:Olver1; @yehorchenko:Bluman; @Kumei; @book; @yehorchenko:FSS]) is defined by the following basis operators: $$\label{yehorchenko:poin}
p_{\mu} = ig_{\mu \nu} {\frac{\partial }{\partial x_{\nu}}}, \qquad J_{\mu
\nu} = x_{\mu} p_{\nu} -x_{\nu} p_{\mu},$$ where $\mu, \nu$ take the values $0,1,\ldots,3$; the summation is implied over the repeated indices (if they are small Greek letters) in the following way: $x_{\nu } x_{\nu} \equiv x_{\nu} x^{\nu} \equiv x^{\nu} x_{\nu} =x_0^2
-x_1^2 - \cdots - x_n^2$, $g_{\mu \nu }= \mathrm{diag}\,(1,-1,\ldots,-1). $
However, summation for all derivatives of the function $u$ is assumed as follows: $u_{\nu } u_{\nu} \equiv u_{\nu} u^{\nu} \equiv u^{\nu} u_{\nu}
=-u_0^2 +u_1^2 + \cdots + u_n^2$.
Unlike the standard convention on summation of the repeated upper and lower indices we consider $x_{\nu}$ and $x^{\nu}$ equal with respect to summation not to mix signs of derivatives and numbers of functions.
**Theorem 1** The nonlinear wave equation (\[yehorchenko:nl wave2\]) is conditionally invariant with the condition (\[yehorchenko:condition\]) under the projective operator $$\begin{gathered}
A_1=\frac 12(x_0-x_3)^2(\partial _0-\partial _3)+(x_0-x_3)(x_1\partial
_1+x_2\partial _2) \nonumber \\
\qquad {}+\frac{imx^2}2(u\partial _u-u^{*}\partial _{u^{*}})+\frac{n-1}%
2(x_0-x_3)(u\partial _u+u^{*}\partial _{u^{*}}). \label{yehorchenko:A1}\end{gathered}$$
To prove Theorem 1 it is sufficient to show that the system (\[yehorchenko:nl wave2\]), (\[yehorchenko:condition\]) is invariant under the operator (\[yehorchenko:A1\]) by means of the classical Lie algorithm.
Our further study aims at construction of other Poincaré-invariant equations possessing the same conditional invariance property.
Example: construction of conditional differential invariants
============================================================
Now we adduce fundamental bases of differential invariants that will be utilised for construction of our example of conditional differential invariants.
First we present a functional basis of differential invariants for the Poincaré algebra (\[yehorchenko:poin\]) of the second order for the complex-valued scalar function $u = u (x_0,x_1,\ldots,x_3).$ It consists of 24 invariants $$\label{yehorchenko:poin inv}
u^r,\qquad R_k\left(u^r_{\mu}, u^1_{\mu\nu} \right),\qquad S_{jk}
\left(u^r_{\mu\nu}, u^1_{\mu\nu}\right).$$
In (\[yehorchenko:poin inv\]) everywhere $k=1,\ldots,4$; $j=0,\ldots,k$. A functional basis of differential invariants for the Galilei algebra (\[yehorchenko:Galilei1\]), mass $m \ne 0$, of the second order for the complex-valued scalar function $\Phi = \Phi (\tau,x_1,\ldots,x_2)$ consists of 16 invariants.
For simplification of the expressions for differential invariants we introduced the following notations: $$\Phi = \exp\phi, \qquad \mathrm{Im}\, \Phi = \arctan \frac{\mathrm{Re}\,\phi%
}{\mathrm{Im}\,\phi}\,.$$
The elements of the functional basis may be chosen as follows: $$\begin{gathered}
\phi + \phi^*, \quad M_1 = 2im \phi_{t} + \phi_{a} \phi_{a},\quad M_1^* ,
\quad M_2 = - m^2 \phi_{tt} + 2im \phi_{a}\phi_{at} + \phi_{a}\phi_{b}
\phi_{ab}, \quad M_2^* , \nonumber \\
S_{jk} (\phi_{ab},\phi_{ab}^*), \quad R^1_j = R_j (\theta_a,\phi_{ab}),
\quad R^2_j = R_j (\theta_a^*,\phi_{ab}),\quad R^3_j = R_j
(\phi_a+\phi_a^*,\phi_{ab}). \label{yehorchenko:Galilei1 inv}\end{gathered}$$ Here $\theta_a = im \phi_{at} + \phi_{a} \phi_{ab}, \phi_{ab}$ are covariant tensors for the Galilei algebra.
A functional basis of differential invariants for the Galilei algebra (\[yehorchenko:Galilei1\]) extended by the dilation operator (\[yehorchenko:D\]) and the projective operator (\[yehorchenko:A\]) may be chosen as follows: $$\begin{gathered}
N_{1}e^{-2(\phi +\phi ^{\ast })},\quad \frac{N_{1}}{N_{1}^{\ast }},\quad
\frac{N_{2}}{N_{1}^{2}},\quad \frac{N_{2}^{\ast }}{(N_{1}^{\ast })^{2}}%
,\quad S_{jk}(\rho _{ab},\rho _{ab}^{\ast }),\quad R_{j}(\rho _{a},\rho
_{ab}), \nonumber \\
R_{j}(\rho _{a}^{\ast },\rho _{ab}),\quad R_{j}(\phi _{a}+\phi _{a}^{\ast
},\rho _{ab})N_{1}^{-1},\quad (\phi _{aa}+\phi _{aa}^{\ast })N_{1}^{-1},
\label{yehorchenko:Galilei2 inv}\end{gathered}$$ where $$N_{1}=M_{1}+\phi _{aa}=2im\phi _{t}+\phi _{aa}+\phi _{a}\phi _{a},\qquad
N_{2}=\frac{1}{n}\phi _{aa}N_{1}+\frac{\phi _{aa}^{2}}{2n}+M_{2}$$ and the covariant tensors have the form $$\rho _{a}=\theta _{a}N_{1}^{-3/2},\qquad \rho _{ab}=\left( \phi _{ab}-\frac{%
\delta _{ab}}{n}\phi _{cc}\right) N_{1}^{-1}.$$
An algorithm for construction of conditional differential invariants may be derived directly from the Definition 4. Such invariants may be found by means of the solution of the system (\[yehorchenko:Gcond\]), (\[yehorchenko:Xcond\]).
We can construct conditional differential invariants of the Poincaré algebra (\[yehorchenko:poin\]) and the projective operator (\[yehorchenko:A1\]) solving the system $${\mathop {A_1}\limits_{2}}F(\mathrm{Inv}_{P})=0,\qquad u_{0}+u_{3}+imu=0,$$ where $\mathrm{Inv}_{P}$ are all differential invariants (\[yehorchenko:poin inv\]) of the Poincaré algebra (\[yehorchenko:poin\]). Using the fact that the ansatz (\[yehorchenko:ansatz\]) is the general solution of the additional condition (\[yehorchenko:condition\]), we can directly substitute this ansatz into differential invariants (\[yehorchenko:poin inv\]). To avoid cumbersome formulae here we did not list expressions for all differential invariants from (\[yehorchenko:poin inv\]).
The expression $\Box u $ transforms into the following: $$\Box u = u (2im \phi_{\tau}+ \phi_{aa} + \phi_{a} \phi_{a}) ,$$ where $N_1$ is an expression entering into expression for differential invariants (\[yehorchenko:Galilei1 inv\]). Further we get $$\begin{gathered}
u_{\mu} u_{\mu}= u^2 ( 2im \phi_t + \phi_{a} \phi_{a}), \nonumber \\
u_{\mu} u_{\nu} u_{\mu \nu} = u^3 (\phi_{a}\phi_{b} \phi_{ab}+ (\phi_{a}
\phi_{a})^2 - m^2 (\phi_{tt} +4 \phi_t^2) + \phi_{a}\phi_{b} \phi_{ab}+
(\phi_{a} \phi_{a})^2 \nonumber \\
\phantom{u_{\mu} u_{\nu} u_{\mu \nu} =}{} - m^2 (\phi_{tt} +4 \phi_t^2) +
2im \phi_{a}\phi_{at} + 4im \phi_t \phi_{a} \phi_{a}),
\label{yehorchenko:reduced basis}\end{gathered}$$ Substituting the ansatz (\[yehorchenko:ansatz\]) to all elements of the fundamental basis (\[yehorchenko:poin inv\]) of second-order differential invariants of the Poincaré algebra similarly to (\[yehorchenko:reduced basis\]), we can obtain reduced basis of differential invariants, that may be used for construction of all equations reducible by means of this ansatz. We can give the following representation of the Poincaré invariants using expressions $M_k$ (\[yehorchenko:Galilei1 inv\]) and $N_k$ (\[yehorchenko:Galilei2 inv\]), where in the expressions for $M_k$, $N_k$ $%
(k=1,2)$ time variable is $\tau=x_0-x_3$: $$\begin{gathered}
\Box u = u N_1,\quad u_{\mu} u_{\mu} = u^2 M_1, \quad u_{\mu} u_{\nu} u_{\mu
\nu} = u^3 \left(M_2 + M_1^2\right), \nonumber \\
u_{\mu \nu} u_{\mu \nu} = u^2 \left( 2M_2 + M_1^2 +
\phi_{ab}\phi_{ab}\right), \nonumber \\
u_{\mu} u^*_{\mu}=\frac{uu^*}{2}\left(M_1+M^*_1-\left(\phi_{a}+\phi^*_{a}%
\right) \left(\phi_{a}+\phi^*_{a}\right)\right).\end{gathered}$$ Here $a$, $b$ take values from 1 to 2.
Whence $$\begin{gathered}
M_1 = u_{\mu} u_{\mu} u^{-2}, \quad \phi_{aa} = N_1 - M_1 = \frac{u \Box u -
u_{\mu} u_{\mu}}{u^2}, \nonumber \\
M_2 = u_{\mu} u_{\nu} u_{\mu \nu} u^{-3} - (u_{\mu} u_{\mu})^2 u^{-4}, \quad
N_1 = \frac{\Box u }{u}, \nonumber \\
N_2 = \frac{1}{n}\phi_{aa} N_1 + \frac{\phi_{aa}^2}{2n} + M_2 = u_{\mu}
u_{\nu} u_{\mu \nu} u^{-3} - (u_{\mu} u_{\mu})^2 u^{-4} \nonumber \\
\qquad {}+\frac{1}{n} \frac{\Box u}{u} \frac{u \Box u - u_{\mu} u_{\mu}}{u^2}
+ \frac{1}{2 n} \frac{(u \Box u - u_{\mu} u_{\mu})^2}{u^4}, \nonumber \\
R_1 (\phi_{a} + \phi_{a}^*, \rho_{ab}) N_1^{-1}=(\phi_{a} +
\phi_{a}^*)(\phi_{a} + \phi_{a}^*) N_1^{-1} \nonumber \\
\qquad =\left(N_1+N_1^*-{\frac{2 }{uu^*}} u_{\mu} u^*_{\mu}\right)N_1^{-1}=%
\frac{u^*\Box u +u\Box u^*-2u_{\mu} u^*_{\mu}}{u^*\Box u }.\end{gathered}$$
We construct Poincaré-invariant conditional differential invariants of the projective operator (\[yehorchenko:A1\]) under the condition (\[yehorchenko:condition\]) using differential invariants (\[yehorchenko:Galilei1 inv\]) $$\begin{gathered}
I_1 = N_1 e^{- 2(\phi + \phi^*)} = \frac{ \Box u}{u(uu^*)^2}, \quad I_2 =
\frac{N_1}{N_1^* } = \frac{ u^* \Box u}{u \Box u^*}, \nonumber \\
I_3 = \frac{N_2}{N_1^2 } = \left(u u_{\mu} u_{\nu} u_{\mu \nu} + \frac{3}{2 n%
} u^2 (\Box u)^2 + \left(\frac{1}{2n}-1\right) (u_{\mu} u_{\mu})^2 -\frac{2}{%
n} u \Box u(u_{\mu} u_{\mu})\right)\! {\left(u^2 (\Box u)^2\right)}^{-1},
\nonumber \\
I_4 =R_1 (\phi_{a} + \phi_{a}^*, \rho_{ab}) N_1^{-1}=\frac{u^*\Box u +u\Box
u^*-2u_{\mu} u^*_{\mu}}{u^*\Box u }.\end{gathered}$$
Whence, we may state that all equations of the form $F(I_1, I_2, I_3, I_4) =
0 $ are conditionally invariant with respect to the operator $A_1$ (\[yehorchenko:A1\]) with the additional condition (\[yehorchenko:condition\]).
Finding similar representations for all elements of the functional basis (\[yehorchenko:Galilei1 inv\]) of the second-order differential invariants of the Galilei algebra (\[yehorchenko:Galilei1\]) extended by the dilation operator (\[yehorchenko:D\]) and the projective operator (\[yehorchenko:A\]), we can construct functional basis of conditional differential operators. Such basis would allow to describe all Poincaré-invariant equations for the scalar complex-valued functions that are conditionally invariant under the operator $A_1$ (\[yehorchenko:A1\]).
Conclusion
==========
The procedure for finding conditional differential invariants outlined above may be used for other cases when the additional condition (\[yehorchenko:Gcond\]) has the general solution that may be used as ansatz, and when a functional basis of the operator (\[yehorchenko:Xcond\]) in the variables involved in such reduction is already known.
Besides finding new conditionally invariant equations, further developments of the ideas presented in this paper may be description of all equations reducible by means of a certain ansatz, and search of methods for restoration of original equations from the reduced equations.
The symmetry of the nonlinear wave equation discussed in the paper may also be interpreted as a hidden symmetry arising as symmetry of the reduced equation. Thus the method described (construction of conditional differential invariants) may also be used for description of equations possessing hidden symmetry (see e.g. [@yehorchenko:Abraham]).
[99]{}
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---
abstract: 'The proof identity problem asks: When are two proofs the same? The question naturally occurs when one reflects on mathematical practice. The problem understandably can be seen as a challenge for mathematical logic, and indeed various perspectives on the problem can be found in the proof theory literature. From the proof theory perspective, the challenge is met by laying down new calculi that eliminate “bureaucracy”; techniques such as normalization and cut-elimination, as well as proof compression, are employed. In this note a new perspective on the proof identity problem is outlined. The new approach employs the concepts and tools of automated theorem proving and complements the rather more theoretical perspectives coming from pure proof theory. The practical approach is illustrated with experiments coming from the TPTP Problem Library.'
author:
- 'Jesse Alama[^1]'
bibliography:
- 'proof-analysis.bib'
title: Proof identity for mere mortals
---
Introduction {#sec:introduction}
============
The proof identity problem asks: *When are two proofs the same?* The question has undoubtedly occurred to anyone who has reflected on mathematical practice. Generally, having multiple proofs of a result is generally regarded as valuable. (For a philosophical perspective, see [@hersh1997prove].) Some results in mathematics receive considerable scrutiny, in the sense that they are given many different proofs. Consider, for instance, Carl Gauss’ many proofs of the quadratic reciprocity theorem in number theory [@lemmermeyer2000reciprocity][^2], or the diversity of proofs of the Pythagorean theorem [^3]. If one teaches mathematics, the experience of grading students’ work is often that one finds that successful solutions to mathematical problems can be clusters: some students seem to solve the problem one way, others another way. A new proof may even be a strong desideratum, such as the case of the four-color theorem, which for a few decades was widely regarded as correct but proved in a discomfiting way [@gonthier2008four].
But what is a “way of proving”? The question is perhaps not well-posed. Mathematical logic gives us some tools for making the problem somewhat more concrete (e.g., the completeness theorem for first-order logic, the wide catalog of available calculi for formally proving theorems). But the underlying notion of identity of proof, even in a formal setting, seems to be somewhat slippery; the proof identity problem might is perhaps just a proxy for the whole field of proof theory.
Perhaps the problem can be addressed indirectly: when is one proof simpler than another? One might perhaps be interested in the simplest proof of a theorem under certain conditions. Here we can take some comfort in the presence of company. Hilbert’s (previously unknown) 24th problem [@thiele2003hilbert] addresses precisely this issue. Already in 1900 Hilbert was thinking about the difficulty:
> The 24th problem in my Paris lecture \[the presentation in 1900 at the International Congress of Mathematicians where Hilbert’s famous problem list was given\] was to be: Criteria of simplicity, or proof of the greatest simplicity of certain proofs. Develop a theory of the method of proof in mathematics in general. Under a given set of conditions there can be but one simplest proof. Quite generally, if there are two proofs for a theorem, you must keep going until you have derived each from the other, or until it becomes quite evident what variant conditions (and aids) have been used in the two proofs. Given two routes, it is not right to take either of these two or to look for a third; it is necessary to investigate the area lying between the two routes.
Hilbert can be considered one of the central figures in the history of proof theory; his student Gentzen gave us natural deduction and sequent calculus, two formalism that surely cannot be ignored in any discussion of the proof identity problem. See [@prawitz2000ideas] for a readable perspective on some central results in proof theory such as normalization in natural deduction and cut-elimination for sequent calculi. In recent years there has been an interest in pushing cut-elimination, which customarily is understood as not applying to non-logical axiom systems (in other words, in general cut-elimination fails as soon as one postulates a non-logical axiom), to the analysis of axiom systems [@negri2008structural; @negri2013proof].
We are thus interested in the problem, loosely understood, of what it means for two proofs to be the same. Hilbert’s suggested approach, maximizing simplicity (however understood), might be followed. Can we make these terms more precise?
At first blush, one immediately faces a difficulty: the *Multiplicity of Proof* problem:
> If there is one proof of a theorem, then there is (probably) another.
When one looks at proofs through a formal lens, one generally encounters a somewhat sharper phenomenon, which we might call the *Infinitude of Proof* problem:
> If there is one proof of a theorem, then there are (in fact) infinitely many.
Thus, consider a Hilbert-style calculus, a formula $\phi$ and a set $X$ of axioms, and a derivation $d$ (a certain kind of sequence) of $\phi$ from $X$. The derivation $d$ witnesses the derivability (which we identify with provability) of $\phi$ from $X$. Monotonicity allows us to add new, unused premises to $X$, simply prepending them to $d$. If one objects to adding new proper axioms, one can simply take new logical axioms (or instances thereof) and prepend them to $d$. We thereby obtain, apparently, a new derivation of $\phi$. In $d$ we may be able to permute certain terms, thereby yielding another (?) derivation of $\phi$ from $X$. No end seems to be in sight.
Combating such multiplicity is a genuine problem. Such “attacks” on the notion of “proof of $\phi$ from premises $X$” can, in part, be addressed through trivial requirements such as:
- ensuring that all members of $X$ are actually used somewhere in the proof (no spurious premises)
- ensuring that from the conclusion there is a path back to all axioms (a stronger form of the first condition)
- permitting no duplication (it is not necessary to have two occurrences of one and the same formula in a derivation)
- identifying derivations that are the same up to a permutation
But such structural conditions are just the tip of the iceberg. One can derive $\phi$ from $\phi \wedge \phi$, and from $\phi \vee \vee$; what if the latter are postulated as axioms? Such conundrums presumably ought to be addressed. One could object that no proof depends on such tricks. But if we want a robust notion of redundancy, we presumably have to address these kinds of obstacles.
To combat such maddening multiplicity, one naturally wants to define and employ a notion of redundancy. Two derivations could be identified if they are the same up to the notion of redundancy. One can employ various syntactic and semantic notions to help define such notions of redundancy.
In two other models of mathematical proof, sequent calculus and natural deduction, the situation seems to be no different. The “if there is one, there are infinitely many” phenomenon also seems to hold sway. One can define notions of normalization. But then one is boxed in from another direction: if we employ normalization of derivations, how do we avoid the problem that theorems are always proved in one way? This also seems intuitively unacceptable. We don’t want to rule out the multiplicity of proofs (or at least, the potential multiplicity of proofs) by identifying all proofs.
Going the other way around, one can design normalization procedures that identify proofs. But then one encounters the notion of normalization in proof theory. Here, the interest is in defining suitable reduction relations among derivations and proving that an arbitrary derivation can be put into a normal form. Even further, one finds interest in reducing derivations to a unique normal form (strong normalization). Such an approach witnesses the following *Canonicity Principle*:
> Every theorem has a canonical proof.
On the theoretical side, there is a complexity problem: normalization and cut-elimination often lead to tremendous blowup. Starting with a “moderately” small natural deduction derivation, one may very well find that its normal form is so large as to be unworkable:
> Faced with two concrete proofs in mathematics—for example, two proofs of the Theorem of Pythagoras, or something more involved—it could seem pretty hopeless to try to decide whether they are identical just armed with the Normalization Conjecture or the Generality Conjecture. But this hopelessness might just be the hopelessness of formalization. We are overwhelmed not by complicated principles, but by sheer quantity. [@Dos03]
(The two conjectures mentioned here, the Normalization Conjecture and the Generality Conjecture, are proposals for addressing the proof identity problem.) Indeed, normalized proofs are often quite large. (Nonetheless, the situation is not always hopeless [@baaz2011methods].) One can design alternative calculi with the hope of cutting down on the tedious “bureaucracy” that generally accompanies any non-trivial formalization effort [@Gugl:05:The-Prob:nu].
Thus it seems that “theory”, that is, pure mathematical logic (proof theory), offers few genuinely practical solutions to the proof identity problem. Sure, one can implement these algorithms, but if one is interested in mathematics (as opposed to just pure logic\[s\]), we need to tamper our expectations. We seek tools that we can apply to many examples, we want to get our hands on normalized derivations, but it seems that the paths down that road are prohibitively expensive. Knuth put it best:
> Of course, computers are not infinitely fast, and our expectations have become inflated even faster than our computational capabilities. We are forced to realize that there are limits beyond which we cannot go. The numbers we can deal with a not only finite, they are very finite, and we do not have the time or space to solve certain problems even with the aid of the fastest computers. [@knuth1976coping]
In this paper another perspective is taken. Towards addressing Hilbert’s 24th problem, we eschew the solutions coming from pure proof theory and turn toward automated theorem provers. Our perspective is not entirely new; L. Wos, for instance, has discussed Hilbert’s 24th problem in the context of automated theorem proving for some time [@wos2003missing; @thiele2002hilbert]
> As we understand more fully the range of mechanical mathematics, we get a clearer view of the relation between complexity and conceptual difficulty in mathematics, since we would probably wish to say that mechanizable pieces, even when highly complex, are conceptually easy. When alternative proofs are fully mechanizable, we obtain also a quantitative measure of the simplicity of mathematical proofs, to supplement our vaguer but richer intuitive concept of simplicity. With the increasing power to formalize and mechanize, we are freed from tedious details and can more effectively survey the content and conceptual core of a mathematical proof. [@wang1963mechanical]
Wang’s programmatic comment was made in 1963, when the field of automated theorem was very young and when computers were also fantastically weaker than they are today. Thanks to decades of work and early efforts to develop practical solutions in this field, we can safely say that Wang’s programmatic comment is more approachable today than it was when he made it.
An important contribution of mathematical logic the design of various metrics for formal proofs, such as Herbrand complexity [@leitsch1997resolution] and their relation to other metrics, such as cut-complexity. Intuitively, Herbrand complexity provides a calculus-independent measure of the complexity of a theorem, and certain proofs of the theorem can come more or less close to witnessing this minimal complexity.
So if we do not accept every theorem has exactly one proof, but we want to turn away from the infinity of proofs that mathematical logic tells us always exist, then how can we proceed? What is the *practical* proof identity problem? Or: what is an appropriate notion of proof such that one can tackle the proof identity problem in a practical way? We want to investigate theorems. The “space” of theorems shouldn’t be infinite, but it shouldn’t be too big, either. Mathematical logic does not give us the answer we want. Where can we turn?
Measurements on refutations {#sec:counting}
===========================
If one is interested in the proof identity problem, one might take notice of the following extract from the manual:
> **Multiple Proofs**
>
> `assign(max_proofs, n). % default n=1, range [-1 .. INT_MAX]`
>
> This parameter tells Prover9 to stop searching when the $n$-th proof has been found.
From a theoretical standpoint, a proof theorist interested in the proof identity problem ought to be stunned by such a statement. The immediate question is: what is the notion of proof employed by ? How can it distinguish between different proofs? Is it at all practical to enumerate different proofs? By what criteria would one know when to stop the enumeration? Aren’t there always infinitely many proofs, if there is even one proof?
The answer is that uses a standard set-of-support search strategy in (essentially) a resolution calculus [@leitsch1997resolution]. From a theoretical view, the calculus is quite simple. As has been standard for decades, to derive $\phi$ from axioms $X$, one considers the clause logic analogue of $X \cup \{ \neg \phi \}$ and attempts to derive a contradiction.
But from the theoretical point of view, this is not yet an answer; the proof identity problem is not solved by (1) switching to clause logic and (2) reducing the arbitrary proof-finding problem to the problem of finding a refutation. In this setting, one can replay the intuitive arguments given in the previous section for why—in the absence of special assumptions—there ought to exist infinitely resolution proofs.
The notion of proof at play behind and similar systems is, however, informed by decades of experience dealing with *notions of redundancy* [@DBLP:books/el/RV01/BachmairG01]. The point is that various “rabbit holes” that theoretically exist can be avoided in this setting [@DBLP:books/el/RV01/Weidenbach01]. The goal is often to find a refutation by generating clauses (that is, drawing inferences). In my automated theorem proving systems based on resolution or related clause-based calculi, by default, when a contradiction (the empty clause) is derived, the search terminates. The principal problem for automated theorem proving is to determine the existence of at least one proof. Derivability is already a very hard problem, confirmed by decades of experience of many researchers.
But if we can find one proof, why not keep going? Just keep deriving inferences until no more inferences can be drawn. Every time the empty clause is derived, simply output the path by which we got to this particular occurrence of the empty clause.
Experiments {#sec:experiments}
===========
We now proceed to consider a variety of specific test cases on which to explore Hilbert’s 24th problem and the attendant proof identity problem. We consider four proof-finding problems coming from the TPTP library [@DBLP:journals/jar/Sutcliffe09]. To facilitate our exploration, we implemented a custom tool that we call . is not a new proof search tool; it is (like [@kinyon2013loops]) essentially just a wrapper around that accomplishes, in one fell swoop, what can easily be done by hand with the suite.
Partition of monoids {#sec:partition}
--------------------
We now consider a problem about partitioning monoids ([[`ALG011-1`]({http://www.cs.miami.edu/~tptp/cgi-bin/SeeTPTP?Category=Problems&Domain=GRA&File={ALG011-1}.p})]{}). We are to show that, if a monoid $M$ is partitioned into (nonempty) subsets $C$ and $D$, it is not possible that both $C \times C \subseteq D$ and $D \times D \subseteq C$.
For this problem, in its automatic mode is able to produce $374$ proofs before running out of possible inferences. This number is obviously quite large, and it is quite unlikely that these proofs are equal interest. But we find that there are $2$ proofs having the minimal length $20$. (The longest proof has length 44.) If we look at these two proofs, one features conclusions (clauses):
- `c(f(A,B)) | -d(A) | c(B)`
- `c(f(a1,f(a1,A))) | c(A)`
where the binary function symbol `f` is the multiplication of the monoid, the unary predicates `c` and `d` represent the partition of the monoid, and the constants `a1` and `a2` are witnesses to the nonemptiness of the partitions (Prolog-style variables are being used, so `A` and `B` are variables). The other minimal length derivation draws the inferences:
- `c(f(a1,f(a1,A))) | -d(A)`
- `-d(f(a1,a2))`
All other clauses in the two derivations are shared. The question we now face is whether these two “combinatorially distinct” derivations are materially different. Notice that the second features a (negative) literal, but the distinctive two inferences drawn in the first proof are both non-literals. This particular problem has the effect that in it `c` and `d` are opposites of one another. Thus, the second distinctive clause of the first proof is, in effect, the same as the first distinctive clause of the first proof. With this understanding, these two proofs are therefore probably the same and we are have one minimal (length) proof.
Tarski-Knaster fixed-point theorem {#sec:tk}
----------------------------------
[[`LAT381+1`]({http://www.cs.miami.edu/~tptp/cgi-bin/SeeTPTP?Category=Problems&Domain=GRA&File={LAT381+1}.p})]{} is a formalization of a Tarski-Knaster result, asserting the uniqueness of suprema in the lattice of subsets of a set. The result is generated from another development by the system [@VLP07].
For this problem, 8 proofs can be fairly quickly found. There may a 9th proof; it appears that hundreds of thousands of given clauses need to be generated before a ninth proof is found.
Among the 8 proofs, the minimal length is 30, and there are 2 proofs having this length and the maximum length is 45 (and there are 4 such proofs). They differ only in the way they derive the empty clause. In other words, the two proofs are identical up to their final steps. If one inspects the precise premises involved in different steps (that is, the premises of the final rule application(s) of the two proofs), they are exactly the same. A difference can be detected, then, only if one were to use, say, ’s expand command to rewrite these proofs in an even more explicit manner than ’s proof objects. One could argue, then, that there is in fact only one shortest proof here.
One can, however, improve on the result by eliminating redundant premises. As mentioned, this particular problem is the result of translating an interactively developed proof. A typical feature of such translations is that numerous additional premises are present in the problem. This does not reflect laziness or disregard for the usual preference in mathematics and logic for parsimonious assumptions; rather, it reflects in the first place the desire to maintain completeness when one translates a development in one logic (or language) into the language of pure FOL (see, e.g., [@DBLP:conf/birthday/UrbanV13; @DBLP:conf/lpar/AlamaKU12]). In short, the translation problem is in focus, not the minimality of the translated premises (which raises its own issues, e.g., the non-uniqueness of a minimal set).
An inspection of [[`LAT381+1`]({http://www.cs.miami.edu/~tptp/cgi-bin/SeeTPTP?Category=Problems&Domain=GRA&File={LAT381+1}.p})]{} with the premise-minimization tool [@alama2012tipi] finds that, of the 16 available premises, a (semantically minimal) subset of 6 can be found that suffice to derive the conjecture. When is applied to the minimized problem, one obtains a sharper result. One can now unqualifiedly show that only $5$ proofs exist (as opposed to making the qualified claim that only 8 could be found). The “space” of possible proof lengths, however, is shifted while at the same time narrower. Thus, the minimum length of the proofs increases but the maximum length decreases: the shortest proof now has length 28 (and there are 3 such proofs), whereas the longest proof has length 29 (and there are 2 such proofs). 27 clauses are held common to all 5 derivations, which gives reason to believe that the differences between the 5 possible proofs are likely quite small. Our results are reminiscent of the approach taken when following what Wos calls cramming [@wos2003cramming].
Chinese remainder theorem {#sec:chinese}
-------------------------
The problem ([[`RNG126+1`]({http://www.cs.miami.edu/~tptp/cgi-bin/SeeTPTP?Category=Problems&Domain=GRA&File={RNG126+1}.p})]{}) is to show a kind of Chinese remainder theorem for rings. As with the Tarski-Knaster-like theorem discussed in Section \[sec:tk\], this problem was generated by the system. Similarly, the problem has a few redundant premises whose removal can give us a different view of the problem. Thus, the problem has 47 premises; a minimal version has only 9 premises.
When working with the original version, we seem to have to impose some limits that might affect the completeness of our method. This is understandable in light of the fact that this is a rather hard problem (its rating in the 6.0.0 version of the TPTP library is 0.70 [@DBLP:journals/aicom/Sutcliffe13], which means that most systems were not able to solve it within the limits of the CASC competition).
If we turn to the minimized version, we are able to relax the limits and let explore the search space with fewer constraints. In the original problem, we found only one proof (presumably there are more, but the nature of the problem and limits of time and space prevent us from doing a complete investigation). Interestingly, only two proofs of the minimized problem are available. One has length 38, the other has length 51. Interestingly, 37 clauses (counting both input and derived clauses) are shared between the two proofs. Since the first has 38 total clauses—including the empty clause—the second proof can be said to be, in effect, an extension of the first but following a different line of reasoning.
Discussion {#sec:discussion}
==========
Limitations of our proof analysis methods should be kept in mind. The experiments were more or less arbitrarily chosen. We chose problems that were, first of all, solvable—fairly quickly (less than a minute)—by . Obviously, if cannot solve a problem, then the kind of proof analysis that is in focus here is, of course, not applicable.
As we saw in Section \[sec:tk\] and \[sec:chinese\], the presence of redundant axioms in a problem influences our analysis. When redundant premises are present, shorter proofs can be found, but the number of proofs can increase, perhaps even tipping the scales so much that infinitely many proofs become available. In such a situation, we have to impose limits on the search (such as on term complexity, the number of clauses to be kept in the set of support, etc.) and cope with the attendant incompleteness of the analysis (that is, accepting the possibility that there may be proofs “beyond the horizon”). If one preprocesses a proof-finding problem by choosing a minimal set of premises, the result is that the length of all proofs might increase (as we saw in Section \[sec:chinese\]), but we might be able to achieve greater satisfaction because the overall search might genuinely “bottom out” (as we saw in Section \[sec:tk\]) by removing various “rabbit holes” that might go down. (Not all rabbit holes can always be eliminated.)
Conclusion {#sec:conclusion}
==========
Towards solving the proof identity problem, we have pointed out that a practical (albeit partial) solution to the problem may be staring us in the face, begging for recognition. If we are dealing with proofs that can be faithfully represented as a finite set of axioms in clause logic, then the door is open (or, to be more modest, may not be closed) to an investigation of the “space” of possible proofs (solutions to the proof-finding problem) that Hilbert suggests.
As often happens in theorem proving, totally automated, universally-applicable methods seem to be elusive, but that doesn’t mean we are justified in shying away from them. More specifically, our approach here relies on the ability of a theorem prover to saturate a set of clauses (in effect, to draw all possible inferences from the clauses); but doing so is, in general, theoretically impossible. We carried out experiments in the TPTP library showing that, in at least some cases, we can practically enumerate all proofs, a result having theoretical interest and which permits concrete applications of “proof analysis”.
Dožen concludes his discussion of two natural deduction-based approaches to the proof identity problem by emphasizing that a serious investigation of the issue requires us to shift perspective:
> The question we have discussed here \[the proof identity problem\] suggests a perspective in logic—or perhaps we may say a dimension—that has not been explored enough. Logicians were, and still are, interested mostly in provability, and not in proofs. This is so even in proof theory. When we address the question of identity of proofs we have certainly left the realm of provability, and entered into the realm of proofs.
Moving into the realm of proofs (as opposed to staying in the realm of provability) requires, it would seem, a new set of skills. The question to which we kept returning—are these two proofs materially different?—needs to be tackled afresh every time it is asked. Even with the heavy lifting being done by an automated theorem prover, we still lack good criteria for going over the finish line and giving a robust answer to the proof identity problem. Perhaps there are no universally applicable methods. In Section \[sec:partial\], for instance, we found two distinct minimal-length proofs but argued in the end that they seem to be the same proof. The argument rested on a symmetry that is built-in to the problem. The argument is admittedly not conclusive. Even if one were to be persuaded by such reasoning, one naturally asks: could this symmetry be detected mechanically?
Echoing Dožen, one might make a parallel remark that in automated theorem proving, at least among those systems that aim to find proofs, the principle focus is derivability, and not the exploration of the space of proofs. This is an understandable focus, given the sheer difficulty of the proof-finding problem to begin with. But one can hope that the explorative tradition that one sees in, e.g., , can be extended to currently world-class systems such as or . For now, we shall merely pose the (rather philosophical) question of whether the Canonicity Principle in Section \[sec:introduction\]—“every theorem has a canonical proof”—is compatible with the everyday mathematical truism (and apparently confirmed, in part, by everyday experience with automated theorem proving systems, as well as the results discussed in Section \[sec:experiments\]) that some theorems can be proved in at least two ways, even from the (exact) same premises.
[^1]: The author was partially supported by FWF grant P25417-G15 (LOGFRADIG) as well as by the Portuguese Science Foundation (FCT) project *The notion of mathematical proof* (PTDC/MHC-FIL/5363/2012).
[^2]: At <http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html> H. Lemmermeyer maintains a list of proofs of the quadratic reciprocity law. At the time of writing, Lemmermeyer’s list has 240 entries.
[^3]: The mathematics site Cut the Knot maintains 99 proofs: <http://www.cut-the-knot.org/pythagoras/index.shtml>
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---
abstract: |
**Öz**
Günümüzde insans[i]{}z araçlarla alan taramas[i]{}, yani bir alan[i]{}n tümünün veya bir k[i]{}sm[i]{}n[i]{}n insans[i]{}z araçlarla en az efor ile dolaş[i]{}lmas[i]{}, alan taramas[i]{}na duyulan ihtiyaç ve insans[i]{}z araçlar[i]{}n kullan[i]{}m[i]{}n[i]{}n artmas[i]{}yla beraber h[i]{}zla önem kazanmaktad[i]{}r. İnsans[i]{}z araçlarla alan taramas[i]{}n[i]{}n, İHA’lar (İnsans[i]{}z Hava Araçlar[i]{}) ile bir alanda keşif yapmaktan robotlar ile may[i]{}nl[i]{} arazilerin may[i]{}nlardan ar[i]{}nd[i]{}r[i]{}lmas[i]{}na, büyük al[i]{}şveriş merkezlerinde yerlerin robotlarla temizlenmesinden büyük arazilerde çim biçmeye kadar pek çok uygulamas[i]{} mevcuttur. Problemin tek araçla alan taramas[i]{}, birden fazla araçla alan taramas[i]{}, çevrimiçi (arazinin nas[i]{}l olduğu daha önceden bilinmiyorsa) alan taramas[i]{} gibi pek çok versiyonu mevcuttur. Ayr[i]{}ca arazide çeşitli büyüklüklerde engeller de bulunabilmektedir. Doğal olarak, bu problem üzerinde birçok araşt[i]{}rmac[i]{} çal[i]{}şmaktad[i]{}r ve günümüze kadar pek çok çal[i]{}şma yap[i]{}lm[i]{}şt[i]{}r. Bu çal[i]{}ş\
-malar[i]{}n çok büyük bir k[i]{}sm[i]{}, MKA (Minimum Kapsama Ağac[i]{}) yaklaş[i]{}-\
m[i]{}n[i]{} kullanmaktad[i]{}r. Bu yaklaş[i]{}mda temel olarak, düzlemsel bir arazi, arac[i]{}n görüş alan[i]{}na göre eş büyüklükte karelere bölünmekte ve bu karele-\
rin merkezleri birer düğüm olarak kabul edilerek araziden kenarlar[i]{} birim ağ[i]{}rl[i]{}kta olan bir çizge elde edilmektedir. Sonra bu çizgenin MKA’s[i]{} bulunup bu MKA’n[i]{}n etraf[i]{} araçlar taraf[i]{}ndan turlanmaktad[i]{}r. Bizim önerdiğimiz metot düzlemsel olmayan arazilerin de insans[i]{}z araçlar ile taranmas[i]{}na çözüm getirmektedir. Biz de çözümde MKA yaklaş[i]{}m[i]{}n[i]{} temel ald[i]{}k, ancak arazi düzlemsel olmad[i]{}ğ[i]{} için çizgedeki kenarlara birim ağ[i]{}rl[i]{}k vermek yerine iki kare aras[i]{}ndaki eğime bağl[i]{} ağ[i]{}rl[i]{}klar verdik. Bu yaklaş[i]{}m ile ayn[i]{} zamanda özellikle İHA’lar için rüzgar[i]{}n şiddeti ve yönü de hesaba kat[i]{}larak bir rota elde edilip alan taramas[i]{} yap[i]{}labilir.\
**Anahtar Kelimeler:** alan taramas[i]{}, çizge teorisi, İHA (İnsans[i]{}z Hava Arac[i]{}), MKA (Minimum Kapsama Ağac[i]{}), rota planlama\
**Abstract**
The importance of area coverage with unmanned vehicles, in other words, traveling an area with an unmanned vehicle such as a robot or an UAV (Unmanned Aerial Vehicle) completely or partially with minimum cost, is increasing with the increase in usage of such vehicles today. Area coverage with unmanned vehicles are used today in exploration of an area with a UAV, sweeping mines with robots, cleaning ground with robots in large shopping malls, mowing lawn in a large area etc. The problem has versions such as area coverage with a single unmanned vehicle, area coverage with multiple unmanned vehicles, on-line area coverage (The map of the area that will be covered is not known before starting the coverage) with unmanned vehicles etc. In addition, the area may have obstacles that the vehicles cannot move over. Naturally, many researches are working on the problem and a lot of researches have been done on the problem until today. Spanning tree coverage is one of the major approaches to the problem. In this approach, at the basic level, the planar area is divided into identical squares according to range of sight of the vehicle, and centers of these squares are assumed to be vertexes of a graph. The vertexes of this graph are connected with the edges with unit costs and after finding MST (Minimum Spanning Tree) of the graph, the vehicle strolls around the spanning tree. The method we propose suggests a way to cover a non-planar area with unmanned vehicles. The method we propose also takes advantage of spanning tree coverage approach, but instead of assigning unit costs to the edges, we assigned a weight to each edge using slopes between vertexes connected with those edges. We have gotten noticeably better results than the results we got when we did not consider the slope between two squares and used classical spanning tree approach.\
**Keywords:** area coverage, graph theory, MST (Minimum Spanning Tree), route planning, UAV (Unmanned Aerial Vehicle)
author:
- Çaglar Seylan
- 'Özgür Sayg[i]{}n Bican'
- Fatih Semiz
bibliography:
- 'usmos.bib'
title: İNSANSIZ ARAÇLARLA DÜZLEMSEL OLMAYAN ALANLARIN TARANMASI
---
GİRİŞ
=====
Günümüzde insans[i]{}z araçlar için en önemli problemlerden birisi alan taramas[i]{}d[i]{}r. Alan taramas[i]{} bir arazinin her yerini dolaşmak olarak tan[i]{}mlanabilir. Alan taramas[i]{} yap[i]{}l[i]{}rken göz önünde bulundurulan en önemli k[i]{}staslardan bir tanesi bu taramay[i]{} olabildiğince verimli yapmakt[i]{}r. Bu amaca ulaşmak için, bir yandan taranan alan maksimize edilmeye çal[i]{}ş[i]{}l[i]{}rken, bir yandan da harcanan zaman ve yak[i]{}t minimize edilmeye çal[i]{}ş[i]{}l[i]{}r. Arazileri verimli bir şekilde taramak için en s[i]{}k kullan[i]{}lan yaklaş[i]{}mlardan bir tanesi kapsama ağac[i]{} yöntemidir. Kapsama ağac[i]{}n[i]{}n tan[i]{}m[i]{} şu şekildedir:
G bağl[i]{} bir çizge olsun. Hem ağaç olan hem de G’nin tüm düğüm noktalar[i]{}n[i]{} kapsayan bir alt çizgeye kapsama ağac[i]{} denir.
Problemin tek araçla alan taramas[i]{}, birden fazla araçla alan taramas[i]{}, çevrimiçi (Taranacak alan[i]{}n haritas[i]{} daha önceden bilinmiyorsa) alan taramas[i]{}, çevrimd[i]{}ş[i]{} alan taramas[i]{} veya engel bulunan arazilerin taranmas[i]{}, engel bulunmayan arazilerin taranmas[i]{} gibi çeşitli varyasyonlar[i]{} mevcuttur.
Günümüzde alan taramas[i]{}n[i]{}n İHA’lar ile bir alanda keşif yapmaktan robotlar ile may[i]{}nl[i]{} arazilerin may[i]{}nlardan ar[i]{}nd[i]{}r[i]{}lmas[i]{}na, büyük al[i]{}şveriş merkezlerinde yerlerin robotlarla temizlenmesinden büyük arazilerde çim biçmeye kadar pek çok uygulamas[i]{} mevcuttur [@yedi], [@dort], [@sekiz].
Bu konuda bugüne kadar yap[i]{}lan çal[i]{}şmalar[i]{}n çok büyük bir k[i]{}sm[i]{} taranacak arazileri düzlemsel olarak ele alm[i]{}şt[i]{}r. Bu çal[i]{}şman[i]{}n amac[i]{}, düzlemsel arazilerde alan taramas[i]{} için geliştirilen mevcut yaklaş[i]{}mlar üzerine geliştirmeler yap[i]{}larak düzlemsel olmayan arazilerin de verimli olarak taranabileceği bir yaklaş[i]{}m türet-\
mektir.
İLGİLİ ÇALIŞMALAR
=================
Alan taramas[i]{}nda, en önemli problemlerden bir tanesi arazinin bilgisayar ortam[i]{}nda nas[i]{}l gösterileceğidir. Daha aç[i]{}k söylemek gerekirse problem, sürekli gerçek dünyan[i]{}n kesikli bilgisayar dünyas[i]{}nda nas[i]{}l gösterilebileceğidir. Çözümler-den bir tanesi A. Elfes taraf[i]{}ndan önerilen araziye, araziyi eş hücrelere bölerek\
yak[i]{}nsamakt[i]{}r [@bir]. Bilinmeyen ve yap[i]{}land[i]{}r[i]{}lmam[i]{}ş alanlar için, Elfes sonar tabanl[i]{} gerçek dünya harita ve navigasyon sistemi önermiştir. Böylece araziyi bilgisayar ortam[i]{}nda gösterebilmek için bir yaklaş[i]{}m bulmuştur. Bu yaklaş[i]{}mda arazi, eş hücrelerden oluşan iki boyutlu bir dizi olarak gösterilmiştir. Hücre, hücreye bir araç girince veya arac[i]{}n alg[i]{}lay[i]{}c[i]{}s[i]{} hücreyi alg[i]{}lay[i]{}nca gezilmiş olarak kabul edilir. 2 boyutlu dizideki tüm hücreler gezildiğinde alan taranm[i]{}ş olur.
İnsans[i]{}z araçlarla alan taramas[i]{}na diğer hücre tabanl[i]{} bir yaklaş[i]{}m ise Y. Gabriely ve E. Rimon taraf[i]{}ndan önerilen kapsama ağac[i]{} yaklaş[i]{}m[i]{}d[i]{}r [@iki]. Bu yaklaş[i]{}mda araziye, arazi eş hücrelere bölünüp daha sonra bu hücrelerden bir çizge elde edilerek yak[i]{}nsanm[i]{}şt[i]{}r. Daha sonra bu çizgenin kapsama ağac[i]{} bulunur ve etraf[i]{} bir ya da birden fazla araçla dolaş[i]{}l[i]{}r. Dolaşma bittiğinde alan taranm[i]{}ş olur.
S. Hert, S. Tiwari ve V. Lumelsky otonom su alt[i]{} araçlar[i]{}na 3 boyutlu su alt[i]{} alanlar[i]{}n[i]{}n taranmas[i]{} için rota plan[i]{} yapan k[i]{}smi kesikli bir algoritma önermiştir [@uc]. Önerilen algoritma çevrimiçidir (Taranacak olan su alt[i]{}ndaki alan tarama başlamadan önce bilinmemektedir). Bu yaklaş[i]{}mdaki hücreler tamamen sabit değildir, enleri sabit fakat boylar[i]{} değişkendir. Bu algoritma ile basit bağlant[i]{}l[i]{} ya da basit olmayan bağlant[i]{}l[i]{} alanlar taranabilmektedir. Algoritman[i]{}n özyinelemeli bir doğas[i]{} vard[i]{}r ve araç alan[i]{} paralel düz çizgiler boyunca zigzag hareketi yaparak tarar. Alan adalar ve koylar içerebilir. Koylara girerken ya da ç[i]{}karken koylar[i]{}n içindeki baz[i]{} alanlar birden fazla taranabilir ya da baz[i]{} alanlar hiç taranmayabilir (Bu tür koylara sapt[i]{}r[i]{}c[i]{} koy denir). Bu tür koylar[i]{} taramak için özel yaklaş[i]{}mlara gerek vard[i]{}r. Alan ada içerdiği zaman, yani alan basit bağlant[i]{}l[i]{} değilse, ayn[i]{} prosedürler üzerinde küçük değişiklikler yap[i]{}larak alan[i]{}n tamamen taranmas[i]{} sağlan[i]{}r.
Alan[i]{}n hücrelere ayr[i]{}lmas[i]{} bir yak[i]{}nsama olmak zorunda değildir, tam da olabilir. Alan tam olarak hücrelere ayr[i]{}l[i]{}rken, öyle farkl[i]{} bölgelere ayr[i]{}l[i]{}r ki bu bölgelerin birleşimi alan[i]{} tam olarak verir [@dort]. Tam olarak hücrelere ay[i]{}rma tekniklerinden bir tanesi alan[i]{} yamuksal şekilde ay[i]{}rmakt[i]{}r. J. R. VanderHeide ve N. S. V. Rao yamuksal şekilde alan ay[i]{}rma tabanl[i]{} çevrimiçi bir alan taramas[i]{} metodu önermiştir [@bes]. Temel olarak, insans[i]{}z araç alan hakk[i]{}nda bilgi toplama faz[i]{}ndan sonra alan[i]{} dikdörtgenlere ay[i]{}r[i]{}r ve her dikdörtgen için ayr[i]{} bir rota planlamas[i]{} yapar. Daha sonra her dikdörtgen için bulunan rotalar[i]{} birleştirir ve tek bir rota elde eder.
ÖNERİLEN YAKLAŞIM
=================
Bugüne kadar önerilen metotlar genelde düz alanlarda alan taramas[i]{} yapmak içindir. Biz bu yaz[i]{}da düzlemsel olmayan, engebeli yüzeyleri taramak için, insans[i]{}z araçlara rota planlamas[i]{} yapan bir yaklaş[i]{}m sunuyoruz. Bizim yaklaş[i]{}m[i]{}m[i]{}z da temel olarak Y. Gabriely ve E. Rimon’un yaklaş[i]{}m[i]{} [@iki] gibi Kapsama Ağac[i]{} Taramas[i]{} yöntemini kullanmaktad[i]{}r. Ancak bu yöntemde, kapsama ağac[i]{} oluş-turulurken, araziden elde edilen çizgenin kenarlar[i]{}na birim ağ[i]{}rl[i]{}k yerine komşu iki düğüm aras[i]{}ndaki eğime bağl[i]{} bir ağ[i]{}rl[i]{}k atan[i]{}r. Daha sonra oluşan çizgenin minimum kapsama ağac[i]{} bulunarak bu ağac[i]{}n etraf[i]{} dolaş[i]{}l[i]{}r.
Arazi, bilgisayarda her bir eleman[i]{} o noktan[i]{}n yüksekliğini (Belirli bir yere göre) belirten bir N x M’lik matris ile gösterilir. Şekil 1’de bu matrise bir örnek görülmektedir.
![Arazi için örnek matris.[]{data-label="Figure:sekil1"}](gnu_plot_tik3/sekil1.png)
İnsans[i]{}z araca bağl[i]{} kameran[i]{}n görüş aç[i]{}s[i]{}n[i]{}n, ya da daha genel bir söylemle araca bağl[i]{} cihaz[i]{}n kapsama alan[i]{}n[i]{}n, matriste bir karelik alana denk geldiği farz edilmektedir. Bu matris N. Hazon’[i]{}n yaklaş[i]{}m[i]{}ndaki gibi her kenar[i]{} iki kare olan (Daha genel bir söylemle arac[i]{}n kapsama alan[i]{} çap[i]{}n[i]{}n iki kat[i]{} uzunluğunda) daha büyük karelere ayr[i]{}l[i]{}r [@alti] (Şekil 2).
![Alt karelere ayr[i]{}lm[i]{}ş matris.[]{data-label="Figure:sekil2"}](gnu_plot_tik3/sekil2.png)
Bu matristen (N/2) x (M/2)’lik yeni bir matris üretilir. Bu matrisin her bir eleman[i]{} eski matrisin 2 x 2’lik karelerinin içindeki elemanlara (Şekil 2’de kal[i]{}n s[i]{}n[i]{}rlarla belirtilen) denk gelmektedir. Yeni matristeki her bir eleman da eski matristeki gibi yükseklik belirtmektedir. Bu yükseklikler eski matristeki 2 x 2’lik karelerdeki 4 eleman[i]{}n aritmetik ortalamas[i]{} al[i]{}narak bulunur ve gerçek araziye yak[i]{}nsama yap[i]{}l[i]{}r. Yani, bir 2 x 2’lik kare içindeki elemanlar H1, H2, H3, H4 ise yeni matriste bu kareye denk gelen eleman[i]{}n değeri Hyeni, bu dört eleman[i]{}n aritmetik ortalamas[i]{}d[i]{}r (Şekil 3).
![Alt matristeki dört eleman[i]{}n aritmetik ortalamas[i]{}.[]{data-label="Figure:sekil3"}](gnu_plot_tik3/sekil3.png)
Yeni matristeki her bir hücre ayn[i]{} zamanda bir düğüm noktas[i]{}d[i]{}r. Her düğüm noktas[i]{} ile, kendisine karş[i]{}l[i]{}k gelen hücrenin komşular[i]{} kenarlar arac[i]{}l[i]{}ğ[i]{}yla bağl[i]{}d[i]{}r (Her bir hücre, kendisinin sağ[i]{}ndaki, solundaki, üstündeki ve alt[i]{}ndaki hücre ile komşudur, çapraz hücreler komşudan say[i]{}lmaz). Kenarlar[i]{}n ağ[i]{}rl[i]{}klar[i]{}, bağlad[i]{}klar[i]{} düğüm noktalar[i]{} aras[i]{}ndaki eğime bağl[i]{} bir fonksiyon arac[i]{}l[i]{}ğ[i]{}yla bulunur. Bu fonksiyon, düğüm noktalar[i]{} aras[i]{}ndaki yatay uzakl[i]{}k d, yükseklikleri aras[i]{}ndaki fark h ise, doğrudan aradaki uzakl[i]{}ğ[i]{} veren Pisagor bağ[i]{}nt[i]{}s[i]{}, yani $\sqrt{d^2 + h^2}$, özellikle İHA’lar için rüzgar[i]{}n yönü ve şiddetine bağl[i]{} bir bağ[i]{}nt[i]{}, ya da amaca göre çok daha karmaş[i]{}k bir bağ[i]{}nt[i]{} olabilir.
Çizge bu şekilde oluşturulduktan sonra minimum kapsama ağac[i]{} bulunur ve bu kapsama ağac[i]{}n[i]{}n etraf[i]{} Y. Gabriely ve E. Rimon’un yapt[i]{}ğ[i]{} yaklaş[i]{}mdaki gibi araçlar taraf[i]{}ndan turlan[i]{}r [@iki]. Yükseklik matrisi Şekil 1’deki gibi olan matriste, doğrudan Pisagor bağ[i]{}nt[i]{}s[i]{}na göre bulunmuş çizgenin, bu yaklaş[i]{}ma göre turlan[i]{}rken ortaya ç[i]{}kan rota şekil 4’te gösterilmiştir.
TEST SONUÇLARI
==============
Testlerde 15 tane girdi kullan[i]{}ld[i]{}. Girdiler, rastgele üretilmiş araziler olup 250x250 matris şeklinde gösterilmiştir. Girdilerdeki arazilere en fazla 30 adet engel koyul-muştur. Testlerde bizim önerdiğimiz yaklaş[i]{}m ile şimdiye kadar düzlemsel arazileri taramada kullan[i]{}lan doğrudan kapsama ağac[i]{} yaklaş[i]{}m[i]{}n[i]{} bu 15 girdiyi her iki yaklaş[i]{}m için de kullanarak karş[i]{}laşt[i]{}rd[i]{}k. Bunu yaparken her birinin ortalama eğimi farkl[i]{} olan girdiler denedik.
Ortalama eğim, bütün komşu hücre ikililerinden elde edilen eğimlerin topla-m[i]{}n[i]{}n çizgedeki kenar say[i]{}s[i]{}na bölümünden elde edilmiştir. İki hücre aras[i]{}ndaki eğim, iki hücrenin ortalama yükseklikleri aras[i]{}ndaki fark[i]{}n bu hücreler aras[i]{}ndaki yatay uzakl[i]{}ğa bölümü olarak tan[i]{}mlanm[i]{}şt[i]{}r.
Testlerde metot, çizgedeki kenarlara ağ[i]{}rl[i]{}k atamada kullan[i]{}lan iki farkl[i]{} ağ[i]{}rl[i]{}k fonksiyonu kullan[i]{}larak denenmiştir. Bu fonksiyonlardan ilki, kenarlara doğrudan Pisagor bağ[i]{}nt[i]{}s[i]{} kullan[i]{}larak bulunan, hücreler aras[i]{}ndaki doğrusal uzakl[i]{}ğ[i]{} bulmaktad[i]{}r. Diğeri ise yine ilk fonksiyon gibi Pisagor bağ[i]{}nt[i]{}s[i]{}n[i]{} kullanarak hücreler aras[i]{}ndaki doğrusal uzakl[i]{}ğ[i]{} bulmaktad[i]{}r. Ancak bunu doğrudan kenarlar[i]{}n ağ[i]{}rl[i]{}ğ[i]{} olarak kullanmak yerine buna, artan eğim ile beraber artan bir ceza puan[i]{} eklemekdedir. Fonksiyonlar ve test sonuçlar[i]{} daha detayl[i]{} olarak aşağ[i]{}da aç[i]{}klanm[i]{}şt[i]{}r.
Doğrudan pisagor bağ[i]{}nt[i]{}s[i]{}n[i]{} kullanan ağ[i]{}rl[i]{}k fonksiyonu
--------------------------------------------------------------------------------
Ortalama yükseklikleri h1 ve h2 olan, aralar[i]{}ndaki yatay uzakl[i]{}k ise d olan hücrelere karş[i]{}l[i]{}k gelen düğümler aras[i]{}ndaki kenarlara, $$w= \sqrt{|h_1-h_2|^2 - d^2}$$
fonksiyonuna göre ağ[i]{}rl[i]{}k atanm[i]{}şt[i]{}r. Girdi numaras[i]{}na göre ortalama eğim, kapsama ağac[i]{} yaklaş[i]{}m[i]{}na göre bulunan ağaçtaki kenarlar[i]{}n ağ[i]{}rl[i]{}klar[i]{} toplam[i]{} ve önerdiğimiz metoda göre (Herhangi bir kapsama ağac[i]{}ndan ziyade minimum kapsama ağac[i]{}na göre) bulunan ağaçtaki kenarlar[i]{}n ağ[i]{}rl[i]{}klar[i]{} toplam[i]{} çizelge 1’de gösterilmiştir. Eğime bağl[i]{} olarak değişen kenar ağ[i]{}rl[i]{}klar[i]{} toplamlar[i]{} çizelge 2’de gösterilmiştir (Dikey eksen ağ[i]{}rl[i]{}klar toplam[i]{}na, yatay eksen ortalama eğime karş[i]{}l[i]{}k gelmektedir.).
![İki metodun denklem (1)’e göre karş[i]{}laşt[i]{}r[i]{}lmas[i]{}.[]{data-label="Figure:sekil4"}](gnu_plot_tik3/sekil4.png)
![İki metodun denklem (1)’e göre karş[i]{}laşt[i]{}r[i]{}lmas[i]{}.[]{data-label="Figure:sekil5"}](gnu_plot_tik3/sekil5.png)
Ek olarak eğime bağl[i]{} ceza puan[i]{} da kullanan ağ[i]{}rl[i]{}k fonksiyonu
-------------------------------------------------------------------------------
Ortalama yükseklikleri h1 ve h2 olan, aralar[i]{}ndaki yatay uzakl[i]{}k ise d olan hücrelere karş[i]{}l[i]{}k gelen düğümler aras[i]{}ndaki kenarlara, $$w= \sqrt{|h_1-h_2|^2 - d^2} * (1 + \frac{|h_1-h_2|}{d})$$
fonksiyonuna göre ağ[i]{}rl[i]{}k atanm[i]{}şt[i]{}r. Bu fonksiyon, ilk fonksiyona yine eğime bağl[i]{} bir ceza puan[i]{} eklenerek elde edilmiştir. Girdi numaras[i]{}na göre ortalama eğim, kapsama ağac[i]{} yaklaş[i]{}m[i]{}na göre bulunan ağaçtaki kenarlar[i]{}n ağ[i]{}rl[i]{}klar[i]{} toplam[i]{} ve önerdiğimiz metoda göre (Herhangi bir kapsama ağac[i]{}ndan ziyade minimum kapsama ağac[i]{}na göre) bulunan ağaçtaki kenarlar[i]{}n ağ[i]{}rl[i]{}klar[i]{} toplam[i]{} çizelge 3’de gösterilmiştir. Eğime bağl[i]{} olarak değişen kenar ağ[i]{}rl[i]{}klar[i]{} toplamlar[i]{} çizelge 4’de gösterilmiştir (Dikey eksen ağ[i]{}rl[i]{}klar toplam[i]{}na, yatay eksen ortalama eğime karş[i]{}l[i]{}k gelmektedir.).
![İki metodun denklem (2)’e göre karş[i]{}laşt[i]{}r[i]{}lmas[i]{}.[]{data-label="Figure:sekil6"}](gnu_plot_tik3/sekil6.png)
![İki metodun denklem (2)’e göre karş[i]{}laşt[i]{}r[i]{}lmas[i]{}.[]{data-label="Figure:sekil7"}](gnu_plot_tik3/sekil7.png)
Yorumlar
--------
İki yaklaş[i]{}m doğrudan Pisagor bağ[i]{}nt[i]{}s[i]{}n[i]{} kullanan ağ[i]{}rl[i]{}k fonksiyonu ile test edildiğinde, önerilen yaklaş[i]{}m diğer yaklaş[i]{}ma göre biraz daha verimli rotalar bulmaktad[i]{}r. Arazinin ortalama eğimi artt[i]{}kça (Özellikle 0.5’e yaklaşt[i]{}kça) aradaki fark daha belirgin hale gelmektedir. Yaklaş[i]{}mlar Pisagor bağ[i]{}nt[i]{}s[i]{}na ek olarak eğime bağl[i]{} ceza puan[i]{} da kullanan ağ[i]{}rl[i]{}k fonksiyonu ile test edildiğinde, yine önerilen yaklaş[i]{}m diğer yaklaş[i]{}ma göre daha verimli rotalar bulmaktad[i]{}r ancak aradaki fark bu sefer daha belirgindir. Arazinin ortalama eğimi artt[i]{}r[i]{}ld[i]{}kça aradaki fark ilk ağ[i]{}rl[i]{}k fonksiyonu kullan[i]{}ld[i]{}ğ[i]{} zamankine göre çok daha belirgin hale gelmektedir. Özellikle ortalama eğim 0.5’e yaklaş[i]{}rken, önerilen yaklaş[i]{}mda toplam ağ[i]{}rl[i]{}k yavaş bir art[i]{}ş göstermiştir, ancak diğer yaklaş[i]{}mda toplam ağ[i]{}rl[i]{}k neredeyse üstel bir biçimde art[i]{}ş göstermiştir.
SONUÇ
=====
Makalede düzlemsel olmayan arazilerin insans[i]{}z araçlarla verimli bir şekilde taranmas[i]{} için yeni bir metot önerilmiştir. Bu metot, temel olarak, kapsama ağac[i]{} yaklaş[i]{}m[i]{}nda bulunan kenarlara birim ağ[i]{}rl[i]{}k yerine eğime bağl[i]{} bir fonksiyona göre atanan ağ[i]{}rl[i]{}klar[i]{} kullanmaktad[i]{}r. Testlerde önerilen metot kapsama ağac[i]{} metodu ile karş[i]{}laşt[i]{}r[i]{}lm[i]{}şt[i]{}r. Genel olarak önerilen metot diğer metoda göre daha verimli sonuçlar verse de kenarlara eğimi daha fazla hesaba katmak için atanan ağ[i]{}rl[i]{}klarda eğimin uzakl[i]{}ğa göre etkisi daha da artt[i]{}r[i]{}ld[i]{}ğ[i]{}nda aradaki fark daha da artmaktad[i]{}r.
Düzlemsel yüzeylerden elde edilen herhangi bir kapsama ağac[i]{} ayn[i]{} zamanda minimum kapsama ağac[i]{} olacağ[i]{}ndan önerilen yaklaş[i]{}mla arada rotan[i]{}n verimi aç[i]{}s[i]{}ndan bir fark olmaz. Ancak yüzey düzlemsel değilse ortalama eğime bağl[i]{} olarak minimum kapsama ağac[i]{} herhangi bir kapsama ağac[i]{}na göre daha verimli sonuçlar verir. Bu nedenle herhangi bir kapsama ağac[i]{}n[i]{}n etraf[i]{} dolaş[i]{}lmaktansa minimum kapsama ağac[i]{}n[i]{}n etraf[i]{} dolaş[i]{}l[i]{}rsa, düzlemsel olmayan arazilerde, tara-ma için daha verimli rotalar elde edilmiş olur.
TEŞEKKÜRLER
===========
Makalenin yazarlar[i]{} olarak ODTÜ-TSK MODSİMMER’e bu makaleye desteğinden dolay[i]{} ve Doç. Dr. Veysi İşler’e yapt[i]{}ğ[i]{} büyük katk[i]{}lar ve yard[i]{}mlardan dolay[i]{} teşekkürü borç biliriz.
|
---
author:
- Claude Bertout
- Françoise Genova
bibliography:
- '5842.bib'
date: 'Received June 16, 2006 / Accepted October 12, 2006'
title: 'A kinematic study of the Taurus-Auriga T association'
---
[This is the first paper in a series dedicated to investigating the kinematic properties of nearby associations of young stellar objects. Here we study the Taurus-Auriga association, with the primary objective of deriving kinematic parallaxes for individual members of this low-mass star-forming region.]{} [We took advantage of a recently published catalog of proper motions for pre-main sequence stars, which we supplemented with radial velocities from various sources found in the CDS databases. We searched for stars of the Taurus-Auriga region that share the same space velocity, using a modified convergent point method that we tested with extensive Monte Carlo simulations.]{} [Among the sample of 217 Taurus-Auriga stars with known proper motions, we identify 94 pre-main sequence stars that are probable members of the same moving group and several additional candidates whose pre-main sequence evolutionary status needs to be confirmed. We derive individual parallaxes for the 67 moving group members with known radial velocities and give tentative parallaxes for other members based on the average spatial velocity of the group. The Hertzsprung-Russell diagram for the moving group members and a discussion of their masses and ages are presented in a companion paper.]{}
Introduction {#IntroductionSection}
============
To accurately determine the two main physical parameters of nearby young stellar objects (YSOs), their age and mass, we must know how far away they are. While determination of distances has been at the heart of astronomical research for many centuries, we have made surprisingly little recent progress in this respect, at least for low-luminosity objects such as the young solar-type T Tauri stars (TTSs). This contrasts with more luminous nearby stars, the targets of the very successful Hipparcos mission, for which accurate parallaxes are now available. Although Hipparcos observed a few low-luminosity pre-main sequence (PMS) stars in nearby star-forming regions , they were clearly a challenge for its small telescope. The situation will improve dramatically with the flight of the Gaia mission, as this satellite will measure the parallaxes and proper motions of millions of faint stars. However, Gaia’s expected launch date is 2012, so it would be useful to make some progress in determining the distances of YSOs in the meantime in order, for example, to better constrain the lifetime of their disks and the timescales of planet formation, two very timely research areas for which a precise determination of YSO ages is urgently required.
Where do we stand today as far as YSO parallaxes are concerned? The post-Hipparcos situation was discussed by , who provided new astrometric solutions of the Hipparcos data for groups of TTSs in various star-forming regions, thus finding average distances to some YSO groups. The new distances generally agreed well with previous estimates of the associated molecular cloud distances based, e.g., on the photometry of a few bright stars enshrouded in reflection nebulosity. For distance determinations of the Taurus star-forming region based on this method, see [@1968AJ.....73..233R] and [@1978ApJ...224..857E].
Although average distances provide valuable information, what we really need for constraining ages and masses by comparing observed stellar properties with evolutionary models are the distances to individual stars of the YSO associations. This is what we are attempting to do in this work, which focuses on the Taurus-Auriga T association. To do so, we use the proper motions of individual stars.
Several proper motion surveys of the Taurus-Auriga star-forming region have been performed in the past, e.g., by [@1979AJ.....84.1872J] and [@1991AJ....101.1050H]. More recently, published an all-sky catalog of proper motions for 1250 YSOs that provides a coherent database for kinematic studies such as the one we are now embarking on. The proper motion database for Taurus-Auriga is briefly discussed in Sect. \[CatalogDescriptionSection\].
The procedure that we use here to derive individual parallaxes is as follows. First, we identify those stars among the Taurus-Auriga confirmed or suspected YSOs that have the same spatial velocity. In other words, we look for the group of stars that *defines* the Taurus-Auriga T association through its common motion in the sky. While young associations are not expected to be gravitationally bound, it is well known that their members all share the same motion in space for several million years before the association dissolves and loses its identity most notably due to the tidal interactions caused by Galactic rotation and encounters with other stellar groups and interstellar clouds [see, e.g., the review by @2002ASPC..285..150B].
To find the likely association members, we developed our own variant of the classic convergent point method. This is described in Sect. \[CPMethodSection\], while Sect. \[MCSims\] presents Monte-Carlo simulations of the moving group search that proved helpful for choosing computational parameters. We then provide lists of kinematic members in Sect. \[KinematicAnalysisSection\], where we also make use of the radial velocity information, when available, to infer parallaxes and associated error bars for individual stars. Stellar radial velocities are known only for a limited sub-sample of association members, but we can compute the average common spatial velocity of this sub-group, which in turn provides a second way of determining kinematic parallaxes for all members of the group by assuming that all stars have the same spatial velocity. These results are discussed in Sect. \[DiscussionSection\].
At that point, we should be armed with the information needed to perform a new age and mass determination for members of the moving group in the T association. This analysis and its astrophysical consequences will be presented in a companion paper (Bertout & Siess, in preparation).
The sample of Taurus-Auriga stellar objects {#CatalogDescriptionSection}
===========================================
The catalog of proper motions for PMS stars contains 217 stars that are in the general area of the Taurus-Auriga star-forming region (which roughly spans the range of coordinates $3^h 50^m$ $\alpha(2000)$ $5^h 10^m$ and $15$$\delta(2000)$ $35$).
Proper motions of Taurus PMS stars
----------------------------------
The upper panel of Fig. \[Fig1\] displays the location of Taurus-Auriga stars contained in the catalog. One recognizes the familiar grouping of YSOs in the vicinity of Taurus molecular cores: L1551 at approximately $\alpha
(2000) = 4^h 32^m$ and $\delta (2000) = 18$. North of L1551 one finds the L1536 and L1529 groups, while in the northwest of L1529 one has the L1495 group and in the northeast the HCL2 group. The few Auriga stars present in the sample are around $\alpha (2000) = 5^h$ and $\delta (2000) = 30$. Individual objects at the periphery of the molecular cloud are mainly weak emission-line TTSs discovered through their X-ray emission.
For comparison, we also show in Fig. \[Fig1\] the stars observed by the Hipparcos satellite in the same region. Because of extinction, the density of Hipparcos stars strongly decreases in the vicinity of the Taurus-Auriga molecular clouds where our targets are located. As we discuss later, only a few PMS stars in the region are common to both the Hipparcos catalog and the catalog of presumably young stars.
The lower panel of Fig. \[Fig1\] shows the proper motion vectors for all objects in the catalog. Although a clear convergent point for these proper motions is not readily apparent, one notices that many proper motion vectors seem to point toward the lower left-hand corner of the figure.
Figure \[Fig2\] presents histograms of the proper motion components $\mu_\alpha \cos \delta$ and $\mu_\delta$, as well as a histogram of the radial velocities for those 127 objects for which we could find a published value (see below). The average values and standard deviations for the proper motions of the full sample are
$\left\{ \begin{array}{lll}
\mu_\alpha \cos \delta & = & 8.20 \pm 14.45 \: \rm{mas/yr} \\
\mu_\delta & = & -20.82 \pm 13.84 \: \rm{mas/yr} \\
\end{array}\right.$
If we then consider the sub-sample of 127 stars with known radial velocities, we have
$\left\{ \begin{array}{lll}
\mu_\alpha \cos \delta & = & 7.16 \pm 8.55 \: \rm{mas/yr} \\
\mu_\delta & = & -20.91 \pm 10.31 \: \rm{mas/yr} \\
v_{rad} & = & 16.03 \pm 6.43 \: \rm{km/s},
\end{array}\right.$
where we note a lower dispersion of the proper motion measurements, due in particular to the fact that radial velocities have been measured primarily in the brightest and most confirmed members of the PMS population. We come back to this point in Sect. \[KinematicAnalysisSection\].
Figure \[Fig3\] displays all proper motion values and their associated uncertainties. Proper motion values cluster around the average values, albeit with considerable scatter, while a few stars have highly discrepant proper motions, often with large error bars. As discussed in Sect. \[KinematicAnalysisSection\], some of the stars included in the catalog are likely to be field stars, which explains the discrepant values. However, a majority of stars with measured radial velocities are confirmed PMS stars.
Caveats
-------
One difficulty with the investigation envisioned here is the apparent similarity between the proper motions of field stars, as represented by the Hipparcos[^1] targets of Fig. \[Fig1\], and the proper motions of our target stars. To illustrate this, we show histograms of the two proper motion components for those Hipparcos stars in Fig. \[Fig4\]. We also depict the proper motion data for these objects, together with their error bars, in Fig. \[Fig5\].
Although the similarities with Figs. \[Fig2\] and \[Fig4\] are obvious, there are some differences between the proper motion distributions that become more evident when one studies the histogram shapes in more detail. We fitted these data with Gaussian curves and while it was easy to fit the proper motions of the objects with a single Gaussian, three different components were needed to fit the Hipparcos data in a satisfactory way. Actually, the three components are easily seen in Fig. \[Fig5\], and Table \[PMfits\] summarizes the properties of the various Gaussian curves that best fitted the data, as well as the squared coefficient of multiple correlation $R^2$ and the reduced $\chi^2$ values for the overall fits. The proper motions are marginally compatible with the proper motion values of Peak \#1 of the Hipparcos proper motions but with much smaller dispersions, and the overall proper motions distributions of the two samples appear quite different. Peaks \#2 and \#3 correspond approximately to the proper motions of the Pleiades and Hyades clusters. We see from Fig. \[Fig1\] that several stars in the catalog appear to be probable members of the Pleiades, while two stars have proper motions consistent with Hyades membership. It is therefore clear that the catalog is contaminated by non pre-main sequence stars to some extent (cf. Sect. \[KinematicAnalysisSection\]).
To further quantify the differences between the proper motion distributions, we performed a Kolmogorov-Smirnov test on the two components of the proper motion for the and Hipparcos samples of stars. We found that the probability of the distributions of $\mu_\alpha \cos \delta$ components for both samples being drawn randomly from the same parent distribution is $8
\cdot 10^{-14}$, while it is $4 \cdot 10^{-7}$ for the $\mu_\delta$ components.
While these differences raise the hope that the following analysis will allow us to recognize the common motion of PMS objects even in the presence of field stars, it is also clear from the above that field stars must exist that share approximately the same proper motions as the sample. Some of them could possibly contaminate the catalog, although it is meant to include – at least in principle – only PMS stars. Because we cannot kinematically distinguish between true members of the association and at least some field stars, it is crucial to scrutinize the catalog for non-PMS objects and to screen out these possible interlopers before we look for a moving group of young stars. We come back to this issue in Sect. \[KinematicAnalysisSection\].
Another worry when dealing with the kinematics of young stars in Taurus is the high degree of stellar multiplicity in that star-forming region, where the duplicity reaches about 49%, a factor 1.9 larger than for solar-type field stars . One of the incentives cited by for preparing a proper motion catalog of PMS objects is that *“dedicated work* \[on objects that have close companions\] *is necessary to derive more reliable proper motions",* and this is indeed what makes this catalog so valuable. Many binaries in Taurus are relatively wide pairs, often with large flux ratios, for which individual proper motion determinations of the components can be done. There are also a number of binaries with sub-arcsecond separations that remain unresolved by and are therefore unlikely to significantly affect their results. The effect of duplicity on the catalog’s proper motions is expected to be most severe for relatively close binaries with a few arcsecond separation that were barely resolved by . These systems are identified in the catalog with a mention AB that we kept in our tables. It indicates that the given proper motion is representative of the binary’s motion.
Radial velocities of Taurus PMS stars
-------------------------------------
The [@1988cels.book.....H] catalog contains all radial velocity values known prior to 1988. In order to access the more recent measurements, we searched the CDS databases using some of the data mining tools available on the CDS site. The search made use of a prototype implementation of the Unified Content Descriptors[^2] in the VizieR database.
We found radial velocity information for only 127 stars of our sample, which is quite surprising given the many investigations of the Taurus YSOs that can be found in the literature. Besides [@1988cels.book.....H], the main sources of radial velocity measurements for Taurus-Auriga stars are [@1949ApJ...110..424J], [@1986ApJ...309..275H], [@1987AJ.....93..907H], [@1988AJ.....96..297W], , and . While we might have missed some data that are not included in the CDS databases or are not known to us otherwise, it is also obvious that investigations of Taurus-Auriga have often focused on the same relatively bright stars in the region, and no systematic, high-precision radial velocity survey has been performed to date in spite of the availability of efficient spectrographs on medium-sized telescopes.
Galactic velocities of the sample
---------------------------------
{width="12cm"}
We now compute the Galactic positions and velocities of the 127 stars for which we know radial velocities, using the method delineated in [@1987AJ.....93..864J]. The Galactic positions are defined on an $XYZ$-grid with the origin at the Sun. There, $X$ points to the Galactic center, $Y$ points in the direction of Galactic rotation, and $Z$ points to the Galactic North pole. The projection of the Galactic velocity on the same grid defines the components $U$, $V$, and $W$.
We plotted the projections of the Galactic velocities onto the three planes $XY$, $XZ$, and $YZ$ in Fig. \[Fig6\], mainly for the purpose of comparison with the results obtained in the following sections. Using the direction of the equatorial coordinate system south-north axis indicated above, one can check that the directions of velocity projections in the $YZ$-plane follow the directions of the proper motion vectors of Fig. \[Fig1\], as expected. Although there is an obvious scatter in the velocity directions, it appears from these projections that a number of stars may have similar space motions, thus providing an incentive for the following investigation.
Figure \[Fig7\] is a 3D cube with sides equal to 60 pc showing the Galactic positions of the Taurus stars when, following , we assume an average distance of 139 pc for the Taurus complex. With this average distance, the stars populate an $XZ$-plane at $X \approx -132$ pc, and one recognizes the T Tauri stellar groupings in this plane. Note that the south-north axis of the equatorial coordinate system runs approximately parallel to the direction of the diagonal from $Z=-70$, $Y=-20$ to $Z=-10$, $Y=40$. The size of the 3D cube was chosen for comparison with the results of Sect. \[KinematicAnalysisSection\].
Finally, we have plotted histograms of the $U$, $V$, $W$, and $V_{rad}$ values in Fig. \[Fig8\]. Their respective average and standard deviation values are
$\left\{ \begin{array}{lll}
U & = & -15.37 \pm 6.00 \: \rm{km/s} \\
V & = & -11.74 \pm 6.50 \: \rm{km/s} \\
W & = & -9.89 \pm 4.94 \: \rm{km/s} \\
V_{space} & = & 23.20 \pm 5.98 \: \rm{km/s}.
\end{array}\right.$
Method of analysis {#CPMethodSection}
==================
We have developed our own variant of the well-known convergent point method for finding members of moving groups that share the same space motion. The original method goes back to [@1916cels.book.....C.] and was further developed by several workers, including [@1950ApJ...112..225B], [@1971MNRAS.152..231J], and more recently by [@1999MNRAS.306..381D], who adapted it so as to take full advantage of the Hipparcos data. We review briefly the method before outlining the variant we developed to deal with the problem at hand.
The classic convergent point method
-----------------------------------
The convergent point method is based on the fact that the proper-motion vectors of a group of stars moving with the same motion in space appear to an observer to converge to a specific point of the sky plane, called the convergent point (CP hereafter).
The two components of the proper motion vector $\vec{\mu}$ in the equatorial coordinate system, $\mu_\alpha \cos \delta$ and $\mu_\delta$, define the tangential velocity vector $\vec{v_{tan}}$ through the relation $$\label{vtan}
\vec{v_{tan}} \equiv \frac{A \: \vec{\mu}}{\pi}$$ where $\pi$ is the parallax in milliarcseconds (mas) and where $A =
4.74047$ km yr/s, the ratio of one astronomical unit in km to the number of seconds in one Julian year, is the constant needed for adjusting the units on both sides of the equation.
If we denote the modulus of the space velocity vector by $
\left|\;\vec{v}\;\right| $ and, for the time being, neglect measurement errors and a possible internal cluster velocity dispersion, we have $$\label{BasicEq}
\left|\;\vec{\mu}\;\right| = \frac{\pi \left|\;\vec{v}\;\right|
\sin \lambda }{A}$$ where $\lambda$ is the angular distance between the position ($\alpha, \delta$) of a given star in the moving group and the position ($\alpha_{CP}, \delta_{CP}$) of the convergent point (see the schematic representation in Fig. \[Fig9\]). Spherical trigonometry tells us that $$\label{lambdadef}
\cos \lambda = \sin \delta \sin \delta_{CP} + \cos \delta \cos
\delta_{CP} \cos(\alpha_{CP}-\alpha).$$
If we now change coordinates (see Fig. \[Fig9\]) and project the proper motion onto the axis joining the star to the CP, on one hand (component $\mu_\parallel$), and onto the perpendicular direction, on the other (component $\mu_\perp$), the proper motion components in both systems are related by $$\left\{ \begin{array}{l} \mu_\parallel = \sin \theta \: \mu_\alpha
\cos
\delta + \cos \theta \: \mu_\delta \\
\mu_\perp = -\cos \theta \: \mu_\alpha \cos \delta + \sin \theta \:
\mu_\delta.
\end{array}
\right.$$ with $$\tan \theta =
\frac{\sin(\alpha_{CP}-\alpha)}{\cos\delta\tan\delta_{CP} -
\sin\delta\cos(\alpha_{CP}-\alpha)}.$$
If the proper motion vector is directed exactly toward the CP, then we also have $$\label{muparaandperp}
\left\{ \begin{array}{l} \mu_\parallel = \frac{\pi v \sin
\lambda}{A} =
\left|\;\vec{\mu}\;\right| \\
\mu_\perp = 0.
\end{array}
\right.$$ In general, the measurement errors $\sigma_\perp$ and velocity dispersion within the moving group $\sigma_{int}$ (expressed in mas/yr) conspire to make $\mu_\perp$ different from zero, but the expectation value of this quantity for any star belonging to a moving group is zero. If one assumes, following [@1999MNRAS.306..381D], that the error-weighted value of $\mu_\perp$, defined as $$\label{tperp}
t_\perp \equiv \frac{\mu_\perp}{\sigma_{tot}}$$ where $$\sigma_{tot} \equiv \sqrt{\sigma_\perp^2+\sigma_{int}^2},$$ is distributed normally with zero mean and unit variance for all stars in the moving group, then the probability distribution for a given combination of $\mu_\perp$ and $\sigma_{tot}$ to occur is $$\label{indivproba}
p_{ind} = \frac{1}{\sqrt{2\pi}}\exp(-\frac{t_\perp^2}{2}).$$ The CP method for finding a moving group within a cluster of $N$ stellar candidates, as implemented by [@1971MNRAS.152..231J], goes through the following steps [see also @1999MNRAS.306..381D].
1. First define a grid $(i, j)$ in the plane of the sky $(\alpha, \delta)$, where $i$ and $j$ vary from 1 to $N_{grid}$. The grid defines candidate CP positions.
2. At each grid point $(i,j)$, use Eq. \[tperp\] to compute the expression $$\label{X2}
X^2 = \sum_{k=1}^N (t_\perp^2)_k$$ where the index $k$ runs over all stars. With $t_\perp^2$ distributed normally, $X^2$ is distributed as $\chi^2$ with $N-2$ degrees of freedom. Minimizing $X^2$ is equivalent to maximizing the likelihood that the computed set of $t_\perp$ occurs, and the most likely CP is therefore the grid point with the lowest $X^2$ value.
3. However, it could still occur that the lowest value of $X^2$ is due to chance rather than to a good fit between observations and the model. To evaluate this possibility, one calculates the probability $p_{min}$ for $X^2$ to be higher than its computed value by chance even if the derived CP is the correct one. This probability is given by the incomplete gamma function $$\label{pmin}
p_{min} = \frac{1}{\Gamma[\frac{1}{2}(N-2)]}\int_{X^2}^\infty
x^{\frac{1}{2}(N-2)-1}\,\exp(-x)\,dx.$$
4. If $p_{min}$ is too low, then one rejects the star with the highest $\left|\,t_\perp\,\right|$, corrects the number of stars and goes back to step 2.
5. One then continues until $p_{min}$ has reached an acceptable value (to be discussed later). When this is done, all remaining stars are defined as *bona fide* group members and the most likely CP candidate in the last iteration is defined as the convergent point of the group.
As noted by [@1999MNRAS.306..381D], the basic equations of the method (Eqs. \[BasicEq\] and \[muparaandperp\]) remain valid if the stars belonging to the association are in a state of uniform expansion. In that case, however, using the proper motion information to determine the CP as we do here implies that the derived velocity $\vec{v}$ is the sum of the actual space motion and the reflex motion of the expansion, and these two motions cannot be disentangled.
Another caveat with the above method is its bias towards low $t_\perp$, which is unavoidable since on principle the CP search favors low values of this parameter. However, not all stars with low $t_\perp$ are necessarily moving group members; both stars with intrinsically small proper motions and stars located at large distances have a low $t_\perp$.
It is therefore customary to eliminate stars with insignificant proper motions prior to doing the analysis, because such objects would not be rejected in the CP search. The criterion for rejection [@1999MNRAS.306..381D] is $$\label{tmin}
t = \frac{\mu}{\sigma_\mu} = \frac {\sqrt{\mu_\alpha^2 \cos^2\delta
+\mu_\delta^2} }{ \sqrt{\sigma^2_{\mu_{\alpha \cos \theta}}
+\sigma^2_{\mu_\delta}}} \leq t_{min}$$ where $t_{min}$ depends on the quality of the data (see below). While this preliminary step eliminates a number of non-group members, it does nothing to correct the bias toward selecting the most distant stars in the group as members.
Modified CP method
------------------
The main difficulty that we encountered while using this method of finding a moving group of stars in Taurus is due to a combination of two properties of this region. First, the Taurus-Auriga star-forming region spans a large volume, but the YSOs are bunched in small groups with vast spaces devoid of stars between them rather than being distributed more evenly on the plane of the sky, as, e.g., older open clusters are. This specific morphology complicates the search for the CP, since the most likely CP position wanders in the plane of the sky depending on the groups of stars that dominate $X^2$ during a given iteration. As a consequence, a large number of possible group members may well be eliminated before $p_{min}$ reaches a reasonable value. Moreover, the remaining stars are the most distant ones, as explained above.
Second, there is a velocity dispersion between the different parts of the Taurus star-forming region of several km/s, which was first noted by [@1979AJ.....84.1872J] and confirmed by subsequent investigators. Although this velocity dispersion is taken explicitly into account in the CP method, it reduces both the computed individual probabilities $p_{ind}$ (Eq. \[indivproba\]) of stars being members of a moving group and the likelihood that the computed set of $t_\perp$ occurs, thus making it more difficult to reach a high $p_{min}$. Extensive Monte-Carlo simulations (cf. Sect. \[MCSims\]), which assumed a group of stars with the same positions as our sample and various sets of parameters, indeed confirmed that both the size of the recovered moving group and the probability of recovering the correct average distance for that moving group decreases with increasing internal velocity dispersion.
One possibility for considering the specificities of our sample would be to use the CP search procedure as described above while lowering the requirement on $p_{min}$ to a value in the range $10^{-1}$. However, we felt uncomfortable with settling for low $p_{min}$ values that do not confirm the reality of the derived moving group. For comparison, [@1999MNRAS.306..381D] chose $p_{min}$ equal to 0.954 in his study of moving groups, also on the basis of Monte-Carlo simulations.
Instead, we devised a modification of the CP method that helped us to find the moving group while giving very low probabilities that the solution was due to chance. The idea is to provide some initial guidance to the code by first guessing the CP position (exploiting the available radial velocity data) and eliminating those stars with the most obviously discrepant proper motions before proceeding with the classic CP search as described above.
For the initial guess of the CP position, we used the sub-sample of stars with known radial velocities $v_{rad}$ and the average, post-Hipparcos Taurus-Auriga parallax value derived by . Starting from Eq. \[vtan\], which defines the two tangential components $v_\alpha$ and $v_\delta$, we converted the velocity components in the equatorial system to a rectangular coordinate system $(x,y,z)$ in which $x$ points towards the vernal equinox, $y$ is directed towards the point on the equator with $\alpha = \pi/2$, and $z$ points towards the Northern equatorial pole. The conversion is done by the following transformation $$\left( \begin{array}{c} v_x \\ v_y \\ v_z \end{array} \right) =
\left( \begin{array}{ccc} -\sin\alpha & -\sin\delta\,\cos\alpha &
\cos\delta\;\cos\alpha \\
\cos\alpha & -\sin\delta\;\sin\alpha & \cos\delta\;\sin\alpha \\
0 & \cos\delta & \sin\delta \end{array} \right) \left(
\begin{array}{c} v_\alpha \\ v_\delta \\ v_{rad} \end{array}
\right).$$ An approximate CP location is then given by $$\left\{ \begin{array}{lll} \alpha_{CP} & = & \arctan
\left(\frac{\overline{\textstyle v_y}}{\overline{\textstyle v_x}}\right) \\ & & \\
\delta_{CP} & = & \arctan \left(\frac{\overline{\textstyle
v_z}}{\left(\overline{\textstyle v_x}^2 + \overline{\textstyle
v_y}^2\right)^{1/2}}\right)
\end{array} \right.$$ where $\overline{v_x}$, $\overline{v_y}$, and $\overline{v_z}$ are the averages of the velocity components over the sample of stars with known radial velocities.
Once this first guess of the CP coordinates was completed, we computed the individual probability of each star being part of a moving group converging to this point by using Eq. \[indivproba\] and eliminated those objects for which $p_{ind}$ was lower than a preset value.
We then proceeded with the usual CP method as described above. Monte-Carlo simulations with various sets of parameters reported in Sect. \[MCSims\] show that the convergence to a highly probable solution is very quick after the initial elimination of discrepant stars. Note also that the overall probability that the group is not due to chance is generally very close to one, and much higher when we perform such an initial screening than when using the usual CP method, which confirms that this simple procedure eliminates those interlopers whose proper motions are not compatible with the derived CP. Conversely, and this is the main drawback of the modified method, more *bona fide* group members are usually eliminated than when using the usual CP method, depending on the choice of computational parameters.
We should not expect this modified CP search method to produce results that are different from the classic method – both methods rely in essence on the same procedure – unless the initial CP guess biases the final result. This can be checked because the second step of the search allows for an iteration of the CP coordinates. In the Monte-Carlo simulations discussed in Sect. \[MCSims\], the final CP coordinates are usually close to the coordinates of the initial guess, thus confirming the lack of appreciable bias. We emphasize that the only advantage to this search strategy over the classic one for the stellar group under investigation is that it often leads to a converged solution, whereas the classic method fails to converge due to the specific properties of Taurus-Auriga discussed above.
Parallax computations
---------------------
Once we have defined a moving group, we can derive the kinematic parallaxes $\pi_{Vrad}$ of individual group members and their uncertainties, if we know their radial velocities $V_{rad}$. We have $$\pi_{Vrad} = \frac{A \mu_\parallel}{V_{rad} \tan \lambda}$$ where $\lambda$ is given by Eq. \[lambdadef\]. The error on $\lambda$ is computed as in [@1999MNRAS.306..381D], and standard error propagation techniques lead to a determination of the uncertainty on $\pi$.
In order to estimate approximate individual parallaxes of group members with unknown radial velocities, we derive the average spatial velocity $V_{group}$ from the Galactic velocity components of the stars with known radial velocities and use the relationship $$\pi_{Vgroup} = \frac{A \mu_\parallel}{V_{group} \sin \lambda}.$$ Again, one can derive the uncertainty on $\pi$ from the uncertainty on the group velocity, $\mu_\parallel$ and $\lambda$. A comparison of both parallax estimates is made in the section describing the results for Taurus-Auriga (Sect. \[KinematicAnalysisSection\]).
This ends the general description of the method used to find the moving group members and their individual parallaxes. In the following section, we present the extensive Monte-Carlo simulations that were used to test our alternate CP method and compare it to the classic one.
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Monte-Carlo simulations of the moving group search {#MCSims}
==================================================
Monte-Carlo simulations are often useful for exploring the role of the various parameters in numerical methods such as the CP search. They are also crucial for gaining confidence in the derived results.
J. [@1999MNRAS.306..381D] performed extensive simulations that convincingly demonstrate the ability of the CP method to find and eliminate interlopers, i.e., field stars that are seen projected against the stellar group but are not members of the moving group. The method also eliminates stars from the moving group that are actual members, and the computation parameters are chosen so as to find the largest number of likely group members.
In our simulations we focused mainly on how the classic CP method and its alternative deal with the large internal velocity dispersion found in our stellar sample, as well as with the relatively large uncertainties in the measured proper motions. These two aspects were not discussed by [@1999MNRAS.306..381D] since he was dealing with groups having low internal velocity dispersion and using accurate Hipparcos proper motions. We also investigated how interlopers affect our results.
Monte-Carlo simulation set-up
-----------------------------
In the following, we compare results obtained using the same data sets by the classic CP method, on one hand, and by its variant with an initial guess of the CP coordinates, on the other.
We constructed our synthetic groups of stars in the same way as [@1999MNRAS.306..381D]. The total number of stars that we used in these simulations was 117, corresponding to the sample common to both the and [@1988cels.book.....H] catalogs. This choice of sample is justified in Sect. \[KinematicAnalysisSection\]. Each synthetic star had the same equatorial coordinates as one of our program stars. For each realization, we randomly chose a group spatial velocity that is shared by all members of the data set. This was done by randomly drawing the velocity modulus in a given interval (typically between 5 and 10 km/s), as well as the two angles $\Theta$ and $\Phi$ defining its direction in the $(X,Y,Z)$ coordinate system. We note in passing that the approximate values of these parameters for the Taurus-Auriga group discussed in Sect. \[KinematicAnalysisSection\] are $V_{group} = 7.3$ km/s, $\Theta = 116^\circ$, and $\Phi = 171^\circ$. We have neglected in this derivation the small correction accounting for the difference between the Galactic rotation values for the Sun and the Taurus region.
In some simulations, we injected a certain percentage of field stars (between 3 and 10% – see discussion in the following section) in order to test the response of the code to their presence. This was done randomly and the proper motion values were assigned by randomly drawing a star from the sample of Hipparcos stars described in Sect. \[CatalogDescriptionSection\]. Since Hipparcos measurement uncertainties are lower than in our sample, we added measurement errors to these data using the procedure described below for moving group members. If the coordinates of the simulated interloper corresponded to a catalog star with known radial velocity, we assigned the observed value of the radial velocity and its uncertainty to the interloper.
Otherwise, stars were considered to be members of the moving group and we thus computed the expected values of the Galactic velocity components from the assumptions described above. We then added a randomly drawn velocity dispersion to each velocity component. These internal motion components were normally distributed with width $\sigma_{int}$. In many simulations, we used $\sigma_{int} \approx
6$ km/s, which is representative of the velocity dispersion of our stellar sample as found in proper motion surveys. The Taurus star-forming region is made up of subgroups of stars, the members of which are sometimes separated by only a few tenths of a parsec in projection. In these subgroups, the velocity dispersion of individual stars is much smaller than the dispersion between the velocities of various subgroups [cf. @1979AJ.....84.1872J]. In the simulations, we therefore assumed that velocities of stars closer than 1 pc in projection were the same except for measurement errors.
Once the velocity dispersion was added to the Galactic velocity components, the values $(U,V,W)$ were corrected for the solar motion, using $(U,V,W)_\odot\footnote{There is a large array of
post-Hipparcos values for the solar motion [see the discussion in
\cite{1998MNRAS.298..387D}], and we chose the values derived from
K0-K5 giants here because our PMS sample includes many stars in this
spectral range, although they are usually luminosity class IV-V
rather than III. These values are not widely different from the
``classical" solar motion values given, e.g., by
\cite{1968gaas.book.....M}.} = (9.88,14.19,7.76)$ km/s . The stellar parallax was then drawn from a normal distribution centered on the average parallax of Taurus with width equal to the standard error computed by : $\pi_{Taurus} = 7.31 \pm 0.49$ mas.
From there, proper motions and radial velocities were computed and measurement errors added to the results. In the simulations reported here, we drew random errors from a normal distribution with width equal to the average uncertainties of the observed proper motions and radial velocities. In other simulations not illustrated here, we used the observational uncertainties given in the catalog for the members of the moving group. The simulation results were similar to those reported here. Note that we assigned a radial velocity value only to those objects that simulated stars with known radial velocity in our data set. Finally, we added the Galactic rotation using the first-order formulae and the same Oort constants as [@1999MNRAS.306..381D].
After constructing such a synthetic data set, we ran the CP search and recorded various simulation parameters $(v_{group}, \Theta,
\Phi)$ and some of the results, notably the number of group members recovered by the search and the average group parallax, which can be directly compared to the known input parameters.
Results
-------
We constructed 500 such data-set realizations when running one Monte-Carlo simulation for a given set of CP-search parameters. Some results are reported below.
Table \[MCResults\] gives the main parameters of the simulation, i.e., the internal velocity dispersion $\sigma_{dis}$ in km/s, the average uncertainty of proper motions measurements $\sigma_{PM}$ in mas/yr, the requested minimum probability that the group is not due to chance, as well as the minimum individual probability of a star being a group member (see Sect. \[CPMethodSection\]). When the classic CP method is used, this parameter is not needed, and the entry is marked as $-$. The simulation results, also summarized in the table, are the probability $p_{\pi}$ of recovering the parallax within the assumed error bars for a given set of simulations, the percentage $f$ of cases for which the moving group was not detected, and the average number of recovered group members in percentage $\bar{N}/N_{total}$.
The main results are illustrated by two figures. Figure \[Fig10\] shows results of the classic CP search for Simulation 1, and Fig. \[Fig11\] shows a set of results obtained using the alternate CP search method, with parameters close to the ones we chose for the actual computations. These two figures have six panels each, and their contents are described in the caption of Fig. \[Fig10\].
Simulation 1 implements the classic CP method for group parameters similar to those of our data set. Table \[MCResults\] shows that the probability of recovering the true parallax of the moving group is only 0.32, and the failure rate $f$ of detecting the moving group is as high as 18%, whereas 84% of the moving group members are recovered when using this method.
Simulations 2 to 7 implement the modified CP method discussed in Sect. \[CPMethodSection\]. Results are given for three different values of the threshold value for the individual probabilities $p_{ind}$ of a star being a group member, for two values of the $(\sigma_{disp}, \sigma_{PM})$ parameter couple, and for three values of $N_{intl}/N_{total}$, the fraction of interlopers among the sample of stars. The results obtained with $p_{ind} = 0.6$ (Simulation 2) roughly correspond to those of the classic CP method in the sense that 84% of the group members are recovered, but the failure rate for finding a moving group is much lower ($<$ 2%), and the probability of recovering the right average parallax is 44%.
As expected, the number of recovered group members decreases with increasing $p_{ind}$, while the probability of recovering the true parallax reaches about 0.5 for $p_{ind} \approx 0.8$.
We should note here that the total number of stars in the sample has little influence on the results, as long as it is sufficiently large for this statistical search method to be meaningful. In addition to the simulations reported here, which used 117 stars, we performed simulations for the total sample of 217 stars. Results with this larger group differed little from those reported in this section, although the probabilities of recovering the right average parallax value were higher by a few %.
One also notes that the probability of detecting the moving group with this method is always very high once $p_{ind} > 0.6$. The method appears tolerant of an interloper fraction up to 10% (Simulations 5 and 6), which barely affects the results. In all cases we investigated, about half of the interlopers were rejected as non-members of the moving group, while the second half remained undetected.
The last simulation considers lower values of the internal velocity dispersion and proper motion measurement errors, and confirmed that the CP search is much more accurate when high quality data are available and the internal velocity dispersion is small. We should nevertheless emphasize here that there is no need to use this alternate CP search method when the internal velocity dispersion and proper motion measurement uncertainties are as low as in this last simulation; the classic CP search method works very well in such cases.
There is no apparent correlation between the average parallax derived from the CP search and the group velocity for the range of velocities (5 to 10 km/s) that we investigated here. For these low group velocities, there is a correlation between the derived parallax and the angles $\Theta$ and $\Phi$, in the sense that the dispersion of recovered parallaxes is smallest for in range of angles for which the contrast between reflex solar motion and group motion is highest.
Results of the alternate CP search method are sensitive to the threshold probability $p_{ind}$, which must be carefully chosen to achieve the best possible compromise between $p_{min}$ and the recovered number of group members. Extensive tests suggest an optimal threshold value in the range $0.7 - 0.8$ for our data set.
Parallaxes are computed in two ways. First, we use the radial velocity information to compute parallaxes for the subset of stars for which this quantity is known. Once this is done, we compute the average group velocity for this subset and then assume that it is the group velocity of all stars in the moving group. Approximate parallaxes can then be computed for all stars in the moving group. We compared the average moving group parallaxes obtained by these two methods (see Figs. \[Fig10\] and \[Fig11\]), and conclude that they give similar results within the computed uncertainties when the recovered average parallaxes are in the correct range (delimited by two vertical lines in the figures). The general bias of the CP method toward lower parallaxes that we discussed earlier is apparent in these figures.
Conclusion
----------
We have presented an alternate CP search method suitable for dispersed stellar groups with large internal velocity dispersions. For such groups, the classic CP method often fails to converge or to give realistic results. The alternate method is much less computationally intensive and nearly always converges when a moving group is present, provided the fraction of interlopers is not too large. However, it tends to eliminate more actual group members than the classic method does. Its use should therefore be limited to cases where the classic method is expected to fail.
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Application to Taurus-Auriga {#KinematicAnalysisSection}
============================
As discussed in Sect. \[CatalogDescriptionSection\], we should expect that an undefined, but probably significant, number of field stars are not distinguishable, as far as their kinematics are concerned, from the PMS stars that we are interested in. A consequence, demonstrated by the simulations reported in Sect. \[MCSims\], is that up to about 50% of those interlopers may remain undetected even though a moving group has been identified and hence may pollute the results. The best way to proceed is thus to preemptively screen out the main sequence stars that may possibly be present in our sample before searching for a moving group among the young stellar population. A critical discussion of the PMS status of the stars included in the catalog is thus in order. This is done in the next subsection, where we divide the Taurus-Auriga sample into two subsets, one for stars that are clearly of PMS nature, and the other for stars in a more uncertain evolutionary state. We then discuss the values chosen for the parameters that enter the computation, present the results of the kinematic analysis for both subsets of stars, and give the list of moving group members with their individual parallaxes.
Refining the PMS star sample
----------------------------
In their proper motion catalog for PMS stars, include a flag indicating the *“classification as given by CDS or found in the literature : T=T Tauri (CTT or WTT), A = HAeBe, Y = YSO, W = WTT, P = PMS, X = X-ray active source (Li 2000, table 2), O = Post T Tauri, L = Emission Line from HBC catalogue, Tc = T Tauri candidate, Wc = WTTs candidate. Ac = HAeBe candidate, Yc = YSOs candidate, Pc = PMS candidate".* It is already apparent from this list that the catalog might well be contaminated by main sequence stars, since many X-ray active main sequence stars exist and also since the evolutionary status of the various “candidates" above is unclear. A closer look at the list of catalogued objects unveils an even more serious problem because most X-ray sources that are detected in the general direction of Taurus are noted as PMS objects, although there is a large body of evidence suggesting that they are, at least in part, active main-sequence stars . Depending on what fraction of the stars detected through their X-ray emission are truly PMS, there could be a sizable number (perhaps up to 30%) of field stars in the sample of “PMS" stars discussed in Sect. \[CatalogDescriptionSection\].
We devised the following strategy to circumvent the problem created by the presence of potential field stars in our sample.
1. We chose a sub-sample of highly probable PMS objects by restricting the sample to stars contained in the [@1988cels.book.....H] catalog of confirmed Orion population members and ran the CP search for that group of 117 objects, thus looking for a *core* Taurus-Auriga moving group.
2. Using the derived CP coordinates, we computed the probability of each star in the full sample of 217 stars being a member of the core moving group and defined as possible additional members those stars whose probability of membership was sufficiently high.
3. We then verified, from a thorough literature search, the PMS status of stars belonging to this extended moving group in order to eliminate the possibly remaining interlopers.
The details of the process are given below.
Choice of computation parameters
--------------------------------
There are five such parameters: the number $N_{grid}$ of trial grid points for the CP search, the value $t_{min}$ that defines the magnitude of insignificant proper motions (see Eq. \[tmin\]), the velocity dispersion $\sigma_{int}$ that determines the basic quantity $t_\perp$ (Eq. \[tperp\]), the threshold probability $p_{min}$ (Eq. \[pmin\]) for deciding that the moving group is not a chance occurrence, and the threshold probability $p_{ind}$ (Eq. \[indivproba\]) of individual stars being considered as possible moving group members in the initial screening described above.
The number of grid points $N_{grid}$ defined over the plane of the sky determines the accuracy of the CP derived position. We typically used a $1000 \times 1000 \: (\alpha, \delta)$ grid for the final models and a $500 \times 500 \: (\alpha, \delta)$ grid for the Monte-Carlo simulations.
Rather than using $t_{min}$ as a truly free parameter, we estimated its value from the data quality, following a remark made by [@1999MNRAS.306..381D], who noted that one should basically reject all stars with proper motion $\mu$ smaller than $3
\sigma_\mu$, i.e.,
$$t_{min} \approx \frac{3
\sigma_\mu}{\sqrt{\sigma^2_{int}+\sigma^2_\mu}}.$$
From the average proper motion error $\sigma_\mu = 6$ mas/yr and a typical velocity dispersion $\sigma_{int} = 6$ km/s, we derived $t_{min}= 1.66$ at the mean distance of Taurus. Using this cut-off value, 11 among the 117 stars of our sample have insignificant proper motions. Table \[InsignificantPM\] lists these stars, which were thus eliminated from the moving group search. (see Sect. \[CPMethodSection\]). Their number in the [@1988cels.book.....H] catalog (HBC) is also given.
In our test computations, we assumed the velocity dispersion $\sigma_{int}$ to be in the range 5 to 6 km/s, in agreement with the value derived by [@1979AJ.....84.1872J] for the velocity dispersion between subgroups of stars in the Taurus cloud. As discussed by these authors, the internal velocity within the stellar groupings themselves is much lower than this value. Note that the proper motion uncertainty $\sigma_{\mu}$, which also enters $t_\perp$, is given in the catalog for each star.
We tested a wide range of values for probability $p_{min}$, which assesses the reality of the moving group. Even for values of $p_{min}$ as high as 0.95, some objects with obviously discrepant galactic velocities were found to pollute our results, so we concluded that a value very close to 1 was needed given the measurement errors and the large internal velocity dispersion in Taurus.
The threshold probability $p_{ind}$ required for individual stars to be considered members of the moving group was typically set between 0.7 - 0.8. As shown by the Monte-Carlo simulations (see Sect. \[MCSims\]), the value of this important parameter largely determines both the final probability $p_{min}$ that the group is not due to chance and the final size of the moving group.
A core Taurus-Auriga moving group {#CoreGroup}
---------------------------------
It is important to realize that the final size of a moving group is not a fixed number but is instead determined by the final realization probability that one decides to adopt. Our approach here was conservative, as we chose to enforce $p_{min} \ge 1 - 10^{-10}$ in order to estimate the size of the Taurus moving group.
We adopted this conservative attitude in spite of the fact that it minimizes the size of the moving group because we are interested less in the number of individual parallaxes that we derive than in the credibility of the moving group. Because moving group members share a common destiny, we will argue in an upcoming paper (Bertout & Siess, in preparation) that the moving group found here is more homogeneous and more significant in a statistical sense than the overall group of Taurus-Auriga YSOs that has been traditionally used for investigations of the global properties of this region.
In this way, we defined a highly probable moving group of 83 stars or stellar systems in Taurus-Auriga for which we could determine individual parallaxes. The corresponding threshold value of $p_{ind}$ is 0.78 for our sample. This procedure is likely to eliminate some *bona fide* moving group members, as the results of Monte-Carlo simulations of Sect. \[MCSims\] show, but it does define a core Taurus moving group that can be regarded as real with a high degree of confidence.
The derived CP coordinates for the moving group are
$\left\{ \begin{array}{lll}
\alpha_{CP} & = & 79.88^\circ \pm 0.01^\circ \\
\delta_{CP} & = & -16.74^\circ \pm 0.02^\circ,
\end{array}
\right.$
which confirms our suspicion while inspecting Fig. \[Fig1\] that the proper motion vectors were often pointing toward the lower left corner of the figure. Note that the high accuracy of the derived CP coordinates was made possible by zooming in on the region surrounding the CP (while keeping the same number of grid points) once it was approximately located.
### Kinematic properties of group members with known radial velocities
Table \[VradPar\] lists the parallaxes for core group members with known radial velocities. Columns 1 and 2 list star names together with their HBC number, while Columns 3 and 4 give the individual parallax and distance (with their uncertainties) computed from the individual stellar radial velocity in addition to the proper motion information. The symbol “:” indicates uncertain values for two stars that will be discussed in Sect. \[DiscussionSection\].
A histogram with bin size equal to 0.6 mas of derived parallaxes for stars with known radial velocities is displayed in Fig. \[Fig15\] together with a Gaussian fit to the data. The peak of the Gaussian curve is at $\pi = 7.14 \pm 0.04$ mas, corresponding to a distance of 140 pc, and its HWHM is equal to $0.68 \pm 0.05$ mas. The reduced $\chi^2$ of the Gaussian fit is 1.16. The apparently normal distribution of the data suggests that no systematic bias affects the derived parallaxes. The average values and standard deviations of the Galactic velocity components $U, V, W,$ and $V_{space}$ are
$\left\{ \begin{array}{lll}
U & = & -16.45 \pm 4.56 \: \rm{km/s} \\
V & = & -13.18 \pm 2.52 \: \rm{km/s} \\
W & = & -10.97 \pm 4.04 \: \rm{km/s} \\
V_{space} & = & 23.95 \pm 5.87 \: \rm{km/s}.
\end{array}
\right.$
Table \[GroupGalVel\] lists the name and HBC number, as well as the values of the derived Galactic coordinates (Columns 3 to 5) and Galactic velocity components with their uncertainties (Columns 6 to 8) for the 67 members of the moving group – both core members and confirmed candidates (see Sect.\[AddCandidates\] below) – with known radial velocities, i.e., those stars for which the derived parallaxes are most accurate. This table can be used to identify individual group members in the figures.
Figure \[Fig12\] displays the projected velocities on the $XY$, $XZ$, and $YZ$ planes for the 67 same group members. Figure \[Fig13\] shows their spatial distribution. The stars are distributed in a $X,Y,Z$ cube with sides of approximately 60pc, and filamentary structure, reminiscent of the Taurus molecular cloud structure, can be seen in all three $(X,Y)$, $(Y,Z)$, and $(X,Z)$ projection planes. In the $(X,Z)$ plane, there is some tendency for the densest subgroups of stars to align roughly in the direction away from the Sun. This is an expected result of the changed parallax values, since the stars must maintain their apparent distribution in the plane of the sky, as seen from Earth. Note that individual stars can be identified by their coordinate values given in Table \[GroupGalVel\]. Finally, Fig. \[Fig14\] displays histograms of the $U, V, W$ galactic velocity components and spatial velocities of the members of the Taurus moving group with known radial velocities.
### Approximate parallaxes for other moving group members
We now assume that all members of the moving group share the same average value of $V_{space}$ derived above. This hypothesis allows us to compute *expected* radial velocities for all stars in the moving group and subsequently to derive tentative parallaxes for those group members.
Figure \[Fig16\] compares the parallax values obtained by both methods and shows that they give similar results within the error bars, but the scatter is large. There is also a bias in the derived parallaxes in the sense that small parallaxes are underestimated, and large parallaxes are overestimated, when determined from the group spatial velocity, as shown by the linear fit to the data (dotted line in Fig. \[Fig16\]). According to this regression curve, the parallax average and standard deviation computed using radial velocities, $6.99 \pm 1.14$ mas, translate to $6.74 \pm 1.44$ mas when using the average spatial velocity for computing the parallax; not only is the resulting parallax smaller on average but its standard deviation is larger. The parallaxes deviating most from the mean in Table \[VspacePar\] should therefore be viewed with extreme caution. A case in point is the couple FY/FZ Tau, two stars separated by less than 20 on the sky whose derived parallaxes are respectively the smallest and the largest of our sample. The result for FY Tau is obviously unrealistic, as both stars are seen in projection on the B18 nebula and have the same line-of-sight visual extinction.
The average parallax and associated standard deviation of stars in Table \[VspacePar\] are $\pi = 7.21 \pm 1.71$ mas, corresponding to an average distance of 139 pc. The average distance of these stars is similar to that of stars for which we know the radial velocities, but the parallax standard deviation is almost twice as large, as anticipated from the regression analysis. Although the derived parallaxes can only be considered as tentative and would be much improved, and their uncertainties reduced, if accurate spectroscopic radial velocities were available for the full sample, this approximation nevertheless may provide a useful first estimate of the distance to those moving group members whose parallax is not too different from the mean (e.g., DG Tau, XZ Tau, and UZ Tau).
Additional group member candidates {#AddCandidates}
----------------------------------
After finding a core moving group among the pre-main sequences stars of Taurus-Auriga, we now look for plausible additional members among stars in the catalog whose evolutionary status is uncertain. We thus now consider the full sample of 217 stars discussed in Sect. \[CatalogDescriptionSection\] and identify those whose space motion is compatible with that of the core moving group.
### Analysis and results
Specifically, we assumed that the CP of the entire moving group has the coordinates found above for the core moving group and computed the individual probability of each star being part of the moving group converging to this point by using Eq. \[indivproba\]. A $p_{min}$ minimum value of 0.91 was chosen because it is the lowest value to allow for the rejection of all those stars in the subgroup of confirmed PMS objects that are not actual members of the core moving group. Stars with a lower $p_{min}$ were thus eliminated, leaving us with 30 stars whose space motion is compatible with that of the core moving group.
Among these 30 objects, 15 have known radial velocities; their parallaxes are given in Table \[CandVradPar\] and the kinematic properties of confirmed members (see below) are listed in Table \[GroupGalVel\]. As previously, we computed the spatial velocity of this sample to compute approximate parallaxes for the remaining 15 stars, and we give these tentative parallaxes in Table \[CandVspacePar\].
### Evolutionary status of candidate members
The ROSAT-detected stars included in Table \[CandVradPar\] were studied in some detail by , who assessed their evolutionary status on the basis of their lithium line strengths. The two other stars, GSC 01262-00421 and V1078 Tau, were studied by [@1988AJ.....96..297W], who concluded, also from their lithium strength, that they were *bona fide* PMS objects. A “y” in last column of Table \[CandVradPar\] indicates that the star is a PMS object on the basis of these two works. We thus find that 9 of of the 15 stars are confirmed PMS objects and thus members of the Taurus-Auriga moving group.
Most stars included in Table \[CandVspacePar\] have not been studied in any kind of detail, so assessing their evolutionary status on the basis of the available information is a challenge. All stars marked with “y?" in the last table column are IRAS point sources [@1989AJ.....97.1451S] and are therefore likely to be associated with circumstellar matter, which strongly hints at PMS status; however, spectroscopic confirmation is unavailable. 1RXS J035330.5+263152 is the X-ray counterpart of a likely member of the Pleiades corona , while RX J0412.8+2442 is unlikely to be young according to . KPNO-Tau 11 is a recently-detected low-mass member of the Taurus star-forming region [@2003ApJ...590..348L], while RX J0431.3+2150 was originally proposed as a PMS object by , and this status was confirmed by . EZ Tau is a flare star and probable low-mass member of the Hyades [@1993MNRAS.265..785R]. 1RXS J050029.8+172400 was classified as a T Tauri star on the basis of its X-ray variability by , but this needs to be confirmed by optical spectroscopy. Finally, there is not enough information to assess the status of RX J0456.6+3150 and 1RXS J051111.1+281353. The second star was classified as T Tauri star by [@2004ChJAA...4..258L] on the basis of its proper motion, but we have seen that this criterion is not discriminatory. Altogether, only 2 of these 15 stars are confirmed members of the moving group, while 8 are possibly additional members.
Note on X-ray selected stars located south of Taurus
----------------------------------------------------
Finally, we searched for putative moving group members in the region south of Taurus where a large population of X-ray sources possibly related to the Taurus-Auriga star-forming region was detected by the ROSAT All-Sky X-ray survey . We thus considered the region $3^h 50^m$ $\alpha(2000)$ $5^h
10^m$ and $0$$\delta(2000)$ $15$, where the catalog lists 47 stars, and did the same analysis as in Sect. \[AddCandidates\] thus searching for additional members of the core moving group defined in Sect. \[CoreGroup\]. In this way, we identified 7 stars whose space motion is compatible with that of the core moving group. These are RX J0357.3+1258, RX J0358.1+0932,RX J0404.4+0519, RX J0450.0+0151, RX J0405.5+0324, RX J0441.9+0537, and RX J0445.3+0914.
Since the kinematic analysis is not sufficient for identifying moving group members with any kind of certainty, we searched the literature to determine the evolutionary status of these objects. The first four objects were studied among other PMS candidates by , who found that RX J0450.0+0151 is a likely PMS star while the other stars are on the main sequence. The remaining stars are part of a sample studied by , who list them as main sequence objects.
With the possible exception of RX J0450.0+0151, for which we derive a parallax of $11.84 \pm 1.33$ mas corresponding to a distance of $84^{+11}_{-9}$ pc, we thus conclude that the X-ray sources located south of Taurus that have proper motions compatible with those of the core Taurus-Auriga moving group are field stars unrelated to the Taurus-Auriga PMS association.
A posteriori assessment of results
----------------------------------
To assess the validity of the results presented above, we performed additional Monte Carlo simulations of the core moving group search using the results derived above. For this, we constructed synthetic data sets as explained in Sect. \[MCSims\]; but instead of drawing a random velocity vector, we used the velocity of the moving group as derived from the CP search, i.e., $V_{group} = 7.3$ km/s, $\Theta
= 116$, and $\Phi = 171$. To define the individual velocity of each synthetic star, we added to the group velocity vector an individual random velocity drawn from a normal distribution with variance $\sigma_{int}$ and derived the resulting proper motions and radial velocities from there. We then added random measurements errors and corrected for the Galactic rotation as explained in Sect. \[MCSims\]. Because the considered sample of 117 Taurus-Auriga stars used in the CP search is made up of confirmed pre-main sequence stars[^3], we did not include interlopers in these simulations.
We constructed 500 such realizations in our simulation, which we ran with the same computational parameters as the actual CP search of Sect. \[KinematicAnalysisSection\]. The results of these simulations are illustrated by Fig. \[Fig17\], which demonstrates that the probability of recovering the average parallax of a moving group with Taurus-Auriga properties is 0.38 and that the average number of stars in the recovered moving group is $85 \pm 5$, whereas we found 83 members for the core moving group in the actual solution. The probability that the CP method cannot find a moving group in this situation is 0.05. The histogram of derived parallaxes can be fitted by a Gaussian curve peaking at $7.06 \pm 0.04$ mas with HWHM equal to $0.86 \pm 0.06$ mas, whereas the input values for these two quantities were respectively 7.31 and 0.49 mas. Our actual computation recovers the average parallax of Taurus because we hit on one of the best possible solutions for the CP coordinates, as shown below.
The derived CP coordinates in the 500 simulations are shown in Fig. \[Fig18\]. As previously discussed, e.g., by [@1999MNRAS.306..381D] CP coordinates fall along the great circle associated with the CP, which is accurately defined by the proper motions, but the precise location of the CP on this great circle is quite uncertain. For comparison, we also plot in Fig. \[Fig18\] the $X^2$ values (as defined by Eq. \[X2\]) that measure the probability of finding the CP at a given location on the plane of the sky. The curves plotted here correspond to the solution found in Sect. \[KinematicAnalysisSection\], and the white star indicates the coordinates of the derived Taurus-Auriga core moving group CP. The topology of the $X^2$ surface seen here is typical of the CP method. A large fraction of the sky forms a high plateau (in terms of $X^2$) cut by a deep valley following the great circle associated with the CP, with an elongated minimum following the great circle at the location of the CP. The height of the plateau and the depth of the valley are directly related to the velocity dispersion and measurements uncertainties in the sense that reducing these values increases the contrast between plateau and valley, thus making it easier to locate the CP precisely.
The lowest $X^2$ contour in Fig. \[Fig18\] corresponds to a value of 13 and includes most CP realizations. With an $X^2$ of 10.6, the coordinates of the CP derived for Taurus-Auriga obviously represent one of the best possible solutions, thus validating the choice of computational parameters used to find it. However, we expect only 73% of the actual moving group members to be recovered in the computation. Therefore, most stars in our core sample are likely to be actual members of the moving group even though they were rejected during the CP search. In order to identify these potentially additional moving group members and derive their individual parallaxes, more precise radial velocities and proper motions will be necessary.
Remarks on positions and parallaxes {#DiscussionSection}
===================================
A first check of our results is provided by a comparison with Hipparcos parallaxes.
Comparison with Hipparcos results
---------------------------------
As already mentioned in Sect. \[CatalogDescriptionSection\], there is little overlap between Taurus YSOs and Hipparcos targets, due mainly to the faintness of most Taurus PMS stars. We found 11 stars among our moving group that have been observed by Hipparcos and listed them in Table \[HIPComp\], together with the parallaxes derived in this work and the Hipparcos parallaxes. The values agree within the error bars except for 3 objects: BP Tau, DF Tau, and UX Tau, for which Hipparcos reports a negative parallax. The discrepant Hipparcos parallaxes of BP Tau and DF Tau were discussed in some detail by , who concluded from a re-analysis of the Hipparcos data that the derived large parallaxes were unlikely to be significant, although they could not be ruled out entirely. also re-computed a more precise parallax of $7.98 \pm 3.15$ mas for RW Aur after rejecting the bad abscissae. We thus conclude that there is a general agreement between the parallaxes derived here and the Hipparcos values except for BP Tau and DF Tau. The values reported in this work for these two objects are much more in line with expectations than the Hipparcos values, since the two stars are clearly associated with the molecular cloud.
We restrict the following discussion to those members of the moving group with measured radial velocities since we derived accurate parallaxes for these objects, whereas the parallaxes derived for other stars from the group’s average spatial velocity are tentative.
Notes on remarkable stars
-------------------------
NTTS 035120+3154 (HBC 352/353) provides an interesting illustration of the current limitations of the study reported here. According to , the SW and NE components of this object are likely to form a physical pair. The proper motions and radial velocities of both stars, given in Table \[NTTS\], agree within the error bars; and yet, we derive distances (see Table \[VradPar\]) that, while they agree with each other within the derived error bars, differ by 34 pc, i.e., 14% of the mean distance value of 242 pc. This uncertainty thus appears representative of what can be done with the current data. Assuming that the radial velocities were equal within 0.1 km/s, one finds that the gap between distances is reduced to 22 pc, or 9% of the mean distance. This increased accuracy is thus within easy reach since it only requires accurate radial velocities. On the other hand, this also shows that without much more accurate proper motions measurements, the uncertainty on derived distances cannot be expected to decrease much below 10%.
NTTS 042835+1700 (HBC 392) appears to be the closest star in the moving group at 106 pc (2.1$\sigma$ from the average moving group distance), while NTTS 043124+1824 (HBC 407) is the farthest away at 360 pc, i.e., 3.7$\sigma$ away from the average. The proper motion values of HBC 407, an apparently single star , are indeed very small, with $\mu_\alpha
\cos \delta = 0 \pm 2$ mas/yr and $\mu_\delta = -7 \pm 2$ mas/yr. Its radial velocity of 18.4 km/s is not very different from the average of 16 km/s and it is therefore difficult to understand how the star could have reached this location in space if it is a true member of the Taurus-Auriga moving group. Its PMS nature has been confirmed by [@1988AJ.....96..297W] and others, and it is thus unlikely to be an interloper. Its discrepant parallax, if true, remains therefore somewhat of a mystery, but we note that the computed uncertainty on its value is particularly high.
CZ Tau (HBC 31) is another interesting case because it is located at less than 20 from DD Tau and has the same proper motion yet its radial velocity is highly discrepant (compared to the Taurus-Auriga average) at 44 km/s. The radial velocity measurements for both objects go back to the classic paper where [@1949ApJ...110..424J] discussed the class of T Tauri stars for the first time, and they are noted as very uncertain in the [@1988cels.book.....H] catalog. The derived parallaxes are therefore also very uncertain so we marked them as such in Table \[VradPar\].
Properties of parallaxes for various YSO sub-classes
----------------------------------------------------
Figure \[Fig19\] displays the members of the moving group as seen in the plane of the sky. The moving group contains 23 classical T Tauri stars (CTTSs) and 43 confirmed weak emission-line T Tauri stars (WTTSs) with known radial velocities. Among the WTTSs, we have 18 X-ray selected WTTSs (XWTTSs) and 25 optically-selected WTTSs (OWTTSs). A comparison of the parallaxes of stars in these subgroups confirms that X-ray selected WTTSs tend to be found on the outskirts of the molecular clouds where extinction is relatively low, while CTTSs (and OWTTSs) are associated with the denser regions of the clouds (see also the parallax histograms of the various YSO populations in Fig. \[Fig20\]).
An interesting, although not unexpected new finding of our investigation is that WTTSs are located not only in front of the clouds, but also at their back, while CTTSs are confined to the central parts of the moving group. This is illustrated by Fig. \[Fig20\], which displays histograms of the CTTS and WTTS parallaxes. All CTTSs are at distances between 126 and 173 pc, while WTTSs[^4] span the range of distances between 106 and 259 pc.
Computing average parallaxes and standard deviations for these various subgroups, we get $$\begin{aligned}
\overline{\pi_{CTTS}} & = & 7.01 \pm 0.67 {\rm mas} \nonumber \\
\overline{\pi_{WTTS}} & = & 7.00 \pm 1.15 {\rm mas} \nonumber \\
\overline{\pi_{OWTTS}} & = & 7.28 \pm 0.65 {\rm mas} \nonumber \\
\overline{\pi_{XWTTS}} & = & 6.60 \pm 1.55 {\rm mas}, \nonumber\end{aligned}$$ where we note that the standard deviation of the XWTTS parallaxes is more than twice as large as those of CTTSs and OWTTSs.
For the sample of CTTSs and OWTTSs with known radial velocities, the individual parallaxes of which are presumably more accurate than those computed from the group spatial velocity, we recover the average parallax value derived by from Hipparcos data. Our analysis thus confirms that the core of the Taurus association, as defined by this CTTS and OWTTS sample, is located at $140^{+14}_{-12}$ pc.
That we are actually observing WTTSs on both sides of the molecular clouds may seem at first to contradict the idea that X-ray selected stars tend to be detected in low-extinction regions. However, one must recall that extinction is very patchy in the Taurus-Auriga star-forming region, characterized by a dense filamentary structure and zones of lower molecular gas density. To test whether XWTTSs are indeed preferentially found along low-extinction lines of sight, we used the extinction maps of [@1998ApJ...500..525S] to derive the line-of-sight (LOS) visual extinction in the direction of each moving group star. Figure \[Fig21\] shows that XWTTSs are on the average located on lines of sight that have lower visual extinctions than OWTTSs and CTTSs. The horizontal dash-dotted line indicates the upper limit of LOS extinction for XWTTSs of 2 mag. Not surprisingly, there is no significant difference between the LOS extinctions of OWTTS and CTTS subgroups. Note that no star farther away than $\approx 180$ pc (except for one star, CZ Tau, the parallax of which is highly uncertain as discussed above) has a LOS extinction higher than 1.5 magnitudes. The upper envelope of the data points in Fig. \[Fig21\] (dotted line) indicates approximately how far one “sees” in the Taurus star-forming region for a given LOS extinction. Lines of sight where the extinction is in the range $2
\leq A_V \leq 20$ mag. become opaque in the optical domain at distances in the range $210 \geq d \geq 120$ pc. From these numbers, and using the conversion factor from extinction to hydrogen column density given by , one finds an average hydrogen density of $\approx 2 \cdot 10^2$ cm$^{-3}$ in Taurus, a value that agrees with expectations for large molecular clouds.
Conclusions {#ConclusionSection}
===========
We have identified a moving group of 94 stars in the Taurus-Auriga association that defines the kinematic properties of the T association. Because the variant of the CP search method that we developed for dealing with the high internal velocity dispersion among Taurus subgroups possibly eliminates a number of potential group members, we detected a *minimum* moving group that may not contain all the kinematically associated stars in the region.
Determination of accurate parallaxes for all moving group members is hampered by the lack of observed high-precision radial velocities for many group members. New radial velocity measurements for all stars of the proper-motion catalog are obviously needed in order to make further progress in deriving accurate parallaxes for individual YSOs in T associations. We therefore encourage observers to make use of the highly efficient and precise spectrographs employed, in particular, for extra-solar planet searches to perform a complete radial velocity survey of Taurus-Auriga and other star-forming regions.
The present result nevertheless represents a first step towards better understanding the distances to Taurus-Auriga PMS stars, and we use these new distances to re-assess the physical properties of members of the Taurus-Auriga moving group in a companion paper (Bertout & Siess, in preparation).
We are grateful to Ulrich Bastian, George Herbig, and Steven Shore for their careful reading of a previous version of the manuscript and for useful comments. We are indebted to an anonymous referee for pointing out that the interloper problem was likely to be much more severe than we had anticipated, which led to a much improved analysis. This research made advanced use of the Centre de Données de Strasbourg facilities in the framework of a test program aiming at improving the ease of use and inter-operability of data mining tools. We acknowledge use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^1]: Note that we assume implicitly throughout this section that Hipparcos stars are reasonable proxies for the fainter field star population that could contaminate the PMS catalog.
[^2]: UCDs are standardized descriptors of astronomical quantities defined by the International Virtual Observatory Alliance; cf. [http://www.ivoa.net/Documents/latest/UCDlist.html.]{}
[^3]: Except for Wa Tau/1 (HBC 408), whose pre-main sequence status is questionable but which was duly eliminated in the CP search.
[^4]: We excluded here the two stars discussed above, CZ Tau and NTTS 043124+1824, because their parallaxes are highly uncertain.
|
---
author:
- '**A.I. Golikov and I.E. Kaporin**'
title: Inexact Newton method for minimization of convex piecewise quadratic functions
---
Introduction {#intro}
============
The present paper is devoted to theoretical and experimental study of novel techniques for incorporation of preconditioned conjugate gradient linear solver into inexact Newton method. Earlier, similar method was successfully applied to optimizaton problems arising in numerical grid generation [@GK99; @GKK04; @Ka03], and here we will consider its application to the numerical solution of piecewise-quadratic unconstrained optimization problems [@GGEN09; @KMPN17; @Ma02; @Ma04]. The latter include such problems as finding the projection of a given point onto the set of nonnegative solutions of an underdetermined system of linear equations [@GGE18] or finding a distance between two convex polyhedra [@Bo89] (and both are tightly related to the standard linear programming problem). The paper is organized as follows. In Section 2, a typical problem of minimization of piecewise quadratic function is formulated. In Section 3, certain technical results are given related to objective functions under consideration. Section 4 describes an inexact Newton method adjusted to the optimization problem. In Section 5, a convergence analysis of the proposed algorithm is given with account of special stopping rule of inner linear conjugate gradient iterations. In Section 6, numerical results are presented for various model problems.
Optimization problem setting {#underdet}
============================
Consider the piecewise-quadratic unconstrained optimization problem $$p_* = \arg\min_{p\in R^m}
\left(\frac12\|(\widehat x + A^{\rm T}p)_+\|^2 - b^{\rm T}p\right),
\label{prob_set}$$ where the standard notation $\xi_+=\max(0,\xi)=(\xi + |\xi|)/2$ is used. Problem (\[prob\_set\]) can be viewed as the dual for finding projection of a vector on the set of nonnegative solutions of underdetermined linear systems of equations [@GGE18; @GGEN09]: $$x_* = \arg\min_{^{Ax=b}_{x\ge 0}} \frac12 \|x-\widehat x\|^2,$$ the solution of which is expressed via $p_*$ as $x_* = (\widehat x + A^{\rm T}p_*)_+$. Therefore, we are considering piecewise quadratic function $\varphi:R^m\rightarrow R^1$ determined as $$\varphi (p) =
\frac12\|(\widehat x + A^{\rm T}p)_+\|^2 - b^{\rm T}p,
\label{phi}$$ which is convex and differentiable. Its gradient $g(p) = {\rm grad}~p$ is given by $$g(p) = A(\widehat x + A^{\rm T}p)_+ - b,
\label{grad}$$ and it has generalized Hessian [@HSN84] $$H(p) =
A{\rm Diag}\left({\rm sign}(\widehat x + A^{\rm T}p)_+\right)A^{\rm T}.
\label{genh}$$ The relation of $H(p)$ to $\varphi(p)$ and $g(p)$ will be explained later in Remark 1.
Taylor expansion of $(\cdot)_+^2$ function
==========================================
The following result is a special case of Taylor expansion with the residual term in integral form.
[LEMMA 1.]{} [*For any real scalars $\eta$ and $\zeta$ it holds $$\frac12((\eta+\zeta)_+)^2 - \frac12(\eta_+)^2 - \zeta\eta_+
= \zeta^2\int_0^1\left(\int_0^1 {\rm sign}(\eta+st\zeta)_+ds\right)tdt.
\label{intexp1}$$* ]{}
[PROOF.]{} Consider $f(\xi)=\frac12(\xi_+)^2$ and note that $f'(\xi)=\xi_+$ and $f''(\xi)={\rm sign}(\xi_+)$ (note that $f''(0)$ can formally be set equal to any finite real number, and w.l.o.g. we use $f''(0)=0$). Inserting this into the Taylor expansion $$f(\eta+\zeta) = f(\eta) + \zeta f'(\eta) +
\zeta^2 \int_0^1 \left(\int_0^1 f''(\eta+st\zeta)ds \right) tdt$$ readily gives the desired result.
[LEMMA 2.]{} [*For any real $n$-vectors $y$ and $z$ it holds $$\frac12\|(y+z)_+\|^2-\frac12\|y_+\|^2-z^{\rm T}y_+
= \frac12 z^{\rm T}{\rm Diag}(d)z,
\label{intexp1a}$$ where $$d = \int_0^1\left(\int_0^1 {\rm sign}(y+stz)_+ds\right)2tdt.
\label{intexp1b}$$* ]{}
[PROOF.]{} Setting in (\[intexp1\]) $\eta=y_j$, $\zeta = z_j$, and summing over all $j=1,\ldots,n$ obviously yields the required formula. Note that the use of scalar multiple 2 within the integral provides for the estimate $\|{\rm Diag}(d)\| \le 1$.
[LEMMA 3.]{} [ *Function (\[phi\]) and its gradient (\[grad\]) satisfy the identity $$\varphi(p+q)-\varphi(p)-q^{\rm T}g(p)
= \frac12 q^{\rm T}A\,{\rm Diag}(d)A^{\rm T}q,
\label{intexp2a}$$ where $$d = \int_0^1\left(\int_0^1
{\rm sign}(\widehat x + A^{\rm T}p + stA^{\rm T}q)_+ds\right)2tdt.
\label{intexp2b}$$* ]{}
[PROOF.]{} Setting in (\[intexp1a\]) and (\[intexp1b\]) $y=\widehat x + A^{\rm T}p$ and $z = A^{\rm T}q$ readily yields the required result (with account of cancellation of linear terms involving $b$ in the left hand side of (\[intexp2a\])).
[REMARK 1.]{} As is seen from (\[intexp2b\]), if the condition $${\rm sign}(\widehat x + A^{\rm T}p + \vartheta A^{\rm T}q)_+
= {\rm sign}(\widehat x + A^{\rm T}p)_+,
\label{nearsol}$$ holds true for any $0\le\vartheta\le 1$, then (\[intexp2a\]) is simplified as $$\varphi(p+q) - \varphi(p) - q^{\rm T}g(p) = \frac12 q^{\rm T}H(p)q,
\label{locquad}$$ where the generalized Hessian matrix $H(p)$ is defined in (\[genh\]). This explains the key role of $H(p)$ in the organization of the Newton-type method considered below. Note that a sufficient condition for (\[nearsol\]) to hold is $$|(A^{\rm T}q)_j| \le |(\widehat x + A^{\rm T}p)_j| \qquad
{\rm whenever} \qquad (A^{\rm T}q)_j (\widehat x + A^{\rm T}p)_j < 0;
\label{nearsol2}$$ that is, if certain components of the increment $q$ are relatively small, then $\varphi$ is exactly quadratic (\[locquad\]) in the corresponding neighborhood of $p$.
Inexact Newton method for dual problem
======================================
As suggests condition (\[nearsol\]) and its consequence (\[locquad\]), one can try to numerically minimize $\varphi$ using Newton type method $p_{k+1} = p_k - d_k$, where $d_k = H(p_k)^{-1}g(p_k)$. Note that by (\[locquad\]) this will immediately give the exact minimizer $p_* = p_{k+1}$ if the magnitudes of $d_k$ components are sufficiently small to satisfy (\[nearsol2\]) taken with $p=p_k$ and $q=-d_k$. However, initially $p_k$ may be rather far from solution, and only gradual improvements are possible. First, a damping factor $\alpha_k$ must be used to guarantee monotone convergence (with respect to the decrease of $\varphi(p_k)$ as $k$ increases). Second, $H(p_k)$ must be replaced by some appropriate approximation $M_k$ in order to provide its invertibility with a reasonable bound for the inverse. Therefore, we propose the following prototype scheme $$p_{k+1} = p_k - \alpha_k M_k^{-1}g(p_k),$$ where $$M_k = H(p_k) + \delta {\rm Diag}(AA^{\rm T}).
\label{reg_hess}$$ The parameters $0< \alpha_k \le 1$ and $0\le \delta \ll 1$ must be defined properly for better convergence. Furthermore, at initial stages of iteration, the most efficient strategy is to use approximate Newton directions $d_k \approx M_k^{-1}g(p_k)$, which can be obtained using preconditioned conjugate gradient (PCG) method for the solution of Newton equation $M_kd_k=g(p_k)$. As will be seen later, it suffices to use any vector $d_k$ which satisfies conditions $$d_k^{\rm T}g_k = d_k^{\rm T}M_kd_k =\vartheta_k^2g_k^{\rm T}M_k^{-1}g_k
\label{newt_dir}$$ with sufficiently separated from zero. For any preconditioning, the approximations constructed by the PCG method satisfy (\[newt\_dir\]), see Section [\[PCG\]]{} below. With account of the Armijo type criterion $$\varphi(p_k - \alpha d_k) \le
\varphi(p_k) - \frac{\alpha}{2} d_k^{\rm T}g(p_k),
\qquad \alpha\in\lbrace{1,~1/2,~1/4,~\ldots\rbrace},
\label{back_trk}$$ where the maximum steplength $\alpha$ satisfying (\[back\_trk\]) is used, the inexact Newton algorithm can be presented as follows:
[*Input:*]{}
$A \in R^{m\times n}$, $b\in R^m$, $\widehat x\in R^m$;
[*Initialization:*]{}
$\delta = 10^{-6}$, $\varepsilon = 10^{-12}$, $\tau = 10^{-15}$
$k_{\max} = 2000$, $l_{\max} = 10$; $p_0 = 0$,
[*Iterations:*]{}
[**for**]{} $k = 0, 1, \ldots, k_{\max}-1$:
$x_k = (\widehat x + A^{\rm T}p_k)_+$
$\varphi_k = \frac12 \|x_k\|^2 - b^{\rm T}p_k$
$g_k = Ax_k - b$
[**if**]{} $(\|g_k\| \le \varepsilon\|b\|)$ [**return**]{} $\lbrace{x_k,p_k,g_k\rbrace}$ [**find**]{} $d_k \in R^m$ [**such that**]{}
$d_k^{\rm T}g_k = d_k^{\rm T}M_kd_k = \vartheta_k^2g_kM_k^{-1}g_k$,
[**where**]{} $M_k = A~{\rm Diag}({\rm sign}(x_k))~A^{\rm T}
+ \delta {\rm Diag}(AA^{\rm T})$
$\alpha^{(0)} = 1$
$p_k^{(0)} = p_k - d_k$
[**for**]{} $l = 0, 1, \ldots, l_{\max}-1$:
$x_k^{(l)} = (\widehat x + A^{\rm T}p_k^{(l)})_+$
$\varphi_k^{(l)} = \frac12 \|x_k^{(l)}\|^2 - b^{\rm T}p_k^{(l)}$
$\zeta_k^{(l)} = \left(\frac12
\alpha^{(l)}d_k^{\rm T}g_k + \varphi_k^{(l)}\right)-\varphi_k$
[**if**]{} ($\zeta_k^{(l)}>\tau|\varphi_k|$) [**then**]{}
$\alpha^{(l+1)} = \alpha^{(l)}/2$
$p_k^{(l+1)} = p_k - \alpha^{(l+1)} d_k$
[**else**]{}
$p_k^{(l+1)} = p_k^{(l)}$
[**go to**]{} NEXT
[**end if**]{}
[**end for**]{}
NEXT: $p_{k+1} = p_k^{(l+1)}$
[**end for**]{} Next we explore the convergence properties of this algorithm.
Convergence analysis of inexact Newton method
=============================================
It appears that Algorithm 1 exactly conforms with the convergence analysis presented in [@Ka03] (see also [@AK01]). For the completeness of presentation and compatibility of notations, we reproduce here the main results of [@Ka03].
Estimating convergence of inexact Newton method
-----------------------------------------------
The main assumptions we need for the function $\varphi(p)$ under consideration are that it is bounded from below, have gradient $g(p)\in R^m$, and satisfies $$\varphi(p+q)-\varphi(p)-q^{\rm T}g(p) \le {\gamma\over 2}q^{\rm T}Mq
\label{uppbnd}$$ for the symmetric positive definite $m\times m$ matrix $M=M(p)$ defined above in (\[reg\_hess\]) and some constant $\gamma\ge 1$. Note that the exact knowledge of $\gamma$ is not necessary for actual calculations. The existence of such $\gamma$ follows from (\[reg\_hess\]) and Lemma 3. Indeed, denoting $D = ({\rm Diag}(AA^{\rm T}))^{1/2}$ and $\widehat A = D^{-1}A$, for the right hand side of (\[intexp2a\]) one has, with account of $\|{\rm Diag}(d)\|\le 1$ and $H(p)\ge 0$, $$A{\rm Diag}(d)A^{\rm T} \le AA^{\rm T}
\le \|\widehat A\|^2{\rm Diag}(AA^{\rm T})
\le \frac{\|\widehat A\|^2}{\delta}
\Bigl(\delta{\rm Diag}(AA^{\rm T})+H(p)\Bigr)
= \frac{\|\widehat A\|^2}{\delta} M;$$ therefore, (\[uppbnd\]) holds with $$\gamma = \|\widehat A\|^2/\delta.
\label{gamma}$$ The latter formula explains our choice of $M$ which is more appropriate in cases of large variations in norms of rows in $A$ (see the examples and discussion in Section \[NumTest1\]).
Next we will estimate the reduction in the value of $\varphi$ attained by the descent along the direction $(-d)$ satisfying (\[newt\_dir\]). One can show the following estimate for the decrease of objective function value at each iteration (here, simplified notations $p=p_k$, $\hat p = p_{k+1}$ etc. are used) with $\alpha=2^{-l}$, where $l=0,1,\ldots$, as evaluated according to (\[back\_trk\]): $$\varphi(\hat p) \le \varphi(p) - \frac{\vartheta^2}{4\gamma} g^{\rm T}M^{-1}g.
\label{g_conv}$$ In particular, if the values of $\vartheta^2$ are separated from zero by a positive constant $\vartheta_{\min}^2$ (lower estimate for $\vartheta$ follows from Section \[PCG\] and an upper bound for , then, with account for $M \le (1+\delta)\|A\|^2 I$ and the boundedness of $\varphi$ from below, it follows $$\sum_{j=0}^{k-1}g_j^{\rm T}g_j
\le \frac{4\gamma(1+\delta)\|A\|^2
(\varphi(p_0)-\varphi(p_*))}{\vartheta_{\min}^2}.$$ Noting that the right hand side of the latter estimate does not depend on $k$, it finally follows that $$\lim_{k\rightarrow\infty}\|g(p_k)\| = 0,$$ where $k$ is the number of the outer (nonlinear) iteration.
Estimate (\[g\_conv\]) can be verified as follows (note that quite similar analysis can be found in [@Ma02]). Using $q = -\beta d$, where $0 < \beta < 2/\gamma$, one can obtain from (\[uppbnd\]) and (\[newt\_dir\]) the following estimate for the decrease of $\varphi$ along the direction $(-d)$: $$\begin{aligned}
\nonumber
\varphi(p-\beta d)
&=& \varphi(p) - \beta d^{\rm T}g +
\left(\varphi(p-\beta d) - \varphi(p) - \beta d^{\rm T}g\right) \\
\label{upp2bnd}
&\le& \varphi(p) -
\left(\beta - {\gamma\over 2}\beta^2\right)d^{\rm T}Md.\\
\nonumber\end{aligned}$$ The following two cases are possible.
[*Case 1.*]{} If the condition (\[back\_trk\]) is satisfied at once for $\alpha=1$, this means that (recall that the left equality of (\[newt\_dir\]) holds) $$\varphi(\hat p) \le \varphi(p) - \frac12 d^{\rm T}Md.$$
[*Case 2.*]{} Otherwise, if at least one bisection of steplength was performed (and the actual steplength is $\alpha$), then, using (\[upp2bnd\]) with $\beta = 2\alpha$, it follows $$\varphi(p) - \alpha d^{\rm T}Md < \varphi(p - 2\alpha d)
\le \varphi(p) - (2\alpha - 2\gamma\alpha^2) d^{\rm T}Md,$$ which readily yields $\alpha > 1/(2\gamma)$. Since we also have $$\varphi(p - \alpha d) \le \varphi(p) - {\alpha\over 2} d^{\rm T}Md,$$ it follows $$\varphi(p - \alpha d) \le \varphi(p) - {1\over 4\gamma} d^{\rm T}Md.
\label{g2conv}$$ Joining these two cases, taking into account that $\gamma\ge 1$, and using the second equality in (\[newt\_dir\]) one obtains the required estimate (\[g\_conv\]).
It remains to notice that as soon as the norms of $g$ attain sufficiently small values, the resulting directions $d$ will also have small norms. Therefore, the case considered in Remark 1 will take place, and finally the convergence of the Newton method will be much faster than at its initial stage.
Linear CG approximation of Newton directions {#stopCG}
--------------------------------------------
Next we relate the convergence of inner linear Preconditioned Conjugate Gradient (PCG) iterations to the efficiency of Inexact Newton nonlinear solver. Similar issues were considered in [@AK01; @GKK04; @KA94; @Ka03].
An approximation $d^{(i)}$ to the solution of the problem $Md=g$ generated on the $i$th PCG iteration by the recurrence $d^{(i+1)} = d^{(i)} + s^{(i)}$ (see Algorithm 2 below) can be written as follows (for our purposes, we always set the initial guess for the solution $d^{(0)}$ to zero): $$d^{(i)} = \sum_{j=0}^{i-1}s^{(j)},
\label{pcg_di}$$ where the PCG direction vectors are pairwise $M$-orthogonal: , . Let also denote the $M$-norms of PCG directions as $\eta^{(j)} = (s^{(j)})^{\T}Ms^{(j)}$, $j=0,1,\ldots,i-1$. Therefore, from (\[pcg\_di\]), one can determine $$\zeta^{(i)} = (d^{(i)})^{\T} M d^{(i)} = \sum_{j=0}^{i-1}\eta^{(j)},$$ and estimate (\[g2conv\]) takes the form $$\varphi_{k+1} \le \varphi_k - {1\over 4\gamma} \sum_{j=0}^{i_k-1}\eta_k^{(j)},$$ where $k$ is the Newton iteration number. Summing up the latter inequalities for $0 \le k \le m-1$, we get $$c_0 \equiv 4\gamma(\varphi_0 - \varphi_*)
\ge \sum_{k=0}^{m-1} \sum_{j=0}^{i_k-1}\eta_k^{(j)}
\label{est_conv}$$ On the other hand, the cost measure related to the total time needed to perform $m$ inexact Newton iterations with $i_k$ PCG iterations at each Newton step, can be estimated as proportional to $$T_m=\sum_{k=0}^{m-1} \left( \epsilon_{\rm CG}^{-1} + i_k\right)
\le c_0\frac{\sum_{k=0}^{m-1}\left( \epsilon_{\rm CG}^{-1} + i_k\right)}
{\sum_{k=0}^{m-1} \sum_{j=0}^{i_k-1}\eta_k^{(j)}}
\le c_0 \max_{k<m}\frac{\epsilon_{\rm CG}^{-1} + i_k}{\sum_{j=0}^{i_k-1}\eta_k^{(j)}}.$$ Here $\epsilon_{\rm CG}$ is a small parameter reflecting the ratio of one linear PCG iteration cost to the cost of one Newton iteration (in particular, including construction of preconditioning and several $\varphi$ evaluations needed for backtracking) plus possible efficiency loss due to early PCG termination. Thus, introducing the function , (here, we omit the index $k$) one obtains a reasonable criterion to stop PCG iterations in the form . Here, the use of smaller values $\varepsilon_{\rm CG}$ generally corresponds to the increase of the resulting iteration number bound. Rewriting the latter condition, one obtains the final form of the PCG stopping rule: $$(\epsilon_{\rm CG}^{-1} + i)\eta^{(i-1)} \le \zeta^{(i)}.
\label{new_stop}$$ Note that by this rule, the PCG iteration number is always no less than 2.
Finally, we explicitly present the resulting formulae for the PCG algorithm incorporating the new stopping rule. Following [@GGEN09], we use the Jacobi preconditioning $$C = ({\rm Diag}(M))^{-1}.
\label{diag_prec}$$ Moreover, the reformulation [@KM11] of the CG algorithm [@HS52; @Ax76] is used. This may give a more efficient parallel implementation, see, e.g., [@GGEN09].
Following [@KM11], recall that at each PCG iteration the $M^{-1}$-norm of the -th residual attains its minimum over the corresponding Krylov subspace. Using the standard PCG recurrences (see Section \[PCG\] below) one can find $d^{(i+1)} = d^{(i)}+Cr^{(i)}\alpha^{(i)}+s^{(i-1)}\alpha^{(i)}\beta^{(i-1)}$. Therefore, the optimum increment $s^{(i)}$ in the recurrence , where and , can be determined via the solution of the following 2-dimensional linear least squares problem: $$\left[_{\beta^{(i)}}^{\alpha^{(i)}}\right] = h^{(i)}
=\arg\min_{h\in{\bf R}^2}\|g-Md^{(i+1)}\|_{M^{-1}}
=\arg\min_{h\in{\bf R}^2}\|r^{(i)}-MV^{(i)} h\|_{M^{-1}}.$$ By redefining and introducing vectors , the required PCG reformulation follows:
[**Algorithm 2.**]{}
$r^{(0)} = -g,$ $d^{(0)} = s^{(-1)} = t^{(-1)} = 0,$ $\zeta^{(-1)} = 0;$
$i=0,1,\ldots,it_{\max}:$
$w^{(i)} = Cr^{(i)},$
$z^{(i)} = Mw^{(i)},$
$\gamma^{(i)} = (r^{(i)})^{\T}w^{(i)},$ $\xi^{(i)} = (w^{(i)})^{\T}z^{(i)},$ $\eta^{(i-1)} = (s^{(i-1)})^{\T}t^{(i-1)},$
$\zeta^{(i)} = \zeta^{(i-1)} + \eta^{(i-1)},$
$((\varepsilon_{\rm CG}^{-1}+i)\eta^{(i-1)} \le\zeta^{(i)})$ [**or**]{} $(\gamma^{(i)} \le \varepsilon_{\rm CG}^2 \gamma^{(0)})$ [**return**]{} $\lbrace d^{(i)}\rbrace$;
($k=0$) [**then**]{}
$\alpha^{(i)} = -\gamma^{(i)}/\xi^{(i)}, \quad \beta^{(i)} = 0;$
,
, ;
$t^{(i)} = z^{(i)}\alpha^{(i)} + t^{(i-1)}\beta^{(i)},$ $r^{(i+1)} = r^{(i)} + t^{(i)},$
$s^{(i)} = w^{(i)}\alpha^{(i)} + s^{(i-1)}\beta^{(i)},$ $d^{(i+1)} = d^{(i)} + s^{(i)}.$
For maximum reliability, the new stopping rule (\[new\_stop\]) is used along with the standard one; however, in almost all cases the new rule provides for an earlier CG termination.
Despite of somewhat larger workspace and number of vector operations compared to the standard algorithm, the above version of CG algorithm enables more efficient parallel implementation of scalar product operations. At each iteration of the above presented algorithm, it suffices to use one [MPI$\_$AllReduce($*$,$*$,3,…)]{} operation instead of two [MPI$\_$AllReduce($*$,$*$,1,…)]{} operation in the standard PCG recurrences. This is especially important when many MPI processes are used and the start-up time for MPI$\_$AllReduce operations is relatively large. For another equivalent PCG reformulations allowing to properly reorder the scalar product operations, see [@DE03] and references cites therein.
Convergence properties of PCG iterations {#PCG}
----------------------------------------
Let us recall some basic properties of the PCG algorithm, see, e.g. [@Ax76]. The standard PCG algorithm (algebraically equivalent to Algorithm 2) for the solution of the problem $Md=g$ can be written as follows (the initial guess for the solution $d_0$ is set to zero): $$\begin{aligned}
&&d^{(0)} = 0, \quad r^{(0)} = g, \quad s^{(0)} = Cr^{(0)};
\nonumber
\\
&&{\bf for} ~ i = 0, 1, \ldots, m-1:
\nonumber
\\
&&\qquad \alpha^{(i)} = (r^{(i)})^TCr^{(i)}/(s^{(i)})^TMs^{(i)},
\nonumber
\\
&&\qquad d^{(i+1)} = d^{(i)} + s^{(i)}\alpha^{(i)},
\nonumber
\\
&&\qquad r^{(i+1)} = r^{(i)} - Ms^{(i)}\alpha^{(0)},
\nonumber
\\
&&\qquad {\bf if} ~ ((r^{(i)})^TCr^{(i+1)} \le
\varepsilon_{\rm CG}^2 (r^{(0)})^TCr^{(0)}) ~ {\bf return} ~ d^{(i+1)}
\nonumber
\\
&&\qquad \beta^{(i)} = (r^{(i+1)})^TCr^{(i+1)}/(r^{(i)})^TCr^{(i)},
\nonumber
\\
&&\qquad s^{(i+1)} = Cr^{(i+1)} + s^{(i)}\beta^{(i)}.
\\
&&{\bf end for}
\nonumber\end{aligned}$$ The scaling property (\[newt\_dir\]) (omitting the upper and lower indices at $d$, it reads $d^{\rm T}g = d^{\rm T}Md$) can be proved as follows. Let $d=d^{(i)}$ be obtained after $i$ iterations of the PCG method applied to $Md=g$ with zero initial guess $d^{(0)}=0$. Therefore, $d \in K_i = {\rm span} \lbrace Cg, CMCg, \dots, (CM)^{i-1}Cg\rbrace$, and, by the PCG optimality property, it holds $$d = \arg\min_{d\in K_i} (g-Md)^TM^{-1}(g-Md).$$ Since $\alpha d\in K_i$ for any scalar $\alpha$, one gets $$(g-\alpha Md)^TM^{-1}(g-\alpha Md) \ge (g-Md)^TM^{-1}(g-Md).$$ Setting here $\alpha=d^Tg/d^TMd$, one can easily transform this inequality as $0 \ge (-d^Tg+d^TMd)^2$, which readily yields (\[newt\_dir\]). Furthermore, by the well known estimate of the PCG iteration error [@Ax76] using Chebyshev polynomials, one gets $$1-\theta^2 \equiv (g-Md)^TM^{-1}(g-Md)/g^TM^{-1}g \le
\cosh^{-2}\left(2i/\sqrt{\kappa}\right)$$ where $$\kappa = {\rm cond}(CM) \equiv \lambda_{\max}(CM)/\lambda_{\min}(CM).$$ By the scaling condition, this gives $$\theta^2 = d^TMd/g^TM^{-1}g \ge \tanh^2\left(2i/\sqrt{\kappa}\right).
\label{pcg_conv}$$ Hence, $0<\theta<1$ and $\theta^2\rightarrow 1$ as the PCG iteration number $i$ grows.
Numerical test results {#NumTest1}
======================
Below we consider two families of test problems which can be solved via minimization of piecewise quadratic problems. The first one was described above in Section \[underdet\] (see also [@GGE18]), while the second coincides with the problem setting for the evaluation of distance between two convex polyhedra used in [@Bo89]. The latter problem is of key importance e.g., in robotics and computer animation.
Test results for 11 NETLIB problems
-----------------------------------
Matrix data from the following 11 linear programming problems (this is the same selection from NETLIB collection as considered in [@KMPN17]), were used to form test problems (\[prob\_set\]). Note that further we only consider the case $\widehat x=0$. Recall also the notation $x_* = (\widehat x + A^Tp_*)_+$. The problems in Table 1 below are ordered by the number of nonzero elements ${\rm nz}(A)$ in $A\in R^{m\times n}$.
[p[2cm]{}p[1cm]{}p[1cm]{}p[1cm]{}p[2cm]{}p[2cm]{}p[2cm]{}]{} name & m & n & nz(A) & $\|x_*\|$ & $\min_i(AA^T)_{ii}$ & $\max_i(AA^T)_{ii}$\
afiro & 27 & 51 & 102 & 634.029569 & 1.18490000 & 44.9562810\
addlittle & 56 & 138 & 424 & 430.764399 & 1.00000000 & 10654.0000\
agg3 & 516 & 758 & 4756 & 765883.022 & 1.00000001 & 179783.783\
25fv47 & 821 & 1876 & 10705 & 3310.45652 & 0.00000000 & 88184.0358\
pds$\_$02 & 2953 & 7716 & 16571 & 160697.180 & 1.00000000 & 91.0000000\
cre$\_$a & 3516 & 7248 & 18168 & 1162.32987 & 0.00000000 & 27476.8400\
80bau3b & 2262 & 12061 & 23264 & 4129.96530 & 1.00000000 & 321739.679\
ken$\_$13 & 28362 & 42659 & 97246 & 25363.3224 & 1.00000000 & 170.000000\
maros$\_$r7& 3136 & 9408 & 144848 & 141313.207 & 3.05175947 & 3.37132546\
cre$\_$b & 9648 & 77137 & 260785 & 624.270129 & 0.00000000 & 27476.8400\
osa$\_$14 & 2337 & 54797 & 317097 & 119582.321 & 18.0000000 & 845289.908\
[p[2cm]{}p[2cm]{}p[2cm]{}p[1.7cm]{}p[1.7cm]{}p[1.7cm]{}]{} name & solver & time(sec) & $\|Ax - b\|_\infty$ & [\#]{}NewtIter & [\#]{}MVMult\
afiro & GNewtEGK & 0.001 & 8.63E–11 & 17 & 398\
–“– & ssGNewton & 0.06 & 6.39E–14 & – & –\
–”– & cqpMOSEK & 0.31 & 1.13E–13 & – & –\
addlittle & GNewtEGK & 0.003 & 6.45E–10 & 22 & 1050\
–“– & ssGNewton & 0.05 & 2.27E–13 & – & –\
–”– & cqpMOSEK & 0.35 & 7.18E–11 & – & –\
agg3 & GNewtEGK & 0.14 & 3.93E–07 & 116 & 9234\
–“– & ssGNewton & 0.27 & 3.59E–08 & – & –\
–”– & cqpMOSEK & 0.40 & 2.32E–10 & – & –\
25fv47 & GNewtEGK & 0.54 & 7.15E–10 & 114 & 32234\
–“– & ssGNewton & 1.51 & 3.43E–09 & – & –\
–”– & cqpMOSEK & 1.36 & 1.91E–11 & – & –\
pds$\_$02 & GNewtEGK & 0.32 & 1.55E–08 & 75 & 8559\
–“– & ssGNewton & 2.30 & 1.40E–07 & – & –\
–”– & cqpMOSEK & 0.51 & 8.20E–06 & – & –\
cre$\_$a & GNewtEGK & 3.36 & 2.64E–09 & 219 & 85737\
–“– & ssGNewton & 1.25 & 4.13E–06 & – & –\
–”– & cqpMOSEK & 0.61 & 2.15E–10 & – & –\
80bau3b & GNewtEGK & 0.27 & 3.33E–09 & 79 & 6035\
–“– & ssGNewton & 0.95 & 1.18E–12 & – & –\
–”– & cqpMOSEK & 0.80 & 2.90E–07 & – & –\
ken$\_$13 & GNewtEGK & 1.41 & 2.70E–08 & 55 & 6285\
–“– & ssGNewton & 9.09 & 4.39E–09 & – & –\
–”– & cqpMOSEK & 2.09 & 1.71E–09 & – & –\
maros$\_$r7& GNewtEGK & 0.10 & 1.18E–09 & 27 & 535\
–“– & ssGNewton & 2.86 & 2.54E–11 & – & –\
–”– & cqpMOSEK & 55.20 & 3.27E–11 & – & –\
cre$\_$b & GNewtEGK & 9.25 & 6.66E–10 & 75 & 24590\
–“– & ssGNewton & 13.20 & 1.62E–09 & – & –\
–”– & cqpMOSEK & 2.31 & 1.61E–06 & – & –\
osa$\_$14 & GNewtEGK & 42.59 & 8.25E–08 & 767 & 104874\
–“– & ssGNewton & 60.10 & 4.10E–08 & – & –\
–”– & cqpMOSEK & 4.40 & 7.82E–05 & – & –\
[p[2.2cm]{}p[2.2cm]{}p[2.1cm]{}p[2.2cm]{}p[2.2cm]{}]{} criterion &$\varepsilon_{CG}$& geom.mean time & arithm.mean time & geom.mean res.\
old & 0.05 & 1.78 & 12.37 & 3.64–09\
old & 0.03 & 1.45 & 12.21 & 3.47–09\
old & 0.01 & 1.35 & 14.91 & 2.11–09\
old & 0.003 & 1.48 & 21.14 & 2.14–09\
old & 0.001 & 1.82 & 29.12 & 3.64–09\
new & 0.003 & 1.66 & 11.12 & 3.46–09\
new & 0.002 & 1.39 & 11.36 & 3.64–09\
new & 0.001 & 1.26 & 10.81 & 4.65–09\
new & 0.0003 & 1.26 & 11.25 & 4.23–09\
new & 0.0001 & 1.33 & 12.47 & 3.60–09\
It is readily seen that 3 out of 11 matrices have null rows, and more than half of them have rather large variance of row norms. This explains the proposed Hessian regularization (\[reg\_hess\]) instead of the earlier construction [@GGE18; @KMPN17] $M_k = H(p_k) + \delta I_m$. The latter is a proper choice only for matrices with rows of nearly equal length, such as [**maros$\_$r7**]{} example or various matrices with uniformly distributed quasirandom entries, as used for testing in [@GGEN09; @KMPN17]. In particular, estimate (\[gamma\]) with $D=I$ would take the form $\gamma=\|A\|^2/\delta$, so the resulting method appears to be rather sensitive to the choice of $\delta$.
In Table 2, the results presented in [@KMPN17] are reproduced along with similar data obtained with our version of Generalized Newton method. It must be stressed that we used the fixed set of tunung parameters $$\delta=10^{-6}, \qquad
\varepsilon=10^{-12}, \qquad
\varepsilon_{\rm CG}=10^{-3}, \qquad
l_{\max} = 10,
\label{def_set}$$ for all problems. Note that In [@KMPN17] the parameter choice for the Armijo procedure was not specified.
In [@KMPN17], the calculations were performed on 5GHz AMD 64 Athlon X2 Dual Core. In our experiments, one core of 3.40 GHz x8 Intel (R) Core (TM) i7-3770 CPU was used, which is likely somewhat slower.
Note that the algorithm of [@KMPN17] is based on direct evaluation of $M_k$ and its sparse Cholesky factorization, while our implementation, as was proposed in [@GGEN09], uses the Jacobi preconditioned Conjugate Gradient iterations for approximate evaluation of Newton directions. Thus, the efficiency of our implementation critically depends on the CG iteration convergence, which is sometimes slow. On the other hand, since the main computational kernels of the algorithm are presented by matrix-vector multiplications of the type $x=Ap$ or $q=A^Ty$, its parallel implementation can be sufficiently efficient.
In Table 2, the abbreviation [**cqpMOSEK**]{} refers to MOSEK Optimization Software package for convex quadratic problems, see [@KMPN17]. The abbreviation [**ssGNewton**]{} denotes the method implemented and tested in [@KMPN17], while [**GNewtEGK**]{} stands for the method proposed in the present paper.
Despite the use of slower computer, our [**GNewtEGK**]{} demonstrates considerably faster performance in 8 cases of 11. Otherwise, one can observe that smaller computational time of [**cqpMOSEK**]{} goes along with much worse residual norm, see the results for problems [**cre$\_$b**]{} and [**osa$\_$14**]{} .
Thus, in most cases the presented implementation of Generalized Newton method takes not too large number of Newton iterations using approximate Newton directions generated by CG iterations with diagonal preconditioning (\[diag\_prec\]) and special stopping rule (\[new\_stop\]).
A direct comparison of efficiency for the standard CG iterations stopping rule (see Algorithm 2 for the notations) and the new one (\[new\_stop\]) is given in Table \[tab3\], where the timing (in seconds) and precision results averaged over the same 11 problems are given. One can see that nearly the same average residual norm $\|Ax-b\|_{\infty}$ can be obtained considerably faster and with less critical dependence on $\varepsilon_{\rm CG}$ when using the new PCG iteration stopping rule.
Evaluating the distance between convex polyhedra {#NumTest2}
------------------------------------------------
Let the two convex polyhedra ${\cal X}_1$ and ${\cal X}_2$ be described by the following two systems of linear inequalities: $${\cal X}_1 = \lbrace x_1: ~A_1^Tx_1\le b_1 \rbrace, \qquad
{\cal X}_2 = \lbrace x_2: ~A_2^Tx_2\le b_2 \rbrace,$$ where $A_1\in R^{s\times n_1}$, $A_2\in R^{s\times n_2}$, and the vectors $x_1, x_2, b_1, b_2$ are of compatible dimensions. The original problem of evaluating the distance between ${\cal X}_1$ and ${\cal X}_2$ is (cf. [@Bo89], where it was solved by the projected gradient method) $$x_*=\arg\min_{A_1^Tx_1\le b_1,A_2^Tx_2\le b_2}\|x_1-x_2\|^2/2,
\quad {\rm where} \quad x=[x_1^T,x_2^T]^T\in R^{2s}.$$ We will use the following regularized/penalized approximate reformulation of the problem in terms of unconstrained convex piecewise quadratic minimization. Introducing the matrices $$A =
\left[
\begin{array}{cc}
A_1 & 0 \\
0 & A_2
\end{array}
\right] \in R^{2s\times(n_1+n_2)},
\qquad
B =
\left[
\begin{array}{cc}
~I_s & -I_s \\
-I_s &~~I_s
\end{array}
\right] \in R^{2s\times 2s},$$ and the vector $b=[b_1^T,b_2^T]^T\in R^{n_1+n_2}$, we consider the problem $$x_*(\varepsilon) = \arg\min_{x\in R^{2s}}
\left(\frac{\varepsilon}{2}\|x\|^2 + \frac{1}{2} x^TBx
+ \frac{1}{2\varepsilon}\|(A^Tx-b)_+\|^2\right),$$ where the regularization/penalty parameter $\varepsilon$ is a sufficiently small positive number (we have used $\varepsilon=10^{-4}$). The latter problem can readily be solved by adjusting the above described Algorithm 1 using $\delta=0$ and the following explicit expressions for the gradient and the generalized Hessian: $$g(x)=\varepsilon x + Bx +\varepsilon^{-1}A(A^Tx-b)_+, \qquad
H(x)=\varepsilon I + B + \varepsilon^{-1}A D(x) A^T,$$ where $D(x)={\rm Diag}({\rm sign}(A^Tx-b)_+)$. When solving practical problems of evaluating the distance between two 3D convex polyhedra determined by their faces (so that $s=3$), the inexact Newton iterations are performed in $R^6$, and the cost of each iteration is proportional to the total number $n=n_1+n_2$ of the faces determining the two polyhedra. In this case, the explicit evaluation of $H(x)$ and the use of its Cholesky factorization is more preferable than the use of the CG method.
Test polyhedrons with $n/2$ faces each were centered at the points $e=[1,1,1]^T$ or $-e$ and defined as $A_1^T(x-e)\le b_1$ and $A_2^T(x+e)\le b_2$ with $b_1=b_2=[1,\ldots,1]\in R^{n/2}$, respectively. The columns of matrices $A_1$ and $A_2$ were determined by $n/2$ quasirandom unit $3$-vectors generated with the use of logistic sequence (see, e.g.[@YBZS10] and references cited therein) $\xi_0=0.4$, $\xi_k=1-2\xi_{k-1}^2$. We used $A_1(i,j)=\xi_{20(i-1+3(j-1))}$ and similar for $A_2$; then the columns of these matrices were normalized to the unit length. The corresponding performance results seem quite satisfactory, see Table \[tab4\]. Note that the lower bound $\|x_1-x_2\|\ge 2\sqrt{3}-2 \approx 1.464101$ always holds for the distance (since the two balls $\|x_1-e\|=1$ and $\|x_2+e\|=1$ are inscribed in the corresponding polyhedrons).
[p[1.8cm]{}p[1.9cm]{}p[1.9cm]{}p[1.9cm]{}p[1.9cm]{}p[1.6cm]{}]{} n & $\|x_1-x_2\|_2$ & $\|A^Tx_*-c\|_{\infty}$ & time(sec) & $\|g(x_*)\|_{\infty}$ & [\#]{}NewtIter\
8 & 0.001815 & 9.69–09 &$<0.001$ & 7.89–13 & 15\
16 & 0.481528 & 8.63–05 &$<0.001$ & 1.27–13 & 3\
32 & 0.795116 & 8.80–05 &$<0.001$ & 1.46–12 & 28\
64 & 1.102286 & 1.32–04 &$<0.001$ & 5.58–13 & 13\
128 & 1.446262 & 1.36–04 &$<0.001$ & 7.12–13 & 17\
256 & 1.449913 & 9.54–05 &$<0.001$ & 4.37–13 & 11\
512 & 1.460197 & 1.31–04 &$ 0.001$ & 8.16–13 & 15\
1024 & 1.460063 & 1.46–04 &$ 0.002$ & 1.09–12 & 14\
2048 & 1.463320 & 1.04–04 &$ 0.005$ & 6.58–13 & 19\
4096 & 1.463766 & 1.26–04 &$ 0.009$ & 3.59–13 & 20\
8192 & 1.463879 & 1.03–04 &$ 0.009$ & 8.32–14 & 12\
16384 & 1.463976 & 7.58–05 &$ 0.009$ & 1.64–12 & 13\
32768 & 1.464046 & 3.28–05 &$ 0.018$ & 1.54–12 & 13\
This work was supported by the Russian Foundation for Basic Research grant No. 17-07-00510 and by Program No. 26 of the Presidium of the Russian Academy of Sciences. The authors are grateful to the anonymous referees and to Prof. V.Garanzha for many useful comments which greatly improved the exposition of the paper.
|
---
abstract: |
The nucleolus offers a desirable payoff-sharing solution in cooperative games thanks to its attractive properties - it always exists and lies in the core (if the core is non-empty), and is unique. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e., the number of players $n \leq 20$). This, however, becomes more challenging for larger games because of the need to form and check the balancedness of possibly exponentially large collections of coalitions, each collection could be of an exponentially large size. We develop a simplifying set of the Kohlberg criteria that involves checking the balancedness of at most $(n-1)$ sets of coalitions. We also provide a method for reducing the size of these sets and a fast algorithm for verifying the balancedness.
Nucleolus; cooperative game; Kohlberg criterion.
author:
- 'Tri-Dung Nguyen'
bibliography:
- 'bibliography.bib'
title: Simplifying the Kohlberg Criterion on the Nucleolus
---
Introduction
============
Cooperative games model situations where players can form coalitions to jointly achieve some objective. Once such example is where entrepreneurs, with possibly complementary skills, consider running a business together. Assuming that the entrepreneurs can jointly run a more successful business than by working individually (or in smaller groups), a natural question is how to divide the reward among the players in such a way that could avoid any subgroup of players to breaking away from the grand coalition in order to increase the total payoff. Solution concepts in cooperative games provide the means to achieving this.
Formally, let $n$ be the number of players and $\N = \{1,2,\ldots,n\}$ be the set of all the players. A *coalition* $\S$ is a subset of the players; i.e., $\S \subseteq \N$. The *characteristic function* $v~:~2^\N \mapsto \R$ maps each coalition to a real number with $v(\S)$ representing the payoff that coalition $\S$ is guaranteed to receive if all players in $\S$ collaborate, despite the actions of the other players. A solution (also called a payoff distribution) of the game $\bx = (x_1,x_2,\ldots,x_n)$ is a way to distribute the reward among the players, with $x_i$ being the share for player $i$.
Given the total payoff $v(\N)$ of the grand coalition, we are interested in *efficient* solutions $\bx$ which satisfy $\sum_{i \in \N} x_i = v(\N)$. Let us denote $\bx(\S) = \sum_{i \in \S} x_i$. For each imputation $\bx$, the *excess value* of a coalition $\S$ is defined as $d(\S, \bx) := v(\S) - \bx(\S)$ which can be regarded as the level of dissatisfaction the players in coalition $\S$ feel over the proposed solution $\bx$. Here, we concern with profit games and assume that it is more desirable to have higher shares. All the results can be extended to cost games either through transforming the characteristic function to the corresponding profit games or by redefining the excess values.
Player $i$ is considered rational if he/she only accepts a share $x_i$ of at least the amount $v(\{i\})$. A group of players $\S$ is considered rational if it only accepts a total share $\bx(\S) := \sum_{i \in \S} x_i$ of at least the amount $v(\S)$ that the group is guaranteed to receive by breaking away from the grand coalition and forming its own coalition; i.e., $d(\S, \bx) \leq 0$.
An *imputation* is an efficient solution that satisfies *individual rationality*; that is, $x_i \geq v(\{i\}), \forall i \in \N$. The *core* of the game is the set of all efficient solutions $\bx$ such that no coalition has the incentive to break away – i.e., satisfying *group rationality* – and hence the solutions are stable. It is, however, possible that there is no solution satisfying this set of conditions, and the core might not exist. In that case, we consider alternative solutions that, although not stable, are least susceptive to deviations. The first such solution concept is called the *least core*, which minimizes the worst level of dissatisfaction among all the coalitions. Note that the least core always exists but might not be unique. We denote $\mathbf{I}$ as the imputation set and $\mathbf{Co}$ as the core of the game.
Among all solutions in the least core, if we also ensure not only the worst dissatisfaction level but also all the dissatisfactions to be lexicographically minimized, we arrive at the concept of the *nucleolus* which is the ‘most stable’ solution in the imputation set. Formally, for any imputation $\bx$, let $\Theta(\bx) =(\Theta_1(\bx),\Theta_2(\bx),\ldots,\Theta_{2^n}(\bx))$ be the vector of all the $2^n$ excess values at $\bx$ sorted in a non-increasing order; i.e., $\Theta_i(\bx) \geq \Theta_{i+1}(\bx)$ for all $1 \leq i < 2^n$. Let us denote $\Theta(\bx) <_L \Theta(\by)$ if there exists $r \leq 2^n$ such that $\Theta_i(\bx) = \Theta_i(\by),\forall 1 \leq i < r$ and $\Theta_r(\bx) < \Theta_r(\by)$. Then $\bnu \in \mathbf{I}$ is the *nucleolus* if $\Theta(\bnu) <_L \Theta(\bx),~\forall \bx \in \mathbf{I}, ~\bx \neq \bnu$. If we relax the condition $\bx, \bnu \in \mathbf{I}$, we arrive at the definition of the prenucleolus.
The nucleolus is one of the most important solution concepts for cooperative games with transferable utilities, and was introduced in 1969 by @schmeidler1969nucleolus as a solution concept with attractive properties - it always exists (if the imputation is non-empty), it is unique, and it lies in the core (if the core is non-empty). Despite the desirable properties that the nucleolus has, its computation is, however, very challenging because the process involves the lexicographical minimization of $2^n$ excess values, where $n$ is the number of players. There are a small number of games whose nucleoli can be computed in polynomial time (e.g., @solymosi1994algorithm, @hamers2003nucleolus, @solymosi2005computing, @potters2006nucleolus [@deng2009finding; @kern2009core]). It has been shown that finding the nucleolus is NP-hard for many classes of games such as the utility games with non-unit capacities (@deng2009finding) and the weighted voting games (@elkind2007computational).
@Kopelowitz1967Computation suggests using nested linear programming (LP) to compute the kernel of a game. This encouraged a number of researchers to study the computation of the nucleolus using linear programming. For example, @kohlberg1972nucleolus presents a single LP with ${\ensuremath{\operatorname{O}\bigl(2^n!\bigr)}}$ constraints which later on is improved by @owen1974note with ${\ensuremath{\operatorname{O}\bigl(4^n\bigr)}}$ constraints (at the cost of having larger coefficients). @puerto2013finding recently introduces a different single LP formulation with ${\ensuremath{\operatorname{O}\bigl(4^n\bigr)}}$ constraints and ${\ensuremath{\operatorname{O}\bigl(4^n\bigr)}}$ decision variables and with coefficients in $\{-1,0,1\}$. The nucleolus can also be found by solving a sequence of LPs. However, either the number of LPs involved is exponentially large (@maschler1979geometric, @sankaran1991finding) or the sizes of the LPs are exponential (@potters1996computing, @derks1996implementing, @Nguyen2016).
While finding the nucleolus is very difficult as shown in the aforementioned literature, @kohlberg1971nucleolus provides a necessary and sufficient condition for a given imputation to be the nucleolus as is described in the subsequent section. This set of criteria is particularly useful in relatively small games (e.g., less than 10 players) or in larger games with special structures which allow us to take an educated guess on the nucleolus. The verification of the criterion, however, becomes time consuming when the number of players exceeds 15, and becomes almost impossible in general cases when the number of players exceeds 25. This is because the criterion requires forming the sets of coalitions of all $2^n$ possible coalitions and iteratively verifying if unions of these sets are balanced. This work aims to resolve these issues and proposes a new set of simplifying criteria.
The key contributions of our work include the following:
- We present a new set of necessary and sufficient conditions for a solution to be the nucleolus in Section \[subsec:simplifiedKohlberg\]. The number of subsets of coalitions to check for balancedness is at most $(n-1)$ (instead of exponentially large).
- The balancedness condition is essentially equivalent to solving a linear program with strict inequalities which are often undesirable in mathematical programming. We provide a solution to this in Section \[subsec:balancedness\_checking\].
- On checking the Kohlberg criterion, we might end up having to store an exponentially large number of coalitions. We provide a method for reducing this to the size of at most $n(n-1)$ in Section \[subsec:simplifiedKohlberg2\].
The Kohlberg criterion for verifying the nucleolus {#subsec:Kohlberg}
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For each efficient payoff distribution $\bx$, @kohlberg1971nucleolus first defines the following sets of coalitions: $T_0(\bx) = \{\{i\},i=1,\ldots,n~:~x_i = v(\{i\})\}$, $H_0(\bx) = \{\N,\emptyset\}$ and $H_k(\bx) = H_{k-1}(\bx) \cup T_k(\bx),k=1,2,\ldots,$ where for each $k \geq 1$, $$\begin{aligned}
T_k(\bx) &=& \displaystyle {\mathop{\rm argmax}}_{\S \not \in H_{k-1}(\bx)} \left\{v(\S) - x(\S)\right\},\quad \epsilon_k(\bx) = \displaystyle \max_{\S \not \in H_{k-1}(\bx)} \left\{v(\S) - x(\S)\right\}. \nonumber\end{aligned}$$
Here, $T_k(\bx)$ includes all coalitions that have the same excess value $\epsilon_k(\bx)$ and $\epsilon_1(\bx) > \epsilon_2(\bx) >\ldots$. The terms ‘collection of coalitions’ and ‘subset of the powerset $2^\N$’ are equivalent and used interchangeably in this paper. We also use the terms ‘collection’ and ‘subset’ as their shorter versions.
For each collection $Q \subseteq 2^\N$, let us denote $|Q|$ as the size of $Q$. We associate each collection $Q$ with a weight vector in $\R^{|Q|}$ with each element denoting the weight of the corresponding coalition in $Q$. Throughout this paper, we use bold fonts for vectors and normal font for scalars.
Let us denote $\be(\S),\S \in \N$, as a binary vector in $\R^n$ with the $i$th element equal to one if and only if player $i$ is in the coalition. With this, for all $\bx \in \R^n$, we have $\bx(\S) = \sum_{i \in \S} x_i = \bx^T \be(\S)$. The concept of balancedness is defined as follows:
A collection of coalitions $Q\subseteq 2^N$ is balanced if there exists a weight vector $\mb{\omega} \in \R^{|Q|}_{>0}$ such that $\be(\N) = \sum_{\S \in Q} \omega_\S \be(\S)$.
Given a collection $T_0 \subseteq 2^N$, a collection $Q \subseteq 2^N$ is called $T_0$-balanced if there exist weight vectors $\mb{\gamma} \in \R^{|T_0|}_{\geq 0}$ and $\mb{\omega} \in \R^{|Q|}_{>0}$ such that $\be(\N) = \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S)$.
Remarks:
- Note that when $T_0 = \emptyset$, the concept of $T_0$-balanced is equivalent to the usual balancedness concept.
- All results in this paper concern with finding the nucleolus. These results and the algorithms can be simplified to finding the pre-nucleolus by setting $T_0 = \emptyset$.
For any collection $Q$ of coalitions, let us define $$Y(Q) = \left\{ \by\in \R^n~:~\by(\S) \geq 0~\forall \S \in Q,~ \by(N) = 0\right\}.$$ We have $Y(Q) \neq \emptyset$ since $\mb{0} \in Y(Q)$. The first key result in @kohlberg1971nucleolus that will be exploited in this work is the following lemma:
\[lemma:Kohlberg\] Given a collection $T_0 \subseteq 2^N$, a collection $T\subseteq 2^N$ is $T_0$-balanced if and only if $\by \in Y(T_0 \cup T)$ implies $\by(\S)=0,\forall \S \in T$.
This result allows the author to define two sets of equivalent properties on a sequence of collections $(Q_0,Q_1,\ldots)$ as:
$(Q_0,Q_1,\ldots)$ has Property I if for all $k \geq 1$, the following claim holds: $\displaystyle \by \in Y(\cup_{j=0}^k Q_j)$ implies $\displaystyle \by(\S) = 0, ~\forall \S \in \cup_{j=1}^k Q_j$.
$(Q_0,Q_1,\ldots)$ has Property II if for all $k \geq 1$, $\displaystyle \cup_{j=1}^k Q_j$ is $Q_0$-balanced.
The main result in @kohlberg1971nucleolus can be summarized in the following theorem:
\[theorem:Kohlberg\] The following three claims are equivalent: (a) $\bx$ is the nucleolus; (b) $(T_0(\bx),T_1(\bx),\ldots)$ has Property I; and (c) $(T_0(\bx),T_1(\bx),\ldots)$ has Property II.
For completeness, we show a slightly different version and proof of Theorem \[theorem:Kohlberg\] by Lemma \[lemma:balancedness2\] in Appendix A.
To appreciate the practicality of the Kohlberg criterion and for convenience in the later development, we first present the algorithmic view of the Kohlberg criterion in Algorithm \[alg:Kohlberg\].
**Input**: Game $G(N,v)$, imputation solution $\bx$ **Output**: Conclude if $\bx$ is the nucleolus **1**. Initialization: Set $H_0 = \{\be_N,\emptyset\}$, $T_0 = \{\{i\},i=1,\ldots,n~:~x_i = v(\{i\})\}$ and $k=1$ **5**. Conclude that $\bx$ **is** the nucleolus.
In this algorithm, we iteratively form the tight sets $T_j,j=0,1,\ldots$ until either all the coalitions are included and we conclude the input solution is the nucleolus (i.e., stopping at step 5) or stop at a point where the union of the tight coalitions is not $T_0$-balanced (in step 4), in which case we conclude that the solution is not the nucleolus. To demonstrate the Kohlberg criterion, we consider the following simple three-player cooperative game:
\[ex1\] Let the characteristic function be: $v(\{1\}) = 1,~ v(\{2\}) = 1,~ v(\{3\}) = 1$, $v(\{1,2\}) = 7$, $v(\{1,3\}) = 4$, $v(\{2,3\}) = 5$, $v(\{1,2,3\}) = 12$. The set of all imputations is: $\mathbf{I} =\{ (x_1, x_2, x_3) ~:~ x_1 + x_2+x_3 = 12,~ x_1 \geq 1,~x_2 \geq 1,~x_3 \geq 1\},$ and the core of the game is: $$\begin{aligned}
\mathbf{Co} =\{ (x_1, x_2, x_3) : && x_1 + x_2+x_3 = 12,~x_1 \geq 1,~x_2 \geq 1, ~x_3 \geq 1,\\
&&x_1 + x_2 \geq 7,~x_1 + x_3 \geq 4, ~x_2 + x_3 \geq 5\}.$$ The least core is the line segment connecting $\bx=(5,4,3)$ and $\by=(3,6,3)$. The nucleolus is $\bnu = (4,5,3)$. At the nucleolus $\bnu$, we can find $T_0 = \emptyset$, $T_1 = \{\{1,2\}\},\{3\}\}$, $T_2 = \{\{1\}\},\{2,3\},\{1,3\}\}$, $\epsilon_1 = -2$, $\epsilon_2 = -3$, and $\epsilon_3 = -4$. Here, $T_1$ is $T_0$-balanced with a weight $\mb{\omega} = (1,1)$. Similarly, $(T_1 \cup T_2)$ is $T_0$-balanced with a weight $\mb{\omega} = (1/2,1/4,1/4,1/2,1/4)$. We can also verify that the Kohlberg criterion does not hold for any $\bx' \neq \bnu$. For example, let $\bx' = 1/2(\bx+\bnu)$. Then $T_1 = \{\{1,2\}\},\{3\}\}$ and $T_2 = \{\{2,3\}\}$. Although $(T_1)$ is $T_0$-balanced, $(T_1 \cup T_2)$ is not.
The simplifying Kohlberg criterion {#sec:simplifiedKohlberg}
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The Kohlberg criterion offers a powerful tool to assess whether a given payoff distribution is the nucleolus by providing both the necessary and sufficient conditions. This often arises in relatively small or well-structured games where a potential candidate for the nucleolus can be easily identified and where checking the balancedness of the corresponding tight sets can be done easily (possibly analytically). For larger games, it is inconvenient to apply the Kohlberg criterion because this could involve forming and checking for the balancedness of an exponentially large number of subsets of tight coalitions, each of which could be of exponentially large size. This section aims to resolve these issues.
Bounding the number of iterations to $(n-1)$ {#subsec:simplifiedKohlberg}
--------------------------------------------
On using linear algebra operators on the collection of coalitions, we slightly abuse the notations and refer each coalition $\S \in 2^{\N}$ interchangeably with its binary vector $\be(\S)$ indicating whether the players are in the coalition. For each collection of coalitions $T$, let us denote $rank(T)$ as the rank of the coalitions in $T$ and $span(T)$ as the collection of all coalitions that lie in the linear span of the coalitions in $T$. The key idea in simplifying the Kohlberg criterion is to note that, once we have obtained and verified the $T_0$-balancedness of $\cup_{j=1}^k T_j$, we do not have to be concerned about all those coalitions that belong to $span(\cup_{j=1}^k T_j)$. In brief, this is because once a collection is $T_0$-balanced, its span is also $T_0$-balanced as is formalized in the following lemma:
\[lemma:balancedness\] From any collection $T_0 \subseteq 2^\N$, the following results hold:
- If a collection $T$ is $T_0$-balanced, then $span(T)$ is also $T_0$-balanced.
- If collections $U,V$ are $T_0$-balanced then $U \cup V$, $span(U) \cup span(V)$ are also $T_0$-balanced.
- If $U$ is $T_0$-balanced and $U \subseteq V$, then $span(U) \cap V$ is also $T_0$-balanced.
\(a) Given that $T$ is $T_0$-balanced, there exists $\mb{\gamma} \in \R_{\geq0}^{|T_0|}$ and $\mb{\omega} \in \R_{>0}^{|T|}$ such that $$\be(\N) = \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S).$$
For any $\S_0 \in span(T)$, there exists $\mb{\beta}$ such that $\be({\S_0}) =\sum_{\S \in T} \beta_S \be(\S)$. Thus, for any $\delta$, we have $$\begin{aligned}
\be(\N) &=& \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S)\\
&=& \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S) + \delta (\be(\S_0) -\sum_{\S \in T} \beta_S \be(\S))\\
&=& \delta \be(\S_0) + \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} (\omega_S-\delta \beta_S) \be(\S).\end{aligned}$$ Since $\mb{\alpha} >0$, we can choose $\delta >0$ which is small enough such that $(\alpha_\S-\delta \beta_\S) > 0,~\forall \S \in T$. Thus, $T\cup\{\S_0\}$ is a $T_0$-balanced collection. Since this holds for all $\S_0 \in span(T)$, we can conclude that $span(T)$ is $T_0$-balanced.
\(b) Given that collections $U,V$ are $T_0$-balanced, there exists $\mb{\gamma},\mb{\omega} \in \R_{\geq0}^{|T_0|}$ and $\mb{\alpha} \in \R_{>0}^{|U|}$, $\mb{\beta} \in \R_{>0}^{|V|}$ such that $$\be({\N}) =\sum_{\S \in T_0} \gamma_S \be(\S) +\sum_{\S \in U} \alpha_S \be(\S) = \sum_{\S \in T_0} \omega_S \be(\S) + \sum_{\S \in V} \beta_S \be(\S).$$ This leads to $$\be({\N}) = \sum_{\S \in T_0} (1/2\gamma_S + 1/2\omega_S) \be(\S) + \sum_{\S \in U} 1/2\alpha_S \be(\S) + \sum_{\S \in V} 1/2 \beta_S \be(\S).$$ Thus $U \cup V$ is also $T_0$-balanced. We can also prove that $span(U) \cup span(V)$ is $T_0$-balanced in a similar way as shown in the proof of part (a).
\(c) The proof is similar to part (a) due to the fact that, for any $\S_0 \in span(U) \cap V$, we have $\S_0 \in span(U)$ and hence $U \cup \S_0$ is also $T_0$-balanced. Thus, $span(U) \cap V$ is $T_0$-balanced. ${\hfill\blacksquare}$
With this result, we can provide an improved Kohlberg criterion as shown in Algorithm \[alg:Modified\_Kohlberg1\].
**Input**: Game G(N,v), imputation solution $\bx$ **Output**: Conclude if $\bx$ is the nucleolus or not **1**. Initialization: Set $H_0 = \{\be_N,\emptyset\}$, $T_0 = \{\{i\},i=1,\ldots,n~:~x_i = v(\{i\})\}$ and $k=1$ **5**. Conclude that $\bx$ **is** the nucleolus.
The main differences between Algorithm \[alg:Modified\_Kohlberg1\] and Algorithm \[alg:Kohlberg\] are: (a) the stopping condition of the while loop has been changed from $H_{k-1} \neq 2^\N$ to $rank(H_{k-1}) < n$, and (b) the search space at step 2 has been changed from $\S \not \in H_{k-1}$ to $\S \not \in span(H_{k-1})$. As a result, we have the following desirable property:
\[thm:algo1\_correctness\] The while-loop in Algorithm \[alg:Modified\_Kohlberg1\] terminates after at most $(n-1)$ iterations and it correctly decides whether a solution is the nucleolus.
First of all, by construction in step 2 of the algorithm, $T_k \cup span(H_{k-1}) = \emptyset$ and hence, by step 3, we have the rank of $H_k = H_{k-1} \cup T_k$ keeps increasing. Therefore, $$n \geq rank(H_k) = rank(H_{k-1} \cup T_k) \geq rank(H_{k-1})+1\geq \ldots \geq rank(H_0)+k =k+1,$$ and hence the algorithm, i.e., the while loop, terminates in at most $(n-1)$ iterations. Here, we note that the algorithm terminates at either step 4 or step 5 with complementary conclusions. Proving that the algorithm correctly decides whether a solution is the nucleolus is equivalent to showing that (a) if $\bx$ is the nucleolus then the algorithm correctly terminates at step 5, and (b) if the algorithm terminates at step 5, then the input solution must be the nucleolus.
Part (a): If $\bx$ is the nucleolus, then $T_1$ must be $T_0$-balanced as a direct result from the Kohlberg criterion (described in Theorem \[theorem:Kohlberg\]). Thus $T_1$ is $T_0$-balanced and the algorithm goes through to step 3 at $k=1$. Suppose, as a contradiction, that the algorithm goes through to step 4, instead of step 5, at some index $k>1$; that is $(\cup_{j=1}^k T_j)$ is not $T_0$-balanced. By Lemma \[lemma:Kohlberg\], there exists $\by \in R^n$ such that $$\begin{aligned}
&&\by(\S) \geq 0, \forall \S \in \cup_{j=0}^k T_j;~ \by(N) = 0;~ \by(\S') > 0, \text{ for some } \S' \in \cup_{j=1}^k T_j. \label{ieq:thm1:a0}\end{aligned}$$ Notice, however, that $\cup_{j=1}^{k-1} T_j$ is $T_0$-balanced by the construction in step 3 of the previous iteration. Therefore $\S' \not \in H_{k-1}$ since, otherwise, the result in Lemma \[lemma:Kohlberg\] is violated. Thus $\S' \in T_k$ and hence (\[ieq:thm1:a0\]) leads to
$$\begin{aligned}
&&(\bx+\by)(\S) \geq \bx(\S), \forall \S \in T_k;~ (\bx+\by)(\S') > \bx(\S'), \text{ for some } \S' \in T_k,\\
&\Rightarrow& d(\S,\bx+\by) \leq d(\S,\bx), \forall \S \in T_k;~ d(\S',\bx+\by) < d(\S',\bx), \text{ for some } \S' \in T_k,\end{aligned}$$ that is, for all coalitions in $T_k$, the corresponding excess values for $(\bx+\by)$ is no greater than that of $\bx$ with at least one strict inequality for some coalition. Thus, $$\begin{aligned}
&& \Phi(\bx+\by) <_{L,T_{k}} \Phi(\bx),\label{ieq:thm1:b} \end{aligned}$$
where, for each collection of coalition $Q$, the subscript $(\cdot_{L,Q})$ is the lexicographical comparison with respect to *only* coalitions in $Q$. Since $H_{k-1}$ is $T_0$-balanced by the construction in step 3 of the previous iteration, $span(H_{k-1})$ is also $T_0$-balanced by Lemma \[lemma:balancedness\]. Thus, $\by(\S) = 0,~ \forall \S \in span(H_{k-1})$ and $$\begin{aligned}
\Phi(\bx+\by) =_{L,span(H_{k-1})} \Phi(\bx).\label{ieq:thm1:a}\end{aligned}$$
From [$(\ref{ieq:thm1:b})$]{} and [$(\ref{ieq:thm1:a})$]{} we have $$\begin{aligned}
\Phi(\bx+\by) <_{L,span(H_{k-1})\cup T_k} \Phi(\bx).\label{ieq:thm1:c}\end{aligned}$$
For all $\S \not \in (span(H_{k-1})\cup T_k)$ we have $v(\S) -\bx(\S) < \epsilon_k$. Thus, there exists $\delta >0$ and small enough such that $$v(\S) -(\bx+\delta \by)(\S) < \epsilon_k,~\forall \S \not \in (span(H_{k-1})\cup T_k).$$
Note that results in [$(\ref{ieq:thm1:a0})$]{} also holds if we scale $\by$ by any positive factor. Thus, $$\begin{aligned}
\Phi(\bx+\delta \by) <_{L,span(H_{k-1})\cup T_k} \Phi(\bx).\end{aligned}$$ In other words, the $|span(H_{k-1})\cup T_k|$ largest excess value of $\bx$ is lexicographically larger than the excess values of $(\bx+\delta \by)$ on these collections of coalitions and the remaining coalitions which means $\bx$ is not the nucleolus. Contradiction!
Part (b): If the algorithm bypassed step 4 and went to step 5, then, $(\cup_{j=1}^k T_j)$ is $T_0$-balanced for all $k$ until $rank(H_{k-1}) = n$. Let $\bz$ be the nucleolus; then by its definition, its worst excess value should be no larger than the worst excess value of $\bx$, which is equal to $\epsilon_1$. Thus, the excess value of $\bz$ over any coalition, including those in $T_1$, must be at most $\epsilon_1$; i.e., $$\begin{aligned}
&&(\bz-\bx)(\S) \geq 0, \forall \S \in T_{1}.\end{aligned}$$ Since $T_1$ is $T_0$-balanced, we have, by Lemma \[lemma:Kohlberg\], $(\bz-\bx)(\S) = 0$ for all $\S \in T_1$ (by noticing also that $(\bz-\bx)(\N) = 0$ and $(\bz-\bx)(\S) \geq 0,\forall \S \in T_0$ from the fact that $\bz$ is an imputation and the construction of $T_0$). Using a similar argument, given that $\bx$ and $\bz$ are lexicographically equivalent on $span(T_1)$ and since $\bz$ is the nucleolus, we also have $(\bz-\bx)(\S) \geq 0, \forall \S \in T_2$. Thus, $$\begin{aligned}
&&(\bz-\bx)(\S) \geq 0, \forall \S \in T_1 \cup T_2.\end{aligned}$$ Again, given that $(T_1 \cup T_{2})$ is $T_0$-balanced, we have, by Lemma \[lemma:Kohlberg\], $(\bz-\bx)(\S) = 0$ for all $\S \in T_1 \cup T_2$. We can continue with this and use an induction argument to show that $(\bz-\bx)(\S) = 0$ for all $\S \in H_{k-1}, k\geq 1$. Given that $rank(H_{k-1}) = n$, we must have $\bx=\bz$ or $\bx$ is the nucleolus. ${\hfill\blacksquare}$
**Remark**: It is noted that step 2 in both Algorithms \[alg:Kohlberg\] and \[alg:Modified\_Kohlberg1\] involves comparing vectors of exponentially large sizes. Indeed we cannot escape from having an exponentially large number of operations because we are (lexicographically) comparing exponentially large vectors. The key finding in Theorem \[thm:algo1\_correctness\], however, is to show that step 2 of Algorithm \[alg:Modified\_Kohlberg1\] is not repeated more than $(n-1)$ times (instead of possibly exponential times in the original Kohlberg criterion described in Algorithm \[alg:Kohlberg\]). Although we cannot escape from having exponential number of operations for games without any structure, there are structured games such as the voting game, the network flow game and the coalitional skill games in which step 2 can be done efficiently. We refer the readers to @Nguyen2016 for more details on this.
Fast algorithm for checking the balancedness {#subsec:balancedness_checking}
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According to the Kohlberg criterion, to check the $T_0$-balancedness of $T$, we need to show the existence (or non-existence) of $\mb{\gamma} \in \R_{\geq0}^{|T_0|}$ and $\mb{\omega} \in \R_{>0}^{|T|}$ such that $$\be(\N) = \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S).$$ This is not a big issue for small-sized $T$ where the inspection of such $(\gamma,\omega)$ can be done easily. @solymosi2015 \[Lemma 3\] provide an approach by solving $|T|$ linear programs as follows. For each $\C \in T$, let $$q_{\C}^* =\left\{\max w_{\C} ~:~ \sum_{\S \in T_0 } \gamma_\S \be(\S)+\sum_{\S \in T} w_\S \be(\S) = \be(\N),~ (\mb{\gamma},\mb{\omega}) \in \R_{\geq 0}^{|T_0|+|T|} \right\}.$$ Then $T$ is $T_0$-balanced if and only if $q_{\C}^*>0,\forall {\C} \in T$. Notice, however, that the collection $T$ appearing in the Kohlberg criterion could be exponentially large, and hence solving all the $|T|$ linear programs is not practical. We present a faster approach that involves at most $rank(T)$ linear programs (this is an upper-bound and, in practice, we often need to solve a much smaller number of LPs). Algorithm \[alg:balancedness\] describes this in details.
**Input**: A collection of coalitions $T$ **Output**: To conclude if $T$ is $T_0$-balanced or not
**1**. Initialization: Set $U = \emptyset$ **5**. Stop the algorithm and conclude that $T$ is $T_0$-balanced.
Algorithm \[alg:balancedness\] correctly decides if $T$ is $T_0$-balanced and it terminates in at most $rank(T)$ iterations.
First of all, the while loop should terminate given that $rank(U)$ keeps increasing via the construction of $U$ in steps 2 and 4; i.e., the set $U$ is kept added with coalitions outside its span. Thus, the algorithm terminates at either step 3 or 5 and we need to prove that the corresponding conclusions are correct. If the algorithm terminates at step 3, then $\norm{\mb{\omega}^*} = 0$ (as otherwise the optimal solution in step 2 should be strictly positive) and hence $T$ is not $T_0$-balanced. If the algorithm terminates at step 5 then, prior to that, we have $rank(U) = rank(T)$ in order for the while loop to terminate. The construction of $U$ in step 4 ensures that $U$ is a $T_0$-balanced set by Lemmas \[lemma:balancedness\]b and \[lemma:balancedness\]c. Thus $T= span(U) \cap T$ is also $T_0$-balanced by Lemma \[lemma:balancedness\]c. In conclusion, the algorithm always terminates with the correct conclusion. ${\hfill\blacksquare}$
Reducing the sizes of the tight sets {#subsec:simplifiedKohlberg2}
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On checking the Kohlberg criterion, we might end up having to store an exponentially large number of coalitions. We provide a method for reducing this to the size of at most $n(n-1)$. We start with the following theoretical results.
\[lemma:balancedness2\] The following results hold
- If $T \subseteq 2^N $ is a $T_0$-balanced set then there exists $R \subseteq T$ with $1 \leq |R| = rank(R) \leq rank(T)$ that is $T_0$-balanced.
- For nonempty $P,Q \subseteq 2^N$ with $Q \cup P$ is a $T_0$-balanced set, there exists a subset $R \subseteq Q$ with $1 \leq |R| = rank(R) \leq rank(Q)$ such that $R \cup P$ is $T_0$-balanced.
$~~~$
- Given that $T$ is $T_0$-balanced, there exists $\mb{\gamma} \in \R_{\geq0}^{|T_0|}$ and $\mb{\omega} \in \R_{>0}^{|T|}$ such that $$\be(\N) = \sum_{\S \in T_0} \gamma_S \be(\S) + \sum_{\S \in T} \omega_S \be(\S).$$ Thus, $$\begin{aligned}
&&0 \neq \frac{1}{\sum_{\S \in T} \omega_S} \left(\be(\N) - \sum_{\S \in T_0} \gamma_S \be(\S)\right) = \sum_{\S \in T} \frac{\omega_S}{\sum_{\S \in T} \omega_S} \be(\S),\end{aligned}$$ i.e., $\frac{1}{\sum_{\S \in T} \omega_S} \left(\be(\N) - \sum_{\S \in T_0} \gamma_S \be(\S)\right) $ belongs to the convex combination of $\{\be(\S)\}_{\S \in T}$. Applying the Caratheodory theorem, there exists a subset $U \subseteq T$ with $rank(U) = |U|= dim(T)$ such that $\frac{1}{\sum_{\S \in T} \omega_S} \left(\be(\N) - \sum_{\S \in T_0} \gamma_S \be(\S)\right) = \sum_{\S \in U} \beta_\S \be(\S)$.
By removing those coefficients $\beta_\S =0$, we obtain a subset $R \subseteq U \subseteq T$ with $rank(R) \leq rank(U)$ that is $T_0$-balanced. Note also that, since $$\frac{1}{\sum_{\S \in T} \omega_S} \left(\be(\N) - \sum_{\S \in T_0} \gamma_S \be(\S)\right) \neq 0,$$ there exists at least a coalition $\S$ with $\beta_\S >0$. Thus $1 \leq rank(R) \leq rank(T)$ and $R$ is $T_0$-balanced. In addition, since $rank(U) = |U|$, we also have $rank(R) = |R|$.
- Since $Q \cup P$ is $T_0$-balanced, there exists $\mb{\gamma} \in \R_{\geq0}^{|T_0|}$, $\mb{\alpha} \in \R_{>0}^{|P|}$ and $\mb{\beta} \in \R_{>0}^{|Q|}$ such that $\be({\N})= \sum_{\S \in T_0} \gamma_\S \be(\S) + \sum_{\S \in P} \alpha_\S \be(\S) + \sum_{\S \in Q} \beta_\S \be(\S)$. Thus, $$\begin{aligned}
&&(\be(\N)-\sum_{\S \in T_0} \gamma_\S \be(\S)-\sum_{\S \in P} \alpha_\S \be(\S))= \sum_{\S \in Q} \beta_\S \be(\S) \neq 0.$$ Using the same argument as in part (a), there exists a subset $Q' \subseteq Q$ with $rank(Q') = |Q'|= dim(Q)$ such that $$\begin{aligned}
&&(\be(\N)-\sum_{\S \in T_0} \gamma_\S \be(\S)-\sum_{\S \in P} \alpha_\S \be(\S))= \sum_{\S \in Q'} \beta_\S \be(\S).\end{aligned}$$ By removing those coalitions $\S \in Q'$ with $\beta_\S=0$, we obtain a non-empty subset $R \subseteq Q'$ such that $R \cup P$ is $T_0$-balanced and $1 \leq |R| = rank(R) \leq rank(Q)$. ${\hfill\blacksquare}$
We denote such a subset $R$ in Theorem \[lemma:balancedness2\]a as $R = rep(T;T_0)$ and subset $R$ in Theorem \[lemma:balancedness2\]b as $R = rep(Q;P,T_0)$. Algorithm \[alg:Modified\_Kohlberg2\] shows the improved Kohlberg Algorithm for verifying if a solution is the nucleolus by replacing each tight set of coalitions by its representation derived in Theorem \[lemma:balancedness2\].
**Input**: Game G(N,v), imputation solution $\bx$ **Output**: Conclude if $\bx$ is the nucleolus or not **1**. Initialization: Set $H_0 = T_0 = \be_N$, $T_0 = \{\{i\},i=1,\ldots,n~:~x_i = v(\{i\})\}$, and $k=1$ **5**. Conclude that $\bx$ is the nucleolus.
The main difference between Algorithm \[alg:Modified\_Kohlberg2\] and Algorithm \[alg:Modified\_Kohlberg1\] is in step 3 where we set $H_k = H_{k-1} \cup rep(T_k; H_{k-1},T_0)$ instead of $H_k = H_{k-1} \cup T_k$. This means we store only a representative of $T_k$ in the subsequent rounds. The correctness of the algorithm can still be proven as presented in the following theorem.
The while-loop in Algorithm \[alg:Modified\_Kohlberg2\] terminates after at most $(n-1)$ iterations and it correctly decides whether a solution is the nucleolus.
After each iteration, we have $R_k \not \subseteq span(H_{k-1})$ and $rank(R_k) \geq 1$ by its construction. Therefore $rank(H_k)=rank(R_k \cup H_{k-1})$ keeps increasing and hence Algorithm \[alg:Modified\_Kohlberg2\] terminates after at most $(n-1)$ iterations. We also note that the algorithm terminates at either step 4 or 5 with complementary conclusions.
Proving that the algorithm correctly decides whether a solution is the nucleolus is equivalent to showing that (a) if $\bx$ is the nucleolus then the Algorithm terminates at step 5, and (b) if the algorithm terminates at step 5, then the input solution must be the nucleolus. We use results from Lemma \[lemma:balancedness\] and Theorem \[lemma:balancedness2\] for this.
The proof for part (a) is still the same as that proof for Theorem 2 since the key property used in that proof was to keep $H_k$ always $T_0$-balanced. This is summarized as follows. If $\bx$ is the nucleolus then $T_1$ is $T_0$-balanced and the algorithm gets through to step 3 at $k=1$. Suppose, on contradiction, that the algorithm terminate at step 4 at some index $k>1$ with $(H_{k-1} \cup T_k)$ not $T_0$-balanced while $H_{k-1}$ is $T_0$-balanced by the construction in step 3 of the previous iteration. Then, by Lemma \[lemma:Kohlberg\], there exists $\by \in R^n$ such that $$\begin{aligned}
&&\by(\S) \geq 0, \forall \S \in T_0 \cup H_{k-1} \cup T_k;~ \by(N) = 0;~ \by(\S') > 0, \text{ for some } \S' \in T_k.\end{aligned}$$ Thus, $$\begin{aligned}
\Phi(\bx+\by) <_{L,span(H_{k-1})\cup T_k} \Phi(\bx).\end{aligned}$$
In addition, for all $\S \not \in (span(H_{k-1})\cup T_k)$ we have $v(\S) -\bx(\S) < \epsilon_k$ by the construction in step 2. Thus there exist $\delta >0$, which is small enough such that $$v(\S) -(\bx+\delta \by)(\S) < \epsilon_k,~\forall \S \not \in (span(H_{k-1})\cup T_k)$$ and $$\begin{aligned}
\Phi(\bx+\delta \by) <_{L,span(H_{k-1})\cup T_k} \Phi(\bx).\end{aligned}$$ In other words, the $|span(H_{k-1})\cup T_k|$ largest excess value of $\bx$ is lexicographically larger than the excess values of $(\bx+\delta \by)$ on these collections of coalitions and the remaining coalitions which means $\bx$ is not the nucleolus. Contradiction.
The proof for part (b) is also the same as that in Theorem 2 where the key property of retaining the rank of $H_k$ increased throughout the algorithm is still preserved. Due to the $T_0$-balancedness of $(H_{k-1} \cup T_k)$, we can use Lemma \[lemma:Kohlberg\] to recursively show that $(\bz-\bx)(\S) = 0$ for all $\S \in H_{k-1} \cup T_k$ where $\bz$ is the nucleolus as follows.
Let $\by = \bz - \bx$. We have $\by(\N) =0$ and $\by(\S) \geq 0, \forall \S \in T_0$ due to the fact that $\bz$ is an imputation and the definition of $T_0$. We also have $\by(\S) \geq 0, \forall \S \in R_1$ due to the fact that $\Phi(\bz) \leq_{L,R_1} \Phi(\bx)$ (or otherwise $\bz$ is not the nucleolus).
Applying result from Lemma \[lemma:Kohlberg\] with a note that $R_1$ is $T_0$-balanced, we have $\by(\S) = 0, \forall \S \in R_1$ which means $\bz$ and $\bx$ are lexicographically equivalent on $R_1$. We will prove by induction that $\by(\S) = 0, \forall \S \in \cup_{j=1}^k R_j$ for all indices $k$ valid in Algorithm \[alg:Modified\_Kohlberg2\]. Suppose this indeed hold for $(k-1)$, i.e. $\by(\S) = 0, \forall \S \in \cup_{j=1}^{k-1} R_j$. In other words, $\bz$ and $\bx$ are lexicographically equivalent on $\cup_{j=1}^{k-1} R_j$ which is the collection of coalitions from which $\bx$ receives the worst excess values. In order for $\bz$ to be at least as good lexicographically as $\bx$, the excess values of $\bz$ on those coalition in $R_k$ must be no worst that those from $\bx$, i.e., $\by(\S) \geq 0, \forall \S \in R_k$. Applying result from Lemma \[lemma:Kohlberg\] with a note that $\cup_{j=1}^k R_j$ is $T_0$-balanced, we must also have $\by(\S) = 0, \forall \S \in R_k$.
Since $rank(H_{k-1}) = n$, we must have $\bx=\bz$ or $\bx$ is the nucleolus. ${\hfill\blacksquare}$
The collection of tight coalitions $\cup_{j=1}^k R_j$ stored by Algorithm \[alg:Modified\_Kohlberg2\] is of size at most $n(n-1)$.
In the proof of Theorem \[lemma:balancedness2\], note that each $R_j$ is constructed as a subset of another full row rank subset, its size is at most $n$ rows. Since the number of iterations involved is at most $(n-1)$, the total size of $\cup_{j=1}^k R_j$ is at most $n(n-1)$. ${\hfill\blacksquare}$
*Remark*: We conjecture that, under mild conditions, the size of $\cup{j=1}^k R_j$ is equal to $(n+k)$. We also conjecture that the algorithms developed can be extended to the case of finding the nucleolus within any polyhedron by replacing the set $T_0$ accordingly. We leave these explorations for future work though.
Conclusion
==========
The Kohlberg criterion proves to be a powerful tool for verifying whether a payoff distribution is the nucleolus in relatively small games. Its application to larger games is, however, rather limited due to the need for repeatedly forming, storing and checking the balancedness of an exponentially large collection of coalitions for an exponentially large number of iterations. In this work, we simplify the Kohlberg criterion to achieve the following desirable properties: (a) the number of iteration is bounded to at most $(n-1)$, (b) the size of collections of coalitions for storage is at most $n(n-1)$. In addition, we provide a fast algorithm for checking the balancedness. It is expected that the findings will boost the applications of the Kohlberg criterion and possibly provide new directions for finding efficient algorithms to compute the nucleolus.
Acknowledge {#acknowledge .unnumbered}
===========
We thank Dr Holger Meinhardt [@meinhardt2017simplifying] for pointing out some of the typos in our ealier working draft of this paper (mostly on handling the $T_0$-balancedness condition which we overlooked when changing the original algorithm for finding the prenucleolus to finding the nucleolus). We however still disagree with Dr Meinhardt’s very strong claim on the correctness of the well-established proof technique used in this paper (such as in Theorem \[thm:algo1\_correctness\]).
Appendix A: Alternative Proof of Kohlberg Criterion {#app:Kohlberg_thm_restate .unnumbered}
===================================================
Let $(T_0,T)$ be two collections of coalitions. For each coalition $C \in T$, let us introduce the following primal LP:
$$\begin{aligned}
P(C):\quad \max_{\mb{\gamma},\omega,\alpha} &&\omega_C \\
s.t. && \sum_{\S \in T_0} \gamma_\S e(\S)+\sum_{\S \in T} \omega_\S e(\S) -\alpha e(N) = 0,\\
&&\mb{\gamma} \geq 0, \omega \geq 0.\end{aligned}$$
The corresponding dual problem is: $$\begin{aligned}
D(C):\quad \min_{y} &&0 \\
s.t. && y^T e(\S) \geq 0,\forall \S \in T_0 \cup T,\\
&& y^T e(\N) = 0,\\
&& y^T e(C) \geq 1.\end{aligned}$$
We have the following results:
\[lemma:balancedness2\] For any given pair of subsets $(T_0,T)$ of the powerset $2^N$, the following are equivalent
- $T$ is $T_0$-balanced if any only if for all $C \in T$, the primal problem $P(C)$ is unbounded.
- For any $C \in T$, the primal problem $P(C)$ is unbounded if any only if the dual $D(C)$ is infeasible.
- The primal problem $D(C)$ is infeasible for all $C \in T$ if any only if $(T_0,T)$ has property II.
- $(T_0,T)$ has property II if and only if $T$ is $T_0$-balanced.
$~~~$ Result in part (d) is what we want to show and this will follows directly if we are able to show (a)-(c). We choose to show both sides of the if and only if statements in part (a)-(c) so that each of these can be viewed as stand-alone results eventhough the proof of the entire lemma only requires one direction such as (a)$\Rightarrow$(b)$\Rightarrow$(c)$\Rightarrow$(d)$\Rightarrow$(a).
- $\Rightarrow$ If $T$ is $T_0$-balanced, there exists weight vectors $\mb{\gamma} \in \R_{\geq 0}^{|T_0|}$,$\mb{\omega} \in \R_{>0}^{|T|}$ such that $$\sum_{\S \in T_0} \gamma_\S e(\S) +\sum_{\S \in T} \omega_\S e(\S) = e(N).$$ For each $C \in T$, we have $(\mb{\gamma},\mb{\omega},1)$ is a feasible solution to $P(C)$ with an objective value of $\omega_C >0$. Since the problem is homogeneous on $(\mb{\gamma},\mb{\omega}, \alpha)$, that is for all $\Delta >0$, we have $(\Delta \mb{\gamma}, \Delta \mb{\omega}, \Delta \alpha)$ is also a feasible solution with an optimal value of $\Delta \omega_C$. Thus, the primal problem is unbounded and hence the dual problem $P(C)$ is infeasible.
$\Leftarrow$ For each $C \in T$, given the primal problem $P(C)$ is unbounded, we can pick a corresponding feasible solution $(\mb{\gamma},\mb{\omega},1)$ with a positive objective value $\omega_C$. Average out all such feasible solutions $(\mb{\gamma},\mb{\omega},1)$, one for each $C \in T$, we would obtain the average weight $(\bar{\mb{\gamma}},\bar{\mb{\omega}})$ that satisfies $\bar{\mb{\gamma}} \geq 0$, $\bar{\mb{\omega}} >0$, and $$\sum_{\S \in T_0} \bar{\gamma}_\S e(\S) +\sum_{\S \in T} \bar{\omega}_\S e(\S) = e(N)$$ Thus, $T$ is $T_0$-balanced.
- We can see that the primal problem is alway feasible at $(\mb{\gamma}=0, \mb{\omega} = 0, \alpha = 0)$. In addition, the problem is homogeneous on $(\mb{\gamma}, \mb{\omega}, \alpha)$ and hence its optimal value is either zero or positive infinitive (unbounded). The dual problem, on the other hand, is either infeasible or with an optima value of zero. From linear programming duality results, it is easy to show that in this case, the primal is unbounded if and only if the dual is infeasible.
- $\Leftarrow$ If $(T_0,T)$ has property II, we have $D(C)$ infeasible for all $C \in T$ by definition of property II.
$\Rightarrow$ If $D(C)$ infeasible for all $C \in T$ then $(T_0,T)$ must have property II since otherwise there exists a $y \in Y(T_0 \cup T)$ and a coalition $C \in T$ such that $y(C) >0$. Thus, we can scale up $y$ by an appropriate factor $\Delta$ such that $\Delta y \in Y(T_0 \cup T)$ and $\Delta y(C) \geq1$. This means the dual problem $D(C)$ is feasible. Contradiction!
${\hfill\blacksquare}$
|
---
abstract: 'We present flux-ratio curves of the [*fold and cusp*]{} (i.e. close multiple) images of six JVAS/CLASS gravitational lens systems. The data were obtained over a period of 8.5 months in 2001 with the Multi-Element Radio-Linked Interferometer Network (MERLIN) at 5–GHz with 50 mas resolution, as part of a MERLIN Key-Project. Even though the time delays between the fold and cusp images are small ($\la$1d) compared to the time-scale of intrinsic source variability, all six lens systems show evidence that suggests the presence of extrinsic variability. In particular, the cusp images of B2045+265 – regarded as the strongest case of the violation of the cusp relation (i.e. the sum of the magnifications of the three cusp images add to zero) – show extrinsic variations in their flux-ratios up to $\sim$40% peak-to-peak on time scales of several months. Its low Galactic latitude of $b\approx -10^\circ$ and a line-of-sight toward the Cygnus superbubble region suggest that Galactic scintillation is the most likely cause. The cusp images of B1422+231 at $b\approx +69^\circ$ do not show strong extrinsic variability. Galactic scintillation can therefore cause significant scatter in the cusp and fold relations of some radio lens systems (up to 10% rms), even though these relations remain violated when averaged over a $\la$1 year time baseline.'
author:
- 'L.V.E. Koopmans'
- 'A. Biggs'
- 'R.D. Blandford'
- 'I.W.A. Browne, N.J. Jackson, S. Mao, P.N. Wilkinson'
- 'A.G. de Bruyn'
- 'J. Wambsganss'
title: 'Extrinsic Radio Variability of JVAS/CLASS Gravitational Lenses'
---
Introduction
============
Cosmological Cold–Dark–Matter (CDM) simulations predict the existence of condensed structures in the halos around massive galaxies (e.g. Klypin et al. 1999; Moore et al. 1999), if the initial power-spectrum does not cut off at small scales and dark matter is cold and not self-interacting. However, observationally, we see at most the high-mass tail of these structures in the form of dwarf galaxies. This raises the question of where most of their less massive ($10^{6}-10^9$ M$_\odot$) counterparts are located. Either (i) these CDM structures have not formed – in conflict with CDM predictions – or (ii) they consist predominantly of dark-matter and baryons have been blown out, preventing star formation altogether, or (iii) baryons are present but have not condensed inside their potential well to form visible stars. If either one of the latter two is the case, the only way to detected them is through their gravitational effect, in particular through dynamics and lensing.
The initial suggestion by Mao & Schneider (1998) that anomalous flux-ratios in the lens system B1422+231 can be caused by small-scale mass substructure in the lens galaxy, was recently extended to a larger – although still limited – sample of gravitational-lens systems with fold and cusp images (Metcalf & Madau 2001; Keeton 2001; Chiba 2002; Metcalf & Zhao 2002; Dalal & Kochanek 2002; Bradač et al. 2002; Keeton et al. 2002). In particular, analyses have focused on the so called normalized “cusp relation”, which says that $R_{\rm
cusp}\equiv \Sigma \mu_i/\Sigma |\mu_i|\rightarrow 0$, for the magnifications $\mu_i$ of the three merging images of a source well inside the cusp (Blandford 1990; Schneider & Weiss 1992). A similar relation holds for the two fold images. These relations are only two of many (in fact $\infty$) scaling laws (Blandford 1990). Because globular clusters and dwarf-galaxies are too few in number to explain the rate of anomalous flux-ratios and cusp relations, this could be used as an argument in favor of CDM substructure as the dominant cause of these apparent anomalies (e.g. Kochanek & Dalal 2003).
If the observed violations of the cusp relation (i.e. $R_{\rm cusp}\ne
0$), as discussed above, are due to substructure on mass scales of $10^6$ to $10^9$ M$_\odot$, the effect should be the same for radio and optical flux ratios (if the latter are available), and it should be constant in time. However, another possible explanation is microlensing of stellar mass objects in combination with a smoothly distributed (dark) matter component (Schechter & Wambsganss 2002). This one does not require the optical and radio flux ratios to behave in the same way, and in particular it predicts the optical flux ratios to [*change*]{} over time scales of years. Finally, there is also the possibility that the flux-density and surface brightness distribution of lensed radio images are affected by the ionized ISM in the lens galaxy and/or our Galaxy, also leading to changes in the apparent value of the cusp relation.
Hence, before one can confidently accept the detection CDM substructure, it requires rigorous testing to see whether observed flux ratios correspond to magnification ratios or whether they can be affected by propagation effects (or microlensing). Here, we make the first coordinated attempt to test the effects of propagation on the observed [*radio*]{} fluxes of lensed images.
In Sections 2 and 3 we present the first results of our MERLIN Key-Project (Biggs et al. 2003) to search for extrinsic variability between fold and cusp images (i.e. close multiple images), based on their flux-density curves. A discussion and conclusions are given in Sect.4.
MERLIN 5–GHz Data
=================
MERLIN 5–GHz data were obtained between Feb 21 and Nov 7 2001. A total of 41 epochs of 24 hours each were obtained, on average once per week. Eight lens systems were observed (i.e. Table 1 plus B1608+656 and B1600+434) of which seven are four-image systems and one is a double. The data acquisition and reduction is described in Biggs et al. (2003), which also presents the flux-density curves of all the lensed images.
In this paper, we focus on the [*flux-ratio*]{} curves. This approach has several advantages when looking for extrinsic variability. The dominant errors on flux-density curves in the radio are those resulting from residual noise in the maps and from multiplicative errors as a result of erroneous flux calibration. Because multiplicative errors are equal for each of the lensed images, they disappear in the flux-ratio curves (not corrected for the time-delays) which should therefore be flat and dominated by noise in the absence of variability.
All presented lens systems also have small time-delays between cusp/fold images ($\la$1d) compared to the time between observations and the time-scale of intrinsic variability as seen in the flux-density curves (Biggs et al. 2003). Hence, intrinsic flux-density variations should effectively occur simultaneously in fold and cusp images and thus disappear in the flux-ratio curves. Throughout this paper we therefore assume that (i) due to the small time-delays between fold/cusp images, intrinsic variability does not affect the flux-ratio curves, (ii) systematic flux-density errors are multiplicative and also do not affect the flux-ratio curves, and (iii) extrinsic variability does not correlate between lensed images.
We exclude the double B1600+434 and the quad B1608+656 from our analysis, which both have non-negligible time-delays (i.e. several weeks to months; see Fassnacht et al. 1999a, 2002; Koopmans et al. 2000; Burud et al. 2000).
Results
=======
Normalized Flux-ratio Curves
----------------------------
In Fig.\[fig:rcurves\] the resulting flux-ratio curves of all images are shown with respect to image A which is often the brightest image. We follow the labeling of these images as published in the literature (e.g. Biggs et al. 2003). Each flux-ratio curve has been normalized to unity by dividing them through the average flux-ratio of all 41 epochs. The errors are the square root of the sum of the two fractional (noise) errors on the flux-densities squared. The flux-density errors are determined from the rms in the residual maps (i.e. the radio maps with the lensed images subtracted).
In Table 1, we list (i) the average flux-ratios and the rms scatter for each image pair, (ii) the reduced–$\chi^2$ values, by assuming that each normalized flux-ratio should be unity in the case of no extrinsic variability and under the assumptions mentioned in Sect.2, and (iii) the values of $R_{\rm cusp}$ (see Mao & Schneider 1998; Keeton et al. 2002), which we discuss further in Sect.4.
Evidence for Extrinsic Variability
----------------------------------
To test for extrinsic variability in the lensed images, on time-scales less than the monitoring period of 8.5 month, we introduce the following method:
Let us designate the normalized light curves of the individual cusp/fold images as $a_n \equiv A/\langle A\rangle$, $b_n \equiv
B/\langle B\rangle$ and $c_n \equiv C/\langle C\rangle$, where their average flux-densities over the 41 epochs are $\langle A\rangle$, $\langle B\rangle$ and $\langle C\rangle$, respectively[^1]
First, the points $(a_n,b_n,c_n)$ are plotted in a three-dimensional Cartesian space, such that multiplicative errors [*and*]{} intrinsic flux-density variations (the latter because of the negligible time-delays) move points parallel to the vector $(1,1,1)$.
Second, each point $(a_n,b_n,c_n)$ is projected on to a two-dimensional plane that is normal to the vector $(1,1,1)$. Hence, the projected points will [*not*]{} move on that plane, either because of intrinsic flux-density variations or multiplicative errors. Both of these are movements perpendicular to the plane and thus translate to the same projected point.
Third, if one defines the $x$–axis, $\hat x$, of this two-dimensional plane to be the projected $a$–axis, $\hat a$, of the three-dimensional space, and $\hat y$ to be perpendicular to $\hat x$ in the same normal plane, one finds the following simple mapping: $$\begin{aligned}
x &=& (2 a_n - b_n -c_n)/\sqrt{6} \nonumber\\
y &=& (b_n - c_n)/\sqrt{2}\end{aligned}$$ or in polar coordinates $$\begin{aligned}
r^2 &=& {x^2 + y^2}\nonumber\\
\theta &=& \arctan(x,y).\end{aligned}$$ Because $\hat a$ projects on to $\hat x$, any uncorrelated extrinsic variations in image A will only result in a movement of a point along $\hat a$ and thus only along the $\hat x$ axis.
Because the 1–$\sigma$ errors on the normalized flux-densities $a_n$, $b_n$ and $c_n$ are known from the observations, one can calculate the corresponding expected 1–$\sigma$ errors on $x$ and $y$. $$\begin{aligned}
\sigma_x^2 & = & {2/3\, \sigma_{a}^2 + 1/6\, \sigma_{b}^2 + 1/6\, \sigma_{c}^2}\nonumber\\
\sigma_y^2 & = & {1/2\, \sigma_{b}^2 + 1/2\, \sigma_{c}^2}\end{aligned}$$ and similarly $$\begin{aligned}
\sigma_r^2 & = & ((2 a_n - b_n -c_n)^2\, \sigma_a^2 + (2 b_n - c_n -a_n)^2\,
\sigma_b^2 + (2 c_n - a_n -b_n)^2\, \sigma_c^2)/(9 r^2).\end{aligned}$$ Notice that the scatter in $x$ will be a combination of the scatter in $a_n$, $b_n$ and $c_n$, if each image behaves independently.
On the other hand, $$\chi^2_r = \frac{1}{\rm DOF}\sum_i{(r_i/\sigma_{r,i})^2}$$ is a direct estimator of the significance of the presence of extrinsic variability on time-scales of $<$8.5 month, [*irrespective of the image(s) it occurs in*]{}. In other words, it does not tell us which image or images exhibit extrinsic variability, only that extrinsic variability is present if $\chi^2_r$ is significantly larger than unity.
The significance of extrinsic variability in individual image is far more difficult to assess. However, we can estimate the level of extrinsic variability in image A, for example, by knowing that the expected variance in that image, due to noise [*and*]{} in the absence of extrinsic variability, should be $${\rm E}\{\langle \sigma_a^2 \rangle\} \approx 3/2\, {\rm var}(x) - 1/2\, {\rm var}(y).$$ If the observed value of $\langle \sigma_a^2 \rangle = (\sum^N_i \sigma_{a,i}^2)/{N}$ is smaller than ${\rm E}\{\langle \sigma_a^2 \rangle\}$, the difference is due to extrinsic variability with an estimated variance of $${\rm var}(a_{\rm ext}) \approx {\rm E}\{\langle \sigma_a^2 \rangle\} - \langle \sigma_a^2 \rangle.$$ The same procedure can be repeated for each of the other images. In Table 2, we have listed the values of $\chi^2_r$ and the values of ${\rm var}(a_{\rm ext},b_{\rm ext},c_{\rm ext})$, if larger than zero (note that ${\rm E}\{\langle \sigma_a^2 \rangle\}$ is an estimate and could therefore be smaller than $\langle \sigma_a^2 \rangle$ when measured from a finite set of observations).
Finally, we further discuss whether correlations between the flux measurements of the merging images could potentially occur. We note, however, that $\sigma_{a/b/c}$ are noise errors as determined from residual maps, i.e. the original maps after we subtract of the best-fit model of the lensed images. The residual radio maps are consistent with noise maps. Since the images are separated by many beam sizes (i.e. resolution elements), the flux measurements of images A, B and C – even though measured from the same map – are independent, except for the multiplicative errors as explained previously. Hence there should be [*no*]{} effect of measurement correlations in Eq.(3) or (4), which could skew our results.
Hence, the technique discussed above is explicitly designed to separate the effects of multiplicative errors, extrinsic variability and noise, and should also be free of measurement correlations. For example, if one were to cross correlate (e.g. using the Spearmann rank correlation) the flux-ratio curves (Fig.1) of a single lens system with each other, one would find that they correlate strongly, even in the absence of extrinsic variability. The reason being that the same noise variations in image A would be introduced in both $(B/A)_{\rm
n}$ and $(C/A)_{\rm n}$. A Spearman rank correlation on flux-ratio curves without extrinsic variability but with similar noise properties and number of epochs confirms this. However, one notices from equations 3 and 4 that any multiplicative error does not affect $\sigma^2_{x/y}$ or $\sigma^2_r$ (where it cancels out) or the projection on the plane that we defined in equations 1 and 2, as previously discussed. In addition, one finds from equations 1, 2 and 4 that if there is no extrinsic variability, $\chi^2_r \rightarrow 1$, whereas the presence of extrinsic variability implies $\chi^2_r > 1$. Hence, $\chi^2_r$ is indeed independent from multiplicative errors and therefore the correct estimator of the significance of the presence of extrinsic variability, in the (shown) absence of measurement correlations.
Individual Lens Systems
-----------------------
Here, we discuss each case, based on their reduced $\chi^2$ values. Image D is not considered because of its faintness and larger inferred time-delay compared with the other images.
### All systems, except B2045+265
Based on the relatively low values of $\chi^2_r$ and the estimated levels of extrinsic variability (Table 2) for the images of B0128+437, B1359+154 and B1422+231 and B1555+375, and the remaining possibility that some minor undetected additive errors could be present, the evidence for extrinsic variability in these four systems is not totally convincing. We exclude these from further discussion.
In the case of B0712+472, the reduced $\chi^2$ values of the $(B/A)_{\rm n}$ flux-ratio curves and also $\chi_r^2$ seem more significant. In Fig.1 we see that a large number of epochs are deviant over the entire observing season. Deviations of the $(C/A)_{\rm n}$ flux-ratio curve from unity are less significant, probably because image C has a larger fractional error than images A and B. Even though there is some evidence in this system for extrinsic variability between the two fold images, we conservatively regard it also as weak and we will concentrate our discussion on B2045+265. In Sect.4, however, we further discuss what a possible reason for some of the higher values of $\chi_r^2$ and extrinsic variability can be.
### B2045+265
In Tables 1 and 2, we see that (i) both the $(B/A)_{\rm n}$ and $(C/A)_{\rm n}$ flux-ratio curves have very high values of the reduced $\chi^2$, reflected also in large rms values, (ii) the estimated rms values of extrinsic variability and the value of $\chi_r^2$ are very large, and (iii) a visual inspection of the $(B/A)_{\rm n}$ and $(C/A)_{\rm n}$ flux-ratio curves shows changes of up to $\sim$40% on time scales of several months. Because the time delays between the cusp images are only a fraction of a day (Fassnacht et al. 1999b), residual intrinsic source variability can not cause these variations.
A more quantitative analysis based on the structure function (e.g. Simonetti et al. 1985) of the flux-ratio curves (indicated by $R(t)$) is shown in Fig.2. The structure function $<D^{(1)}(\tau)>=<[R(t+\tau)-R(t)]^2>$ quantifies the average rms fluctuations (squared) between two points on the same flux-ratio curve, separated by a time lag $\tau$. A lower value of $<D^{(1)}(\tau)>$ means a stronger correlation (assuming no errors). Fig.2 shows that, even though $<D^{(1)}(\tau)>$ fluctuates considerably, it continues to increase toward longer lags. Around $\tau\sim 150$ d, the rms suddenly decreases considerably, suggesting possible long-term correlated variations in the flux-ratios on that time-scale. If $<D^{(1)}(\tau)>$ increases beyond $\tau\ga 200$ days, flux-density variations of several tens of percent on a time-scale of $\ga 1$yr could be present as well. However, we note that the overlap of the flux-ratio curves becomes smaller for longer lags and consequently the errors become larger. Longer observations are required to make stronger statements about the longer time lags. Even so, similar fluctuations of the structure function are seen in other scintillating sources (e.g. Dennett-Thorpe & de Bruyn 2003).
Several reliability checks of the extrinsic variations of the cusp images of B2045+265 are called for: First we note that the lensed images are of roughly equal brightness and – within a factor about two – as bright as the images in B0128+437, B1359+154, B0712+472 and B1555+375. Hence there is no indication that the observed flux-ratio variations are related to the faintness or brightness of the lensed images. Second, there are no problems with the closeness between the cusp images ($\ga 0.3''$) and the separation of their flux densities because of the high resolution of the MERLIN radio maps ($\sim$50 mas). Hence the fluxes of the three images are fully independent. Third, we have calculated the Spearman rank correlation coefficients ($r_{\rm S}$) between each of its images A, B and C and those of the other five lens systems. This leads to 45 independent values of $r_{\rm S}$ (i.e. noise does not introduce correlation in this case), which on average should tend to zero. We find $<r_{\rm
S}>$=0.0024 and an rms of 0.153. The theoretical expectation value of the rms value is $1/\sqrt{N-1}$=0.151, where $N=45$ in our case. Hence, we recover the expectation values of both the average and rms. This shows that any correlation between the images can not be the result of obvious systematic errors in the data-reduction process, in the creation of the flux-ratio curves, or in our analysis.
Hence, we confidently conclude that the cusp images of B2045+265 show strong evidence for the presence of extrinsic variability.
Discussion & Conclusions
========================
We have presented the flux-ratio curves of six gravitational lens systems, each composed of 41 epochs taken over a period of 8.5 months in 2001 with MERLIN at 5 GHz, as part of a MERLIN Key-Project. The systems were chosen to have merging cusp or fold images, such that the time-delays between these images are negligible ($\la$1d) compared to the time-scale of intrinsic variability and the rate at which the light curves are sampled. The flux-ratio curves should therefore be void of intrinsic variability and multiplicative errors. The main goal of our program was to find additional cases of extrinsic variability other than radio-microlensing in B1600+434 (Koopmans & de Bruyn 2000, 2003).
We find some statistical evidence for extrinsic variability in all six lens systems, based on reduced $\chi_r^2$ values larger than unity (Sect.3.2; Tables 1 & 2). Residual intrinsic variations – due to the finite time delays – or small additive error are unlikely to be the cause of this, but can not fully be excluded yet. The high resolution of MERLIN also ensures negligible correlations between the fluxes of the merging images. The evidence for B0128+437, B1359+154, B1422+231 and B1555+375 is fairly marginal. The case for B0712 is stronger, however, and this object clearly deserves further study. The best case is B2045+265, which we discuss further below.
Even though radio-microlensing cannot be excluded, we think at this point that Galactic scintillation is the more likely cause of some of the higher values of $\chi^2$ (Tables 1 & 2). Indeed, [*all*]{} compact extragalactic radio sources should show refractive scintillation at some level. At wavelengths of 5 GHz and for image sizes $\sim$1 mas, the expected rms fluctuations due to scintillation, in a typical line-of-sight out of the Galactic plane, are a few percent (e.g. Walker 1998, 2001), which are comparable to the observed flux-density errors.
One gravitational lens systems, B2045+265, shows unambiguous evidence for extrinsic variability, based on the reduced $\chi^2$ values significantly larger than unity (Tables 1 & 2) and visually apparent long-term variations in its flux-ratio curves (Fig.1). One possible explanation for the variations is radio microlensing similar to B1600+434 (Koopmans & de Bruyn 2000, 2003). However, because B2045+265 has a Galactic latitude $b \approx -10^\circ$ and is the lowest Galactic latitude system in our sample, Galactic refractive scintillation is the more likely explanation.
To examine this, first we naively use the revised electron-density model of our Galaxy by Cordes & Lazio (2003). This model gives a scattering measure of $8\times 10^{-4}$ kpc m$^{-20/3}$, an angular broadening at 5 GHz of 50 $\mu$as and a transition frequency of 22 GHz between the weak and strong scattering regimes. If we choose the source size to be 250 $\mu$as, we find a modulation index of 7% (from Walker 1998, 2001) or an rms scatter of $\sim$10% in the flux-ratio curves (as observed; Table 1), and a typical variability time-scale of $\sim$1 week for an effective transverse velocity (i.e. the velocity of the ISM, earth, local and solar peculiar motions combined) of the medium of 50 kms$^{-1}$. Note however that the time-scale of variability might vary with time of the year due to the earth’s motion (e.g. Dennett-Thorpe & de Bruyn 2000, 2002).
Refractive scintillation could therefore explain the observed extrinsic variations up to a time-scale of possibly several weeks in B2045+265 for reasonable lensed-images sizes. However, the structure function shows correlated variations on time-scales that are much longer. These could either indicate modification(s) of the Kolmogorov spectrum of density fluctuations that was assumed in the above calculation or a very low transverse velocity of the medium, i.e. 10 kms$^{-1}$. If there is more power in the spectrum on larger scales, or a cutoff on smaller scales, fluctuations will become stronger on longer time scales (e.g. Blandford et al. 1986; Romani, Narayan & Blandford 1986; Goodman et al. 1987). Such large-scale electron density waves might also explain the apparent fluctuations in the observed structure function (Fig.2).
On further examination, however, we find that B2045+265 is very close, if not seen through, the Cygnus “superbubble” region (see Fig.6 in Fey, Spangler & Mutel 1989), making our analysis based on the model in Cordes & Lazio (2003) rather uncertain. This region has considerably enhanced scattering measures and if this is the case for B2045+265 as well, it would strongly support Galactic scintillation as the cause of the observed flux-density variations. The complexity of such regions, where turbulence in the ionized ISM presumably originates, could be the reason why we see large-amplitude fluctuations in the flux ratios with time scales that are not expected from simple Kolmogorov turbulence models (see also e.g. J1819+3845; Dennett-Thorpe & de Bruyn 2000, 2002).
Finally, it is interesting to note that B2045+265 has the strongest and most significant violation of the cusp relation of all known lens systems (Keeton et al. 2003). Even so, the values of $R_{\rm cusp}$ of the systems discussed in this paper (see Table 1) agree with those in Keeton et al. (2003). However, the strong observed variations in the flux-ratio curves should caution against the use of both flux-ratios and values of $R_{\rm cusp}$ (Sect.1) – derived from single-epoch observations – even if the inferred time-delays are only a few hours!
Whether the violation of the cusp relation in B2045+265, averaged over 8.5 months (Table 1), and Galactic refractive scintillation and/or scattering is completely coincidental, is not clear at this point. At any instant in time, however, large scale electron-density fluctuations in the Galactic ISM can focus [*or*]{} defocus the images with long time scales of variability – as is apparent from our observations – probably even more so toward regions of enhanced turbulence (i.e. the Cygnus region). CDM substructure mostly focuses the images. It is interesting to note that B0712+472, probably the system with second-best evidence for extrinsic variability in our sample, also has a low Galactic latitude $b=+23^\circ$.
While the observations reported in this paper do not contradict the exciting conclusion that CDM substructure might have been detected within the central regions of lens galaxies, they do suggest that extrinsic, refractive effects are also of importance and that it is imperative to carry out further, multi-frequency monitoring to distinguish them from achromatic, gravitational effects.
[LVEK acknowledges the support from an STScI Fellowship grant. RDB is supported by an NSF grant AST–0206286. MERLIN is a National Facility operated by the University of Manchester at Jodrell Bank Observatory on behalf of PPARC. LVEK thanks Chris Kochanek and Neal Dalal for suggestions that improved the presentation of this work. We thank the referee for helping to clarify the manuscript.]{}
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$<r_{(B/A)}>$ $<r_{(C/A)}>$ $<r_{(D/A)}>$ $\chi^2$/DOF $R_{\rm cusp}$ (ABC)
----------- --------------- --------------- --------------- -------------- ----------------------
B0128+437 0.584(0.029) 0.520(0.029) 0.506(0.032) 1.8/1.9/2.4 0.445 (0.018)
B0712+472 0.843(0.061) 0.418(0.037) 0.082(0.035) 4.8/3.2/8.0 0.255 (0.030)
B1359+154 0.580(0.039) 0.782(0.031) 0.193(0.031) 1.9/0.9/1.2 0.510 (0.024)
B1422+231 1.062(0.009) 0.551(0.007) 0.024(0.006) 1.8/2.0/1.5 0.187 (0.004)
B1555+375 0.620(0.039) 0.507(0.030) 0.086(0.024) 3.4/2.1/2.4 0.417 (0.024)
B2045+265 0.578(0.059) 0.739(0.073) 0.102(0.025) 8.2/10.9/2.9 0.501 (0.035)
[Table 1.— The flux-ratios of each image pair. The rms scatter in the flux-ratio is indicated between parentheses, calculated from the 41 epochs. The reduced $\chi^2$ values are listed as well, calculated on the basis that each normalized flux-ratio curve should be unity and that there is no variability. In addition, the values of $R_{\rm cusp}$ (see Sect.1) and its rms (between parentheses) are listed (e.g. Mao & Schneider 1998; Keeton et al. 2002).]{}
rms($a_{\rm ext}$) rms($b_{\rm ext}$) rms($c_{\rm ext}$) $\chi^2_r/{\rm DOF}$
----------- -------------------- -------------------- -------------------- ----------------------
B0128+437 2.9% 1.9% 2.5% 3.3
B0712+472 4.8% 4.2% 4.8% 6.2
B1359+154 1.0% 4.6% – 2.8
B1422+231 – 0.6% 0.9% 3.7
B1555+375 3.3% 4.2% 3.0% 5.3
B2045+265 6.1% 7.0% 7.2% 17.1
[ Table 2.— The estimated rms levels of extrinsic variability in images A, B and C. The reduced values of $\chi^2_r$ are given to indicate the significance of the presence of extrinsic variability in the combined set of images. The dashes indicate that the estimated variance was smaller than zero (see Sect.3.2 for more details).]{}
[Fig. 1.— Continued]{}
[Fig. 1.— Continued]{}
[^1]: We use the notation $A$, $B$, $C$ and $a_n$, $b_n$, $c_n$ to indicate both the light curves as a whole, as well as their individual flux-density values.
|
---
abstract: 'This work is a continuation of studies presented in the papers arXiv: 0911.5597, 1003.4523. In the work it is demonstrated that with the use of one and the same parameter deformation may be described for several cases of the General Relativity within the scope of both the Generalized Uncertainty Principle (UV-cutoff) and the Extended Uncertainty Principle (IR-cutoff). All these cases have a common thermodynamic interpretation of the corresponding gravitational equations. Consideration is given to the possibility for extension of the obtained results to more general cases. Possible generalization of the uncertainty relation for the pair (cosmological constant, “space-time volume”), where the cosmological constant is regarded as a dynamic quantity at high and low energies is analyzed.'
---
[**Deformed Quantum Field Theory, Thermodynamics at Low and High Energies, and Gravity. II**]{}\
A.E.Shalyt-Margolin [^1]\
*National Center of Particles and High Energy Physics, Bogdanovich Str. 153, Minsk 220040, Belarus*
PACS: 03.65, 05.20\
Keywords: quantum field theory with UV and IR cutoff, gravitational thermodynamics, deformed gravity
Introduction
============
In the last decade numerous works devoted to a Quantum Field Theory (QFT) at Planck’s scale [@Planck1]–[@Planck3] have been published(of course, the author has no pretensions of being exhaustive in his references). This interest stems from the facts that (i) at these scales it is expected to reveal the effects of a Quantum Gravity (QG), and this still unresolved theory is intriguing all the researchers engaged in the field of theoretical physics; (ii) modern accelerators, in particular LHC, have the capacity of achieving the energies at which some QG effects may be exhibited.\
Now it is clear that ÷òî a Quantum Field Theory (QFT) at Planck’s scales, and possibly at very large scales as well, undergoes changes associated with the appearance of additional parameters related to (i) a minimal length (on the order of the Planck’s length)and (ii)a minimum momentum. As this takes place, the corresponding parameters are naturally considered as deformation parameters, i.e. the related quantum theories are considered as a high-energy deformation (at Planck’s scales) and a low-energy deformation (IR-cutoff), respectively, of the well-known quantum field theory, the latter being introduced in the corresponding high- and low-energy limits and exact to a high level. The deformation is understood as an extension of a particular theory by inclusion of one or several additional parameters in such a way that the initial theory appears in the limiting transition [@Fadd].\
Most natural approach to the introduction of the above-mentioned parameters is to treat a quantum field theory with the Generalized Uncertainty Principle (GUP) [@Ven1]–[@Kim1] and with the Extended Uncertainty Principle (EUP), respectively [@Bole:05]–[@Kim1]. In the case of GUP we easily obtain a minimal length on the order of the Planck’s $l_{min}\sim l_{p}$ and the corresponding high-energy deformation of well-known QFT–QFT with GUP. It should be noted that QFT with GUP at Planck’s scales (Early Universe) is attested in many works (for example [@Ven1]–[@Magg1])/ Even if we disregard the works devoted to a string theory, still remaining a tentative one, GUP is quite naturally derived from the gedanken experiment [@GUPg1]–[@Ahl1].\
On the other hand, GUP has no way in the spaces with large length scales (for example (A)dS). For such spaces, e.g., in [@Bole:05],[@Park] the Extended Uncertainty Principle has been introduced (find its exact definition below) giving an absolute minimum in the uncertainty of the momentum.\
The problem is to find whether there are cases when the deformations generated by GUP and EUP are defined by the same parameter. By author’s opinion this is the case for Gravity modified (deformed) within GUP and EUP, when the corresponding initial theory has a “thermodynamic interpretation” [@Jac1]–[@Cai1]. Specifically, the deformation parameter $\alpha=l_{min}^{2}/x^{2}$,$l_{min}\sim l_{p}$,$0<\alpha\leq1/4$ where $x$ is the measuring scale, introduced by the author in a series of works [@shalyt1]–[@shalyt13] meets the above requirements.\
Note that this parameter has been introduced to study the deformation of QFT at Planck’s scale, although the deformation per se, associated with a high-energy modification of the density matrix, was “minimal” in that it presented no noncommutativity operators related to different spatial coordinates $$\label{UDPl}
[X_{i},X_{j}]\neq 0, i\neq j$$ and hence “limited” as in the end it failed to lead to GUP. Nevertheless, the corresponding deformation parameter in some way is universal.\
This paper continues the studies, described in [@shalyt-aip] –[@shalyt-gravity] (the latter in particular), of the fundamental quantities in “thermodynamic interpretation” of gravity [@Jac1]–[@Cai1] for GUP and EUP deformations of the latter. Compared to the works [@shalyt-gravity], the results from which are used in this paper, the important results associated with EUP are put forward together with the demonstration that GUP and EUP have the same deformation parameter, at least in this context.\
The structure of this work is as follows. In Section 2 it is shown that the deformation of the fundamental thermodynamic quantities for black holes within GUP and EUP may be interpreted with the use of the same parameter. In Section 3, within the scope of a dynamic model for the cosmological constant $\Lambda$ (vacuum energy density), GUP is studied for the pair ($\Lambda,V$) [@shalyt-aip],[@shalyt-entropy2], where $V$ – is the “space-time volume”. In this Section consideration is given to the possible existence of EUP for this pair, i.e. to a possible extension of the Uncertainty Principle to the pair in the IR region, and hence to the possible substantiation of the proper (coincident with the experimental) value for $\Lambda$. In Section 4 the results of Section 2 are applied to Einstein’s Equations for space with horizon and to Friedmann’s Equations. It is demonstrated that in both cases their deformation (in the first case within GUP and in the second case within EUP) may be interpreted with the use of the same small dimensionless parameter having a known variability domain.\
And, finally, in Section 5 the problems of further investigations are discussed, some final comments are given.
Universal Deformation Parameter in Gravitational Thermodynamics with GUP and EUP
================================================================================
In this Section the Gravitational Thermodynamics (GT) is understood as thermodynamics of spaces with horizon [@Padm11],[@Padm13].
Gravitational Thermodynamics with GUP
-------------------------------------
We use the notation and principal results from [@Park]. So, GUP is of the form $$\label{UDP2.1}
\Delta x_{i}\Delta p_{j}\geq \hbar \delta_{ij} [ 1 + \alpha^{\prime 2}
l^2_{p}
\frac{(\Delta p_{i})^{2}}{\hbar ^2}]$$ and, since $ \Delta x_{i}\Delta p_{j}>\hbar \delta_{ij}$, we have $$\label{UDP2.2}
\Delta x_{i}\Delta p_{i}\geq \hbar \delta_{ij} [ 1 +
\alpha^{\prime 2}
\frac{l^2_{p}}{\Delta x_{i}^{2}}
\frac{(\Delta p_{i})^{2}\Delta x_{i}^{2}}{\hbar ^2}]>\hbar
\delta_{ij}[ 1 + \frac{1}{4}\alpha_{\Delta x_{i}}],$$ where $\alpha_{\Delta x_{i}}$ – parameter $\alpha$ corresponding to $\Delta x_{i}$, $l_{min}=2\alpha^{\prime}l_{p}$. Besides, as distinct from [@Park], for the dimensionless factor in GUP, instead of $\alpha$, we use $\alpha^{\prime}$ to avoid confusion with the deformation parameter.\
In this terms the uncertainty in moment is given by the nonstrict inequality $$\label{UDP2.4}
2\hbar(\alpha_{\Delta x_{i}}\Delta
x_{i})^{-1}[1-\sqrt{1-\alpha_{\Delta x_{i}}}]\leq\Delta p_{i}\leq
2\hbar(\alpha_{\Delta x_{i}}\Delta
x_{i})^{-1}[1+\sqrt{1-\alpha_{\Delta x_{i}}}].$$ But for the quantities determining GT in terms of $\alpha$ one can derive exact expressions. Indeed, in terms of $\alpha$ the GUP-modification (or rather GUP-deformation)is easily obtained for the Hawking temperature [@acs]–[@Nou],[@Park],[@Kim1] that has been computed in the asymptotically flat $d$ - dimensional space for a Schwarzshild black hole with a metric given by $$\label{UDP2.5}
ds^2=-N^2 dt^2 +N^{-2} dr^3 +r^2 d \Omega^2_{d-2},$$ where $$N^2= 1-\frac{16 \pi G M}{(d-2) \Omega_{d-2} r^{d-3}},$$ $\Omega_{d-2}$ is the area of the unit sphere $S^{n-2}$, and $r_{+}$ is the uncertainty in the emitted particle position by the Hawking effect, expressed as $$\label{UDP2.6}
\Delta x_{i}\approx r_{+}$$ and being nothing else but a radius of the event horizon. In this case the deformation parameter $\alpha$ arises naturally. Actually, modification of the Hawking temperature is of the form, see formula (10) in [@Park] $$\label{UDP2.7}
T_{GUP}=(\frac{d-3}{4\pi})\frac{\hbar r_{+}}{2\alpha^{\prime
2}l^{2}_{p}}[1-(1-\frac{4\alpha^{\prime 2}l_{p}^{2}}{r_{+}^{2}})^{1/2}]$$ and may be written in a natural way as $$\label{UDP2.71}
T_{GUP}=2(\frac{d-3}{4\pi})\frac{\hbar}{r_{+}} \alpha_{r_{+}}
^{-1} [1-(1-\alpha_{r_{+}})^{1/2}],$$ where $\alpha_{r_{+}}$- parameter $\alpha$ associated with $r_{+}$. It is clear that $T_{GUP}$ is actually the deformation $T_{Hawk}$ – black hole temperature for a semiclassical case [@Hawk3]. In such a manner compared to $T_{Hawk}$ $T_{GUP}$ is additionally dependent only on the dimensionless small deformation parameter $\alpha_{r_{+}}$.\
The dependence of the black hole entropy on $\alpha_{r_{+}}$ may be derived in a similar way. For a semiclassical approximation of the Bekenstein-Hawking formula [@Bek1],[@Hawk3] $$\label{UDP2.8}
S=\frac{1}{4}\frac{A}{l^{2}_{p}},$$ where $A$ – surface area of the event horizon, provided the horizon event is of radius $r_+$, $A\sim r^{2}_+$ and (\[UDP2.8\]) is clearly of the form $$\label{UDP2.81}
S=\sigma \alpha^{-1}_{r_{+}},$$ where $\sigma$ is some dimensionless denumerable factor. The general formula for quantum corrections [@mv] given as $$\label{UDP2.9}
S_{GUP} =\frac{A}{4l_{p}^{2}}-{\pi\alpha^{\prime 2}\over 4}\ln
\left(\frac{A}{4l_{p}^{2}}\right) +\sum_{n=1}^{\infty}c_{n}
\left({A\over 4 l_p^2} \right)^{-n}+ \rm{const}\;,$$ where the expansion coefficients $c_n\propto \alpha^{\prime
2(n+1)}$ can always be computed to any desired order of accuracy [@mv], may be also written in the general case as a Laurent series in terms of $\alpha_{r_{+}}$ $$\label{UDP2.91}
S_{GUP}=\sigma \alpha^{-1}_{r_{+}}-{\pi\alpha^{\prime 2}\over
4}\ln (\sigma \alpha^{-1}_{r_{+}}) +\sum_{n=1}^{\infty}(c_{n}
\sigma^{-n}) \alpha^{n}_{r_{+}}+ \rm{const}.$$ In what follows the representation in terms of the deformation parameter $\alpha$ is referred to as [**$\alpha$-representation**]{}.
Gravitational Thermodynamics with EUP
-------------------------------------
Let us consider QFT with EUP [@Park]. In this case we obtain QFT with $p_{min}$. Obviously, there is no minimal length $l_{min}$ in QFT with EUP whatsoever but we assume that QFT with GUP is valid. At the present time for such an assumption we can find solid argumentation [@GUPg1]–[@Ahl1]. As will be shown later, in this case the fundamental quantities may be also expressed in terms of $\alpha$. Hereinafter we use [**a small dimensionless parameter**]{} $$\label{EUP01}
\alpha_{\widetilde{l}}=\frac{l^{2}_{or}}{\widetilde{l}^{2}},$$ where $l_{or}\equiv l_{original}=2\alpha^{\prime}l_{p}$, $\alpha^{\prime}$–dimensionless constant on the order of unity from GUP (\[UDP2.1\]), and it is suggested that $$\label{EUP02}
2l_{or}\leq \widetilde{l}\quad, i.e \quad
0<\alpha_{\widetilde{l}}\leq 1/4.$$ Similar to the previous Section, it is convenient to use the principal results of [@Park] (sections 3,4). Then EUP in (A)dS space takes the form $$\label{EUP}
\Delta x_{i}\Delta p_{j}\geq \hbar \delta_{ij} [1 + \beta^{2}
\frac{(\Delta x_{i})^{2}}{\l^2}],$$ where $l$ is the characteristic, large length scale $l\gg l_{p}$ and $\beta$ is a dimensionless real constant on the order of unity [@Park]. From EUP there is an absolute minimum in the momentum uncertainty??? $$\label{EUP0}
\Delta p_{i}\geq \frac{2\hbar \beta}{l} ,$$ EUP (\[EUP\]) may be rewritten as $$\label{EUP1}
\Delta x_{i}\Delta p_{j}\geq \hbar \delta_{ij} [1 + \beta^{2}
\frac{(\Delta x_{i})^{2}}{\l_{or}^2}\frac{\l_{or}^2}{\l^2}]=\hbar
\delta_{ij} [ 1 + \beta^{2} \alpha_{l}\alpha^{-1}_{\Delta x_{i}}].$$ Considering that in a theory with fixed $l\gg l_{p}$ $$\label{EUP2}
\alpha_{l}=const\ll 1,$$ (\[EUP\]),(\[EUP1\]) may be written as $$\label{EUP3}
\Delta x_{i}\Delta p_{j}\geq \hbar \delta_{ij}
[ 1 + \beta^{2}\alpha_{l} \alpha^{-1}_{\Delta x_{i}}]=\hbar
\delta_{ij} [ 1 + \widetilde{\beta}^{2}\alpha^{-1}_{\Delta
x_{i}}],$$ where $\beta$ is redetermined as $$\label{EUP3.1}
\beta\mapsto\widetilde{\beta}=\sqrt{\alpha_{l}}\beta.$$ However, in this case $\beta$ may be left as it is, whereas $\alpha$ may be redetermined because $\alpha^{-1}_{\Delta x_{i}}$ in (\[EUP1\]),(\[EUP3\]) is not a small parameter. In consequence we can redetermine $\alpha$ as $$\label{EUP4}
\widetilde{\alpha}_{\Delta x_{i}}=\alpha_{l} \alpha^{-1}_{\Delta
x_{i}},$$ where $\widetilde{\alpha}_{\Delta x_{i}}$ is now a small parameter.\
Owing to such a [**duality**]{}, EUP (\[EUP\]),(\[EUP1\]) may be rewritten in terms of a new small parameter $\widetilde{\alpha}$ similar to $\alpha$ as follows: $$\label{EUP5}
\Delta x_{i}\Delta p_{j}\geq \hbar \delta_{ij}
[ 1 + \beta^{2} \widetilde{\alpha}_{\Delta x_{i}}].$$ Then in analogy with [@Park] (Section 3), for Hawking temperature of the $d$-dimensional Schwarzshild-AdS black hole with the metric function we have $$\label{EUP6}
N^2= 1+ \frac{r^2}{l^2_{AdS}}-\frac{16 \pi G M}{(d-2) \Omega_{d-2}
r^{d-3} }$$ in the metric of(\[UDP2.5\]) and the cosmological constant $\Lambda=-(d-1)(d-2)/2 l^2_{AdS}$.\
Therewith the $\alpha$-representation of the Hawking temperature $T_{EUP}$ [@Park] (formula (15)) takes the form $$\label{EUP7}
T_{EUP(AdS)}=(\frac{d-3}{4\pi}){\frac{\hbar }{r^{2}_{+}}}[ 1+
\frac{(d-1)}{(d-3)}\alpha^{-1}_{r_{+}}\alpha_{l_{AdS}}]=(\frac{d-3}{4\pi}){\frac{\hbar }{r^{2}_{+}}}[ 1+
\frac{(d-1)}{(d-3)}\widetilde{\alpha}_{r_{+}}].$$ In the same way we can easily obtain the $\alpha$-representation of the Hawking temperature for a Schwarzshild-AdS black hole and for a combined case ((formula (28) from the [@Park])) of GUP and EUP – (GEUP) $$\begin{aligned}
\label{EUP9}
T_{GEUP(AdS)}=2(\frac{d-3}{4\pi})\frac{\hbar}{r_{+}}
\alpha_{r_{+}} ^{-1}
[1-\sqrt{1-\alpha_{r_{+}}[1+\frac{\alpha_{l_{AdS}}(d-1)}{(d-3)}\alpha^{-1}_{r_{+}}]}]\nonumber \\
=2(\frac{d-3}{4\pi})\frac{\hbar}{r_{+}} \alpha_{r_{+}} ^{-1}
[1-\sqrt{\frac{d-3-\alpha_{l_{AdS}}(d-1)}{(d-3)}-\alpha_{r_{+}}}],\end{aligned}$$ i.e. in the general case we get a Laurent series from $\alpha$.\
Similarly, we can obtain the $\alpha$-representation for the corresponding value of $T_{GEUP(dS)}$ ((formula (32) from [@Park]) in the de Sitter (dS) space by the substitution $l^{2}_{AdS}\rightarrow -l^{2}_{dS}.$\
Note that, as it has been indicated in [@shalyt-aip], [@shalyt-entropy2], $\alpha_{r_{+}} ^{-1}$ has one more interesting feature $$\label{comm6}
\alpha_{r_{+}} ^{-1}\sim {r_{+}}^{2}/l_{p}^{2}\sim S_{BH}.$$ Here $S_{BH}$ is the Bekenstein-Hawking semiclassical black hole entropy with the characteristic linear size $r_{+}$. For example, in the spherically symmetric case $r_{+}=R$ - radius of the corresponding sphere with the surface area $A$, and $$\label{comm7}
A=4\pi r_{+}^{2},S_{BH}=A/4l_{p}^{2}=\frac{\pi}{4\alpha^{\prime
2}} \alpha_{r_{+}}^{-1}.$$ In [@Kim1] GUP and EUP are combined by the principle called the Symmetric Generalized Uncertainty Principle (SGUP): $$\label{SGUP1}
\Delta x \Delta p \ge \hbar \left( 1 + \frac{(\Delta x)^2}{L^2} +
\l^2 \frac{(\Delta p)^2}{\hbar^2} \right),$$ where $l\ll L$ and $l$ defines the limit of the UV-cutoff (not being such up to a constant factor as in the case of GUP).Then a minimal length is determined as
$\Delta x_{\rm min} =
2l/\sqrt{1-4\l^2/L^2}$,
whereas $L$ defines the limit for IR-cutoff i. e. we have a minimum momentum
$\Delta p_{\rm min} = 2\hbar/(L\sqrt{1-4\l^2/L^2)}$.
And using the Euclidian action formalism by Gibbons and Hawking [@Hawk4], in [@Kim1] the corresponding correction of the Hawking temperature for an ordinary(not A(dS)) Schwarzshild-black hole is computed. This correction is given as $T_{SGUP}$. In the notation of this work
$\Delta x_{\rm min} =
2l/\sqrt{1-4\alpha_{l}^{-1}\alpha_{L}}=2l/\sqrt{1-4\widetilde{\alpha_{l}}}$,
where $\widetilde{\alpha_{l}}$–small parameter introduced in conformity with (\[EUP4\]). We can easily obtain the $\alpha$-representation for $T_{SGUP}$ that is completely similar to the $\alpha$-representation of $T_{GEUP(AdS)}$.\
It should be noted that in the realistic theories $l\sim l_{p}$, and it is obvious that $(\sqrt{1-4\l^2/L^2)}\approx 1$. Thus, $\Delta
x_{\rm min}\approx 2l \sim 2l_{p}$ and hence in this case we get a minimal length that is much the same (to within $\alpha^{\prime}$) as in the case of GUP. It is seen that, with due regard for the requirement $l\ll L$, $\Delta p_{\rm min}$ is derived close (to within $\beta$) to $\Delta p_{\rm min}$ (\[EUP0\]) in a theory with EUP.\
\
The question arises as to [**what for all these manipulations with writing and rewriting of the already derived expressions in the $\alpha$-representation are necessary**]{}.
[**2.1**]{} Owing to this procedure, we can draw the conclusion that all the quantities within the scope of the stated problem are dependent on one and the same deformation parameter $\alpha$ that is small, dimensionless (discrete in the case of GUP), and varying over the given interval. And, provided the infrared cutoff $l$ is defined, we have
$\alpha_{l}=\alpha_{min}=l^{2}_{or}/l^{2}\leq\alpha\leq 1/4$ and $l_{or}\equiv l_{original}\sim l_{p}$.
If we primordially consider a theory with GUP only, then $l_{or}\equiv l_{min}$. But in the arbitrary case it is required that $l_{or}=2\alpha^{\prime}l_{p}$, where $\alpha^{\prime}$ is a certain dimensionless constant on the order of unity.\
The property of discreteness is retained for $\alpha$ in the cases when only GUP (without generalizations)is valid because in this case the length seems to be quantized, the lengths being considered from $2l_{min}$ rather than from $l_{or}=l_{min}$ as a singularity arises otherwise [@shalyt2]–[@shalyt9].\
\
[**2.2**]{} Actually, all the quantities may be represented as a Laurent series in terms of $\alpha$, and a solution of the problem at hand may be understood as finding of the members in this series.\
\
[**2.3**]{} When the problem has separate solutions for the cases including the UV- and IR-cutoffs, we can consider expansion in each of the cases in terms of their own small parameters: $\alpha$ in the case of UV-cutoff and $\widetilde{\alpha}$ in the case of IR-cutoff, where $\widetilde{\alpha}$ [**is a duality**]{} of $\alpha$
$\widetilde{\alpha}_{\widetilde{l}}=\alpha_{l}
\alpha^{-1}_{\widetilde{l}},$
$l$ determines, to within a factor on the order of unity, the characteristic system’s size, and $l\gg l_{p}$.
The Cosmological Constant Problem and QFT with GUP and SGUP
===========================================================
In this section it is assumed that $\Lambda$ may be varying in time. Generally speaking, $\Lambda$ is referred to as a constant just because it is such in the equations, where it occurs: Einstein equations [@Einst]. But in the last few years the dominating point of view has been that $\Lambda$ is actually a dynamic quantity, now weakly dependent on time [@Ran1]–[@Shap]. It is assumed therewith that, despite the present-day smallness of $\Lambda$ or even its equality to zero, nothing points to the fact that this situation was characteristics for the early Universe as well. Some recent results [@Min1]–[@Min4] are rather important pointing to a potentially dynamic character of $\Lambda$. Specifically, of great interest is the Uncertainty Principle derived in these works for the pair of conjugate variables $(\Lambda,V)$: $$\label{CC1}
\Delta\Lambda\, \Delta V \sim \hbar,$$ where $\Lambda$ is the vacuum energy density (cosmological constant). It is a dynamic value fluctuating around zero; $V$ is the space-time volume. Here the volume of space-time $V$ results from the Einstein-Hilbert action $S_{EH}$ [@Min2]: $$\label{CC2}
\Lambda \int d^{4}x \sqrt{-g}=\Lambda V$$ where (\[CC2\]) is the term in the $S_{EH}$. In this case the notion of conjugation is well-defined, but approximate, as implied by the expansion about the static Fubini–Study metric (Section 6.1 of [@Min1]). Unfortunately, in the proof per se (\[CC1\]), relying on the procedure with a non-linear and non-local Wheeler–de-Witt-like equation of the background-independent Matrix theory, some unconvincing arguments are used, making it insufficiently rigorous (Appendix 3 of [@Min1]). But, without doubt, this proof has a significant result, though failing to clear up the situation.\
In[@shalyt-aip],[@shalyt-entropy2], [@shalyt-gup] the Heisenberg Uncertainty Relation for the pair $(\Lambda,V)$ (\[CC1\]) has been generalized to GUP $$\label{CC8}
\Delta V\geq \frac{\hbar}{\Delta
\Lambda}+\alpha_{\Lambda}^{\prime} t_{p}^2
\overline{V}_{p}^{2}\frac{\Delta \Lambda}{ \hbar}$$ or that is the same $$\label{CC8.1}
\Delta V\Delta \Lambda \geq \hbar(1+\alpha_{\Lambda}^{\prime}
t_{p}^2 \overline{V}_{p}^{2}\frac{(\Delta
\Lambda)^{2}}{\hbar^{2}}).$$ where $\alpha_{\Lambda}^{\prime}$ is a new constant and $\overline{V}_{p}=l_{p}^{3}.$\
In the case of UV - limit: $t\rightarrow t_{min}$,$\Delta\Lambda$ becomes significant $$\label{CC10}
\lim\limits_{t\rightarrow
t_{min}}\overline{V}=\overline{V}_{min}\sim
\overline{V}_{p}=l_{p}^{3}; \lim\limits_{t\rightarrow
t_{min}}V=V_{min}\sim V_{p}=l_{p}^{3}t_{p},$$ where $\overline{V}$ – spatial part of ${V}.$\
The existence of $V_{min}\sim V_{p}$ directly follows from GUP for the pair $(p,x)$ (\[UDP2.1\])and GUP for the pair $(E,t)$ [@shalyt3],[@shalyt9] as well as from solutions of the quadratic inequalities(\[CC8\]),(\[CC8.1\]).\
So, (\[CC8\]) is nothing else but $$\label{CC11}
\Delta V \geq \frac{\hbar}{\Delta
\Lambda}+\alpha_{\Lambda}^{\prime} V_{p}^{2}\frac{\Delta
\Lambda}{\hbar}.$$ And in the case of UV – cutoff we have $$\label{CC12}
\lim\limits_{t\rightarrow t_{min}}\Lambda \equiv \Lambda_{UV} \sim
\Lambda_{p}\equiv\hbar/V_{p}=E_{p}/\overline{V}_{p}.$$\
It is easily seen that in this case $\Lambda_{UV} \sim m_{p}^{4}$, in agreement with the value obtained using a standard (i.e. without super-symmetry and the like) quantum field theory [@Zel1],[@Wein1]. Despite the fact that $\Lambda $ at Planck’s scales (referred to as $\Lambda_{UV} $) is also a dynamic quantity, it is not directly related to the familiar $\Lambda $ because the latter, as opposed to the first one, is derived from Einstein’s equations $$\label{CC13}
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = 8\pi G_N \left( -
\Lambda g_{\mu \nu} + T_{\mu \nu} \right).$$ However, Einstein’s equations (\[CC13\]) are not valid at the Planck scales and hence $\Lambda_{UV}$ may be considered as some high-energy generalization (deformation) of the conventional cosmological constant in the low-energy limit.\
The problem is whether a correct generalization of GUP for the pair $(\Lambda,V)$ (\[CC8.1\]) to the Symmetric Generalized Uncertainty Principle (SGUP) of the form given by (\[SGUP1\]) is possible. If the answer is positive, a theory also includes $\Lambda_{min}$ that may be referred to as $\Lambda_{IR}$ in similarity with $\Lambda_{UV}$. Then, similar to Ò (\[SGUP1\]), an additional term defining the IR-cutoff must be of the form $$\label{SGUP-cc}
\Omega_{IR}=\frac{(\Delta V)^2}{\widetilde{V}^2},$$ where $\widetilde{V}$ - certain space-time volume effectively specifying the IR-limit of the observable part of the Universe with the spatial part $\widetilde{\overline{V}}\sim L^{3}$; $L$ – radius of the observable part of the Universe. Now it is known that $L\approx 10^{28}$. Clearly, the introduction of an additional term of the form (\[SGUP-cc\]) into the right-hand side of (\[CC8.1\]) leads to $\Lambda_{IR}\ll \Lambda_{UV}$ and might lead to the value of $\Lambda$ close to the experimental value $\Lambda_{exp}$ [@Dar1].\
Note that the Holographic Principle [@Hooft1]–[@Sussk1] used to the Universe as a whole [@Sussk1] gives $\Lambda_{exp}$ [@Bal]. In [@shalyt14],[@shalyt-vac],[@shalyt-aip],[@shalyt-entropy2] it has been demonstrated that the $\alpha$–representation ($\alpha$–deformation) of QFT with GUP plays a significant role. In particular, consider $$\label{CC14}
\Lambda_{exp}\approx \alpha_{L} \Lambda_{UV},$$ where $L\approx 10^{28}$.(\[CC14\]) is like (\[EUP4\]). But the Holographic Principle imposes strict restrictions on the number of degrees of freedom in the Universe, and hence for us it is important to study the inferences of introducing the additional term of the form (\[SGUP-cc\]) in (\[CC8.1\]).
GUP, EUP, and General Relativity Deformation
============================================
In this Section we use the previously obtained results for some cases of high- energy and low-energy deformation of GR. Specifically, we demonstrate that in the cases when the [**Thermodynamics Approach**]{} [@Jac1]–[@Cai1] is applicable to the General Relativity the deformation of GR with GUP and EUP may be a natural result of the $\alpha$-representation.
$\alpha$–Representation of Einstein’s Equations for space with horizon
----------------------------------------------------------------------
Let us consider $\alpha$-representation and high energy $\alpha$-deformation of the Einstein’s field equations for the specific cases of horizon spaces (the point (c) of Section 4). In so doing the results of the survey work ([@Padm13] p.p.41,42)are used. Then, specifically, for a static, spherically symmetric horizon in space-time described by the metric $$\label{GT9}
ds^2 = -f(r) c^2 dt^2 + f^{-1}(r) dr^2 + r^2 d\Omega^2$$ the horizon location will be given by simple zero of the function $f(r)$, at $r=a$.\
It is known that for horizon spaces one can introduce the temperature that can be identified with an analytic continuation to imaginary time. In the case under consideration ([@Padm13], eq.(116)) $$\label{GT10}
k_BT=\frac{\hbar cf'(a)}{4\pi}.$$ Therewith, the condition $f(a)=0$ and $f'(a)\ne 0$ must be fulfilled.\
Then at the horizon $r=a$ Einstein’s field equations $$\label{GT11}
\frac{c^4}{G}\left[\frac{1}{ 2} f'(a)a - \frac{1}{2}\right] = 4\pi
P a^2$$ may be written as the thermodynamic identity ([@Padm13] formula (119)) $$\label{GT12}
\underbrace{\frac{{{\hbar}} cf'(a)}{4\pi}}_{\displaystyle{k_BT}}
\ \underbrace{\frac{c^3}{G{{\hbar}}}d\left( \frac{1}{ 4} 4\pi a^2 \right)}_{
\displaystyle{dS}}
\ \underbrace{-\ \frac{1}{2}\frac{c^4 da}{G}}_{
\displaystyle{-dE}}
= \underbrace{P d \left( \frac{4\pi}{ 3} a^3 \right) }_{
\displaystyle{P\, dV}}$$ where $P = T^{r}_{r}$ is the trace of the momentum-energy tensor and radial pressure. In the last equation $da$ arises in the infinitesimal consideration of Einstein’s equations when studying two horizons distinguished by this infinitesimal quantity $a$ and $a+da$ ([@Padm13] formula (118)).\
Now we consider (\[GT12\]) in a new notation, expressing $a$ in terms of the corresponding deformation parameter $\alpha$. Hereinafter in this Section we write $\alpha$ instead of $\alpha_{a}$ as we consider the same $a$. Then we have $$\label{GT13}
a=l_{min}\alpha^{-1/2}.$$ Therefore, $$\label{GT14}
f'(a)=-2l^{-1}_{min}\alpha^{3/2}f'(\alpha).$$ Substituting this into (\[GT11\]) or into (\[GT12\]), we obtain in the considered case of Einstein’s equations in the “$\alpha$–representation” the following: $$\label{GT16}
\frac{c^{4}}{G}(-\alpha f'(\alpha)-\frac{1}{2})=4\pi
P\alpha^{-1}l^{2}_{min}.$$ Multiplying the left- and right-hand sides of the last equation by $\alpha$, we get $$\label{GT16.1}
\frac{c^{4}}{G}(-\alpha^{2}f'(\alpha)-\frac{1}{2}\alpha)=4\pi
Pl^{2}_{min}.$$ But since usually $l_{min}\sim l_{p}$ (that is just the case if the Generalized Uncertainty Principle (GUP) is satisfied), we have $l^{2}_{min}\sim l^{2}_{p}=G\hbar/c^{3}$. When selecting a system of units, where $\hbar=c=1$, we arrive at $l_{min}\sim l_{p}=\surd
G$, and then (\[GT16\]) is of the form $$\label{GT16.A}
-\alpha^{2}f'(\alpha)-\frac{1}{2}\alpha=4\pi P\vartheta^{2}G^{2},$$ where $\vartheta=l_{min}/l_{p}$. L.h.s. of (\[GT16.A\]) is dependent on $\alpha$. Because of this, r.h.s. of (\[GT16.A\]) must be dependent on $\alpha$ as well, i. e. $P=P(\alpha)$.
[**Analysis of $\alpha$-Representation of Einstein’s Equations**]{}
Now let us get back to (\[GT12\]). In [@Padm13] the low-energy case has been considered, for which ([@Padm13] p.42 formula (120)) $$\label{GT17.A}
S=\frac{1}{ 4l_p^2} (4\pi a^2) = \frac{1}{ 4} \frac{A_H}{ l_p^2}; \quad E=\frac{c^4}{ 2G} a
=\frac{c^4}{G}\left( \frac{A_H}{ 16 \pi}\right)^{1/2},$$ where $A_H$ is the horizon area. In our notation (\[GT17.A\]) may be rewritten as $$\label{GT17.A1}
S= \frac{1}{4}\pi\alpha^{-1}; \quad E=\frac{c^4}{2G} a
=\frac{c^4}{G}\left( \frac{A_H}{ 16 \pi}\right)^{1/2}=\frac{\vartheta}{2\surd G}\alpha^{1/2}.$$ We proceed to two entirely different cases: low energy (LE) case and high energy (HE) case. In our notation these are respectively given by
A)$\alpha\rightarrow 0$ (LE), B)$\alpha\rightarrow 1/4$ (HE),\
C)$\alpha$ complies with the familiar scales and energies.
The case of C) is of no particular importance as it may be considered within the scope of the conventional General Relativity.\
Indeed, in point A)$\alpha\rightarrow 0$ is not actually an exact limit as a real scale of the Universe (Infrared (IR)-cutoff $l_{max}\approx 10^{28}cm$), and then
$\alpha_{min}\sim l_{p}^{2}/l^{2}_{max}\approx 10^{-122}$.
In this way A) is replaced by A1)$\alpha\rightarrow \alpha_{min}$. In any case at low energies the second term in the left-hand side (\[GT16.A\]) may be neglected in the infrared limit. Consequently, at low energies (\[GT16.A\]) is written as $$\label{GT16.LE}
-\alpha^{2}f'(\alpha)=4\pi P(\alpha)\vartheta^{2}G^{2}.$$ Solution of the corresponding Einstein equation – finding of the function $f(\alpha)=f[P(\alpha)]$ satisfying(\[GT16.LE\]). In this case formulae (\[GT17.A\]) are valid as at low energies a semiclassical approximation is true. But from (\[GT16.LE\])it follows that $$\label{GT16.solv}
f(\alpha)=-4\pi \vartheta^{2}G^{2}\int
\frac{P(\alpha)}{\alpha^{2}}d\alpha.$$ On the contrary, knowing $f(\alpha)$, we can obtain $P(\alpha)=T^{r}_{r}.$\
[**Possible High Energy $\alpha$-Deformation of General Relativity**]{}
Let us consider the high-energy case B). Here two variants are possible.
[**I. First variant**]{}.
In this case it is assumed that in the high-energy (Ultraviolet (UV))limit the thermodynamic identity (\[GT12\]) is retained but now all the quantities involved in this identity become $\alpha$-deformed. This means that they appear in the $\alpha$-representation with quantum corrections and are considered at high values of the parameter $\alpha$, i.e. at $\alpha$ close to 1/4. In particular, the temperature $T$ from equation (\[GT12\]) is changed by $T_{GUP}$ (\[UDP2.71\]), the entropy $S$ from the same equation given by semiclassical formula (\[GT17.A\]) is changed by $S_{GUP}$ (\[UDP2.91\]), and so forth:
$E\mapsto E_{GUP}, V\mapsto V_{GUP}$.
Then the high-energy $\alpha$-deformation of equation (\[GT12\]) takes the form $$\label{GT8.GUP}
k_{B}T_{GUP}(\alpha)dS_{GUP}(\alpha)-dE_{GUP}(\alpha)=P(\alpha)dV_{GUP}(\alpha).$$ Substituting into (\[GT8.GUP\]) the corresponding quantities\
$T_{GUP}(\alpha),S_{GUP}(\alpha),E_{GUP}(\alpha),V_{GUP}(\alpha),P(\alpha)$ and expanding them into a Laurent series in terms of $\alpha$, close to high values of $\alpha$, specifically close to $\alpha=1/4$, we can derive a solution for the high energy $\alpha$-deformation of general relativity (\[GT8.GUP\]) as a function of $P(\alpha)$. As this takes place, provided at high energies the generalization of (\[GT12\]) to (\[GT8.GUP\]) is possible, we can have the high-energy $\alpha$-deformation of the metric. Actually, as from (\[GT12\]) it follows that $$\label{GT8.GUP1}
f'(a)=\frac{4\pi k_{B}}{\hbar c}T=4\pi k_{B}T$$ (considering that we have assumed $\hbar=c=1$), we get $$\label{GT8.GUP2}
f'_{GUP}(a)=4\pi k_{B}T_{GUP}(\alpha).$$ L.h.s. of (\[GT8.GUP2\]) is directly obtained in the $\alpha$-representation. This means that, when $f'\sim T$, we have $f'_{GUP}\sim T_{GUP}$ with the same factor of proportionality. In this case the function $f_{GUP}$ determining the high-energy $\alpha$-deformation of the spherically symmetric metric may be in fact derived by the expansion of $T_{GUP}$, that is known from (\[UDP2.71\]), into a Laurent series in terms of $\alpha$ close to high values of $\alpha$ (specifically close to $\alpha=1/4$), and by the subsequent integration.\
It might be well to remark on the following.\
\
[**4.1.1**]{} As on going to high energies we use (GUP), $\vartheta$ from equation (\[GT16.A\])is expressed in terms of $\alpha^{\prime}$–dimensionless constant from GUP (\[UDP2.1\]):$\vartheta=2\alpha^{\prime}.$\
\
[**4.1.2**]{} Of course, in all the formulae including $l_{p}$ this quantity must be changed by $G^{1/2}$ and hence $l_{min}$ by $\vartheta G^{1/2}=2\alpha^{\prime} G^{1/2}.$\
\
[**4.1.3**]{} As noted in the end of subsection 6.1, and in this case also knowing all the high-energy deformed quantities $T_{GUP}(\alpha),S_{GUP}(\alpha),E_{GUP}(\alpha),V_{GUP}(\alpha)$, we can find $P(\alpha)$ at $\alpha$ close to 1/4.\
\
[**4.1.4**]{} Here it is implicitly understood that the Ultraviolet limit of Einstein’s equations is independent of the starting horizon space. This assumption is quite reasonable. Because of this, we use the well-known formulae for the modification of thermodynamics and statistical mechanics of black holes in the presence of GUP [@acs]–[@Nou],[@Park],[@Kim1].\
\
[**4.1.5**]{} The use of the thermodynamic identity (\[GT8.GUP\]) for the description of the high energy deformation in General Relativity implies that on going to the UV-limit of Einstein’s equations for horizon spaces in the thermodynamic representation (consideration) we are trying to remain within the scope of [**equilibrium statistical mechanics**]{} [@Balesku1] ([**equilibrium thermodynamics**]{}) [@Bazarov]. However, such an assumption seems to be too strong. But some grounds to think so may be found as well. Among other things, of interest is the result from [@acs] that GUP may prevent black holes from their total evaporation. In this case the Planck’s remnants of black holes will be stable, and when they are considered, in some approximation the [**equilibrium thermodynamics**]{} should be valid. At the same time, by author’s opinion these arguments are rather weak to think that the quantum gravitational effects in this context have been described only within the scope of [**equilibrium thermodynamics**]{} [@Bazarov].\
\
[**II. Second variant**]{}.\
According to the remark of [**4.1.5**]{}, it is assumed that the interpretation of Einstein’s equations as a thermodynamic identity (\[GT12\]) is not retained on going to high energies (UV–limit), i.e. at $\alpha\rightarrow 1/4$, and the situation is adequately described exclusively by [**non-equilibrium thermodynamics**]{} [@Bazarov],[@Gyarm]. Naturally, the question arises: which of the additional terms introduced in (\[GT12\]) at high energies may be leading to such a description?\
In the [@shalyt-gup],[@shalyt-aip] it has been shown that in case the cosmological term $\Lambda$ is a dynamic quantity, it is small at low energies and may be sufficiently large at high energies. In the right-hand side of (\[GT16.A\]) in the $\alpha$–representation the additional term $GF(\Lambda(\alpha))$ is introduced: $$\label{GT16.B}
-\alpha^{2}f'(\alpha)-\frac{1}{2}\alpha=4\pi P\vartheta^{2}G^{2}-GF(\Lambda(\alpha)),$$ where in terms of $F(\Lambda(\alpha))$ we denote the term including $\Lambda(\alpha)$ as a factor. Then its inclusion in the low-energy case (\[GT11\])(or in the $\alpha$ -representation (\[GT16.A\])) has actually no effect on the thermodynamic identity (\[GT12\])validity, and consideration within the scope of equilibrium thermodynamics still holds true. It is well known that this is not the case at high energies as the $\Lambda$-term may contribute significantly to make the “process” non-equilibrium in the end [@Bazarov],[@Gyarm].\
Is this the only cause for violation of the thermodynamic identity (\[GT12\]) as an interpretation of the high-energy generalization of Einstein’s equations? Further investigations are required to answer this question.
$\alpha$–Representation for Friedmann Equations with GUP and EUP
----------------------------------------------------------------
Thermodynamic interpretation of Section 4 has been also developed for Friedmann Equations (FEs) of the Friedmann-Robertson-Walker (FRW) Universe in [@Cai1]. In the process it is taken into consideration that in the FRW space-time, where the metric is given by the formula $$\begin{aligned}
ds^2=-dt^2+a^2(\frac{dr^2}{1-kr^2}+r^2d\Omega_{n-1}^2),\end{aligned}$$ and $d\Omega_{n-1}^2$ denotes a line element of the ($n-1$)-dimensional unit sphere, $a$ is the scale factor, $k$ is the spatial curvature constant, there is a dynamic [**apparent horizon**]{}, the radius of which is as follows: $$\begin{aligned}
\tilde{r}_A=\frac{1}{\sqrt{H^2+k/a^2}},\end{aligned}$$ where $H\equiv \dot{a}/a$ is the Hubble parameter.\
FEs in [@Cai1] have been derived proceeding from the assumption that [**apparent horizon**]{} is endowed with the associated entropy and temperature such the event horizon in the black hole case $$\begin{aligned}
S=\frac{A}{4G},~~~~~T=\frac{1}{2\pi \tilde{r}_A}\end{aligned}$$ and from the validity of the first low of thermodynamics $$\begin{aligned}
dE=TdS.\end{aligned}$$\
In [@FRW1] with the use of this thermodynamic interpretation of FEs the modifications of GUP and EUP (or more precisely the GUP and EUP deformations) of FEs have been obtained. It is clear that these (GUP and EUP)–deformed FEs may be written in the form of the $\alpha$–representation. For simplicity, let us consider the case $n=3$.\
Then for GUP the formula (26) from [@FRW1] takes the form $$\begin{aligned}
(\dot{H}-\frac{k}{a})[1+\pi\alpha^{\prime 2}l_p^2\frac{1}{A}+
2(\pi \alpha^{\prime 2}l_p^2)^2\frac{1}{A^2}\nonumber\\
+\sum_{d=3}c_d(4\pi\alpha^{\prime 2}
l_p^2)^{2d}\frac{1}{A^d}]=-4\pi G(\rho+p),\label{fr3},\end{aligned}$$ whereas in the $\alpha$–representation its form is more elegant $$\begin{aligned}
(\dot{H}-\frac{k}{a})[1+\frac{1}{16}\alpha_{\tilde{r}_A}
+\frac{1}{32}\alpha^{2}_{\tilde{r}_A}
\nonumber\\
+\sum_{d=3}\frac{c_d}{4^{d}}\alpha^{d}_{\tilde{r}_A}]=-4\pi
G(\rho+p),\label{fr31}.\end{aligned}$$ Also, more elegant is $\alpha$–representation of the second Friedmann Equation (formula (27) from [@FRW1]) $$\begin{aligned}
\frac{8\pi G^{2}}{3}\rho= \frac{\alpha_{\tilde{r}_A}}{4\pi
\alpha^{\prime 2}}[\pi+\frac{1}{32}\alpha_{\tilde{r}_A}
+\frac{1}{96}\alpha^{2}_{\tilde{r}_A}
\nonumber\\
+\sum_{d=3}\frac{c_d}{4^{d}(d+1)}\alpha^{d}_{\tilde{r}_A}],\label{fr4},\end{aligned}$$ with the assumption that $\hbar=c=1$.\
It is obvious that therewith familiar FEs appear at low energies, i.e. at $\alpha_{\tilde{r}_A}\ll 1/4$.\
In the nontrivial high-energy case one can obtain the solution for FE and, in particular $\rho,H,p$ as a series in terms of $\alpha$ close to 1/4.\
In the case of EUP the $\alpha$–representation of the deformed FE [@FRW1] seems to be even simpler. Specifically, using (\[EUP1\]) – (\[EUP5\]), one can derive deformed first Friedmann equation of the form $$\begin{aligned}
(\dot{H}-\frac{k}{a^2})(1+\frac{\beta^2}{\pi
l^2}A)=(\dot{H}-\frac{k}{a^2})(1+\frac{\beta^2}{\pi l^2}\frac{4\pi
\tilde{r}_A^{2}}{l^{2}_{or}}l^{2}_{or})\nonumber\\
=(\dot{H}-\frac{k}{a^2})(1+4\beta^2\alpha^{-1}_{\tilde{r}_A}\alpha_{l_{or}})
=(\dot{H}-\frac{k}{a^2})(1+4\beta^2\widetilde{\alpha}_{\tilde{r}_A})=-4\pi
G(\rho+p),\end{aligned}$$ where, as expected, the deformation parameter $\widetilde{\alpha}_{\tilde{r}_A}$ is small.\
In a similar way we can obtain the $\alpha$–representation of the EUP-deformation for the second Friedmann equation.
Some Comments and Problems of Interest
======================================
In this Section some comments are given and some problems are stated.\
\
Ñ1. The Laurent series expansion in terms of $\alpha$ is asymmetric for UV and IR cutoffs. Indeed, as in the general case the variability domain $0<\alpha\leq1/4$, in the UV-cutoff when $\alpha\approx 1/4$ the contribution is made by $\alpha$-terms both with positive and with negative powers, while in the IR-cutoff ($\alpha\ll 1/4$) only the $\alpha$-terms with negative powers will be significant.\
\
C2. The external constant $\alpha^{\prime}$ in the cases, where $l_{or}\neq l_{min}$ (EUP or SGUP is the case), is not found in the final expressions, being reduced due to the substitution of(\[EUP4\]).\
\
Several questions remain to be answered and necessitate further investigations.\
\
Q1) How far the $\alpha$-representation may be extended for the General Relativity? As shown in this work, such a representation exists for the General Relativity at High and Low Energies when the [**Thermodynamic Approach**]{} [@Jac1]–[@Cai1] is applicable or, that is the same, the Thermodynamic Interpretation is the case. It is interesting whether the extension of the $\alpha$-representation to the general case both at High and Low Energies is possible. The problem is whether, in some or other way, the general case may be reduced to the well-known ones.\
\
Q2) Considering Q1), for High Energy the problem is whether there is an effective description of the space-time foam [@Wheel]–[@Isham] in terms of $\alpha$. The results of [@Gar2] suggest that such a description should be existent.\
\
Q3) Proceeding from the results of E.Verlinde [@Verl], the problem is whether the High-Energy deformation of the Entropic Force is obtainable. Provided the answer of Q1) positive, the problem concerns the form of this deformation in terms of $\alpha$: we must find its $alpha$-representation.\
Note that the notion of Entropic Force, however, without the introduction of the term per se has been proposed by T.Padmanabhan in Conclusion of his paper [@Padm-new] earlier than by E.Verlinde.
Conclusion
==========
In the case the problems stated in the previous Section will be solved positively, the small dimensionless discrete parameter $\alpha$ must be at once introduced in Íigh-Energy Thermodynamics and Gravity, without its appearance in the low-energy limit at the scales under study. At the same time, at large scales GR has not been subjected to verification too [@Tur]. The availability of Dark Matter and Dark Energy is a strong motivation for the IR-modification of GR [@Dvali]– [@Rub]. The deformation of the General Relativity due to EUP seems to be one of the IR-modifications of Gravity possible. In this case an analysis of such a deformation in terms of the parameter $\alpha$, of the corresponding variability domain, and the like may be important for studies of the IR-modified (IR-deformed) General Relativity.
Acknowledgments
===============
I am grateful to Prof. Sabine Hossenfelder (Stockholm, Sweden) for the information about a number of interesting works in the field of physics at Planck’s scales and for her support that has stimulated my research efforts. Besides, I would like to thank Prof. Rong-Jia Yang (Baoding, China) for his support.
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[^1]: E-mail: a.shalyt@mail.ru; alexm@hep.by
|
---
abstract: |
This work is focused on the doubly nonlinear equation $$\nonumber
\ptt u+\partial_{xxxx}u + \big(p-\|\partial_x
u\|_{L^2(0,1)}^2\big)\partial_{xx}u+ \pt u +k^2u^+= f,$$ whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness $k^2$. When the ends are pinned, longterm dynamics is scrutinized for arbitrary values of axial load $p$ and stiffness $k^2$. For a general external source $f$ we prove the existence of bounded absorbing sets. When $f$ is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.
address:
- ' $^1$ Dipartimento di Matematica e Informatica, Università degli studi di Salerno, Italy and INFN, Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Napoli, Italy'
- '$^2$Dipartimento di Matematica, Università degli studi di Brescia, Italy'
author:
- 'Ivana Bochicchio$^1$, Claudio Giorgi$^2$, Elena Vuk$^2$'
title: |
Longterm Damped Dynamics\
of the Extensible Suspension Bridge
---
Introduction
============
The model equation
------------------
In this paper, we scrutinize the longtime behavior of a nonlinear evolution problem describing the damped oscillations of an extensible elastic bridge of unitary natural length suspended by means of flexible and elastic cables. The model equation ruling its dynamics can be derived from the standard modeling procedure, which relies on the basic assumptions of continuous distribution of the stays’ stiffness along the girder and of the dominant truss behavior of the bridge (see, for instance, [@K-NYB]).
In the pioneer papers by McKenna and coworkers (see [@LMK; @MKW; @MKW1]), the dynamics of a suspension bridge is given by the well known damped equation $$\label{BRIDGE} \ptt u+\partial_{xxxx}u + \pt u +k^2u^+= f,$$ where $u=u(x,t):[0,1]\times\R\to\R$, accounts for the downward deflection of the bridge in the vertical plane, and $u^+$ stands for its positive part, namely, $$u^+=
\begin{cases}
u \qquad \text{if } u\geq0,\\
0 \qquad \text{if } u<0.
\end{cases}$$ Our model is derived here by taking into account the midplane stretching of the road bed due to its elongation. As a consequence a geometric nonlinearity appears into the bending equation. This is achieved by combining the pioneering ideas of Woinowsky-Krieger on the extensible elastic beam [@W] with equation . Setting for simplicity all the positive structural constants of the bridge equal to 1, we have $$\label{BEAM} \ptt u+\partial_{xxxx}u + \big(p-\|\partial_x
u\|_{L^2(0,1)}^2\big)\partial_{xx}u+ \pt u +k^2u^+= f,$$ where $f=f(x,t)$ is the (given) vertical dead load distribution. The term $-k^2u^+$ models a restoring force due to the cables, which is different from zero only when they are being stretched, and $\pt u$ accounts for an external resistant force linearly depending on the velocity. The real constant $p$ represents the axial force acting at the ends of the road bed of the bridge in the reference configuration. Namely, $p$ is negative when the bridge is stretched, positive when compressed.
As usual, $u$ and $\pt u$ are required to satisfy initial conditions as follows $$\label{BEAMIC}
\begin{cases}
u(x,0)=u_0(x), & x\in[0,1],\\
\pt u(x,0)={u}_1(x), & x\in[0,1].
\end{cases}$$ Concerning the boundary conditions, we consider here the case when both ends of the bridge are pinned. Namely, for every $t\in\R$, we assume $$\label{BEAMBC}
u(0,t)=u(1,t)=\partial_{xx}u(0,t)=\partial_{xx}u(1,t)=0.$$ This is the simpler choice. However, other types of boundary conditions with fixed ends are consistent with the extensibility assumption as well; for instance, when both ends are clamped, or when one end is clamped and the other one is pinned. We address the reader to [@GPV] for a more detailed discussion. Assuming , the domain of the differential operator $\partial_{xxxx}$ acting on $L^2(0,1)$ is $$\D(\partial_{xxxx})=\{w\in H^4(0,1) : w(0)=w(1)=\partial_{xx}w(0)=\partial_{xx}w(1)=0\}.$$ This operator is strictly positive selfadjoint with compact inverse, and its discrete spectrum is given by $\lambda_n=n^4\pi^4$, $n\in\N$. Thus, ${\lambda_1=\pi^4}$ is the smallest eigenvalue. Besides, the peculiar relation
$$(\partial_{xxxx})^{1/2}=-\partial_{xx}$$ holds true, with Dirichlet boundary conditions and $$\D(-\partial_{xx})=H^2(0,1)\cap H^1_0(0,1).$$ Hence, if pinned ends are considered, the initial-boundary value problem – can be described by means of a single operator $A=\partial_{xxxx}$, which enters the equation at the powers 1 and 1/2. Namely, $$\partial_{tt} u+Au+ \pt u- \big(p-\|u\|^2_1\big)A^{1/2}u+k^2u^+=
f,$$ where $\| \cdot \|_1$ is the norm of $H^1_0(0,1)$. This fact is particularly relevant in the analysis of the [*critical buckling load*]{} $p_{\rm c}$, that is, the magnitude of the compressive axial force $p>0$ at which buckled stationary states appear.
As we shall show throughout the paper, this model leads to exact results which are rather simple to prove and, however, are capable of capturing the main behavioral dynamic characteristics of the bridge.
Earlier contributions
---------------------
In recent years, an increasing attention was payed to the analysis of buckling, vibrations and post-buckling dynamics of nonlinear beam models, especially in connection with industrial applications [@LG; @NP] and suspension bridges [@AGR1; @AH]. As far as we know, most of the papers in the literature deal with approximations and numerical simulations, and only few works are able to derive exact solutions, at least under stationary conditions (see, for instance, [@BGV; @BoV; @CZGP; @GV]). In the sequel, we give a brief sketch of earlier contributions on this subject.
In the fifties, Woinowsky-Krieger [@W] proposed to modify the theory of the dynamic Euler-Bernoulli beam, assuming a nonlinear dependence of the axial strain on the deformation gradient. The resulting motion equation, $$\label{BEAMW} \ptt u+\partial_{xxxx}u + \big(p-\|\partial_x
u\|_{L^2(0,1)}^2\big)\partial_{xx}u = 0,$$ has been considered for hinged ends in the papers [@B1; @D], with particular reference to well-posedness results and to the analysis of the complex structure of equilibria. Adding an external viscous damping term $ \pt u$ to the original conservative model, it becomes $$\label{DAMPED} \ptt u+\partial_{xxxx}u+ \pt u+ \big(p-\|\partial_x
u\|_{L^2(0,1)}^2\big)\partial_{xx}u= 0.$$ Stability properties of the unbuckled (trivial) and the buckled stationary states of have been established in [@B; @D1] and, more formally, in [@RM]. In particular, if $p<p_{\rm c}$, the exponential decay of solutions to the trivial equilibrium state has been shown. The global dynamics of solutions for a general $p$ has been first tackled in [@HAL] and improved in [@EM], where the existence of a global attractor for subject to hinged ends was proved relying on the construction of a suitable Lyapunov functional. In [@CZ] previous results are extended to a more general form of the nonlinear term by virtue of a suitable decomposition of the semigroup first introduced in [@GPV].
A different class of problems arises in the study of vibrations of a suspension bridge. The dynamic response of suspension bridges is usually analyzed by linearizing the equations of motion. When the effects of extensibility of the girder are neglected and the coupling with the main cable motion is disregarded, we obtain the well-known Lazer-McKenna equation . Free and forced vibrations in models of this type, both with constant and non constant load, have been scrutinized in [@AH] and [@CJ]. The existence of strong solutions and global attractors for has been recently obtained in [@ZMS].
In certain cases Lazer-McKenna’s model becomes inadequate and the effects of extensibility of the girder have to be taken into account. This can be done by introducing into the model equation a geometric nonlinear term like that appearing in . Such a term is of some importance in the modeling of cable-stayed bridges (see, for instance, [@K-NYB; @Vir]), where the elastic suspending cables are not vertical and produce a well-defined axial compression on the road bed.
Several studies have been devoted to the nonlinear vibrational analysis of mechanical models close to . Abdel-Ghaffar and Rubin [@AGR; @AGR1] presented a general theory and analysis of the nonlinear free coupled vertical-torsional vibrations of suspension bridges. They developed approximate solutions by using the method of multiple scales via a perturbation technique. If torsional vibrations are ignored, their model reduces to . Exact solutions to this problem, at least under stationary conditions, have been recently exhibited in [@GV].
Outline of the paper
--------------------
In the next Section 2, we formulate an abstract version of the problem. We observe that its solutions are generated by a solution operator $S(t)$, which turns out to be a strongly continuous semigroup in the autonomous case. The existence of an absorbing set for the solution operator $S(t)$ is proved in Section 3 by virtue of a Gronwall-type Lemma. Section 4 is focused on the autonomous case and contains our main result. Namely, we establish the [*existence of the regular global attractor*]{} for a general $p$. In particular, we prove this by appealing to the existence of a Lyapunov functional and without requiring any assumption on the strength of the dissipation term. A characterization of the global attractor is given in terms of the steady states of the system –. First, we proceed with some preliminary estimates and prove the exponential stability of the system provided that the axial force $p$ is smaller than $p_c$. Finally, the smoothing property of the semigroup generated by the abstract problem is stated via a suitable decomposition first devised in [@GPV].
The Dynamical System
====================
In the sequel we recast problem - into an abstract setting in order to establish more general results.
Let $(H,\l\cdot,\cdot\r,\|\cdot\|)$ be a real Hilbert space, and let $A:\D(A)\Subset H\to H$ be a strictly positive selfadjoint operator with compact inverse. For $r\in\R$, we introduce the scale of Hilbert spaces generated by the powers of $A$ $$H^r=\D(A^{r/4}),\qquad \l u,v\r_r=\l A^{r/4}u,A^{r/4}v\r,\qquad
\|u\|_r=\|A^{r/4}u\|.$$ When $r=0$, the index $r$ is omitted. The symbol $\l\cdot,\cdot\r$ will also be used to denote the duality product between $H^r$ and its dual space $H^{-r}$. In particular, we have the compact embeddings $H^{r+1}\Subset H^r$, along with the generalized Poincaré inequalities $$\label{POINCARE}
\lambda_1\|u\|_r^4\leq \|u\|_{r+1}^4,\qquad\forall u\in H^{r+1},$$ where $\lambda_1>0$ is the first eigenvalue of $A$. Finally, we define the product Hilbert spaces $$\H^r=H^{r+2}\times H^r.$$ For $p\in\R$, we consider the following abstract Cauchy problem on $\H$ in the unknown variable $u=u(t)$, $$\label{ASTRATTO}
\begin{cases}
\partial_{tt} u+Au+ \pt u- \big(p-\|u\|^2_1\big)A^{1/2}u+k^2u^+=
f(t),\quad t>0,
\\
u(0)\,=\,u_0, \quad \pt u(0)\,=\,u_1 \ .
\end{cases}$$ Problem - is just a particular case of the abstract system , obtained by setting $H=L^2(0,1)$ and $A=\partial_{xxxx}$ with the boundary condition .
The following well-posedness result holds.
\[EU\] Assume that $f\in L^1_{\rm loc}(0,T; H).$ Then, for all initial data $z=(u_0,u_1)\in\H$, problem admits a unique solution $$(u(t),\pt u(t))\in\C(0,T; \H)\,,$$ which continuously depends on the initial data.
We omit the proof of this result, which is based on a standard Galerkin approximation procedure (see, for istance [@B; @B1]), together with a slight generalization of the usual Gronwall lemma. In particular, the uniform-in-time estimates needed to obtain the global existence are exactly the same we use in proving the existence of an absorbing set.
In light of Proposition \[EU\], we define the [*solution operator*]{} $$S(t)\in\C(\H,\H),\qquad\forall t\geq 0,$$ as $$z=(u_0,u_1)\mapsto S(t)z=(u(t),\pt u(t)).$$ Besides, for every $z\in\H$, the map $t\mapsto S(t)z$ belongs to $\C(\R^+,\H)$. Actually, it is a standard matter to verify the joint continuity $$(t,z)\mapsto S(t)z\in\C(\R^+\times\H,\H).$$
\[semigroup\] In the autonomous case, namely when $f$ is time-independent, the semigroup property $$S(t+\tau)=S(t)S(\tau)$$ holds for all $t,\tau\geq 0$. Thus, $S(t)$ is a strongly continuous semigroup of operators on $\H$ which continuously depends on the initial data: for any initial data $z\in\H$, $S(t)z$ is the unique weak solution to , with related norm given by $$\mathcal{E}(z)\,=\,\|z\|_\mathcal{H}^2\,=\,\|u\|_2^2+\|v\|^2.$$
For any $z=(u, v)\in\H$, we define the [*energy*]{} corresponding to $z$ as $$\label{eq:Energia}
E(z)=\mathcal{E}(z)+\frac12\big(\|u\|_1^2-p\big)^2+k^2\|u^+\|^2,$$ and, abusing the notation, we denote $E(S(t)z)$ by $E(t)$ for each given initial data $z\in \H$. Multiplying the first equation in by $\pt u$, because of the relation $$k^2\l u^+, \pt u \r= \frac {k^2}{2} \frac{\,d}{dt}\,(\| u^+\|^2),$$ we obtain the [*energy identity*]{} $$\label{E} \ddt E+2\|\partial_t u\|^2=2\l \pt u,f\r.$$ In particular, for every $T>0$, there exists a positive increasing function $\Q_T$ such that $$\label{TFIN} E(t)\leq \Q_T(E(0)),\qquad\forall t\in [0,T].$$
The Absorbing Set
=================
It is well known that the absorbing set gives a first rough estimate of the dissipativity of the system. In addition, it is the preliminary step to scrutinize its asymptotic dynamics (see, for instance, [@TEM]). Here, due to the joint presence of geometric and cable-response nonlinear terms in , a direct proof of the existence of the absorbing set via [*explicit*]{} energy estimates is nontrivial. Indeed, the double nonlinearity cannot be handled by means of standard arguments, as either in [@MKW] or in [@ZMS]. Dealing with a given time-dependent external force $f$ fulfilling suitable translation compactness properties, a direct proof of the existence of an absorbing set is achieved here by means of a generalized Gronwall-type lemma devised in [@GGP].
An absorbing set for the solution operator $S(t)$ (referred to the initial time $t=0$) is a bounded set $\BB_{\H} \subset \H$ with the following property: for every $R\geq 0$, there is an [*entering time*]{} $t_R\geq 0$ such that $$\bigcup_{t\geq t_R}S(t)z\subset \BB_{\H},$$ whenever $\|z\|_\H\leq R$. In fact, we are able to establish a more general result.
\[thmABS\] Let $f\in L^\infty(\R^+,H)$, and let $\pt f$ be a translation bounded function in $L^2_{\rm loc}(\R^+,H^{-2})$, that is, $$\label{tb} \sup_{t\geq 0}\int_t^{t+1}\|\pt f(\tau)\|^2_{-2}
\d\tau=M<\infty.$$ Then, there exists $R_0>0$ with the following property: in correspondence of every $R\geq 0$, there is $t_0=t_0(R)\geq 0$ such that $$E(t)\leq R_0,\qquad \forall t\geq t_0,$$ whenever $E(0)\leq R$. Both $R_0$ and $t_0$ can be explicitly computed.
We are able to establish Theorem \[thmABS\], leaning on the following Lemma.
\[superl\] Let $\Lambda:\R^+\to\R^+$ be an absolutely continuous function satisfying, for some $M\geq 0$, $\eps>0$, the differential inequality $$\ddt\Lambda(t)+\eps \Lambda(t)\leq \varphi(t),$$ where $\varphi:\R^+\to\R^+$ is any locally summable function such that $$\sup_{t\geq 0}\int_t^{t+1}\varphi(\tau)\d \tau\leq M.$$ Then, there exist $R_1>0$ and $\gamma>0$ such that, for every $R\geq 0$, it follows that $$\Lambda(t)\leq R_1,\qquad\forall t\geq R^{1/\gamma}(1+\gamma M)^{-1},$$ whenever $\Lambda(0)\leq R$. Both $R_1$ and $\gamma$ can be explicitly computed in terms of $M$ and $\eps$.
Here and in the sequel, we will tacitly use several times the Young and the Hölder inequalities, besides the usual Sobolev embeddings. The generic positive constant $C$ appearing in this proof may depend on $p$ and $\|f\|_{L^\infty(\R^+,H)}$.
On account of , by means of the functional $$\label{elle}
\L(z)=E(z)-2\l u,f\r,$$ we introduce the function $$\L(t)=E(t)-2\l u(t),f(t)\r,$$ which satisfies the differential equality $$\label{derivata}
\ddt \L+2\|\partial_t u\|^2=-2\l u,\pt f \r.$$ Because of the control $$\label{derivataf}
2|\l u,\pt f\r| \leq \frac12 \|u\|_2^2 + 2\| \pt f\|^2_{-2},$$ we obtain the differential inequality $$\ddt \L+2\|\partial_t u\|^2 \leq \frac 12 E + 2\| \pt f\|^2_{-2}.$$ Next, we consider the auxiliary functional $\Upsilon(z)=\l u,v\r$ and, regarding $\Upsilon(S(t)z)$ as $\Upsilon(t)$, we have $$\ddt\Upsilon+\Upsilon + \|u\|_2^2+\big(\|u\|_1^2-p\big)^2
+p\big(\|u\|_1^2-p\big) + k^2\|u^+\|^2 - \l
u,f\r=\|\pt u\|^2.$$ Noting that $$\frac12\big(\|u\|_1^2-p\big)^2
+p\big(\|u\|_1^2-p\big)=\frac12\|u\|_1^4-\frac12 p^2,$$ we are led to $$\ddt\Upsilon + \Upsilon + \frac12\|u\|_1^4 + \|u\|_2^2 + k^2\|u^+\|^2+\frac12\big(\|u\|_1^2-p\big)^2 - \l u,f\r
= \|\pt u\|^2+\frac12 p^2.$$ Precisely, we end up with $$\label{UNODUE}
\ddt\Upsilon + \Upsilon + \frac12 E\leq \frac32 \|\partial_t
u\|^2+\frac{1}{2 \lambda_1}\|f\|^2+\frac{1}{2}p^2.$$ Finally, we set $$\Lambda(z)=\L(z)+\Upsilon(z)+C,$$ where $ C= \frac{2}{\lambda_1}\|f\|^2+\frac{1}{2\lambda_1} + \frac{|p|}{2\sqrt {\lambda_1}}$. We first observe that $\Lambda(z)$ satisfies $$\label{CTRL} \frac12 \E(z)\leq \frac12 E(z)\leq\Lambda(z)\leq 2 E(z)+c.$$ In order to estimate $\Lambda$ from below, a straightforward calculation leads to $$\begin{aligned}
\Lambda(z)&\geq &E(z) - 2 |\l u,f\r | - | \Upsilon(z)| + C
\\
&\geq& E(z) - \frac12\|u\|_2^2 - \frac 12\|v\|^2- \frac14\big(\|u\|_1^2-p\big)^2 -2 \|f\|_{-2}^2 - \frac{1}{2\lambda_1} - \frac{|p|}{2\sqrt {\lambda_1}} + C
\\
&=&\frac 12 \|u\|_2^2 + \frac 12 \|v\|^2 + \frac14\big(\|u\|_1^2-p\big)^2 + k^2\|u^+\|^2 - 2 \|f\|_{-2}^2 - \frac{1}{2\lambda_1} - \frac{|p|}{2\sqrt {\lambda_1}} + C
\\
&\geq& \frac 12 E(z) - \frac{2}{\lambda_1}\|f\|^2-\frac{1}{2\lambda_1} - \frac{|p|}{2\sqrt {\lambda_1}} +C\geq \frac 12 E(z),\end{aligned}$$ where we take advantage of $$\begin{aligned}
| \Upsilon(z)|&\leq \|u\|\|v\| \leq \frac{1}{\root 4 \of{ \lambda_1}} \|u\|_1\|v\| \leq \frac 12\|v\|^2 + \frac{1}{2\sqrt {\lambda_1}} \|u\|_1^2 \\
&\leq \frac 12\|v\|^2 + \frac{1}{2\sqrt {\lambda_1}}\big(\|u\|_1^2-p\big) + \frac{|p|}{2\sqrt {\lambda_1}} \\
&\leq \frac 12\|v\|^2 + \frac14 \big(\|u\|_1^2-p\big)^2 + \frac{|p|}{2\sqrt {\lambda_1}} + \frac{1}{2\lambda_1}.\end{aligned}$$ The upper bound for $\Lambda$ can be easly achieved as follows $$\begin{aligned}
\Lambda(z)&\leq E(z) + \|u\|_2^2 + \|v\|^2+ \frac12 \|u\|_1^4 + \|f\|_{-2}^2 + \frac{1}{32\lambda_1} + C \\
&\leq 2E(z) + \frac{1}{\lambda_1}\|f\|^2 + \frac{1}{32\lambda_1} + C \leq 2E(z) + c,\end{aligned}$$ by virtue of $$\label{Ups2}
| \Upsilon(z)|\leq \|u\|\|v\| \leq \frac{1}{\root 4 \of{ \lambda_1}} \|u\|_1\|v\| \leq \|v\|^2 + \frac{1}{4\sqrt {\lambda_1}} \|u\|_1^2 \leq \|v\|^2 + \frac12 \|u\|_1^4 + \frac{1}{32\lambda_1}.$$ Going back to differential equation and making use of and , the function $\Lambda(t)= \Lambda (S(t)z)$ satisfies the identity $$\ddt\Lambda + \frac {\Lambda}{2} + \frac {\Upsilon}{2} + \frac12\|u\|_2^2 + \frac12\|\partial_tu\|^2 +
\frac14\|u\|_1^4 + \frac12k^2\|u^+\|^2 =
-2\l u,\partial _tf\r +\frac{p^2}{2},$$ and, as a consequence, we obtain the estimate $$\ddt\Lambda + \frac {\Lambda}{2} + \frac 12 \big(\Upsilon + \|u\|_2^2 + \|\partial_tu\|^2 +
\frac12\|u\|_1^4 +4 \l u,\partial _tf\r \big) \leq \frac{p^2}{2}.$$ Now, using and , we have $$\ddt\Lambda + \frac {\Lambda}{2} \leq 2\| \partial _tf\|_{-2}^2 + c,$$ where $c= \frac {1}{16 \lambda_1} + \frac{p^2}{2}$. Thus, by virtue of and , Lemma \[superl\] yields $$E(t) \leq 2\Lambda(t) \leq 2R_1(M,c).$$
If the set of stationary solutions to shrinks to a single element, the subsequent asymptotic behavior of the system becomes quite simple. Indeed, this occurs when $p< p_c=\sqrt{\lambda_1}$. If this the case, the only trivial solution exists and is exponentially stable, as it will be shown in Section 4. The more complex and then attractive situation occurs when the set of steady solutions contains a large (possibly infinite) amount of elements. To this end, we recall here that the set of the bridge stationary-solutions (equilibria) has a very rich structure, even when $f=0$ (see [@GV]).
The Global Attractor
====================
In the remaining of the paper, we simplify the problem by assuming that the external force $f$ is time-independent. In which case, the operator $S(t)$ is a strongly continuous semigroup on $\H$ (see Remark\[semigroup\]). Having been proved in Sect.3 the existence of the absorbing set $\BB$, we could then establish here the existence of a global attractor by showing that the semigroup $S(t)$ admits a bounded absorbing set in a more regular space and that it is uniformly compact for large values of $t$ (see, for instance, [@TEM Theor.1.1]). In order to obtain asymptotic compactness, the $\alpha$-contraction method should be employed (see [@HAL] for more details). If applied to , however, such a strategy would need a lot of calculations and, what is more, would provide some regularity of the attractor only if the dissipation is large enough (see [@EM], for instance).
Noting that in the autonomous case problem becomes a [*gradient system*]{}, there is a way to overcome these difficulties by using an alternative approach which appeals to the existence of a Lyapunov functional in order to prove the existence of a global attractor. This technique has been successfully adopted in some recent papers concerning some related problems, just as the longterm analysis of the transversal motion of extensible viscoelastic [@GPV] and thermoelastic [@GNPP] beams.
We recall that the global attractor $\A$ is the unique compact subset of $\H$ which is at the same time
- attracting: $$\lim_{t\to\infty}\boldsymbol{\delta}(S(t)\BB,\A)\to 0,$$ for every bounded set $\BB\subset\H$, where $\boldsymbol{\delta}$ denotes the usual Hausdorff semidistance in $\H$;
- fully invariant: $$S(t)\A=\A,\qquad\forall t\geq 0.$$
We address the reader to the books [@CV; @HAL; @TEM] for a detailed presentation of the theory of attractors.
\[MAIN\] The semigroup $S(t)$ acting on $\H$ possesses a connected global attractor $\A$ bounded in $\H^2$. Moreover, $\A$ coincides with the unstable manifold of the set ${\mathcal S}$ of the stationary points of $S(t)$, namely, $$\A=
\Big\{z(0): z \text{ is a complete (bounded) trajectory of } S(t)
: \lim_{t\to \infty}\|z(-t)-{\mathcal
S}\|_{\H}=0\Big\}.$$
Due to the regularity and the invariance of $\A$, we observe that $S(t)z$ is a strong solution to whenever $z\in\A$.
The set ${\mathcal S}$ of the bridge equilibria under a vanishing lateral load consists of all the pairs $(u,0)\in \H$ such that the function $u$ is a weak solution to the equation $$Au - \big(p-\|u\|^2_1\big)A^{1/2}u+k^2u^+=0.$$ In particular, $u$ solves the following boundary value problem on the interval $[0,1]$ $$\label{STATICBEAM}
\begin{cases}
\partial_{xxxx}u+\big(b\,\pi^2-\|\partial_{x}u\|^2_{L^2(0,1)}\big)\partial_{xx}u+\kappa^2\pi^4u^+= 0,\\
\noalign{\vskip1mm} u(0)=u(1)=\partial_{xx}u(0)=\partial_{xx}u(1)=0,
\end{cases}$$ where we let $k=\kappa\pi^2$, $\kappa\in\R$, and $p=b\,\pi^2$, $b\in\R$. It is then apparent that ${\mathcal S}$ is bounded in $H^2(0,1)\cap H^1_0(0,1)$ for every $b,\kappa\in\R$.
When $\kappa=0$, a general result has been established in [@CZGP] for a class of non-vanishing sources. In [@BGV; @BoV], the same strategy with minor modifications has been applied to problems close to , where the term $u^+$ is replaced by $u$ (unyielding ties).
The set of buckled solutions to problem is built up and scrutinized in [@GV]. In order to have a finite number of solutions, we need all the bifurcation values to be distinct. This occurrence trivially holds when $\kappa=0$, because of the spectral properties of the operator $\partial_{xxxx}$. On the contrary, for general values of $\kappa$, all critical values “moves" when $\kappa$ increases, as well as in [@BoV]. Hence, it may happen that two different bifurcation values overlap for special values of $\kappa$, in which case they are referred as [*resonant values*]{}.
![The bifurcation picture for $ \kappa=1$.[]{data-label="Fig_1"}](bifurcation.eps){width="15cm" height="5cm"}
Assuming that $\kappa=1$, for istance, Fig. 1 shows the bifurcation picture of solutions in dependence on the applied axial load $p=b\pi^2$. In particular, $u_0=0$ and $$u_1^\pm(x)=A_1^\pm\sin(\pi x),
\qquad
A_1^-=-\sqrt{2(b-1)\,},
\ A_1^+=\sqrt{2(b-2)\,}.$$
The Lyapunov Functional and preliminary estimates
-------------------------------------------------
We begin to prove the existence of a Lyapunov functional for $S(t)$, that is, a function $\L\in C(\H,\R)$ satisfying the following conditions:
- $\L(z)\to\infty$ if and only if $\|z\|_{\H}\to\infty$;
- $\L(S(t)z)$ is nonincreasing for any $z\in\H$;
- $\L(S(t)z)=\L(z)$ for all $t>0$ implies that $z\in{\mathcal S}$.
\[LYAP\] If $f$ is time-independent, the functional $\L$ defined in is a Lyapunov functional for $S(t)$.
Assertion (i) holds by the continuity of $\L$ and by means of the estimates $$\frac12 E(z) - c \leq\L(z)\leq \frac 32 E(z)+c.$$ Using , we obtain quite directly $$\label{LiapDis}
\frac{d}{dt}\L(S(t)z) =-2\|\pt u(t)\|^2 \leq 0,$$ which proves the decreasing monotonicity of $\L$ along the trajectories departing from $z$. Finally, if $\L(S(t)z)$ is constant in time, we have that $\pt
u=0$ for all $t$, which implies that $u(t)$ is constant. Hence, $z=S(t)z=(u_0,0)$ for all $t$, that is, $z\in{\mathcal S}$.
The existence of a Lyapunov functional ensures that $E(t)$ is bounded. In particular, bounded sets have bounded orbits.
Till the end of the paper, $Q: \R^+_0\to\R^+$ will denote a [*generic*]{} increasing monotone function depending explicity only on $R$ and implicity on the structural constants of the problem. The actual expression of $Q$ may change, even within the same line of a given equation.
\[energia-limitata\] Given $f\in H$, for all $t>0$ and initial data $z\in \H$ with $\|z\|_{\H}\leq R$, $$\label{energy} \E(t)\leq Q(R).$$
Inequality ensures that $$\nonumber
\begin{split}
\L(t)=\L(S(t)z) \leq \L(z)\leq Q(R) \ , \quad \forall t \geq 0.
\end{split}$$ Moreover, taking into account that $$\left\| u(t)\right\| ^{2} \leq \frac{1}{\lambda_1}\left\| u(t)\right\|_{2} ^{2} \leq \frac{1}{ \lambda_1}\E(t),$$ we obtain the estimate $$\L(t) \geq \E(t) - 2\left\langle f\,,\,u(t)\right\rangle \geq \E(t) - 2 \left\| f\right\|_{-2} ^{2} - \frac 12 \left\| u(t)\right\|_{2} ^{2} \geq
\frac 12 \E(t) - \frac{2}{\lambda_1} \left\|
f\right\| ^{2}.$$ Finally, we have $$\E(t)\leq 2\L(t) +
\frac{4}{\lambda_1} \left\| f\right\| ^{2} \leq 2Q(R) + \frac{4}{\lambda_1} \left\| f\right\| ^{2} = Q(R).$$
\[FunzionaleF\] Let $p <\, \sqrt{\lambda_1}$ and $
\mathcal F_p(u)\,=Au\,-\,p A^{\frac{1}{2}}u\,
$. Then $$\left\langle \mathcal F_p(u)\,,\,u\right\rangle \geq C(p) \left\| u\right\| _{2}^{2},$$ where $$\label{Cp}
C(p)=
\begin{cases}
1, \qquad\qquad\quad p\leq 0 \\
\left(1- \frac {p}{\sqrt{\lambda_1}}\right), \quad 0<p<\sqrt{\lambda_1}.
\end{cases}$$
Because of the identity $$\left\langle \mathcal F_p(u)\,,\,u\right\rangle = \left\| u\right\|
_{2}^{2}-\,p \left\| u\right\| _{1}^{2},$$ the thesis is trivial when $p \leq 0$. On the other hand, when $0<p<\sqrt{\lambda_1}$ we have $$\left\langle \mathcal F_p(u)\,,\,u\right\rangle = \left\| u\right\|
_{2}^{2}-p \, \left\| u\right\| _{1}^{2}\,\geq \left( 1-\frac{p
}{\sqrt{\lambda_1}}\right) \left\|
u\right\| _{2}^{2} .$$
We are now in a position to prove the following
\[exp-stab\] When $f\,=0$, the solutions to – decay exponentially, i.e. $$\E(t)\,\leq\,c_0\,\E(0)\,e^{- c t}$$ with $c_0$ and $c$ suitable positive constants, provided that $p\,<\, \sqrt{\lambda_1}$.
Let ${\Phi}$ be the functional $$\Phi(z)= \E(z) + \varepsilon \Upsilon(z) - \frac 12 p^2,$$ where the constant $$\label{asterisco}
\varepsilon = {\rm min} \{\lambda _{1}C(p), 1\}$$ is positive provided that $p <\sqrt{\lambda_1}$. In view of applying Lemma \[FunzionaleF\], we remark that $$\Phi =\left\langle \mathcal F_p(u)\,,\,u\right\rangle+\left\|
\partial_t u\right\| ^{2}+\frac{1}{2}\left\| u\right\| _{1}^{4}+\varepsilon
\left\langle u,\partial_t u\right\rangle+k^2\left\| u^+\right\|
^{2}.$$ The first step is to prove the equivalence between $\mathcal{E}$ and $\Phi$, that is $$\label{Phi_bounds}
\frac \varepsilon{2\,\lambda_1}\,\mathcal{E}\leq \Phi\leq Q(\|z\|_{\H})\,\,\mathcal{E}\,.$$ By virtue of , and Lemma \[FunzionaleF\] the lower bound is provided by $$\Phi\geq \left( C(p) -\frac{\varepsilon }{2\,\lambda _{1}
}\right) \left\| u\right\| _{2}^{2}+\left( 1-\frac{\varepsilon }{2\,}\right) \left\|
\partial_t u\right\| ^{2} \geq \frac \varepsilon{2\,\lambda_1}\,\E\,.$$
On the other hand, by applying Young inequality and using (\[POINCARE\]), we can write the following chain of inequalities which gives the upper bound of $\Phi$. $$\begin{aligned}
\Phi &\leq&
\left( C(p)+\frac{k^2}{ \lambda
_{1}}+\frac{1}{2 \,\lambda _{1}}\right) \left\| u\right\|
_{2}^{2}+\left( 1+\frac{\varepsilon ^{2}}{2}\right) \left\|
\partial_t u\right\| ^{2}- p \left\| u\right\|
_{1}^{2}+\frac{1}{2}\left\| u\right\| _{1}^{4}\leq
\\
&\leq &\left( 1+ C(p)+\frac{k^2}{ \lambda _{1}}+\frac{1}{2 \,\lambda
_{1}}+\frac{\varepsilon ^{2}}{2}\right) \E + \left\| u\right\|
_{1}^{2}\left(\frac{1}{2}\left\| u\right\| _{1}^{2}-p\right) .\end{aligned}$$ In particular, from and we find $$\Phi\leq \left( 2+\frac{k^2}{ \lambda _{1}}+\frac{1}{2 \,\lambda
_{1}}+\frac{\varepsilon ^{2}}{2}+\frac{Q(\|z\|_{\H})}{ \sqrt{\,\lambda
_{1}}} \,\right) \E = Q(\|z\|_{\H})\,\,\mathcal{E}.$$ The last step is to prove the exponential decay of $\Phi$. To this aim, we obtain the identity $$\nonumber
\frac{d}{dt}\, \Phi+\varepsilon \Phi+2\left( 1
-\varepsilon \right) \left\| \partial_t u\right\|
^{2}+\frac{\varepsilon }{2}\left\| u\right\| _{1}^{4}+\varepsilon
\left( 1 -\varepsilon \right) \left\langle \partial_t
u\,,\,u\right\rangle =0,$$ where $\eps$ is given by . Exploiting the Young inequality and , we have $$\nonumber
\frac{d}{dt}\, \Phi+\varepsilon \Phi+ \left(
1 -\varepsilon \right) \left\| \partial_t u\right\| ^{2}\leq
\frac{\varepsilon^2\left( 1 -\varepsilon \right)}{4\lambda_{1}
}\,\left\| u\right\| ^{2}_{2} \ \leq \frac{\varepsilon\left( 1 -\varepsilon \right)}{2}\,\Phi,$$ from which it follows $$\nonumber
\frac{d}{dt}\, \Phi+ \frac{\varepsilon\left( 1 +\varepsilon \right)}{2} \Phi\leq 0.$$ Letting $c = {\varepsilon\left( 1 +\varepsilon \right)}/{2}$, by virtue of Lemma \[superl\] (with $M=0$) and we have $$\frac \varepsilon{2\,\lambda_1}\,\E(t)\leq\Phi\left( t\right) \leq \Phi\left( 0\right)
\,e^{- \,c\,t}\leq Q(\|z\|_{\H})\,\E\left( 0\right)\,e^{- \,c\,t}.$$ The thesis follows by putting $c_{0}=2\,\lambda_1Q(\|z\|_{\H})/\varepsilon$.
The existence of a Lyapunov functional, along with the fact that ${\mathcal S}$ is a bounded set, allow us prove the existence of the attractor by showing a suitable (exponential) asymptotic compactness property of the semigroup, which will be obtained exploiting a particular decomposition of $S(t)$ devised in [@GPV] and following a general result (see [@CP], Lemma 4.3), tailored to our particular case.
The Semigroup Decomposition
---------------------------
By the interpolation inequality $\|u\|_1^2\leq \|u\|\|u\|_2$ and (\[POINCARE\]) it is clear that $$\label{gamma} \frac12\|u\|_2^2\leq
\|u\|_2^2-p\|u\|_1^2+\alpha\|u\|^2\leq m\|u\|_2^2,$$ provided that $\alpha>0$ is large enough and for some $m=m(p,\alpha)\geq 1$.
Again, $R>0$ is fixed and $\|z\|_{\H}\leq R$. Choosing $\alpha>0$ such that holds, according to the scheme first proposed in [@GPV], we decompose the solution $S(t)z$ into the sum $$S(t)z=L(t)z+K(t)z,$$ where $$L(t)z=(v(t),\pt v(t))\qquad\text{and}\qquad
K(t)z=(w(t),\pt w(t))$$ solve the systems $$\label{DECAY}
\begin{cases}
\ptt v+Av+\pt v-
(p-\|u\|^2_1)A^{1/2}v+\alpha v= 0,\\
\noalign{\vskip1.5mm} \left(v(0),\pt v(0)\right)=z,
\end{cases}$$ and $$\label{CPT}
\begin{cases}
\ptt w+Aw+\pt w-
(p-\|u\|^2_1)A^{1/2}w-\alpha v+k^2u^+= f,\\
\noalign{\vskip1.5mm} (w(0),\pt w(0))=0.
\end{cases}$$
The next three lemmas show the asymptotic smoothing property of $S(t)$, for initial data bounded by $R$. We begin to prove the exponential decay of $L(t)z$. Then, we prove the asymptotic smoothing property of ${K}(t)$.
\[lemmaDECAY\] There is $\omega=\omega(R)>0$ such that $$\|L(t)z\|_{\H}\leq Ce^{-\omega t}.$$
After denoting $$\E_0(t)=\E_0(L(t)z)=\|L(t)z\|_{\H}^2= \|v(t)\|^2_2+\|\pt v(t)\|^2,$$ we set $\Phi_0(t)=\Phi_0(L(t)z,u(t))$, where $u(t)$ is the first component of $S(t)z$ and $$\Phi_0(L(t)z,u(t))=\E_0(L(t)z)- p \|v(t)\|_1^2+ (\alpha + \frac 12) \|v(t)\|^2
+\|u(t)\|^2_1\|v(t)\|^2_1+\l\pt v(t),v(t)\r.$$ In light of Lemma \[energia-limitata\] and inequalities (\[gamma\]), we have the bounds $$\label{stima-3} \frac12\E_0\leq \Phi_0\leq Q(R)\E_0.$$ Now, we compute the time-derivative of $\Phi_0$ along the solutions to system (\[DECAY\]) and we obtain $$\frac{d}{dt}\Phi_0+\Phi_0=2\l\pt
u,A^{1/2}u\r\|v\|^2_1\leq Q(R)\|\pt u\|\Phi_0.$$ The exponential decay of $\Phi_0$ is entailed by exploiting the following Lemma \[lemmaINT\] and then applying Lemma 6.2 of [@GPV]. From the desired decay of $\E_0$ follows.
\[lemmaINT\] For any $\eps > 0$ $$\int\limits_{\tau }^{t}\left\|
\partial_t u\left( s\right) \right\| ds\leq \eps (t-\tau
)+\frac \eps 4+ \frac{Q(R)}{\eps }\,,$$ for every $t\geq\tau\geq0.$
After integrating over $(\tau, t)$ and taking into account, we obtain $$\frac 12 \E(S(t)z) -2 \frac{\|f\|^2}{\lambda_1} \leq \L(S(t)z) + 2 \int\limits_{\tau }^{t}\left\|
\partial_t u\left( s\right) \right\|^2 ds = \L(S(\tau)z) \leq \L(z).$$ It follows $$\int\limits_{\tau }^{t}\left\|
\partial_t u\left( s\right) \right\|^2 ds \leq Q(R),$$ which, tanks to the Hölder inequality, yields $$\int\limits_{\tau }^{t}\left\| \partial_t u\left( s\right)
\right\| ds\leq
\eps\sqrt{t-\tau } + \frac{Q(R)}{\eps} \leq
\eps (t-\tau
)+\frac \eps 4+ \frac{Q(R)}{\eps },$$ for any $\eps>0$.
The next result provides the boundedness of $K(t)z$ in a more regular space.
[[(see [@GPV], Lemma 6.3)]{}]{} \[lemmaCPT\] The estimate $$\|K(t)z\|_{\H^2}\leq Q(R)$$ holds for every $t\geq 0$.
As well as in [@GPV], we use here the interpolation inequality $$\|w\|_3^2\leq\|w\|_2\|w\|_4.$$ Jointly with $\|w\|_2\leq Q(R)$ (which follows by comparison from (\[energy\]) and Lemma \[lemmaDECAY\]), this entails $$\label{interp}
p\|w\|_3^2\leq \frac12\E_1+Q(R),$$ where $$\E_1(t)=\E_1(K(t)z)=\|K(t)z\|_{\H^2}^2=\|w(t)\|^2_4+\|\pt w(t)\|_2^2.$$ Letting $$\Phi_1=\E_1
+(\|u\|^2_1-p)\|w\|^2_3+\l\pt w,Aw\r-2\l f,A
w\r + 2 k^2 \l u^+,A w\r ,$$ we have the bounds $$\label{Phi1_bounds}
\frac13\E_1-Q(R)\leq \Phi_1\leq Q(R)\E_1+Q(R).$$ Taking the time-derivative of $\Phi_1$, we find $$\begin{aligned}
&\frac{d}{dt}\Phi_1+\Phi_1=
2\l\pt u,A^{1/2}u\r\|w\|^2_3+ 2\alpha \l A^{1/2}v,A^{1/2}\pt w\r +\\
&\quad+\Big[\alpha \l A^{1/2}v,A^{1/2}w\r-\l f,Aw\r\Big] - k^{2} \left\langle Aw,u^{+}\right\rangle
+2k^{2}\left\langle Aw,\partial _{t}u^{+}\right\rangle \ .\end{aligned}$$ Using (\[energy\]) and (\[interp\]), we control the rhs by $$\frac18\E_1+Q(R)\sqrt{\E_1}\,+Q(R)\leq\frac14\E_1 +Q(R) \leq
\frac{3}{4}\Phi_1 +Q(R) ,$$ and we obtain $$\frac{d}{dt}\Phi_1+\frac{1}{4}\Phi_1 \leq Q(R).$$ Since $\Phi_1(0)=0$, the standard Gronwall lemma yields the boundedness of $\Phi_1$. Then, by virtue of , we obtain the desired estimate for $\E_1$.
By collecting previous results, Lemma 4.3 in [@CP] can be applied to obtain the existence of the attractor $\A$ and its regularity. Within our hypotheses and by virtue of the decomposition -, it is also possible to prove the existence of regular exponential attractors for $S(t)$ with finite fractal dimension in $\H$. This can be done by a procedure very close to that followed in [@GPV]. Since the global attractor is the [*minimal*]{} closed attracting set, we can conclude that the fractal dimension of $\A$ in $\H$ is finite as well.
[Acknowledgments]{}
The authors are indebted to the anonymous referees for their valuable remarks and comments.
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|
---
author:
- |
$^{,a,b}$, Ichiro Adachi$^c$, Hideyuki Kawai$^b$, Shohei Nishida$^c$ and Takayuki Sumiyoshi$^d$\
Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agency (JAXA), Sagamihara, Japan\
Department of Physics, Chiba University, Chiba, Japan\
Institute of Particle and Nuclear Studies (IPNS), High Energy Accelerator Research Organization (KEK), Tsukuba, Japan\
Department of Physics, Tokyo Metropolitan University, Hachioji, Japan\
E-mail:
title: |
Recent progress in the development of large area\
silica aerogel for use as RICH radiator in the\
Belle II experiment
---
Introduction
============
A proximity-focusing aerogel-based ring-imaging Cherenkov (A-RICH) counter is installed in the forward end cap of the Belle II detector [@cite1], which is currently being upgraded at the High Energy Accelerator Research Organization (KEK), Japan. The A-RICH counter uses silica aerogel Cherenkov radiators with a refractive index ($n$) of approximately 1.05 to identify charged pions and kaons and has a separation capability greater than 4$\sigma $ at momenta up to 4 GeV/$c$. When filling the large (3.5 m$^2$) end-cap region with aerogel radiators, it is necessary to minimize the number of aerogel tiles (i.e. maximize the aerogel dimensions) to reduce tile boundaries, because the number of detected photoelectrons decreases at the boundaries. To achieve this goal, we developed a method to produce large-area transparent aerogel tiles with no cracks.
For our purposes, aerogels may be produced in one of the two ways, i.e. by the modernized conventional method [@cite2] or by the pin-drying [@cite3] method. With the pin-drying method, more transparent aerogels can be produced than with the conventional method. However, in view of the result of prototypes, we decided to use the conventional method to mass produce large-area aerogel tiles for the detector [@cite4], because this method proved to be more cost-effective and provided a high yield of large-area crack-free aerogel tiles.
We next developed an aerogel-radiator tiling scheme [@cite4] for the end cap of the Belle II detector. The cylindrical end cap will be filled with 124 segmented dual-layer-focusing aerogel combinations (248 tiles in all) based on a multilayer-focusing radiator scheme [@cite5]. This tiling scheme calls for 18 $\times $ 18 $\times $ 2 cm$^3$ large-area aerogel tiles with $n$ = 1.045 and 1.055 for the upstream and downstream layers, respectively. The aerogel tiles with different refractive indices were manufactured separately and will be stacked during the installation stage. With a water jet cutter, the aerogel tiles will be trimmed in a fan shape to fit the cylindrical end-cap geometry (concentric layers 1 to 4, counting from the center of the end cap), making the best use of the hydrophobic features. In an electron-beam test performed at the Deutsches Elektronen-Synchrotron (DESY), we verified that the $K$–$\pi$ separation capability of a prototype A-RICH counter exceeded 4$\sigma $ at 4 GeV/$c$, where we used a prototype focusing combination aerogel tile trimmed with the water jet cutter.
Status of mass production and optical characterization
======================================================
From September, 2013 to May, 2014, the Japan Fine Ceramics Center collaborating with Mohri Oil Mill Co., Ltd mass produced the aerogels for the actual A-RICH counter. Since November 2013, a total of 449 tiles have been delivered to KEK. Figure \[fig:fig1\] shows the first aerogel sample that was delivered. In parallel, we characterized the aerogels by visually checking them and measuring their refractive index, transmission length and density using the methods described in Ref. [@cite2]. At the end of May 2014, 239 tiles have been optically characterized.
![First aerogel sample delivered. The refractive index and transmission length were 1.0444 and 56 mm, respectively. The tile measured 18 $\times $ 18 $\times $ 2 cm$^3$.[]{data-label="fig:fig1"}](tipp2014_proc_arich_fig1.eps){width="53.00000%"}
To date, we have confirmed 95 (70) tiles with $n$ = 1.045 ($n$ = 1.055) as good samples (i.e. candidates for installation into the actual detector). The crack-free yield after supercritical drying was 91% (outperforming our target of 80%), and 69% of the 239 tiles were undamaged and fulfilled our transparency requirements. After characterizing all the aerogel tiles delivered, we should have over 300 good ones. Figure \[fig:fig2\] shows the results of the optical measurements. The modernized conventional method led to aerogel tiles with well-controlled refractive indices and optimal transmission lengths. Note that once aerogels are trimmed by the water jet, we cannot remeasure their refractive index because the water jet machined surface strongly scatters the laser beam used in the measurement. Finally, we determined that the weight (density) of the tiles was useful to distinguish the upstream and downstream aerogels, in addition to management by an identification number.
![Transmission length at 400 nm as a function of refractive index. The refractive index was measured with a 405-nm semiconductor laser.[]{data-label="fig:fig2"}](tipp2014_proc_arich_fig2.eps){width="51.00000%"}
Mock-up installation test
=========================
In March, 2014, we tested the water jet method of trimming the mass-produced aerogel tiles. Tatsumi Kakou Co., Ltd. machined a total of eight tiles (four tiles for each refractive index) with a water jet cutter. The aerogels were trimmed to fit the shape of the second and third concentric layers. Dimensioning errors were less than 0.5% compared with the design (true size was smaller than design size in most cases).
In April, 2014, we tested the installation of the aerogel tiles. A partial mock-up of the aerogel support structure (made of 0.1- to 0.5-mm-thick aluminium sheets) was prepared to hold the aerogel tiles on edge in the end cap (Figure \[fig:fig3\]). The mock-up consisted of five boxes shaped to match the second and third concentric layers in which the aerogel tiles will be installed. One by one, we carefully placed the water-jet-trimmed aerogel tiles in the boxes. The installation was successful because there was sufficient margin between the aerogel tiles and the boxes.
![Partial mock-up of aerogel support structure. Water-jet-trimmed two-layer aerogel combinations were installed in the bottom left box (second concentric layer) and in the upper center box (third concentric layer).[]{data-label="fig:fig3"}](tipp2014_proc_arich_fig3.eps){width="53.00000%"}
Conclusion
==========
We developed large-area, transparent aerogel tiles for use as radiators in the A-RICH counter of the Belle II experiment. For the actual detector, the aerogel tiles were mass produced and are now being optically characterized. In addition, we prepared a partial mock-up of the aerogel support structure, in which we installed several aerogel tiles.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the members of the Belle II A-RICH group for their assistance. We are also grateful to the Japan Fine Ceramics Center, Mohri Oil Mill Co., Ltd. and Tatsumi Kakou Co., Ltd. for their contributions to mass producing the aerogel tiles and water jet machining. This study was partially supported by a Grant-in-Aid for Scientific Research (A) (No. 24244035) from the Japan Society for the Promotion of Science (JSPS). M. Tabata was supported in part by the Space Plasma Laboratory at ISAS, JAXA.
[99]{} T. Abe *et al.*, *Belle II technical design report*, *KEK Rep.* 2010-1, 2010 \[[1011.0352]{}\].
M. Tabata *et al.*, *Hydrophobic silica aerogel production at KEK*, *NIMA* [**668**]{} (2012) 64 \[[1112.3121]{}\].
M. Tabata *et al.*, *Recent progress in silica aerogel Cherenkov radiator*, *Phys. Proc.* [**37**]{} (2012) 642 \[[1203.4060]{}\].
M. Tabata *et al.*, *Silica aerogel radiator for use in the A-RICH system utilized in the Belle II experiment*, *NIMA*, (2014) in press \[[1406.4564]{}\].
T. Iijima *et al.*, *A novel type of proximity focusing RICH counter with multiple refractive index aerogel radiator*, *NIMA* [**548**]{} (2005) 383 \[[physics/0504220]{}\].
|
---
abstract: 'The dynamics of the current sheet of a Plasma Focus device is simulated with a two-dimensional model, in the radial expansion and the axial acceleration phase. The simulation considers the free expansion of the current sheet in hydrogen gas, without cathodes, and the comparison of experimental data with cathodes and without cathodes, using the parameters of Sumaj Lauray Plasma Focus 720 J.'
author:
- Esthefano Morales Campaña
- Héctor Silva Zúñiga
title: 'Two-dimensional modeling of a free expansion Plasma Focus, applied to the Sumaj Lauray 720-J device with and without external electrodes'
---
Introduction
============
The plasma focus (PF) is a type of plasma generation system originally developed as a fusion power device from the early 1960s. The original concept was developed by NV Filippov [@filippov], who noticed the effect while working in the first pinch machines in the USSR and by Mather [@mather] in the US. The basic design derives from the z-pinch concept, both the PF and the pinch use large electrical currents that pass through a gas to make it ionize in a plasma and then pinch themselves to increase the density and temperature of the plasma. Most devices use two concentric cylinders and form the pinch at the end of the central cylinder and as PF can also be found only with the internal cylinder as in the work of J. González [@Gonzalez]. The PF device is a known source of high temperature transient dense plasmas of n $\sim$ $10^{19}$ cm$^{-3}$ and T $\sim 1$keV which in 1980 conducted an extensive study of other phenomena generated by the PF, such as ion emissions and electrons, the emission of hard and soft x-rays, using gases such as hydrogen, argon, nitrogen, among other gases, making it useful for several applications in many different fields, such as lithography, radiography, images, activation analysis, radioisotope production , etc. Being a source of dense hot plasma, intense energy rays, etc., the PF device also shows tremendous potential for applications in plasma nanoscience and plasma nanotechnology [@Rawat] [@soto]. A PF device is a type of pinch discharge, where a high voltage pulse is applied between the coaxial cylindrical electrodes at low pressure. This generates a current sheet (CS) accelerated by Lorentz’s force, beginning to rise between the electrodes. When the plasma reaches the upper end of the internal electrode (anode), the movement continues towards the center focusing the plasma in a small region forming a column of high density and temperature (pinch phase). At this point the plasma column collapses releasing beams of ions, electrons, hard and soft x-rays, or neutrons when Deuterium or Tritium is used as the discharge gas. These devices use a switch called spark-gap, which consists of two conductive electrodes separated by a space normally filled with gas that allows the circuit to close by means of a current arc activated by a high voltage pulse generated by an auxiliary circuit called trigger which discharges the energy from the capacitor bank to the vacuum chamber [@Silva]. The characterization of the discharge can be done with photo scintillator multicolors, which through the photoelectric effect and a process of multiplication by secondary emission is able to capture the x-rays and neutrons emitted by the plasma, another type of detector is the collector probes of charge like the faraday cups, which are capable of detecting the charged particles generated in the plasma and a mass spectrometer can also be used. The mass spectrometer is a device that is used to separate by means of electric and / or magnetic fields, ions that have a different charge-to-mass ratio within a sample in order to identify them. The description of the movement of the plasma comes from the decade of the 70s [@Potter] [@Maxon], one of the most used models to describe the dynamics and compression of the plasma in a discharge of the PF device is the snow-plow one-dimensional model of Sing Lee [@Lee1] [@Lee2]. This type of model does not describe the temporal part of each process as the breakdown, lift-off and axial phase. However, two-dimensional models can be found that do not imply greater additional complexity, which incorporate the length of the insulator and allow obtaining information more adjusted to the experimental reality in small plasma focus devices. To study the times involved and the dynamics of CS in the early phases of the PF discharge, we have developed a two-dimensional model similar to that developed by González. We have implemented it in FORTRAN code, to simulate the operating conditions of the SUMAJ LAURAY PF device (E = 720 J), available at the University of Antofagasta.
Model description and experimental set-up {#section2}
=========================================
This model considers a plasma cylinder with cylindrical geometry that expands freely, without an external electrode (cathodes). We assume that the CS has an infinitely thin thickness and infinite conductivity, the cylinder is composed of the lateral cylindrical surface and an upper ring disc.
In the phase of radial expansion and radial-axial acceleration, the expansion of the CS is proposed, which is attached to the insulator (after the breakdown phase) and is driven by the Lorentz force $(\textbf{J}x\textbf{B})$ causing the radial expansion of the cylindrical surface and axial movement outward of the annular disc. In the radial direction, it sweeps the gas with an efficiency $fm_{r}$ and in the axial direction with an efficiency $fm_{z}$. The terms $fm_{r}$ and $fm_{z}$ are the mass fraction swept in the radial and axial direction due to the displacement of the cylindrical surface and the annular disk respectively.
Figure \[fig1\] shows a representative general scheme of the Mather type plasma focus with the outer and inner electrodes. The capacitor bank is represented by the capacity Co, the inductance of the system by Lo, the spark-gap by S-G. The internal electrode is the anode and the external electrodes are the cathodes, separated by the cylindrical insulator.
Figure \[fig2\] shows the general equivalent circuit of the plasma focus device. The inductance formed by the plasma current sheet and the electrodes is represented as a time-dependent inductance Lp (t). We assume that only a fraction of the total current (I) flows through the CS ($f_{c} \cdot I$), due to the leakage of current in another region of the discharge. Then, the current factor $f_{c}$ is the term that determines the fraction of current flowing through the CS. The processes that generate the leakage current, (1-$f_{c}$)$\cdot$I, are represented by the resistance RL.
Model equations
===============
The movement equations are obtained for the axial direction and for the radial direction of the plasma cylinder, two acceleration equations of the CS are obtained, in addition two equations are obtained for the variation of the axial ($M_{z}$) and radial mass ($M_{r}$) with respect to time. The equation describing the plasma focus circuit is taken as an LC circuit with an additional voltage that delivers the effect of the simulation of the Spark-Gap plus the electrical breakdown of the plasma. Using the magnetic vacuum permeability ($\mu_{0}$), the density of the gas ($\rho$) and the radius of the anode (a), we can write the following equations: $$\label{eqn:1}
\normalsize{\frac{d}{dt}[(L_{0}+f_{c} L_{p}(t))I(t)]=(V_{0}-V_{sg}(t))-\int_{0}^{t} \frac{I(t')}{C_{0}}dt'}$$ $$\label{eqn:2}
\normalsize{\frac{d}{dt}{\left\lbrace{M_{z}(t)\frac{dz}{dt}}\right\rbrace}=\frac{\mu_{0}}{4\pi}ln(\frac{r(t)}{a})(f_{c} I(t))^{2}}$$ $$\label{eqn:3}
\normalsize{\frac{d}{dt}{\left\lbrace{M_{r}(t)\frac{dr}{dt}}\right\rbrace}=\frac{\mu_{0}}{4\pi}\frac{z(t)}{r(t)}(f_{c}I(t))^{2}}$$ $$\label{eqn:4}
\normalsize{\frac{dM_{z}}{dt}=\pi\rho(r(t)^{2}-a^{2})f_{mz}\frac{dz}{dt}}$$ $$\label{eqn:5}
\normalsize{\frac{dM_{r}}{dt}=2\pi\rho r(t)z(t)f_{mr}\frac{dr}{dt}}$$ where, $$\label{eqn:6}
\normalsize{L_{p}(t)=\frac{\mu_{0}}{2\pi}ln(\frac{r(t)}{a})z(t)}$$ and, the Spark-Gap voltage is obtained of Bruzzone model [@Bruzzone]. The initial conditions assume a thin plasma sleeve on the insulator, that begin in rest. With the volume of this plasma sleeve and the gas density, we obtain the initial mass of $M_{r}$ and $M_{z}$. The initial current and its derivative are zero. The initial conditions of the variables for higher derivatives can be obtained from the proposed equations.
Sumaj Lauray Plasma Focus {#section3}
=========================
The experimental data are obtained from the plasma focus device of the University of Antofagasta, called Sumaj Lauray (405-1125 \[J\]). With a Rogowski coil, the arrival time of the CS at the end of the anode can be measured. To measure the arrival of the CS, at different axial positions, anodes of different lengths were used. Hydrogen was used as a filling gas in the PF discharge, with a pressure range of 1 to 5 \[Torr\]. The statistic of ten measurements, for each anode length and gas pressure, was considered in the analysis of experimental data. To simplify the simulation boundary conditions, the PF was operated without the cathode bars and also experimental data were taken with the cathode bars. The main features and working conditions of this PF device are presented in the following table:
Symbol Parameter Value
--------- ------------------------------- ----------------
$E$ Plasma focus energy $720$ J
$I_{0}$ Maximum current $109,1$ kA
$C_{0}$ Capacitor bank capacity $3.6$ $\mu F$
$L_{0}$ Initial inductance $121$ nH
$L_{c}$ Base Inductance of the canyon $12$ nH
$a$ Anode radius $6$ mm
$b$ Cathode radio $20$ mm
$z_{a}$ Insulator length $26$ mm
$r_{a}$ Insulation radius $9$ mm
$p$ Filling pressure $1$ a $5$ Torr
: Experiment and model data table.
\[tabla:4.1\]
Results and discussion {#paf}
======================
The model presented consists of the following phases; the Spark-Gap, breakdown and lift-off phase, radial expansion and the axial phase, developing the simulation in fortran language. The results obtained are governed by the studied geometry, a straight cylindrical geometry of the plasma is established which is an idealization of the plasma movement. The initial occupied voltage is 20 kV, the voltage associated with the potential drop differs from the pressure of the filling gas, when there is more pressure in the vacuum chamber the electrical breakdown can begin before 20 kV, for the reason of that the increase in pressure makes the electrical breakdown more unstable and does not reach 20 kV. One of the experimental errors that can be treated are the measurements at a certain voltage, this instrumental (systematic) error has a variation of $\pm$ 0.5 kV, which gives us a certain margin of error in the statistics made with the 10 shots. This margin of error is identified at the position of the points in the axial position graph with respect to time. 10 shots are made for different lengths of the cannon, $dI/dt$ curves are obtained which clearly identifies the pinch, where the time of arrival at the end of the anode is obtained. The same procedure is performed for lengths of 40, 60, 70, 80, 90, 101 and 120 mm for pressures of 1–5 Torr. The Spark-Gap voltage modeling allows the temporal adjustment of the $dI/dt$ simulation to the signal measured by the Rogowski coil in the PF device. We can estimate the current factor ($f_{c}$) from the plasma voltage $(V_{p}=d(f_{c}L_{p}(t)\cdot I(t))/dt)$, when the CS has reached the mouth of the anode. These estimates give us values of: $f_{c}=0.77$ to $0.79$ for pressures of 1 to 5 Torr. Therefore, the mass drag factors, which allowed adjusting the experimental data of Z v/s t, were: $fm_{r}$= 0.82 to 0.79, and $fm_{z}$= 0.18 to 0.21 approximately, for the same pressure range [@Hawat].\
Figure \[fig3\] shows the comparison of the data of the axial position of the current sheet for pressures of 1 to 5 Torr of filling pressure, which had already been taken previously in the Plasma Focus Sumaj Lauray. The trend of experimental data with cathodes follows the curves of free expansion simulations (without cathodes), which indicates that the current sheet behaves in the same way, either a cathode model or without cathodes. This trend in the dynamics of the dynamics of CS in simulation without cathodes, compared with data with cathodes and without cathodes, clearly shows that during the process in which the plasma is generated and accelerated by Lorentz force through There is no variation of the canyon, therefore, what affects the production of the ion particle beam, hard or soft x-rays are the initial and final phases of the PF.
The radial position of the CS according to the proposed model, as shown in Figure \[fig4\], tells us that the CS expands radially to values greater than 35 mm in radius of the plasma focus device, which indicates that for the cathode model, the plasma that passes between the cathodes could expand to these limits or simply the radial movement can be restricted by the cathodes without affecting the speed it carries. The figure shows the radii reached until the fourth period when $dI/dt=0$ for the different pressures from 1 to 5 Torr.
The speed of the CS is important for the study of the production of ions, see figure \[fig5\], where it is shown that for 1 Torr the speed reached close to $12$ $cm/\mu s$ at the time of $\sim 850$ $ns$, this indicates that there are less amount of mass dragged in the axial position. In Figure \[fig6\], it is observed that the speeds in the radial position of the system is lower than the speeds of the axial position, this is affected by the amount of mass that is dragged during the CS process.
Figure \[fig7\] shows the plasma inductance generated during the CS processes. The initial plasma inductance is 2.1 nH, this initial inductance is independent of pressure, since it only depends on the cylinder geometry of the generated plasma. The curves of the temporal derivative of the current in Figure \[fig8\], are observed the simulation and experimental data for the 3 Torr pressure, which is adjusted by the current and mass drag factors.
Figure \[fig9\] shows the amount of axial mass in units of micrograms, for the pressure of 1 Torr as shown in the graph it reaches a value of 3 $\mu g$ and for the pressure of 5 Torr a value of 24 $\mu g$ is reached. In Figure \[fig10\], it is observed that in the radial position a greater amount of mass is obtained in the order of tens and hundreds of micrograms. This checks the values found in the radial and axial speeds in the figures shown above.
The current factor is obtained from the experimental data semi-empirically, see figure \[fig11\], using the average of the 10 shots taken for each length of the cannon at each pressure. The trend of the current factor gives us the necessary information to understand that during the ionization process of the Hydrogen gas for each pressure is different, while the pressure increases the current factor tends to unity and this clearly affects the factor of axial and radial mass drag. The data used for comparison between experimental data with cathodes (CC) and without cathodes (SC), with simulation data, only the pressure range of 1 to 5 Torr is used. The axial mass drag factor is obtained after obtaining the current factor, since the current factor is obtained experimentally and among the axial and radial mass drag factors the one that infers the most in the equations is the axial drag factor. This factor is obtained from the simulation settings, compared to the least squares adjustment of the experimental data, see figure \[fig12\].
Concluding remarks {#section7}
==================
We have implemented a two-dimensional model to simulate the dynamics of the current sheet of a Plasma Focus device. This model considers only the breakdown, liff-off and axial phases of the Plasma Focus discharge, and considers the start of the discharge in the insulator. The model considers the free expansion of CS in hydrogen gas, without cathodes. The simulation was implemented with a FORTRAN code, using the parameters of the Plasma Focus device of the University of Antofagasta, Sumaj Lauray 720 J as a reference. This model provides us with a powerful tool to facilitate the interpretation of experimental data and for the design of low energy PF devices. Future work considers improving simulations and extending the model to the dense plasma phase.
Acknowledgements
================
This work has been funded by FONDECYT grant 1130787 and Esthefano Morales Campaña thanks the financial support of the project ANT-1856 at the Universidad de Antofagasta, Chile.
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S. Lee, *J. Fusion Energ.* **33**, 235–241 (2014).
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|
---
author:
- '**© 2011 �. A. S. Saburova, D. V. Bizyaev , A. V. Zasov**'
title: 'Do Disk Galaxies with Abnormally Low Mass-to-Light Ratios Exist?'
---
We performed the photometric B, V and R observations of nine disk galaxies that were suspected in having abnormally low total mass-to-light (M/L) ratios for their observed color indices. We use our surface photometry data to analyze the possible reasons for the anomalous M/L. We infer that in most cases this is a result of errors in photometry or rotational velocity, however for some galaxies we cannot exclude the real peculiarities of the galactic stellar population. The comparison of the photometric and dynamical mass estimates in the disk shows that the low M/L values for a given color of disks are probably real for a few our galaxies: NGC 4826 (Sab), NGC 5347 (Sab), and NGC 6814 (Sb). The small number of such galaxies suggests that the stellar initial mass function is indeed universal, and that only a small fraction of galaxies may have a non-typical low-mass star depleted initial mass function. Such galaxies require more careful studies for understanding their star formation history.\
[**Key words:**]{} galaxies, galactic disks, surface photometry, stellar initial mass function.
[**PACS codes:**]{} 98.52.Nr, 98.52.Sw, 98.62.Ai, 98.62.Lv, 98.62.Qz, 98.62.Hr
------------------------------------------------------------------------
\
[$^*$ E-mail:$<$saburovaann@gmail.com$>$]{}\
[$^{**}$ E-mail: $<$dmbiz@apo.nmsu.edu$>$]{}\
[$^{***}$ E-mail: $<$a.v.zasov@gmail.com$>$]{}
INTRODUCTION
============
The mass-to-light ($M/L$) ratio is a very important parameter that is determined by galactic stellar population (the distribution of stars in age, mass, and, to a lesser extent, stellar metallicity), dust (through affecting the galaxy’s luminosity) and the relative mass of its nonstellar components. The latters include the dark matter, which is usually comparable in mass to the stellar component within the galaxy’s optical boundaries (see, e.g., Khoperskov et al. 2010; Zasov et al. 2011; Bizyaev and Mitronova 2009), as well as the gas whose mass fraction can be significant in late-type galaxies. The stellar population models computed for certain accepted initial mass function (IMF) obviously predict $M/L_B$ and $M/L_V$ to be of the order of 1-10 solar units in the visible bands, and $M/L_R$ or $M/L_K \sim
0.5-2$ solar units in the near infrared, in dependence of the relative number of young stars (and therefore the galaxy’s color) and the adopted IMF. The presence of the dark matter or the gas along with internal extinction can only increase the overal galactic $M/L$. Therefore, those galaxies whose total $M/L$ estimated within the optical boundaries looks lower than that provided by the stellar population modeling using the universal IMF (even ignoring any dark halo) are of especial interest. Such candidate galaxies were selected from the sample of objects with known rotational velocities by Saburova et al. (2009).
Since the reliable luminosity, color, and rotational velocity estimates are available not for all selected objects, the conclusion about the $M/L$ anomaly for each specific galaxy should be verified. An unusually low $M/L$ for a given color index, if confirmed, can point to the existence of stellar population with a non-standard IMF: such a galaxy must have a low relative number of stars with a mass less than the solar one because these stars determine the total mass of stellar population once have a little affect on the photometric features. This would imply the existence of special conditions for the formation of the the bulk of its stars. Note that the questions about the IMF universality and the possibility of a non-typical IMF shape are being actively discussed in the literature (see, e.g., Gilmore 2001; Kroupa 2002; Hoversten and Glazebrook 2008; Meureret al. 2009; Bastian et al. 2010; Dabringhausen et al. 2010).
In this paper we present the results of our photometry for nine galaxies selected from Saburova et al (2009) that were previously identified as presumably having low total $M/L$ ratios within the optical radius $R_{25}$ (the radius of the 25 $^m/\Box''$ isophote in $B$) and analyze some possible sources of errors in the observed quantities. We also compare the maximum rotation velocity of the disk component $v_{max~disk}$, expected for a ”normal” stellar population with the maximum observed rotation velocity $v_{max}$. As a reference, we employ the model values of the mass-to-light ratios for the stellar population with Salpeter IMF following the evolutionary models by Bell & de Jong (2001). The condition $v_{max~disk}
> v_{ max}$ suggests that the stellar population model with a regular IMFs is inapplicable and that the the real stellar disk is lighter than it is predicted by the stellar population models even if non-stellar components (dark matter and gas) are ignored. To verify the total luminosity estimate as well as the photometric parameters: color and inclination of the disk, we performed photometric observations of nine galaxies in which we suspect the abnormally low $M/L$.
OBSERVATIONS AND DATA REDUCTION
===============================
Photometric observations of the nine galaxies selected by Saburova et al. (2009) as galaxies with presumably low $M/L$ were conducted at the 0.5-m Apache Point Observatory telescope in the B,V, and R bands. We observed the galaxies during photometric moonless nights in June, July, and October of 2009. Table \[table1\] shows the observing log.[^1] The observing data were reduced in a standard way using MIDAS software package and were corrected for the bias and flat field. The telescope’s wide field of view ($40 '$) allowed the proper sky background subtraction using the starless areas in the field. The photometric standard stars from Landolt (1992, 2009) were observed during the same nights as the galaxies. The foreground stars were removed from the galactic images and replaced by the mean fluxes from the adjacent regions.
---------- --------------------- ------- -------- ------- ------
Galaxy Date of observation D, Mpc
(1) (2) (4)
B V R
NGC1569 15.10.09 3X500 3X300 3X300 2.2
NGC4016 14.06.09 500 200 300 49.2
NGC4016 15.06.09 2X500 2X300 2X300 49.2
NGC4214 15.06.09 2X500 2X300 2X300 7
NGC4826 15.06.09 2X500 2X300 2X300 7
NGC5347 15.06.09 500 300 300 32.5
NGC5347 21.06.09 500 300 300 32.5
NGC5921 15.06.09 500 300 300 25.2
NGC5921 21.06.09 2X500 2X300 2X300 25.2
NGC6814 21.06.09 2X500 2X300 2X300 20.7
NGC7743 21.06.09 500 300 300 24.7
UGC03685 15.10.09 3X500 3X300 3X300 26.8
---------- --------------------- ------- -------- ------- ------
: Log of observations. \[table1\]
RESULTS OF PHOTOMETRIC OBSERVATIONS
===================================
Table \[table2\] shows the derived total magnitudes and $(B-V)$ colors uncorrected for extinction. The total magnitudes were calculated from the fluxes in an aperture close to $R_{25}$ size. For the comparison, the table shows the apparent $B$ magnitudes and $(B-V)$ colors from Hyperleda database.[^2] As follows from Table \[table2\], our total magnitudes and those from Hyperleda agree well. NGC 4016 for which the NED database[^3] provides apparently erroneous value of $m_B$ corresponding to unrealistically low $(B-V)=0.03$ is an exception. We see a good agreement between the magnitudes and colors for our different nights (no more than $0.1^m$ and $0.04^m$ for the magnitudes and colors, respectively).
[Galaxy]{}
------------ ---------------- ------ ----------------- ------ ------
[this paper]{} Leda [this paper ]{} Leda
NGC1569 0.83 0.83 11.8 11.8 10.4
NGC4016 0.32 14.5 13.8 13.8
NGC4214 0.47 0.46 10.4 10.2 9.5
NGC4826 0.84 0.84 9.37 9.30 7.98
NGC5347 0.75 0.76 13.5 13.4 12.1
NGC5921 0.69 0.66 11.7 11.7 10.5
NGC6814 0.9 0.8 12.2 12.1 10.6
NGC7743 0.92 0.90 12.5 12.4 11
UGC03685 0.7 12.8 13.1 11.6
: Total magnitudes and colors. \[table2\]
Along from the total magnitudes we obtained the radial profiles of isophotal flattening ($b/a$) in the R-band images azimuthally averaged in elliptical rings (see Fig. 1). The radial surface brightness profiles in all three $BVR$-bands are plotted in Fig. 2. The brightness profiles were corrected for extinction in the Galaxy using Schlegel et al. (1998), but they were not corrected for the disk orientation and internal extinction[^4] (the profiles of NGC 1569 were not corrected for the extinction in the Galaxy due to the uncertainty in the correction, see below). The radial brightness and color profiles are shown in Figs. 2 and 3. The ranges of errors in Figs. 2 and 3 were calculated from the errors in the fluxes as in Vader & Chaboyer (1994) using the formula: $$\label{2}
\delta I = \sqrt{N_{tot}+(\delta n_{sky}A)^2},$$ where $N_{tot}$ is the flux from the galaxy in aperture $A$, and $\delta n_{sky}$ is the standard deviation from the mean sky background determined in small apertures near the galaxy.
To calculate the stellar disk luminosity, we decomposed the images of the galaxies into components using the BUDDA code developed by de Souza et al. (2004) (version 2.2 which allows for the bulge, disk, bar, and central source components). In our analysis we evaluate the following components: the exponential disk $\mu_d(r)=\mu_0+1.086r/r_d$, a bulge, and a bar with brightness distributed according to the Sersic law $\mu_b(r)=\mu_e+c_n((r/r_e)^{1/n}-1)$. The decomposition results for the R-band profiles are shown in Figs. 4 and 5, respectively. The structural parameters determined for different bands are given in Table. \[table3\]. It contains the following data:\
(1) galaxy name;\
(2) the radial scale length of disk in arcsec;\
(3) the central surface brightness of the disk in $^m/\Box''$;\
(4) the effective radius of the bulge in arcsec;\
(5) the effective surface brightness of the bulge in $^m/\Box''$;\
(6) the Sersic index for the bulge;\
(7) the effective radius of the bar in arcsec;\
(8) the effective surface brightness of the bar in $^m/\Box''$;\
(9) the Sersic index for the bar;\
(10) the photometric band.\
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The images were not successfully decomposed for two galaxies: NGC 1569 and NGC 4214, both are without noticeable bulges, because their disk brightness profiles can be very poorly described by an exponential law. We show no data on these galaxies in Table \[table3\]). Nevertheless their observed profiles were considered when we photometrically estimated the disk contribution to the total rotation curve.
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The quality of the image decomposition is clearly demonstrated in Fig. 6 for NGC 5437. The image of this galaxy before and after the subtraction of the model image generated by the BUDDA code is shown (the model contains disk, bulge and bar). Both images have the same contrast. Only a faint ring and regions of enhanced brightness at the bar ends are noticeable in the residual image.
The BUDDA code reliably estimates the structural parameters of disks (see Figs. 4- 6, Table \[table4\]), although, according to Gadotti (2008), the bulge-to-disk luminosity ratio can be systematically overestimated by 5% while the relative disk luminosity can be systematically underestimated. Below we also consider the total $M/L$ ratios independent of the photometric decomposition results, along with the mass-to-light ratios for the disks.
[Galaxy]{} $r_d$, [$''$]{} $\mu_0$, [$^m/\Box''$]{} $r_{e}$, [$''$]{} $\mu_{e}$, [$^m/\Box''$]{} $n_{bulge}$ $r_{e ~bar}$, [”]{} $\mu_{e ~bar}$, [$^m/\Box''$]{} $n_{bar}$ [Band]{}
------------- ------------------------- -------------------------- ------------------------- ---------------------------- ------------------------ ------------------------ --------------------------------- -------------------------- ----------
1 2 3 4 5 6 7 8 9 10
[NGC4016]{} [10.2 $\pm 0.27$]{} [20.6 $\pm 0.05$]{} B
[10.35 $\pm 0.27$]{} [20.3 $\pm 0.05$]{} V
[10.4 $\pm 0.25$]{} [20.0 $\pm 0.04$]{} R
[NGC4826]{} [59.8 $\pm 2.51$]{} [19.7 $\pm 0.08$]{} [11.0 $\pm 0.50$]{} [20.3 $\pm 0.1$]{} [4.33 $\pm 0.49$]{} B
[59.2 $\pm 1.91$]{} [18.8 $\pm 0.07$]{} [12.2 $\pm 0.55$]{} [19.5 $\pm 0.08$]{} [5.05 $\pm 0.48$]{} V
[58.2 $\pm 1.42$]{} [18.3 $\pm 0.05$]{} [16.7 $\pm 0.59$]{} [19.4 $\pm 0.07$]{} [6.3 $\pm 0.51$ ]{} R
[NGC5347]{} [25.4 $\pm 3.84$]{} [22.5 $\pm 0.2$]{} [5.74 $\pm 0.29$]{} [21.7 $\pm 0.07$]{} [2.61 $\pm 0.18$]{} [35.2 $\pm 5.56$]{} [24.3 $\pm 0.07$]{} [0.693 $\pm 0.47$ ]{} B
[25.0 $\pm 3.35$]{} [21.9 $\pm 0.20$]{} [7.51 $\pm 0.46$]{} [21.1 $\pm 0.11$]{} [4.45 $\pm 1.31$]{} [32.9 $\pm 4.01$]{} [23.3 $\pm 0.11$]{} [0.68 $\pm 0.33$]{} V
[26.0 $\pm 2.3$]{} [21.6 $\pm 0.15$]{} [9.61 $\pm 0.43$]{} [20.9 $\pm 0.07$]{} [3.15 $\pm 0.21$]{} [31.8 $\pm 2.47$]{} [22.7 $\pm 0.07$]{} [0.69 $\pm 0.18$]{} R
[NGC6814]{} [30.3 $\pm 2.87$]{} [21.5 $\pm 0.15$]{} [8.92 $\pm 0.89$]{} [22.3 $\pm 0.11$]{} [3.46 $\pm 0.19$]{} B
[29.6 $\pm 1.88$]{} [20.5 $\pm 0.10$]{} [12.5 $\pm 1.03$]{} [21.8 $\pm 0.1$]{} [3.52 $\pm 0.17$]{} V
[29.9 $\pm 1.21$]{} [20.0 $\pm 0.08$]{} [20.3 $\pm 1.10$]{} [21.7 $\pm 0.07$]{} [3.85 $\pm 0.11$]{} R
[NGC7743]{} [25.8 $\pm 4.05$]{} [21.2 $\pm 0.21$]{} [6.81 $\pm 0.56$]{} [21.2 $\pm 0.1$]{} [2.8 $\pm 0.09$]{} [5.56 $\pm 0.46$]{} [21.8 $\pm 0.09$]{} [0.69 $\pm 0.28$]{} B
[28.6 $\pm 3.12$]{} [20.7 $\pm 0.17$]{} [10.4 $\pm 0.95$]{} [20.9 $\pm 0.11$]{} [3.02 $\pm 0.16$]{} [3.54 $\pm 0.1$]{} [20.9 $\pm 0.11$]{} [0.75 $\pm 0.06$]{} V
[29.5 $\pm 2.46$]{} [20.3 $\pm 0.13$]{} [11.3 $\pm 0.77$]{} [20.5 $\pm 0.08$]{} [3.24 $\pm 0.12$]{} [9.99 $\pm 2.33$]{} [22.5 $\pm 0.08$]{} [0.737 $\pm 0.52$]{} R
[NGC5921]{} [43.6 $\pm 9.54$]{} [23.4 $\pm 0.29$]{} [11.1 $\pm 0.65$]{} [21.4 $\pm 0.12$]{} [3.65 $\pm 0.04$]{} [77.1 $\pm 6.71$]{} [23.5 $\pm 0.12$]{} [0.68 $\pm 0.2$]{} B
[55.6 $\pm 5.29$]{} [22.1 $\pm 0.1$]{} [14.7 $\pm 0.55$]{} [21.3 $\pm 0.06$]{} [3.28 $\pm 0.09$]{} [72.9 $\pm 6.08$]{} [23.4 $\pm 0.06$]{} [0.68 $\pm 0.19$]{} V
[60.7 $\pm 3.9$]{} [22.1 $\pm 0.12$]{} [18.54 $\pm 0.45$]{} [20.9 $\pm 0.03$]{} [3.93 $\pm 0.09$]{} [72.2 $\pm 2.91$]{} [22.7 $\pm 0.03$]{} [0.64 $\pm 0.12$]{} R
[UGC3685]{} [25.6 $\pm 1.91$]{} [21.5 $\pm 0.12$]{} [5.34 $\pm 0.69$]{} [23.4 $\pm 0.25$]{} [3.8 $\pm 2.45$]{} [3.34 $\pm 0.17$]{} [20.4 $\pm 0.25$]{} [0.96 $\pm 0.26$]{} B
[24.12 $\pm 1.3$]{} [20.8 $\pm 0.10$]{} [6.95 $\pm 0.4$]{} [21.8 $\pm 0.11$]{} [4.59 $\pm 0.99$]{} [4.14 $\pm 0.32$]{} [20.9 $\pm 0.11$]{} [1.69 $\pm 0.69$]{} V
[30.8 $\pm 1.44$]{} [20.80 $\pm 0.05$]{} [9.56 $\pm 0.38$]{} [21.68 $\pm 0.07$]{} [3.08 $\pm 0.29$]{} [8.62 $\pm 0.87$]{} [21.0 $\pm 0.07$]{} [0.84 $\pm 0.12$]{} R
: Structural parameters of the galaxies\[table3\]
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COMPARISON WITH OTHER PUBLICATIONS
==================================
We compare the surface brightness profiles with results by other authors. In Fig. 7, the profiles of all galaxies (except NGC 1569) were corrected for extinction in the Galaxy, but were not corrected for the disk inclination. Fig. 7 shows good agreement between our estimates and published results. An exception is NGC 1569, which exhibits a significant discrepancy at the periphery between our R-band brightness profile and those from Swaters & Balcells (2002) and Stil & Israel (2002). This discrepancy may be explained by diferent procedures we used for subtraction of numerous foreground stars for this galaxy. There is also a noticeable discrepancy at the periphery between our R-band profile of NGC 6814 and that from de Robertis et al. (1998). However, there is a good agreement with the profile from Sánchez-Portal et al. (2000) observed in the same band (see Fig. 7). In the remaining cases, the difference between our profiles and those of other authors does not exceed $0.3^m$. This is comparable to the typical errors in the surface brightness estimates far from the center.
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COMPARISON OF THE PHOTOMETRIC AND DYNAMICAL DISK MASS ESTIMATES
===============================================================
We choose two conditions that allow us to suspect the existence of a discrepancy between the photometric and dynamic mass estimates of the stellar population of a galaxy:\
(i) The photometric $M/L$ of stellar population of a galaxy within some radius (taken within four radial scale lengths $r_d$) obtained from the total color index, exceeds the dynamical estimate of $M/L$ determined from the relation $(M/L)_{dyn}=4v_{max}^2r_d/(GL_{tot})$, where $L_{tot}$ is the total luminosity (including bulge) and $v_{max}$ is the maximum rotational velocity of the galaxy.\
(ii) The maximum rotational velocity of the disk component $v_{max~disk}$ estimated from the photometric disk mass and scale length exceeds the observed maximum rotational velocity of the galaxy $v_{max}$. The value of $v_{max~disk}$ can be determined from the radial surface density profile $\sigma(R)$ obtained from the brightness distribution and the model $M/L$ ratio estimated from the measured color index. The rotation curve in spiral galaxies is usually a constant beyond $r=(1.5-2)r_d$. Hence we assume that $v_{max}$ is close to the rotation velocity of galaxy at $r\approx 2r_d$, where the disk component of rotation curve approaches $v_{max~disk}$.
For all galaxies except NGC 4214 and NGC 1569 the disk $M/L$ ratio was assumed to be constant along the radius, while the radial change of $M/L$ in the two mentioned objects was determined from the color profiles. For an exponential disk density distribution adopted for all galaxies except NGC 4214 and NGC 1569, $v_{max~disk}\approx 0.623(GM_d/r_d)^{0.5}$, where $M_d=2\pi
G\sigma_0r_d^2$ is the disk mass and $\sigma_0$ is the central surface density of the disk.\
Galaxies with abnormally low $M/L$, which is at least a factor of 3 lower than the expected ratio for a given $(B-V)_0$ color, are rare - they account for about 3% of the number of galaxies of S0 and later types which have the measured rotational velocities (from the H[I]{} line width) and $(B-V)_0$ colors (see Fig. 1 in Saburova et al. 2009). In fact, their number may be considerably smaller. There are two factors which may be responsible for the discrepancy between the dynamical and photometric disk mass estimates:\
(i) errors in the estimates of observed quantities, such as the circular velocity, the inclination, the distance to the galaxy, the luminosity and color;\
(ii) inconsistency of the galaxy’s stellar population with the evolutionary population synthesis model used for the $M/L$ estimate: the ignorance of starburst, low metallicity Z of the stellar population, and/or an anomalous stellar IMF. Below we consider each of these factors in our galaxies.\
Note that all galaxies of our sample except NGC 1569 and NGC 4016 follow the sequence of $(B-V)-(V-R)$ colors for normal galaxies derived by Buta & Williams (1995). This suggests the absence of significant peculiarities in the star formation history and confirms the applicability of population synthesis models (Bell and de Jong 2001). The star formation in NGC 1569 is very active, then we apply the $M/L$ — color relation including a starburst. The second galaxy, NGC 4016, was included in the sample due to the unrealistically low $(B-V)_0$ given by NED, but it is not confirmed in our work, so we conclude that this is not a galaxy with an abnormally low $M/L$ (see below).\
[**NGC1569**]{} is a Magellanic-type irregular dwarf starburst galaxy. Active star formation in this object is probably associated with tidal interaction (Grocholski et al. 2008). According to Recchi et al. (2006), the chemical abundance in this galaxy is consistent with the assumption that there were long-lasting episodes of quiet star formation completed by series of starbursts. Since NGC 1569 lies near the Galactic plane, the extinction correction is very uncertain for it: the correction coefficients in Schlegel et al. (1998) and Burstein and Heiles (1982) for the $B$ band differ by $1^m$. This galaxy is a probable cause of the low mass-to-light ratio estimate. Therefore, we can adopt the photometric $Ks$-band profile from Vaduvescu et al. (2006), along with the $B, V, R$ magnitudes, in order to estimate the disk mass from the photometry and its contribution to the rotation curve. Using the $Ks$ band also allows us to reduce the effects of intense star formation on the model estimate of the mass-to-light ratio.\
The ionized gas velocity field of NGC 1569 obtained within the framework of the GHASP project (Epinat et al. 2008) covers the central parts of the galaxy and shows no evidence of rotation. However, the rotation curve determined by Stil and Israel (2002) in the H[I]{} line allows the maximum rotational velocity $v_{max}$ to be estimated. We compared $v_{max}$ with the maximum rotational velocity of the disk component of the rotation curve $v_{max~disk}$ by estimating the disk mass from photometric data. The disk density profile needed for this purpose was calculated from the $Ks$ and $R$ brightness profiles as well as from the $(V -R)$ color distribution and the total $ (J-K)$ color obtained by Vaduvescu et al. (2006). We determined the expected $M/L_R$ and $M/L_K$ ratios for the disk (see Table \[table4\]) from the color indices following the models of Bell and de Jong (2001) calculated for a modified Salpeter IMF with the inclusion of a starburst.[^5] Taking into account the radial brightness profiles in these photometric bands, we obtained the radial disk surface density profile and the corresponding values of $v_{max~disk}$ (see Table \[table4\]). It follows that $v_{max~disk}$ determined from $Ks$ photometry turns out to be close to the observed maximum rotational velocity. Hence we may conclude that this object does not have an abnormally low $M/L$. Note that the mass of its dark halo within the optical boundaries should be small compared to the disk mass, as suggested by the closeness of the $v_{max~disk}$ and $v_{max}$ estimates.\
[**NGC4016**]{} is an SBd-type galaxy. We included it in our sample due to its unrealistically low color index $(B-V)=0.03$ in the NED, which we do not confirm (see Table \[table2\]). The rotational velocity $v_{max~disk}$ determined from the photometric density estimate using our color and luminosity measurements turns out to be lower than the total rotational velocity $v_{max}$ (see Table \[table4\]). This gives no grounds for attributing it to objects with anomalous $M/L$.\
[**NGC4214**]{}, just as NGC 1569, is a Magellanic type irregular galaxy with a bar. The main problem related to the mass estimation for this galaxy is an unreliably determined inclination (see Fig. 2). The axial ratio of NGC 4214 changes along the radius from 0.7 to 0.94, which corresponds to the inclinations $i_1=45^o$ and $i_2=20^o$. The rotational velocity $v_{max~disk}$ found from the R-band brightness profile and the $(B - R)$ and $(B - V )$ colors for a modified Salpeter IMF, including a starburst similar to that adopted for NGC 1569, is compared in Table \[table4\] with the observed rotational velocity $v_{max}$ of the galaxy for both inclination angles. As follows from our estimates, $v_{max}$ may be either higher or lower than $v_{max~disk}$ if to choose $i_2=20^o$ or $i_1=45^o$, respectively. Obviously, the observed flattening of the galaxy’s inner part is distorted by the presence of a bar, so the peripheral values of $b/a$ are preferable for the inclination estimate. However, the contradiction between the photometric and dynamical models is retained even at $i_2=20^o$: in this case, in spite of $v_{max}>v_{max~disk}$, the radius at which the disk component of the rotation curve, obtained from photometry data, has a maximum ($R \approx 2$ kpc) turns out to be a factor of 3 smaller than the radius where the observed rotation curve derived by Allsopp (1979) reaches its maximum. Thus, the disk with the photometrically calculated density distribution is inconsistent with the observed rotation curve. This conclusion remains valid even if we take into account the low linear resolution of the H[I]{} data in Allsopp (1979) ($\Delta R \approx 2$ kpc). However, there remains the possibility that the discrepancy between $v_{max~disk}$ and the rotation velocity in the central part of the galaxy can be associated with noncircular motions of gas due to the presence of a bar. Therefore, although an anomalous stellar composition in NGC 4214 remains quite possible, the available data do not allow a reliable $M/L$ estimate to be obtained for this galaxy. A more detailed study of its velocity field is required.\
[**NGC4826**]{} is an Sab-type galaxy with a thick dust lane northeast of the nucleus. The rotation of the outer gas disk in NGC 4826 is opposite to that of the inner one (Braun et al. 1994). A detailed study of the $H_{\alpha}$ kinematics shows that the transition between the two disks occurs near the dust lane at a distance $50''<r<70''$ from the center (Rubin 1994). The velocity $v_{max~disk}$ calculated from the R-band photometric parameters of the disk (see Table \[table3\]) and the $(B-V)$ and $(B-R)$ colors beyond the dust lane turns out to be higher than the maximum rotational velocity $v_{max}$ (see Table \[table4\]). This conclusion remains valid even if we use the model by Bell and de Jong (2001) with lower metallicity ($Z =
0.008$). Thus, our photometric data agree with the abnormally low $M/L$ ratio in this galaxy.\
[**NGC 5347**]{} is an Sab-type galaxy with a bar. The $PV-$ diagram constructed from optical observations has a large scatter of points (Marquez et al. 2004) and, therefore, does not allow $v_{max}$ to be determined reliably. However, the galaxy’s rotational velocity estimated from the H[I]{} line width ($W_{20}$ from the RC3 catalog by de Vaucouleurs et al. 1991) turns out to be lower than that for the disk component of the rotation curve determined from the radial $R-$band surface brightness profile (after the subtraction of the bulge contribution) using the color indices of the galaxy’s outer regions (see Table \[table4\]). Thus, an abnormally low mass-to-light ratio for the disk can actually take place in this case. This object requires a careful study and, first of all, it is necessary to obtain a more extended rotation curve.\
[**NGC 5921**]{} is an Sbc-type galaxy with a noticeable bar surrounded by a ring. As for NGC 4214, the inclination estimates for this galaxy are contradictory. According to Hernandez et al. (2005), the isophotes ellipticity leads to $i=36.5^o$. Our photometric data give $i=43^o$ (see Fig. 2), which agrees with the inclination determined kinematically from $H_{\alpha}$ observations (Hernandez et al. 2005). In Table \[table4\], the rotational velocity $v_{max~disk}$ determined photometrically in the same way as for NGC 5347 (see above) is compared with the two values of rotation velocity $v_{max}$ found from $H_{\alpha}$ measurements for $i=43^o$ and $i=36.5^o$ for radial distance corresponding to $v_{max~disk}$. As we see from Table \[table4\], for $i=36.5^o$ $v_{max~disk}$ is lower than $v_{max}$, while the opposite conclusion is true for $i=43^o$. Since our inclination estimates agree with the latter value, an abnormally low mass-to-light ratio for the disk of NGC 5921 remains quite possible. However, an underestimation of the circular rotational velocity caused by the bar cannot be ruled out. This is suggested by the fact that the rotational velocity estimated from the H[I]{} line width (from the RC3 catalog) for $i=43^o$ is $v_{HI}=136$ $km s^{-1}$, which exceeds $v_{max~disk}$ (see Table \[table4\]). Thus, the abnormally low $M/L$ for NGC 5921 is most likely related to noncircular motions and not to stellar composition anomalies.\
[**NGC 6814**]{} is an SABb-type galaxy with a ring. The inclination of NGC 6814 is low (the photometric axial ratio gives $i=17^o$). However, $i=8^o$ is required for $v_{max}$ determined from the rotation curve in the H[I]{} line to become approximately equal to $v_{max~disk}$. Such a low inclination does not correspond to the photometric estimates (see Fig. 2). The presence of a bar hardly may change the result, because its presence is evident only in the central region ($R < 1$ kpc), while the disk component of the rotation curve has a maximum at $R \approx 3$ kpc. Therefore, the conclusion about an abnormally low $M/L$ of this galaxy remains preferable.\
[**NGC 7743**]{} is an S0-a galaxy with a bar. As in the previous two cases, the abnormally low $M/L$ for the disk may be caused by the influence of the bar on the rotation velocity. It may also partially be attributed to the galaxy’s low metallicity ($Z = 0.008$ according to Katkov et al. 2011). In Table \[table4\], the two values of $v_{max~disk}$ obtained photometrically for two different metallicities are compared with $v_{max}$ reached at $R = 5$ kpc. For $Z = 0.008$, the velocities $v_{max~disk}$ and $v_{max}$ turn out to be close (see Table \[table4\]). Consequently, as has been noted above, the abnormally low $M/L$ in NGC 7743 is not evident if to take into account a low metallicity of the stellar population and the influence of the bar on the $v_{max}$ estimate.\
[**UGC 03685**]{} is an Sb-type galaxy. Just as NGC 6814 and NGC 5921, it has a bar and a ring, while having a small inclination. The inclination determined from the $H_{\alpha}$ kinematics is very uncertain ($i=12^o\pm 16$) (Epinat et al. 2008). According to our photometric data, $i=40^o\pm 15$ (see Fig. 2), which is consistent with the NED ($33^o$) and Hyperleda ($55^o$) estimates. In Table \[table4\], the disk rotational velocity $v_{max~disk}$ estimated photometrically is compared with the total rotational velocities $v_{max}$ corresponding to $i_1=12^o$ and $i_2=40^o$. The conclusion about an abnormally low mass-to-light ratio for the disk of UGC 03685 is confirmed if we take the photometrically estimated inclination $i_2$ and is not confirmed for $i_1$. If we take into account the large error in the kinematic inclination estimate, then the conclusion about an abnormally low $M/L$ remains possible, though uncertain.\
Table \[table4\] contains the following data:\
(1) galaxy name;\
(2) maximum rotational velocity of the disk-related component of the rotation curve $v_{max~disk}$ determined photometrically based on the Bell, de Jong (2001) models;\
(3) maximum rotational velocity $v_{max}$ derived from direct observations (for NGC 7743, NGC 5921, NGC 6814, and UGC 3685, this is the velocity of rotation at the distance $R=2r_d$, where the disk contribution to the rotation curve is maximal);\
(4) reference to the source of the rotation curve;\
(5) note;\
(6) photometrically determined mass-to-light ratio for the stellar population of the disk (in the R- band) from the model by Bell, de Jong (2001);\
(7) total dynamical mass-to-light ratio for the galaxy in the R-band within the optical radius $R = 4r_d$ (for NGC 1569 and NGC 4214 it is within $R_{25}$); for NGC 1569 the second row of the table gives the total $M/L_{Ks}$ ratio;\
(8) total mass-to-light ratio for the galaxy $M/L_R$ calculated from the models by Bell, de Jong (2001) and the total $(B- R)_0$ color (for NGC 1569, the second row gives $M/L_{Ks}$ calculated from the total $(J- Ks)_0$ color).\
-------------- --------------------------------- ---------------------------- ----------------------- ------------ ------------- --------------- ---------------
[Galaxy]{} $v_{max~disk}$, [km s]{}$^{-1}$ $v_{max}$, [km]{} $s^{-1}$ Ref. Note $(M/L_R)_d$ $M/L_{R~dyn}$ $M/L_{R~mod}$
(1) (2) (3) (4) (5) (6) (7) (8)
[NGC1569]{} 60 43 [@stil] R 0.16 0.146 0.26
[NGC1569]{} 45 43 [@stil] Ks 0.49 0.8 0.49
[NGC4016]{} 73 78 [@ngc4016] — 0.8 1.53 0.86
[NGC4214]{} 75 42 [@ngc4214] $i=40^o$ 0.8 0.3 0.73
[NGC4214]{} 75 79 [@ngc4214] $i=20^o$ 0.8 1.06 0.73
[NGC4826]{} 256 154 [@ngc4826] $Z=0.02$ 2.25 1.06 2.24
[NGC4826]{} 241 154 [@ngc4826] $Z=0.008$ 2 1.06 2.02
[NGC5347]{} 111 56 [@rc3],[@Marquez2004] $Z=0.02$ 1.5 0.65 2.00
[NGC5347]{} 103 56 [@rc3],[@Marquez2004] $Z=0.008$ 1.3 0.65 1.79
[NGC5921]{} 139 111 [@ngc5921] $i=43^o$ 1.5 1.75 1.77
[NGC5921]{} 139 127 [@ngc5921] $i=36.5^o$ 1.5 2.3 1.77
[NGC6814]{} 148 120 [@ngc6814] — 1.75 1.07 2.17
[NGC7743]{} 135 118 [@katkov] $Z=0.02$ 2.28 2.3 2.37
[NGC7743]{} 125 118 [@katkov] $Z=0.008$ 2 2.3 2.15
[UGC03685]{} 86 101 [@ugc3685],[@Ghasp] $i=12^o$ 1.25 3.37 1.69
[UGC03685]{} 86 33 [@ugc3685],[@Ghasp] $i=40^o$ 1.25 0.35 1.69
-------------- --------------------------------- ---------------------------- ----------------------- ------------ ------------- --------------- ---------------
: Comparison of $v_{max~disk}$ with $v_{max}$ and $M/L$ estimates \[table4\]
CONCLUSIONS
===========
We present the results of our surface photometry in the B, V and R bands for nine disk galaxies in which the discrepancy between low M/L ratio and color of stellar population is suspected. We obtain the photometric profiles, the radial profiles of the color indices, the position angle, and the flattening of isophotes. We decomposed the images using the estimation of photometric parameters for individual components (bulge, bar and disk) of the galaxies. We apply two methods for the dynamical and photometric disk mass estimates: by checking whether the colors agree with the total $M/L$ ratios of the galaxies, which may be considered as an upper limit of $M/L$ for the stellar population, and by comparing the observed rotational velocities of the galaxies with the maximum circular velocity of the disk expected from its photometric parameters. Our results show that there are no obvious contradictions between the dynamical and photometric mass estimates in most considered objects because the abnormally low dynamical $M/L$ estimates can be naturally explained by the uncertainty in determining the rotational velocity or by the errors in photometry. At the same time our photometric data for NGC 4826, NGC 6814, NGC 5347, and with lesser confidence for UGC 03685 and NGC 4214 suggest that these galaxies may actually be too ”light” for their luminosity, and hence may have the IMF defficient by low-mass stars. The presence of dark matter in the galaxies which was ignored in our study would only increase the discrepancy between the dynamical and photometric $M/L$ estimates. The galaxies we discuss require a more complete study: first of all, more careful measurements of the rotational velocity far from the center are needed.
The low values of the mass-to-light ratio in galaxies, even if this is not a result of anomalous stellar population, suggest the low dark-to-luminous matter mass fraction in them.\
ACKNOWLEDGMENTS
===============
We wish to thank R. Swaters who kindly provided the brightness profile of NGC 1569. We also wish to thank R.E. de Souza and D.A. Gadotti for the opportunity to use the BUDDA code. We are grateful to Hyperleda support team for the opportunity to use this database. Based on observations obtained with the Apache Point Observatory 0.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium.\
This work was supported by Russian Foundation for Basic Research, grant 11-02-12247.
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G.R. Meurer O.I. Wong, J.H. Kim et al., Astrophys. J., **695**, 765 (2009) S. Recchi, G. Hensler, L. Angeretti, F. Matteucci, Astron. Astrophys., **445**, 875 (2006) V. C. Rubin, Astron. J., **107**, 173 (1994) A. S. Saburova, E. S. Shaldenkova, A. V. Zasov, Astronomy reports, **53**, 801 (2009) M. Sánchez-Portal, A.I. Díaz, R. Terlevich et al., MNRAS, **312**, 2S (2000) R. A. Swaters & M. Balcells, Astron. Astrophys., **390**, 863 (2002) J. M. Stil, F. P. Israel, Astron. Astrophys., **392**, 473 (2002) D.B. Fisher, N. Drory, Astron. J., **136**, 773 (2008) O. Hernandez, C. Carignan1, P. Amram, L. Chemin and O. Daigle, MNRAS, **360**, 1201 (2005) E.A. Hoversten, K. Glazebrook, Astrophys. J., **675**, 163 (2008) Khoperskov A., Bizyaev D., Tiurina N., Butenko, M. Astron. Nachr. **331**, 731 (2010) D. J. Schlegel, D. P. Finkbeiner, M. Davis, Astrophys. J., **500**, 525 (1998) H. R. Schmitt, A. L. Kinney, Astrophys. J.Suppl. Ser., **128**, 479 (2000) N. J. Allsopp, MNRAS, **188**, 765 (1979) B. Epinat, P. Amram, M. Marcelin et al., MNRAS, **388**, 500 (2008) THIRD REFERENCE CATALOGUE OF BRIGHT GALAXIES, G. de Vaucouleurs, A. de Vaucouleurs, H. G. Corwin, Volume 1-3, XII, 2069 pp., Springer-Verlag Berlin Heidelberg New York (1991) Hyperleda: http://leda.univ-lyon1.fr/ Ned: http://www.ned.ipac.caltech.edu/
[^1]: The listed distances correspond to the Hubble constant $H =
75 km s^{-1} Mpc^{-1}$. For the nearby galaxy NGC 1569, we used the distance from Stil & Israel (2002).
[^2]: http://leda.univ-lyon1.fr/.
[^3]: http://www.ned.ipac.caltech.edu/.
[^4]: Note that extinction inside the galaxies only slightly affects position of points on $M/L$–color diagram relative to the model evolutionary dependencies because it both redden the colors and increases the $M/L$, so it remains close to the expected value for a given color.
[^5]: We took the starburst duration 0.5 Gyr and the fraction of young stars is 10% of the total mass of the stellar population.
|
---
abstract: 'In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial $\mathcal{R}$-transform is constructed and various operator-valued conditionally bi-free limit theorems are studied.'
address:
- 'Department of Mathematics and Statistics, Queen’s University, Jeffrey Hall, Kingston, Ontario, K7L 3N6, Canada'
- 'Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada'
author:
- Yinzheng Gu and Paul Skoufranis
nocite: '[@*]'
title: 'Conditionally bi-free independence with amalgamation'
---
Introduction
============
The notion of conditionally free (c-free for short) independence was introduced in [@BLS1996] as a generalization of the notion of free independence to two-state systems. In our previous paper [@GS2016] we introduced the notion of conditionally bi-free (c-bi-free for short) independence in order to study the non-commutative left and right actions of algebras on a reduced c-free product simultaneously. Thus conditional bi-freeness is an extension of the notion of bi-free independence [@V2014] to two-state systems. Moreover [@GS2016] introduced c-$(\ell, r)$-cumulants and demonstrated that a family of pairs of algebras in a two-state non-commutative probability space is conditionally bi-free if and only if mixed $(\ell, r)$- and c-$(\ell, r)$-cumulants vanish.
In [@V1995] Voiculescu generalized his own notion of free independence by replacing the scalars with an arbitrary algebra thereby obtaining the notion of free independence with amalgamation (see also [@S1998] for the combinatorial aspects). For c-free independence, the generalization to an amalgamated setting over a pair of algebras was done by Popa in [@P2008] (see also [@M2002]). On the other hand, the framework for generalizing bi-free independence to an amalgamated setting was suggested by Voiculescu in [@V2014]\*[Section 8]{} and the theory was fully developed in [@CNS2015-2].
The main goal of this paper is to extend the notion of c-bi-free independence to an amalgamated setting over a pair of algebras. Furthermore, we demonstrate that the combinatorics of conditionally bi-free probability and bi-free probability with amalgamation, which are governed by the lattice of bi-non-crossing partitions, are specific instances of more general combinatorial structures.
Including this introduction this paper contains nine sections which are structured as follows. Section \[sec:prelims\] briefly reviews some of the background material pertaining to conditionally bi-free probability and bi-free probability with amalgamation from [@CNS2015-1; @CNS2015-2; @GS2016]. In particular, the notions bi-non-crossing partitions and diagrams, their lateral refinements and cappings, interior and exterior blocks, ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability spaces, operator-valued bi-multiplicative functions, and the operator-valued bi-free moment and cumulant functions are recalled.
Section \[sec:c-bi-free-defn\] introduces the structures studied within conditionally bi-free independence with amalgamation. We define the notion of a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ (see Definition \[BBncpsBD\]), demonstrate a representation of ${{\mathcal{A}}}$ as linear operators on a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states (see Theorem \[Embedding\]), and define the notion of conditionally bi-free independence with amalgamation over $({{\mathcal{B}}}, {{\mathcal{D}}})$ thereby generalizing conditionally bi-free independence to the operator-valued setting and bi-free independence with amalgamation to the two-state setting.
Section \[sec:pairs-of-fns\] introduces the notion of an operator-valued conditionally bi-multiplicative pair of functions (see Definition \[CondBiMulti\]). Each such pair consists of two functions where the first function is operator-valued bi-multiplicative (see [@CNS2015-2]\*[Definition 4.2.1]{}) and the second function is defined via a certain rule using the first function. Furthermore, operator-valued conditionally bi-free moment and cumulant pairs (see Definitions \[CBFMomentPair\] and \[OpVCBFCumulants\]) are introduced and shown to be operator-valued conditionally bi-multiplicative.
Sections \[sec:moment-express\] and \[sec:additivity\] provide alternate characterizations of conditionally bi-free independence with amalgamation. More precisely, Section \[sec:moment-express\] demonstrates through Theorem \[MomentFormulae\] that a family of pairs of ${{\mathcal{B}}}$-algebras in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if certain moment expressions with respect to ${{\mathbb{E}}}$ and ${{\mathbb{F}}}$ are satisfied. On the other hand, Section \[sec:additivity\] demonstrates through Theorem \[VanishingEquiv\] that a family of pairs of ${{\mathcal{B}}}$-algebras is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if their mixed operator-valued bi-free and conditionally bi-free cumulants vanish.
Section \[sec:additional\] provides additional properties such as the vanishing of operator-valued conditionally bi-free cumulants when a left or right ${{\mathcal{B}}}$-operator is input, how c-bi-free independence over $({{\mathcal{B}}}, {{\mathcal{D}}})$ can be deduced from c-free independence over $({{\mathcal{B}}}, {{\mathcal{D}}})$ under certain conditions, and how operator-valued conditionally bi-free cumulants involving products of operators may be computed.
In Section \[sec:R-transform\], an operator-valued conditionally bi-free partial $\mathcal{R}$-transform is constructed as the operator-valued analogue of the conditionally bi-free partial $\mathcal{R}$-transform (see [@GS2016]\*[Definition 5.3]{}). As with the operator-valued bi-free partial $\mathcal{R}$-transform (see [@S2015]\*[Section 5]{}), the said transform is also a function of three ${{\mathcal{B}}}$-variables, and a formula relating it to the moment series is proved using combinatorics. Finally, in Section \[sec:limit-thms\], operator-valued c-bi-free distributions are discussed and various operator-valued c-bi-free limit theorems are studied.
Preliminaries {#sec:prelims}
=============
In this section, we review the necessary background on conditionally bi-free probability and operator-valued bi-free probability required for this paper.
Conditionally bi-free probability
---------------------------------
We recall several definitions and results relating to conditionally bi-free probability. For more precision, see [@GS2016].
Let $({{\mathcal{A}}}, \varphi, \psi)$ be a two-state non-commutative probability space; that is, ${{\mathcal{A}}}$ is a unital algebra and $\varphi, \psi: {{\mathcal{A}}}\to {{\mathbb{C}}}$ are unital linear functionals. A *pair of algebras* in ${{\mathcal{A}}}$ is an ordered pair $(A_\ell, A_r)$ of unital subalgebras of ${{\mathcal{A}}}$.
A family $\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$ of pairs of algebras in a two-state non-commutative probability space $({{\mathcal{A}}}, \varphi, \psi)$ is said to be *conditionally bi-freely independent* (or *c-bi-free* for short) with respect to $(\varphi, \psi)$ if there is a family of two-state vector spaces with specified state-vectors $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \xi_k, \varphi_k)\}_{k \in K}$ and unital homomorphisms $$\ell_k: A_{k, \ell} \to {{\mathcal{L}}}({{\mathcal{X}}}_k) {\quad\text{and}\quad}r_k: A_{k, r} \to {{\mathcal{L}}}({{\mathcal{X}}}_k)$$ such that the joint distribution of $\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$ with respect to $(\varphi, \psi)$ is equal to the joint distribution of the family $$\{(\lambda_k \circ \ell_k(A_{k, \ell}), \rho_k \circ r_k(A_{k, r}))\}_{k \in K}$$ in ${{\mathcal{L}}}({{\mathcal{X}}})$ with respect to $(\varphi_\xi, \psi_\xi)$, where $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \xi, \varphi) = *_{k \in K}({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \xi_k, \varphi_k)$.
In general, a map $\chi: \{1, \dots, n\} \to \{\ell, r\}$ is used to designate whether the $k^{\mathrm{th}}$ operator in a sequence of $n$ operators is a left operator (when $\chi(k) = \ell$) or a right operator (when $\chi(k) = r$), a map $\omega: \{1, \dots, n\} \to I \sqcup J$ is used to designate the index of the $k^{\mathrm{th}}$ operator, and a map $\omega: \{1, \dots, n\} \to K$ is used to designate from which collection of operators the $k^{\mathrm{th}}$ operator hails from.
Given $\omega: \{1, \dots, n\} \to I \sqcup J$ for non-empty disjoint index sets $I$ and $J$, we define the corresponding map $\chi_\omega: \{1, \dots, n\} \to \{\ell, r\}$ by $$\chi_\omega(k) = \begin{cases}
\ell &\text{if } \omega(k) \in I\\
r &\text{if } \omega(k) \in J
\end{cases}.$$ Given a map $\omega: \{1, \dots, n\} \to K$, we may view $\omega$ as a partition of $\{1, \dots, n\}$ with blocks $\{\omega^{-1}(\{k\})\}_{k \in K}$. Thus $\pi \leq \omega$ denotes $\pi$ is a refinement of the partition induced by $\omega$.
For the basic definitions and combinatorics of bi-free probability that will be used in this paper, we refer the reader to [@CNS2015-1; @CNS2015-2; @MN2015; @V2014] or the summary given in [@GS2016]\*[Section 2]{}. Particular attention should be paid to:
- the set ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ of bi-non-crossing partitions with respect to $\chi: \{1, \dots, n\} \to \{\ell, r\}$, and the minimal and maximal elements $0_\chi$ and $1_\chi$ of ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ (see [@CNS2015-2]\*[Definition 2.1.1]{});
- for $m, n \geq 0$ with $m + n \geq 1$, $1_{m, n}$ denotes $1_{\chi_{m, n}}$ where $\chi_{m, n}: \{1, \dots, m + n\} \to \{\ell, r\}$ is such that $\chi_{m, n}(k) = \ell$ if $k \leq m$ and $\chi_{m, n}(k) = r$ if $k > m$;
- the Möbius function $\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}$ on the lattice of bi-non-crossing partitions (see [@CNS2015-1]\*[Remark 3.1.4]{});
- the total ordering $\prec_\chi$ on $\{1, \dots, n\}$ and the notion of $\chi$-interval induced by $\chi: \{1, \dots, n\} \to \{\ell, r\}$ (see [@CNS2015-2]\*[Definition 4.1.1]{});
- the set ${{\mathcal{L}}}{{\mathcal{R}}}(\chi, \omega)$ of shaded ${{\mathcal{L}}}{{\mathcal{R}}}$-diagrams corresponding to $\chi: \{1, \dots, n\} \to \{\ell, r\}$ and $\omega: \{1, \dots, n\} \to K$, and the subsets ${{\mathcal{L}}}{{\mathcal{R}}}_k(\chi, \omega)$ ($1 \leq k \leq n$) of ${{\mathcal{L}}}{{\mathcal{R}}}(\chi, \omega)$ with exactly $k$ spines reaching the top (see [@CNS2015-1]\*[Section 2.5]{});
- the notion $\leq_{\mathrm{lat}}$ of lateral refinement (see [@CNS2015-1]\*[Definition 2.5.5]{});
- the family $\{\kappa_\chi: {{\mathcal{A}}}^n \to {{\mathbb{C}}}\}_{n \geq 1, \chi: \{1, \dots, n\} \to \{\ell, r\}}$ of $(\ell, r)$-cumulants (see [@MN2015]\*[Definition 5.2]{}).
Inspired by the ‘vanishing of mixed $(\ell, r)$-cumulants’ characterization of bi-free independence and the ‘vanishing of mixed free and c-free cumulants’ characterization of c-free independence, we introduced in [@GS2016]\*[Subsection 3.3]{} the family of c-$(\ell, r)$-cumulants using bi-non-crossing partitions that are divided into two types. More precisely, a block $V$ of a bi-non-crossing partition $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ is said to be *interior* if there exists another block $W$ of $\pi$ such that $\min_{\prec_\chi}(W) \prec_\chi \min_{\prec_\chi}(V)$ and $\max_{\prec_\chi}(V) \prec_\chi \max_{\prec_\chi}(W)$, where $\min_{\prec_\chi}$ and $\max_{\prec_\chi}$ denote the minimum and maximum elements with respect to $\prec_\chi$. A block of $\pi$ is said to be *exterior* if it is not interior. The family $$\{{{\mathcal{K}}}_\chi: {{\mathcal{A}}}^n \to {{\mathbb{C}}}\}_{n \geq 1, \chi: \{1, \dots, n\} \to \{\ell, r\}}$$ of *c-$(\ell, r)$-cumulants* of a two-state non-commutative probability space $({{\mathcal{A}}}, \varphi, \psi)$ is recursively defined by $$\varphi(a_1\cdots a_n) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)}{{\mathcal{K}}}_{\pi}(a_1, \dots, a_n),$$ where $${{\mathcal{K}}}_{\pi}(a_1, \dots, a_n) = \left(\prod_{\substack{V \in \pi\\V\,\mathrm{interior}}}\kappa_{\chi|_V}((a_1, \dots, a_n)|_V)\right)\left(\prod_{\substack{V \in \pi\\V\,\mathrm{exterior}}}{{\mathcal{K}}}_{\chi|_V}((a_1, \dots, a_n)|_V)\right),$$ for all $n \geq 1$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, and $a_1, \dots, a_n \in {{\mathcal{A}}}$.
Furthermore, as noticed in [@GS2016]\*[Section 4]{}, in order to obtain a moment formula for conditionally bi-free independence, additional sets of shaded diagrams and terminology are required.
\[ShadedDiagrams\] Let $n \geq 1$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, and $\omega: \{1, \dots, n\} \to K$ be given.
1. For $0 \leq k \leq n$, let ${{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)$ denote the set of all diagrams that can be obtained from ${{\mathcal{L}}}{{\mathcal{R}}}_k(\chi, \omega)$ under later refinement (i.e., cutting spines that do not reach the top). For $D' \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)$ and $D \in {{\mathcal{L}}}{{\mathcal{R}}}_k(\chi, \omega)$, write $D \geq_{\mathrm{lat}}D'$ if $D'$ can be obtained by laterally refining $D$. Moreover, let $${{\mathcal{L}}}{{\mathcal{R}}}^{\mathrm{lat}}(\chi, \omega) = \bigcup_{k = 0}^n{{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega).$$
2. Let $0 \leq k \leq n$ and $D \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)$. A diagram $D'$ is said to be a *capping* of $D$, denoted $D \geq_{\mathrm{cap}}D'$, if $D' = D$ or $D'$ can be obtained by removing spines from $D$ that reach the top. Let ${{\mathcal{L}}}{{\mathcal{R}}}_m^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ denote the set of all diagrams with $m$ spines reaching the top that can be obtained by capping some $D \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)$ with $k \geq m$. Moreover, let $${{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega) = \bigcup_{m = 0}^n{{\mathcal{L}}}{{\mathcal{R}}}_m^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega).$$
3. For $D \in {{\mathcal{L}}}{{\mathcal{R}}}_m^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$, let $|D| = (\text{number of blocks of } D) + m$.
4. Let $0 \leq m \leq n$, $k \geq m$, $D \in {{\mathcal{L}}}{{\mathcal{R}}}_k(\chi, \omega)$, and $D' \in {{\mathcal{L}}}{{\mathcal{R}}}_m^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$. We say that $D$ *laterally caps* to $D'$, denoted $D \geq_{{\mathrm{lat}}{\mathrm{cap}}} D'$, if there exists $D'' \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)$ such that $D \geq_{\mathrm{lat}}D''$ and $D'' \geq_{\mathrm{cap}}D'$.
Suppose $a_1, \dots, a_n$ are elements in a two-state non-commutative probability space $({{\mathcal{A}}}, \varphi, \psi)$, and $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ with blocks $V_1, \dots, V_p$ whose spines do not reach the top and $W_1, \dots, W_q$ whose spines reach the top. Writing $V_i = \{r_{i, 1} < \cdots < r_{i, s_i}\}$ and $W_j = \{r_{j, 1} < \cdots < r_{j, t_j}\}$, we define $$\varphi_D(a_1, \dots, a_n) = \prod_{i = 1}^p\psi(a_{r_{i, 1}}\cdots a_{r_{i, s_i}})\prod_{j = 1}^q\varphi(a_{r_{j, 1}}\cdots a_{r_{j, t_j}}).$$ Under the above notation, the following moment type characterization and vanishing of mixed cumulants characterization were established in [@GS2016]\*[Theorems 4.1 and 4.8]{}.
\[CBFMoments\] A family $\{(A_{k, \ell}, A_{k, r})\}_{k \in K}$ of pairs of algebras in a two-state non-commutative probability space $({{\mathcal{A}}}, \varphi, \psi)$ is c-bi-free with respect to $(\varphi, \psi)$ if and only if $$\label{psi-Moment}
\psi(a_1\cdots a_n) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)}\left[\sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \leq \sigma \leq \omega}}\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}(\pi, \sigma)\right]\psi_\pi(a_1, \dots, a_n)$$ and $$\label{phi-Moment}
\varphi(a_1\cdots a_n) = \sum_{D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)}\left[\sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}(\chi, \omega)\\D' \geq_{{\mathrm{lat}}{\mathrm{cap}}} D}}(-1)^{|D| - |D'|}\right]\varphi_D(a_1, \dots, a_n)$$ for all $n \geq 1$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, and $a_1, \dots, a_n \in {{\mathcal{A}}}$ with $a_k \in A_{\omega(k), \chi(k)}$.
Equivalently, for all $n \geq 2$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, and $a_k$ as above, we have $$\kappa_\chi(a_1, \dots, a_n) = {{\mathcal{K}}}_\chi(a_1, \dots, a_n) = 0$$ whenever $\omega$ is not constant.
Bi-free probability with amalgamation
-------------------------------------
Now we recall bi-free probability in an amalgamated setting. Since our constructions for operator-valued conditionally bi-free independence in Section \[sec:c-bi-free-defn\] are very similar, we shall only present the essential concepts. Please refer to [@CNS2015-2]\*[Section 3]{} or the summary given in [@S2015]\*[Section 2]{} for complete details. In particular, the following definitions and results will be generalized:
- a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a specified ${{\mathcal{B}}}$-valued state $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p})$ (see [@CNS2015-2]\*[Definition 3.1.1]{});
- the free product with amalgamation over ${{\mathcal{B}}}$ of a family $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \mathfrak{p}_k)\}_{k \in K}$ of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules with specified ${{\mathcal{B}}}$-valued states (see [@CNS2015-2]\*[Construction 3.1.7]{});
- a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space $({{\mathcal{A}}}, {{\mathbb{E}}}, \varepsilon)$ with left and right algebras ${{\mathcal{A}}}_\ell$ and ${{\mathcal{A}}}_r$ (see [@CNS2015-2]\*[Definition 3.2.1]{});
- any ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability can be represented on a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a specified ${{\mathcal{B}}}$-valued state (see [@CNS2015-2]\*[Theorem 3.2.4]{}).
Furthermore, in order to discuss operator-valued bi-free probability, one needs the correct notions for moment and cumulant functions, which we now review in greater depth.
\[BiMulti\] Let $({{\mathcal{A}}}, \mathbb{E}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space and let $$\Psi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ be a function that is linear in each ${{\mathcal{A}}}_{\chi(k)}$. We say that $\Psi$ is *operator-valued bi-multiplicative* if for every $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $Z_k \in {{\mathcal{A}}}_{\chi(k)}$, $b \in {{\mathcal{B}}}$, and $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, the following four conditions hold.
1. Let $$q = \max\{k \in \{1, \dots, n\} \, \mid \, \chi(k) \neq \chi(n)\}.$$ If $\chi(n) = \ell$, then $$\Psi_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_nL_b) = \begin{cases}
\Psi_{1_\chi}(Z_1, \dots, Z_{q - 1}, Z_qR_b, Z_{q + 1}, \dots, Z_n) &\text{if } q \neq -\infty\\
\Psi_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_n)b &\text{if } q = -\infty
\end{cases}.$$ If $\chi(n) = r$, then $$\Psi_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_nR_b) = \begin{cases}
\Psi_{1_\chi}(Z_1, \dots, Z_{q - 1}, Z_qL_b, Z_{q + 1}, \dots, Z_n) &\text{if } q \neq -\infty\\
b\Psi_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_n) &\text{if } q = -\infty
\end{cases}.$$
2. Let $p \in \{1, \dots, n\}$, and let $$q = \max\{k \in \{1, \dots, n\} \, \mid \, \chi(k) = \chi(p), k < p\}.$$ If $\chi(p) = \ell$, then $$\Psi_{1_\chi}(Z_1, \dots, Z_{p - 1}, L_bZ_p, Z_{p + 1}, \dots, Z_n) = \begin{cases}
\Psi_{1_\chi}(Z_1, \dots, Z_{q - 1}, Z_qL_b, Z_{q + 1}, \dots, Z_n) &\text{if } q \neq -\infty\\
b\Psi_{1_\chi}(Z_1, Z_2, \dots, Z_n) &\text{if } q = -\infty
\end{cases}.$$ If $\chi(p) = r$, then $$\Psi_{1_\chi}(Z_1, \dots, Z_{p - 1}, R_bZ_p, Z_{p + 1}, \dots, Z_n) = \begin{cases}
\Psi_{1_\chi}(Z_1, \dots, Z_{q - 1}, Z_qR_b, Z_{q + 1}, \dots, Z_n) &\text{if } q \neq -\infty\\
\Psi_{1_\chi}(Z_1, Z_2, \dots, Z_n)b &\text{if } q = -\infty
\end{cases}.$$
3. Suppose that $V_1, \dots, V_m$ are $\chi$-intervals ordered by $\prec_\chi$ which partition $\{1, \dots, n\}$, each a union of blocks of $\pi$. Then $$\Psi_\pi(Z_1, \dots, Z_n) = \Psi_{\pi|_{V_1}}((Z_1, \dots, Z_n)|_{V_1})\cdots\Psi_{\pi|_{V_m}}((Z_1, \dots, Z_n)|_{V_m}).$$
4. Suppose that $V$ and $W$ partition $\{1, \dots, n\}$, each a union of blocks of $\pi$, $V$ is a $\chi$-interval, and $$\min_{\prec_\chi}(\{1, \dots, n\}), \max_{\prec_\chi}(\{1, \dots, n\}) \in W.$$ Let $$p = \max_{\prec_\chi}\left(\left\{k \in W \, \mid \, k \prec_\chi \min_{\prec_\chi}(V)\right\}\right) {\quad\text{and}\quad}q = \min_{\prec_\chi}\left(\left\{k \in W \, \mid \, \max_{\prec_\chi}(V) \prec_\chi k\right\}\right).$$ Then $$\begin{aligned}
\Psi_\pi(Z_1, \dots, Z_n) &= \begin{cases}
\Psi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{p - 1}, Z_pL_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{p + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(p) = \ell\\
\Psi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{p - 1}, R_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}Z_p, Z_{p + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(p) = r
\end{cases}\\
&= \begin{cases}
\Psi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{q - 1}, L_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}Z_q, Z_{q + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(q) = \ell\\
\Psi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{q - 1}, Z_qR_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{q + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(q) = r
\end{cases}.\end{aligned}$$
Given an operator-valued bi-multiplicative function, conditions $(1)$ to $(4)$ above are reduction properties which allows one to move ${{\mathcal{B}}}$-operators around and, more importantly, to compute the values on arbitrary bi-non-crossing partitions based on its values on full non-crossing partitions.
Finally, the two most important operator-valued bi-multiplicative functions in the theory, called operator-valued bi-free moment and cumulant functions, are defined as follows.
\[E-pi\] Let $({{\mathcal{A}}}, \mathbb{E}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space. For $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $Z_1, \dots, Z_n \in {{\mathcal{A}}}$, define ${{\mathbb{E}}}_\pi(Z_1, \dots, Z_n) \in {{\mathcal{B}}}$ recursively as follows: Let $V$ be the block of $\pi$ that terminates closest to the bottom, so $\min(V)$ is largest among all blocks of $\pi$.
1. If $\pi$ contains exactly one block (that is, $\pi = 1_\chi$), define ${{\mathbb{E}}}_{1_\chi}(Z_1, \dots, Z_n) = {{\mathbb{E}}}(Z_1\cdots Z_n)$.
2. If $V = \{k + 1, \dots, n\}$ for some $k \in \{1, \dots, n - 1\}$ (so $\min(V)$ is not adjacent to any spine of $\pi$), define $${{\mathbb{E}}}_\pi(Z_1, \dots, Z_n) = \begin{cases}
{{\mathbb{E}}}_{\pi|_{V^\complement}}(Z_1, \dots, Z_kL_{{{\mathbb{E}}}_{\pi|_V}(Z_{k + 1}, \dots, Z_n)}) &\text{if } \chi(\min(V)) = \ell\\
{{\mathbb{E}}}_{\pi|_{V^\complement}}(Z_1, \dots, Z_kR_{{{\mathbb{E}}}_{\pi|_V}(Z_{k + 1}, \dots, Z_n)}) &\text{if } \chi(\min(V)) = r\\
\end{cases}.$$
3. Otherwise, $\min(V)$ is adjacent to a spine. Let $W$ denote the block of $\pi$ corresponding to the spine adjacent to $\min(V)$ and let $k$ be the smallest element of $W$ that is larger than $\min(V)$. Define $${{\mathbb{E}}}_\pi(Z_1, \dots, Z_n) = \begin{cases}
{{\mathbb{E}}}_{\pi|_{V^\complement}}((Z_1, \dots, Z_{k - 1}, L_{{{\mathbb{E}}}_{\pi|_V}((Z_1, \dots, Z_n)|_V)}Z_k, Z_{k + 1}, \dots, Z_n)|_{V^\complement}) &\text{if } \chi(\min(V)) = \ell\\
{{\mathbb{E}}}_{\pi|_{V^\complement}}((Z_1, \dots, Z_{k - 1}, R_{{{\mathbb{E}}}_{\pi|_V}((Z_1, \dots, Z_n)|_V)}Z_k, Z_{k + 1}, \dots, Z_n)|_{V^\complement}) &\text{if } \chi(\min(V)) = r\\
\end{cases}.$$
\[MomentCumulant\] Let $({{\mathcal{A}}}, \mathbb{E}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space. The *operator-valued bi-free moment and cumulant functions* on ${{\mathcal{A}}}$ are $${{\mathcal{E}}}, \kappa: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ defined by $${{\mathcal{E}}}_\pi(Z_1, \dots, Z_n) = {{\mathbb{E}}}_\pi(Z_1, \dots, Z_n) {\quad\text{and}\quad}\kappa_\pi(Z_1, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma \leq \pi}}{{\mathcal{E}}}_\sigma(Z_1, \dots, Z_n)\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}(\sigma, \pi)$$ for all $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$.
A substantial amount of effort was taken in [@CNS2015-2]\*[Sections 5 and 6]{} to show that both ${{\mathcal{E}}}$ and $\kappa$ are operator-valued bi-multiplicative.
Conditionally bi-free families with amalgamation {#sec:c-bi-free-defn}
================================================
In this section, we develop the structures to discuss conditionally bi-free independence with amalgamation. To begin, we need an analogue of a two-state vector space with a specified state-vector.
A *$\mathcal{B}$-$\mathcal{B}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states* is a quadruple $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$, where $\mathcal{B}$ and $\mathcal{D}$ are unital algebras such that $1 := 1_{{\mathcal{D}}}\in \mathcal{B} \subset \mathcal{D}$, $\mathcal{X}$ is a direct sum of $\mathcal{B}$-$\mathcal{B}$-bimodules $\mathcal{X} = \mathcal{B} \oplus \mathcal{X}^\circ$, $\mathfrak{p}: \mathcal{X} \to \mathcal{B}$ is the linear map $\mathfrak{p}(b \oplus \eta) = b$, and $\mathfrak{q}: \mathcal{X} \to \mathcal{D}$ is a linear $\mathcal{B}$-$\mathcal{B}$-bimodule map such that $\mathfrak{q}(1 \oplus 0) = 1$.
Given a $\mathcal{B}$-$\mathcal{B}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$, we have $$\mathfrak{p}(b_1\cdot x\cdot b_2) = b_1\mathfrak{p}(x)b_2 {\quad\text{and}\quad}\mathfrak{q}(b_1\cdot x\cdot b_2) = b_1\mathfrak{q}(x)b_2$$ for all $b_1, b_2 \in {{\mathcal{B}}}$ and $x \in {{\mathcal{X}}}$. Moreover, let ${{\mathcal{L}}}({{\mathcal{X}}})$ denote the set of linear operators on ${{\mathcal{X}}}$, and recall from [@CNS2015-2]\*[Definition 3.1.3]{} that the operators $L_b, R_b \in {{\mathcal{L}}}({{\mathcal{X}}})$ are defined by $$L_b(x) = b\cdot x {\quad\text{and}\quad}R_b(x) = x\cdot b$$ for all $b \in {{\mathcal{B}}}$ and $x \in {{\mathcal{X}}}$. In addition, the *left and right algebras* of ${{\mathcal{L}}}({{\mathcal{X}}})$ are the unital subalgebras ${{\mathcal{L}}}_\ell({{\mathcal{X}}})$ and ${{\mathcal{L}}}_r(X)$ defined by $${{\mathcal{L}}}_\ell(X) = \{Z \in {{\mathcal{L}}}({{\mathcal{X}}}) \, \mid \, ZR_b = R_bZ\text{ for all }b \in {{\mathcal{B}}}\}$$ and $${{\mathcal{L}}}_r({{\mathcal{X}}}) = \{Z \in {{\mathcal{L}}}({{\mathcal{X}}}) \, \mid \, ZL_b = L_bZ\text{ for all }b \in {{\mathcal{B}}}\},$$ respectively.
As we are interested in conditionally bi-free independence with amalgamation, we need two expectations on ${{\mathcal{L}}}({{\mathcal{X}}})$, one onto ${{\mathcal{B}}}$ and one to ${{\mathcal{D}}}$.
Given a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$, define the unital linear maps $\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}: {{\mathcal{L}}}({{\mathcal{X}}}) \to {{\mathcal{B}}}$ and $\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}: {{\mathcal{L}}}({{\mathcal{X}}}) \to \mathcal{D}$ by $$\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z) = \mathfrak{p}(Z(1 \oplus 0)){\quad\text{and}\quad}\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z) = \mathfrak{q}(Z(1 \oplus 0))$$ for all $Z \in {{\mathcal{L}}}({{\mathcal{X}}})$. We call $\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ and $\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ the *expectations* of ${{\mathcal{L}}}({{\mathcal{X}}})$ to ${{\mathcal{B}}}$ and $\mathcal{D}$, respectively.
There are specific properties of these expectations we wish to model.
\[Expectations\] Let $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states. We have $$\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(L_{b_1}R_{b_2}Z) = b_1\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z)b_2,\quad\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZL_b) = \mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZR_b)$$ and $$\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(L_{b_1}R_{b_2}Z) = b_1\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z)b_2,\quad\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZL_b) = \mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZR_b)$$ for all $b_1, b_2, b \in \mathcal{B}$ and $Z \in {{\mathcal{L}}}({{\mathcal{X}}})$.
The results regarding ${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ were shown in [@CNS2015-2]\*[Proposition 3.1.6]{}. Moreover, it is immediate that $\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZL_b) = \mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(ZR_b)$ for all $b \in {{\mathcal{B}}}$ and $Z \in {{\mathcal{L}}}({{\mathcal{X}}})$ as $L_b(1 \oplus 0) = R_b(1 \oplus 0)$. Finally, since $\mathfrak{q}$ is a linear ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule map, we have $$\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(L_{b_1}R_{b_2}Z) = \mathfrak{q}(L_{b_1}R_{b_2}Z(1 \oplus 0)) = b_1\mathfrak{q}(Z(1 \oplus 0))b_2 = b_1{{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z)b_2$$ for all $b_1, b_2 \in {{\mathcal{B}}}$ and $Z \in {{\mathcal{L}}}({{\mathcal{X}}})$.
Given the above definition and proposition, we extend the notion of a two-state non-commutative probability space $({{\mathcal{A}}}, \varphi, \psi)$ to the operator-valued setting as follows. Note this is also a natural extension of the notion of a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space $({{\mathcal{A}}}, {{\mathbb{E}}}, \varepsilon)$ from [@CNS2015-2]\*[Definition 3.2.1]{} to the two-state setting.
\[BBncpsBD\] A *${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations* is a quadruple $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$, where ${{\mathcal{A}}}$, $\mathcal{B}$, and $\mathcal{D}$ are unital algebras such that $1 := 1_{{\mathcal{D}}}\in \mathcal{B} \subset \mathcal{D}$, $\varepsilon: \mathcal{B} \otimes \mathcal{B}^{\mathrm{op}} \to {{\mathcal{A}}}$ is a unital homomorphism such that $\varepsilon|_{\mathcal{B} \otimes 1}$ and $\varepsilon|_{1 \otimes \mathcal{B}^{\mathrm{op}}}$ are injective, and $\mathbb{E}: {{\mathcal{A}}}\to \mathcal{B}$ and $\mathbb{F}: {{\mathcal{A}}}\to \mathcal{D}$ are unital linear maps such that $$\mathbb{E}(\varepsilon(b_1 \otimes b_2)Z) = b_1\mathbb{E}(Z)b_2,\quad\mathbb{E}(Z\varepsilon(b \otimes 1)) = \mathbb{E}(Z\varepsilon(1 \otimes b))$$ and $$\mathbb{F}(\varepsilon(b_1 \otimes b_2)Z) = b_1\mathbb{F}(Z)b_2,\quad\mathbb{F}(Z\varepsilon(b \otimes 1)) = \mathbb{F}(Z\varepsilon(1 \otimes b))$$ for all $b_1, b_2, b \in \mathcal{B}$ and $Z \in {{\mathcal{A}}}$. Moreover, the unital subalgebras ${{\mathcal{A}}}_\ell$ and ${{\mathcal{A}}}_r$ of ${{\mathcal{A}}}$ defined by $${{\mathcal{A}}}_\ell = \{Z \in {{\mathcal{A}}}\, \mid \, Z\varepsilon(1 \otimes b) = \varepsilon(1 \otimes b)Z\text{ for all }b \in \mathcal{B}\}$$ and $${{\mathcal{A}}}_r = \{Z \in {{\mathcal{A}}}\, \mid \, Z\varepsilon(b \otimes 1) = \varepsilon(b \otimes 1)Z\text{ for all }b \in \mathcal{B}\}$$ will be called the *left and right algebras* of ${{\mathcal{A}}}$ respectively.
As with the bi-free case (see [@CNS2015-2]\*[Remark 3.2.2]{}), if $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p}, \mathfrak{q})$ is a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states, then we see via Proposition \[Expectations\] that $({{\mathcal{L}}}({{\mathcal{X}}}), {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}, {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}, \varepsilon)$ is a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations where $\varepsilon: {{\mathcal{B}}}\otimes {{\mathcal{B}}}^{\mathrm{op}} \to {{\mathcal{L}}}({{\mathcal{X}}})$ is defined by $\varepsilon(b_1 \otimes b_2) = L_{b_1}R_{b_2}$. Moreover, the following result demonstrates that any ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations can be represented as linear operators on some $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p}, \mathfrak{q})$. Hence Definition \[BBncpsBD\] is the natural extension of [@CNS2015-2]\*[Definition 3.2.1]{}. As such, we will write $L_b$ and $R_b$ instead of $\varepsilon(b \otimes 1)$ and $\varepsilon(1 \otimes b)$ and refer to these as left and right ${{\mathcal{B}}}$-operators, respectively.
\[Embedding\] If $({{\mathcal{A}}}, \mathbb{E}_{{\mathcal{A}}}, \mathbb{F}_{{\mathcal{A}}}, \varepsilon)$ is a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, then there exist a $\mathcal{B}$-$\mathcal{B}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$ and a unital homomorphism $\theta: {{\mathcal{A}}}\to {{\mathcal{L}}}({{\mathcal{X}}})$ such that $$\begin{gathered}
\theta(L_{b_1}R_{b_2}) = L_{b_1}R_{b_2},\quad\theta({{\mathcal{A}}}_\ell) \subset {{\mathcal{L}}}_\ell({{\mathcal{X}}}),\quad\theta({{\mathcal{A}}}_r) \subset {{\mathcal{L}}}_r({{\mathcal{X}}}), \\
\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathbb{E}_{{\mathcal{A}}}(Z),{\quad\text{and}\quad}\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathbb{F}_{{\mathcal{A}}}(Z)\end{gathered}$$ for all $b_1, b_2 \in {{\mathcal{B}}}$ and $Z \in {{\mathcal{A}}}$.
As shown in the proof of [@CNS2015-2]\*[Theorem 3.2.4]{}, consider ${{\mathcal{X}}}= {{\mathcal{B}}}\oplus {{\mathcal{Y}}}$ as a vector space over $\mathbb{C}$ where $${{\mathcal{Y}}}= \ker(\mathbb{E}_{{\mathcal{A}}})/\mathrm{span}\{ZL_b - ZR_b \, \mid \, Z \in {{\mathcal{A}}}, b \in {{\mathcal{B}}}\}.$$ Define $\theta: {{\mathcal{A}}}\to {{\mathcal{L}}}({{\mathcal{X}}})$ by $$\theta(Z)(b) = {{\mathbb{E}}}_{{\mathcal{A}}}(ZL_b) \oplus \pi(ZL_b - L_{{{\mathbb{E}}}_{{\mathcal{A}}}(ZL_b)}),\quad b \in {{\mathcal{B}}},$$ and $$\theta(Z)(\pi(Y)) = {{\mathbb{E}}}_{{\mathcal{A}}}(ZY) \oplus \pi(ZY - L_{{{\mathbb{E}}}_{{\mathcal{A}}}(ZY)}),\quad Y \in \ker({{\mathbb{E}}}_{{\mathcal{A}}}),$$ where $\pi: \ker({{\mathbb{E}}}_{{\mathcal{A}}}) \to {{\mathcal{Y}}}$ denotes the canonical quotient map. It was shown in [@CNS2015-2]\*[Theorem 3.2.4]{} that $\theta$ is a unital homomorphism and ${{\mathcal{X}}}$ is a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule via $$b\cdot\xi = \theta(L_b)(\xi){\quad\text{and}\quad}\xi\cdot b = \theta(R_b)(\xi)$$ for all $b \in {{\mathcal{B}}}$ and $\xi \in {{\mathcal{X}}}$. Thus we can define a specified ${{\mathcal{B}}}$-valued state $\mathfrak{p}$ on ${{\mathcal{X}}}$ by $\mathfrak{p}(b \oplus \pi(Y)) = b$ for all $b \in {{\mathcal{B}}}$ and $\pi(Y) \in {{\mathcal{Y}}}$. Using this specified ${{\mathcal{B}}}$-valued state, we obtain that $\theta({{\mathcal{A}}}_\ell) \subset {{\mathcal{L}}}_\ell({{\mathcal{X}}})$, $\theta({{\mathcal{A}}}_r) \subset {{\mathcal{L}}}_r({{\mathcal{X}}})$, and $\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathbb{E}_{{\mathcal{A}}}(Z)$.
On the other hand, since ${{\mathbb{F}}}_{{\mathcal{A}}}(ZL_b - ZR_b) = 0$ for all $Z \in {{\mathcal{A}}}$ and $b \in {{\mathcal{B}}}$, there exists a unique linear map $\widetilde{\mathfrak{q}}: {{\mathcal{Y}}}\to {{\mathcal{D}}}$ such that ${{\mathbb{F}}}_{{\mathcal{A}}}|_{\ker({{\mathbb{E}}}_{{\mathcal{A}}})} = \widetilde{\mathfrak{q}} \circ \pi$. Let $\mathfrak{q}: {{\mathcal{X}}}\to {{\mathcal{D}}}$ be the linear map defined by $$\mathfrak{q}(b \oplus \pi(Y)) = b + \widetilde{\mathfrak{q}} \circ \pi(Y),\quad b \in {{\mathcal{B}}},\quad\pi(Y) \in {{\mathcal{Y}}}.$$ Then $\mathfrak{q}(1 \oplus 0) = 1$ and $$\begin{aligned}
\mathfrak{q}(b_1\cdot(b \oplus \pi(Y))\cdot b_2) &= \mathfrak{q}(\theta(L_{b_1})\theta(R_{b_2})(b \oplus \pi(Y)))\\
&= \mathfrak{q}(\theta(L_{b_1})(bb_2 \oplus \pi(R_{b_2}Y)))\\
&= \mathfrak{q}(b_1bb_2 \oplus \pi(L_{b_2}R_{b_2}Y))\\
&= b_1bb_2 + {{\mathbb{F}}}_{{\mathcal{A}}}(L_{b_1}R_{b_2}Y)\\
&= b_1(b + {{\mathbb{F}}}_{{\mathcal{A}}}(Y))b_2\\
&= b_1\mathfrak{q}(b \oplus \pi(Y))b_2\end{aligned}$$ for all $b_1, b_2, b \in {{\mathcal{B}}}$ and $\pi(Y) \in {{\mathcal{Y}}}$. Therefore, the quadruple $({{\mathcal{X}}}, {{\mathcal{Y}}}, \mathfrak{p}, \mathfrak{q})$ is a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states. Finally, we have $${{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathfrak{q}(\theta(Z)(1 \oplus 0)) = \mathfrak{q}({{\mathbb{E}}}_{{\mathcal{A}}}(Z) \oplus \pi(Z - L_{{{\mathbb{E}}}_{{\mathcal{A}}}(Z)})) = {{\mathbb{E}}}_{{\mathcal{A}}}(Z) + {{\mathbb{F}}}_{{\mathcal{A}}}(Z - L_{{{\mathbb{E}}}_{{\mathcal{A}}}(Z)}) = {{\mathbb{F}}}_{{\mathcal{A}}}(Z)$$ for all $Z \in {{\mathcal{A}}}$.
The next step is to extend the construction of the free product with amalgamation over ${{\mathcal{B}}}$ of a family $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \mathfrak{p}_k)\}_{k \in K}$ of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules with specified ${{\mathcal{B}}}$-valued states (see [@CNS2015-2]\*[Construction 3.1.7]{}) to the current framework.
\[Construction\] Let $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$ be a family of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules with pairs of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states. The *c-free product of $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$ with amalgamation over $({{\mathcal{B}}}, {{\mathcal{D}}})$* is defined to be the ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule with a pair of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p}, \mathfrak{q})$, where ${{\mathcal{X}}}= {{\mathcal{B}}}\oplus {{\mathcal{X}}}^\circ$, ${{\mathcal{X}}}^\circ$ is the ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule $${{\mathcal{X}}}^\circ = \bigoplus_{n \geq 1}\left(\bigoplus_{k_1 \neq \cdots \neq k_n}{{\mathcal{X}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{X}}}_{k_n}^\circ\right)$$ with left and right actions of ${{\mathcal{B}}}$ on ${{\mathcal{X}}}^\circ$ defined by $$b\cdot(x_1 \otimes \cdots \otimes x_n) = (b \cdot x_1) \otimes x_2 \otimes \cdots \otimes x_n {\quad\text{and}\quad}(x_1 \otimes \cdots \otimes x_n)\cdot b = x_1 \otimes \cdots \otimes x_{n - 1} \otimes (x_n \cdot b),$$ respectively, $\mathfrak{p}: {{\mathcal{X}}}\to {{\mathcal{B}}}$ is the linear map $\mathfrak{p}(b \oplus \eta) = b$, and $\mathfrak{q}: {{\mathcal{X}}}\to {{\mathcal{D}}}$ is the linear ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule map such that $\mathfrak{q}(1 \oplus 0) = 1$ and $$\mathfrak{q}(x_1 \otimes \cdots \otimes x_n) = \mathfrak{q}_{k_1}(x_1)\cdots\mathfrak{q}_{k_n}(x_n)$$ for $x_1 \otimes \cdots \otimes x_n \in {{\mathcal{X}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{X}}}_{k_n}^\circ$ (note $\mathfrak{q}$ is well-defined as each $\mathfrak{q}_k$ is a linear ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule map).
For every $k \in K$, let $$V_k: {{\mathcal{X}}}\to {{\mathcal{X}}}_k \otimes_{{{\mathcal{B}}}} \left({{\mathcal{B}}}\oplus \bigoplus_{n \geq 1}\left(\bigoplus_{\substack{k_1 \neq \cdots \neq k_n\\k_1 \neq k}}{{\mathcal{X}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{X}}}_{k_n}^\circ\right)\right)$$ be the standard ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule isomorphism, and define the left representation $\lambda_k: {{\mathcal{L}}}({{\mathcal{X}}}_k) \to {{\mathcal{L}}}({{\mathcal{X}}})$ by $$\lambda_k(Z) = V_k^{-1}(Z \otimes I)V_k$$ for $Z \in {{\mathcal{L}}}({{\mathcal{X}}}_k)$. Similarly, let $$W_k: {{\mathcal{X}}}\to \left({{\mathcal{B}}}\oplus \bigoplus_{n \geq 1}\left(\bigoplus_{\substack{k_1 \neq \cdots \neq k_n\\k_n \neq k}}{{\mathcal{X}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{X}}}_{k_n}^\circ\right)\right) \otimes_{{{\mathcal{B}}}} {{\mathcal{X}}}_k$$ be the standard ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule isomorphism, and define the right representation $\rho_k: {{\mathcal{L}}}({{\mathcal{X}}}_k) \to {{\mathcal{L}}}({{\mathcal{X}}})$ by $$\rho_k(Z) = W_k^{-1}(I \otimes Z)W_k$$ for $Z \in {{\mathcal{L}}}({{\mathcal{X}}}_k)$. For the exact formulae of how $\lambda_k(Z)$ and $\rho_k(Z)$ act on ${{\mathcal{X}}}$, we refer to [@CNS2015-2]\*[Construction 3.1.7]{}. Note also that $${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\lambda_k(Z)) = {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\rho_k(Z)) = {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}}_k)}(Z) {\quad\text{and}\quad}{{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\lambda_k(Z)) = {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\rho_k(Z)) = {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}}_k)}(Z)$$ for all $Z \in {{\mathcal{L}}}({{\mathcal{X}}}_k)$.
It is clear that that all of the above discussions hold if ${{\mathcal{B}}}= {{\mathcal{D}}}$. However, the more general setting that ${{\mathcal{B}}}\subset {{\mathcal{D}}}$ is desired due to a result of Boca [@B1991]. Indeed, suppose $\{{{\mathcal{A}}}_k\}_{k \in K}$ is a family of unital $C^*$-algebras containing ${{\mathcal{B}}}$ as a common $C^*$-subalgebra with $1_{{{\mathcal{A}}}_k} \in {{\mathcal{B}}}$, ${{\mathcal{D}}}$ is a unital $C^*$-algebra with $1_{{\mathcal{D}}}\in {{\mathcal{B}}}\subset {{\mathcal{D}}}$, and each ${{\mathcal{A}}}_k$ is endowed with two positive conditional expectations $\Psi_k: {{\mathcal{A}}}_k \to {{\mathcal{B}}}$ and $\Phi_k: {{\mathcal{A}}}_k \to {{\mathcal{D}}}$ such that ${{\mathcal{A}}}_k = {{\mathcal{B}}}\oplus {{\mathcal{A}}}_k^\circ$, where ${{\mathcal{A}}}_k^\circ = \ker(\Psi_k)$, as a direct sum of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules.
Let ${{\mathcal{A}}}= (*_{{\mathcal{B}}})_{k \in K}{{\mathcal{A}}}_k$ be the free product of $\{{{\mathcal{A}}}_k\}_{k \in K}$ with amalgamation over ${{\mathcal{B}}}$ (which can be identified as $${{\mathcal{A}}}= {{\mathcal{B}}}\oplus \bigoplus_{n \geq 1}\left(\bigoplus_{k_1 \neq \cdots \neq k_n}{{\mathcal{A}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{A}}}_{k_n}^\circ\right)$$ as ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules), let $\Psi = (*_{{{\mathcal{B}}}})_{k \in K}\Psi_k$ be the amalgamated free product of $\{\Psi_k\}_{k \in K}$, and let $\Phi: {{\mathcal{A}}}\to {{\mathcal{D}}}$ be the unital linear ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodule map defined by $$\Phi(a_1 \otimes \cdots \otimes a_n) = \Phi_{k_1}(a_1)\cdots\Phi_{k_n}(a_n)$$ for $a_1 \otimes \cdots \otimes a_n \in {{\mathcal{A}}}_{k_1}^\circ \otimes_{{{\mathcal{B}}}} \cdots \otimes_{{{\mathcal{B}}}} {{\mathcal{A}}}_{k_n}^\circ$. It is well-known that $\Psi$ is positive (see, e.g., [@S1998]\*[Theorem 3.5.6]{}). On the other hand, it follows from [@B1991]\*[Theorem 3.1]{} that $\Phi$ is also positive, which is the main motivation for our setting.
With Definition \[BBncpsBD\] and Construction \[Construction\] complete, we can define the notion of conditionally bi-free independence with amalgamation as follows.
\[defn:op-c-bi-free-definition\] Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. A *pair of ${{\mathcal{B}}}$-algebras* in $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ is a pair $({{\mathcal{C}}}_\ell, {{\mathcal{C}}}_r)$ of unital subalgebras of ${{\mathcal{A}}}$ such that $$\varepsilon({{\mathcal{B}}}\otimes 1) \subset {{\mathcal{C}}}_\ell \subset {{\mathcal{A}}}_\ell {\quad\text{and}\quad}\varepsilon(1 \otimes {{\mathcal{B}}}^{\mathrm{op}}) \subset {{\mathcal{C}}}_r \subset {{\mathcal{A}}}_r.$$
A family $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ of pairs of ${{\mathcal{B}}}$-algebras in $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ is said to be *conditionally bi-free with amalgamation over $({{\mathcal{B}}}, {{\mathcal{D}}})$* (or *c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$* for short) if there is a family of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules with pairs of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states $\{(\mathcal{X}_k, \mathcal{X}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$ and unital homomorphisms $$\ell_k: {{\mathcal{A}}}_{k, \ell} \to {{\mathcal{L}}}_\ell(\mathcal{X}_k){\quad\text{and}\quad}r_k: {{\mathcal{A}}}_{k, r} \to {{\mathcal{L}}}_r(\mathcal{X}_k)$$ such that the joint distribution of $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ with respect to $(\mathbb{E}, \mathbb{F})$ is equal to the joint distribution of the family $$\{(\lambda_k \circ \ell_k({{\mathcal{A}}}_{k, \ell}), \rho_k \circ r_k({{\mathcal{A}}}_{k, r}))\}_{k \in K}$$ in ${{\mathcal{L}}}({{\mathcal{X}}})$ with respect to $(\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}, \mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})})$, where $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p}, \mathfrak{q}) = (*_{{\mathcal{B}}})_{k \in K}(\mathcal{X}_k, \mathcal{X}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)$.
It will be an immediate consequence of Theorem \[MomentFormulae\] below that the above definition does not depend on a specific choice of representations. Moreover, it follows immediately from the definition that if $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then the family $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$ (respectively $\{{{\mathcal{A}}}_{k, r}\}_{k \in K}$) of left ${{\mathcal{B}}}$-algebras (respectively right ${{\mathcal{B}}}$-algebras) is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$.
Operator-valued conditionally bi-free pairs of functions {#sec:pairs-of-fns}
========================================================
In order to study operator-valued conditional bi-free independence we must extend the notion of operator-valued bi-multiplicative functions to pairs of functions.
Operator-valued conditionally bi-multiplicative pairs of functions
------------------------------------------------------------------
We begin with an observation, which will be useful later.
\[Decomposition\] If $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ is a bi-non-crossing partition for some $\chi: \{1, \dots, n\} \to \{\ell, r\}$, then there exists a unique partition $V_1, \dots, V_m$ of $\{1,\ldots, n\}$ into $\chi$-intervals such that each $V_k$ a union of blocks of $\pi$ and such that $\min_{\prec_\chi}(V_k)$ and $\max_{\prec_\chi}(V_k)$ are in the same block of $\pi$ for each $k \in \{1, \dots, m\}$. Furthermore, by reordering if necessary, we may assume $\max_{\prec_\chi}(V_k) \prec_\chi \min_{\prec_\chi} (V_{k+1})$ for all $k$. For example, if $\pi$ has the following bi-non-crossing diagram $$\begin{aligned}
\begin{tikzpicture}[baseline]
\node[left] at (0,5.5) {1};
\draw[black,fill=black] (0,5.5) circle (0.05);
\node[left] at (0,5) {2};
\draw[black,fill=black] (0,5) circle (0.05);
\node[right] at (3.2,4.5) {3};
\draw[black,fill=black] (3.2,4.5) circle (0.05);
\node[left] at (0,4) {4};
\draw[black,fill=black] (0,4) circle (0.05);
\node[right] at (3.2,3.5) {5};
\draw[black,fill=black] (3.2,3.5) circle (0.05);
\node[left] at (0,3) {6};
\draw[black,fill=black] (0,3) circle (0.05);
\node[left] at (0,2.5) {7};
\draw[black,fill=black] (0,2.5) circle (0.05);
\node[right] at (3.2,2) {8};
\draw[black,fill=black] (3.2,2) circle (0.05);
\node[left] at (0,1.5) {9};
\draw[black,fill=black] (0,1.5) circle (0.05);
\node[right] at (3.2,1) {10};
\draw[black,fill=black] (3.2,1) circle (0.05);
\node[right] at (3.2,0.5) {11};
\draw[black,fill=black] (3.2,0.5) circle (0.05);
\node[right] at (3.2,0) {12};
\draw[black,fill=black] (3.2,0) circle (0.05);
\draw[thick, black] (0,5.5) -- (1.6,5.5) -- (1.6,3) -- (0,3);
\draw[thick, black] (0,5) -- (0.8,5) -- (0.8,4) -- (0,4);
\draw[thick, black] (3.2,4.5) -- (2.4,4.5) -- (2.4,2) -- (3.2,2);
\draw[thick, black] (2.4,2) -- (2.4,1) -- (3.2,1);
\draw[thick, black] (0,2.5) -- (1.6,2.5) -- (1.6,0.5) -- (3.2,0.5);
\draw[thick, black] (0,1.5) -- (0.8,1.5) -- (0.8,0) -- (3.2,0);
\draw[thick, dashed, black] (0,6) -- (0,-.5) -- (3.2, -.50) -- (3.2,6);
\end{tikzpicture}\end{aligned}$$ then $V_1 = \{\{1, 6\}, \{2, 4\}\}$, $V_2 = \{\{7, 11\}, \{9, 12\}\}$, and $V_3 = \{\{3, 8, 10\}, \{5\}\}$ where $\min_{\prec_\chi}(V_1)=1$, $\max_{\prec_\chi}(V_1) = 6$, $\min_{\prec_\chi}(V_2)= 7$, $\max_{\prec_\chi}(V_2)=11$, $\min_{\prec_\chi}(V_3) = 10$, and $\max_{\prec_\chi}(V_3)=3$. Note that the blocks $V_k' \subset V_k$ containing $\min_{\prec_\chi}(V_k)$ and $\max_{\prec_\chi}(V_k)$ are the exterior blocks of $\pi$.
\[CondBiMulti\] Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, and let $$\Psi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ and $$\Phi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{D}}}$$ be a pair of functions that are linear in each ${{\mathcal{A}}}_{\chi(k)}$. It is said that $(\Psi, \Phi)$ is an *operator-valued conditionally bi-multiplicative pair* if for every $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $Z_k \in {{\mathcal{A}}}_{\chi(k)}$, $b \in {{\mathcal{B}}}$, and $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, $\Psi$ satisfies conditions $(1)$ to $(4)$ of Definition \[BiMulti\] (i.e., $\Psi$ is operator-valued bi-multiplicative), and $\Phi$ satisfies conditions $(1)$ to $(3)$ of Definition \[BiMulti\] and the following modification of condition $(4)$: Under the same notation with the additional assumption that $\min_{\prec_\chi}(\{1, \dots, n\})$ and $\max_{\prec_\chi}(\{1, \dots, n\})$ are in the same block of $\pi$, we have $$\begin{aligned}
\Phi_\pi(Z_1, \dots, Z_n) &= \begin{cases}
\Phi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{p - 1}, Z_pL_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{p + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(p) = \ell\\
\Phi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{p - 1}, R_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}Z_p, Z_{p + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(p) = r
\end{cases}\\
&= \begin{cases}
\Phi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{q - 1}, L_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}Z_q, Z_{q + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(q) = \ell\\
\Phi_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{q - 1}, Z_qR_{\Psi_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{q + 1}, \dots, Z_n\right)|_{W}\right) &\text{if } \chi(q) = r
\end{cases}.\end{aligned}$$
Note the additional assumption that $\min_{\prec_\chi}(\{1, \dots, n\})$ and $\max_{\prec_\chi}(\{1, \dots, n\})$ are in the same block of $\pi$ guarantees that $W$ contains an exterior block of $\pi$ and $V$ is a union of interior blocks of $\pi$.
As with operator-valued bi-multiplicative functions, one may reduce $\Phi_\pi(Z_1, \dots, Z_n)$ to an expression involving $\Psi_{1_\chi}$ and $\Phi_{1_\chi}$ for various $\chi: \{1, \dots, m\} \to \{\ell, r\}$. For example, if $\pi$ is the bi-non-crossing partition from Remark \[Decomposition\] and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$, then $$\Phi_\pi(Z_1, \dots, Z_{12}) = \Phi_{\pi|_{V_1}}(Z_1, Z_2, Z_4, Z_6)\Phi_{\pi|_{V_2}}(Z_7, Z_9, Z_{11}, Z_{12})\Phi_{\pi|_{V_3}}(Z_3, Z_5, Z_8, Z_{10})$$ by condition $(3)$ of Definition \[BiMulti\], which can be further reduced to $$\Phi_{1_{2, 0}}\left(Z_1L_{\Psi_{1_{2, 0}}(Z_2, Z_4)}, Z_6\right)\Phi_{1_{1, 1}}\left(Z_7L_{\Psi_{1_{1, 1}}(Z_9, Z_{12})}, Z_{11}\right)\Phi_{1_{0, 3}}\left(Z_3, R_{\Psi_{1_{0, 1}}(Z_5)}Z_8, Z_{10}\right)$$ by the modified condition $(4)$ of Definition \[CondBiMulti\].
Operator-valued conditionally bi-free moment pairs
--------------------------------------------------
In this subsection, we define the operator-valued conditionally bi-free moment pair $({{\mathcal{E}}}, {{\mathcal{F}}})$ and show that it is operator-valued conditionally bi-multiplicative.
\[CBFMomentPair\] Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. The *operator-valued conditionally bi-free moment pair* on ${{\mathcal{A}}}$ is the pair of functions $${{\mathcal{E}}}: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ and $${{\mathcal{F}}}: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{D}}}$$ where ${{\mathcal{E}}}$ is the operator-valued bi-free moment function on ${{\mathcal{A}}}$ and ${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n)$ for $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$ is defined as follows.
1. If $\pi$ contains exactly one block (that is, $\pi = 1_\chi$), define ${{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) = {{\mathbb{F}}}(Z_1\cdots Z_n)$.
2. If $V_1, \ldots, V_n$ are the blocks of $\pi$, each $V_k$ is a $\chi$-intervals (thus all exterior), and $\max_{\prec_\chi}(V_k) \prec_\chi \min_{\prec_\chi}(V_{k+1})$ for all $k$, define $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n) = {{\mathcal{F}}}_{\pi|_{V_1}}((Z_1, \dots, Z_n)|_{V_1})\cdots {{\mathcal{F}}}_{\pi|_{V_m}}((Z_1, \dots, Z_n)|_{V_m})$$ and apply step $(3)$ to each piece.
3. Apply a similar recursive process as in Definition \[E-pi\] to the interior blocks of $\pi$ as follows: Let $V$ be the interior block of $\pi$ that terminates closest to the bottom. Then
- If $V = \{k + 1, \dots, n\}$ for some $k \in \{1, \dots, n - 1\}$, then $\min(V)$ is not adjacent to any spine of $\pi$ and define $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n) = \begin{cases}
{{\mathcal{F}}}_{\pi|_{V^\complement}}(Z_1, \dots, Z_kL_{{{\mathcal{E}}}_{\pi|_V}(Z_{k + 1}, \dots, Z_n)}) &\text{if } \chi(\min(V)) = \ell\\
{{\mathcal{F}}}_{\pi|_{V^\complement}}(Z_1, \dots, Z_kR_{{{\mathcal{E}}}_{\pi|_V}(Z_{k + 1}, \dots, Z_n)}) &\text{if } \chi(\min(V)) = r\\
\end{cases}.$$
- Otherwise, $\min(V)$ is adjacent to a spine. Let $W$ denote the block of $\pi$ corresponding to the spine adjacent to $\min(V)$ and let $k$ be the smallest element of $W$ that is larger than $\min(V)$. Define $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n) = \begin{cases}
{{\mathcal{F}}}_{\pi|_{V^\complement}}((Z_1, \dots, Z_{k - 1}, L_{{{\mathcal{E}}}_{\pi|_V}((Z_1, \dots, Z_n)|_V)}Z_k, Z_{k + 1}, \dots, Z_n)|_{V^\complement}) &\text{if } \chi(\min(V)) = \ell\\
{{\mathcal{F}}}_{\pi|_{V^\complement}}((Z_1, \dots, Z_{k - 1}, R_{{{\mathcal{E}}}_{\pi|_V}((Z_1, \dots, Z_n)|_V)}Z_k, Z_{k + 1}, \dots, Z_n)|_{V^\complement}) &\text{if } \chi(\min(V)) = r\\
\end{cases}.$$
Again, let $\pi$ be the bi-non-crossing partition from Remark \[Decomposition\] and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$. Then $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_{12}) = {{\mathbb{F}}}\left(Z_1L_{{{\mathbb{E}}}(Z_2Z_4)}Z_6\right){{\mathbb{F}}}\left(Z_7L_{{{\mathbb{E}}}(Z_9Z_{12})}Z_{11}\right){{\mathbb{F}}}\left(Z_3R_{{{\mathbb{E}}}(Z_5)}Z_8Z_{10}\right).$$ In general, the rule is ‘one uses ${{\mathcal{E}}}$ to reduce the interior blocks and then factors ${{\mathcal{F}}}_\pi$ according to the remaining exterior blocks.’
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. The operator-valued conditionally bi-free moment pair $({{\mathcal{E}}}, {{\mathcal{F}}})$ on ${{\mathcal{A}}}$ is operator-valued conditionally bi-multiplicative.
The fact that the operator-valued bi-free moment function ${{\mathcal{E}}}$ on ${{\mathcal{A}}}$ is operator-valued bi-multiplicative is the main result of [@CNS2015-2]\*[Section 5]{}. It is also clear that the function ${{\mathcal{F}}}$ satisfies conditions $(1)$, $(2)$, and $(3)$ of Definition \[BiMulti\].
To demonstrate the modified condition $(4)$ in Definition \[CondBiMulti\], the proof relies on the techniques observed in [@CNS2015-2]\*[Subsection 5.3]{} which show that the function ${{\mathcal{E}}}$ satisfies condition $(4)$ of Definition \[BiMulti\]. In particular, we refer the reader to the proofs of [@CNS2015-2]\*[Lemmata 5.3.1 to 5.3.4]{} for additional details in that which follows. Under the same assumptions and notation, first note that the special case of the assertion holds under the additional assumption of [@CNS2015-2]\*[Lemma 5.3.1]{}; that is, there exists a block $W_0 \subset W$ of $\pi$ such that $$p, q, \min_{\prec_\chi}(\{1, \dots, n\}), \max_{\prec_\chi}(\{1, \dots, n\}) \in W_0.$$ Indeed, suppose $\chi(p) = \ell$ (the other case is similar), and note that $W_0$ is the only exterior block of $\pi$. By the same arguments as in the proof of [@CNS2015-2]\*[Lemma 5.3.1]{}, we have $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n) = {{\mathcal{F}}}_{\pi|_{W}}\left(\left(Z_1, \dots, Z_{p - 1}, Z_pL_{{{\mathcal{E}}}_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{p + 1}, \dots, Z_n\right)|_{W}\right)$$ for all three possible cases, i.e., $\chi(q) = \ell$; $\chi(q) = r$ and $p < q$; $\chi(q) = r$ and $p > q$.
To verify the modified condition $(4)$ in full generality, we examine the proof of [@CNS2015-2]\*[Lemma 5.3.4]{}. Suppose $\chi(p) = \ell$ (the other case is similar), and note that under the additional assumption of the modified condition $(4)$ that there exists a block $W_0 \subset W$ of $\pi$ such that $\min_{\prec_\chi}(\{1, \dots, n\}), \max_{\prec_\chi}(\{1, \dots, n\}) \in W_0$, the block $W_0$ is always the only exterior block of $\pi$. Let $$\alpha = \max_{\prec_\chi}\left(\left\{k \in W_0 \, \mid \, k \preceq_\chi p\right\}\right), \quad \beta = \min_{\prec_\chi}\left(\left\{k \in W_0 \, \mid \, q \preceq_\chi k\right\}\right),$$ and let $U = \{k \, \mid \, \alpha \prec_\chi k \prec_\chi \beta\}$. Thus $U$ is a union of blocks of $\pi$. Let $\overline{W_0} = U^{\complement}$. Then, by the special case above (with $U$ being the $\chi$-interval), we have $${{\mathcal{F}}}_\pi(Z_1, \dots, Z_n) = {{\mathcal{F}}}_{\pi|_{\overline{W_0}}}\left(\left(Z_1, \dots, Z_{\alpha - 1}, Z_\alpha L_{{{\mathcal{E}}}_{\pi|_{U}}\left((Z_1, \dots, Z_n)|_{U}\right)}, Z_{\alpha + 1}, \dots, Z_n\right)|_{\overline{W_0}}\right).$$ Since ${{\mathcal{E}}}$ is operator-valued bi-multiplicative, we have $$Z_\alpha L_{{{\mathcal{E}}}_{\pi|_{U}}\left((Z_1, \dots, Z_n)|_{U}\right)} = Z_pL_{{{\mathcal{E}}}_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}L_{{{\mathcal{E}}}_{\pi|_{U \setminus V}}\left((Z_1, \dots, Z_n)|_{U \setminus V}\right)}$$ if $\alpha = p$, and $${{\mathcal{E}}}_{\pi|_{U}}\left((Z_1, \dots, Z_n)|_{U}\right) = {{\mathcal{E}}}_{\pi|_{U \setminus V}}\left(\left(Z_1, \dots, Z_{p - 1}, Z_pL_{{{\mathcal{E}}}_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{p + 1}, \dots, Z_n\right)|_{U \setminus V}\right)$$ otherwise. Since $W = \overline{W_0} \cup (U \setminus V)$, the assertion follows from applying the special case above in the opposite direction.
Operator-valued conditionally bi-free cumulant pairs
----------------------------------------------------
In this subsection, we recursively define the operator-valued conditionally bi-free cumulant pair $(\kappa, \mathcal{K})$ using the pair $({{\mathcal{E}}}, {{\mathcal{F}}})$ from the previous subsection and show that it is also operator-valued conditionally bi-multiplicative.
\[OpVCBFCumulants\] Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations and let $({{\mathcal{E}}}, {{\mathcal{F}}})$ be the operator-valued conditionally bi-free moment pair on ${{\mathcal{A}}}$. The *operator-valued conditionally bi-free cumulant pair* on ${{\mathcal{A}}}$ is the pair of functions $$\kappa: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ and $${{\mathcal{K}}}: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{D}}}$$ where $\kappa$ is the operator-valued bi-free cumulant function on ${{\mathcal{A}}}$ and ${{\mathcal{K}}}$ is recursively defined as follows.
1. If $n = 1$, then ${{\mathcal{K}}}_{1_{1,0}}(Z_\ell) = {{\mathcal{F}}}_{1_{1,0}}(Z_\ell)$ for $Z_\ell \in {{\mathcal{A}}}_\ell$ and ${{\mathcal{K}}}_{1_{0,1}}(Z_r) = {{\mathcal{F}}}_{1_{0,1}}(Z_r)$ for $Z_r \in {{\mathcal{A}}}_r$.
2. Fix $n \geq 2$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$. If $\pi \neq 1_\chi$, then let $V_1, \dots, V_m$ be the partition of $\pi$ as described in Remark \[Decomposition\]. We define $${{\mathcal{K}}}_\pi(Z_1, \dots, Z_n) = {{\mathcal{K}}}_{\pi|_{V_1}}((Z_1, \dots, Z_n)|_{V_1})\cdots {{\mathcal{K}}}_{\pi|_{V_m}}((Z_1, \dots, Z_n)|_{V_m}),$$ where each ${{\mathcal{K}}}_{\pi|_{V_k}}((Z_1, \dots, Z_n)|_{V_k})$ is defined as follows. Let $V'_k \subset V_k$ be the block containing $\min_{\prec_\chi}(V_k)$ and $\max_{\prec_\chi}(V_k)$, let $V \subset V_k \setminus V_k'$ be the block which terminates closest to the bottom (compared to other blocks of $V_k$). If $p = \max_{\prec_\chi}\left(\left\{j \in V_k \, \mid \, j \prec_\chi \min_{\prec_\chi}(V)\right\}\right)$ define $$\begin{aligned}
&{{\mathcal{K}}}_{\pi|_{V_k}}((Z_1, \dots, Z_n)|_{V_k})\\
&= \begin{cases}
{{\mathcal{K}}}_{\pi|_{V_k \setminus V}}\left(\left(Z_1, \dots, Z_{p - 1}, Z_pL_{\kappa_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}, Z_{p + 1}, \dots, Z_n\right)|_{V_k \setminus V}\right) &\text{if } \chi(p) = \ell\\
{{\mathcal{K}}}_{\pi|_{V_k \setminus V}}\left(\left(Z_1, \dots, Z_{p - 1}, R_{\kappa_{\pi|_{V}}\left((Z_1, \dots, Z_n)|_{V}\right)}Z_p, Z_{p + 1}, \dots, Z_n\right)|_{V_k \setminus V}\right) &\text{if } \chi(p) = r
\end{cases}.\end{aligned}$$ Repeat this process until the only remaining block of $V_k$ is $V'_k$.
3. Otherwise $\pi = 1_\chi$ and define $${{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_n) = {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \neq 1_\chi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_n).$$
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. The operator-valued conditionally bi-free cumulant pair $(\kappa, {{\mathcal{K}}})$ on ${{\mathcal{A}}}$ is operator-valued conditionally bi-multiplicative.
The fact that the operator-valued bi-free cumulant function $\kappa$ on ${{\mathcal{A}}}$ is operator-valued bi-multiplicative was proved in [@CNS2015-2]\*[Section 6]{}. Moreover, it is easy to see that the function ${{\mathcal{K}}}$ satisfies condition $(3)$ of Definition \[BiMulti\].
For condition $(1)$ of Definition \[BiMulti\] we will proceed by induction on $n$ to show that condition $(1)$ holds in greater generality. To be specific, we will demonstrate that condition $(1)$ holds whenever $1_\chi$ is replaced with $\pi$. To proceed, note the base case where $n = 1$ is trivial. For the inductive step, suppose the assertion holds for all $1 \leq n_0 \leq n - 1$, $\chi_0: \{1, \dots, n_0\} \to \{\ell, r\}$, and $\pi_0 \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_0)$. Suppose $\chi : \{1,\ldots, n\} \to \{\ell, r\}$ and that $\chi(n) = \ell$ (as the other case is similar). If $q = -\infty$, then $\chi: \{1, \dots, n\} \to \{\ell\}$ is the constant map, and thus for each $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, $n$ necessarily belongs to an exterior block of $\pi$. Since ${{\mathcal{K}}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b)$ factors according to the exterior blocks of $\pi$, we have ${{\mathcal{K}}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b) = {{\mathcal{K}}}_\pi(Z_1, \dots, Z_n)b$ if $\pi \neq 1_\chi$ by the induction hypothesis. Thus $$\begin{aligned}
{{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_nL_b) &= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_nL_b) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \neq 1_\chi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n)b - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \neq 1_\chi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_n)b\\
&= {{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_n)b.\end{aligned}$$
If $q \neq -\infty$ and $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ such that $\pi \neq 1_\chi$, then as $n$ and $q$ are adjacent with respect to $\prec_\chi$, we have the following possible cases:
1. $n, q \in V$ such that $V$ is an interior block of $\pi$;
2. $n, q \in V$ such that $V$ is an exterior block of $\pi$;
3. $n \in V_1$ and $q \in V_2$ such that $n= \max_{\prec_\chi}(V_1) \prec_\chi \min_{\prec_\chi}(V_2) = q$, and both of $V_1$ and $V_2$ are interior blocks of $\pi$;
4. $n \in V_1$ and $q \in V_2$ such that $n= \max_{\prec_\chi}(V_1) \prec_\chi \min_{\prec_\chi}(V_2)= q$, and both of $V_1$ and $V_2$ are exterior blocks of $\pi$;
5. $n \in V_1$ and $q \in V_2$ such that $\min_{\prec_\chi}(V_2) \prec_\chi \min_{\prec_\chi}(V_1)$ (thus $V_1$ is interior with respect to $V_2$), and $V_2$ is an interior block of $\pi$;
6. $n \in V_1$ and $q \in V_2$ such that $\min_{\prec_\chi}(V_2) \prec_\chi \min_{\prec_\chi}(V_1)$ (thus $V_1$ is interior with respect to $V_2$), and $V_2$ is an exterior block of $\pi$;
7. $n \in V_1$ and $q \in V_2$ such that $\max_{\prec_\chi}(V_2) \prec_\chi \max_{\prec_\chi}(V_1)$ (thus $V_2$ is interior with respect to $V_1$), and $V_1$ is an interior block of $\pi$;
8. $n \in V_1$ and $q \in V_2$ such that $\max_{\prec_\chi}(V_2) \prec_\chi \max_{\prec_\chi}(V_1)$ (thus $V_2$ is interior with respect to $V_1$), and $V_1$ is an exterior block of $\pi$.
Since $\pi \neq 1_\chi$, cases $(i)$, $(ii)$, $(iii)$, $(v)$, $(vi)$, $(vii)$, and $(viii)$ follow from the induction hypothesis and from the fact that $\kappa$ is operator-valued bi-multiplicative. For case $(iv)$, $V_1 \subset \chi^{-1}(\{\ell\})$ and $V_2 \subset \chi^{-1}(\{r\})$, so the result follows from the $q = -\infty$ situation (and the proof where $\chi(n) = r$ which must be run simultaneously with induction). Therefore, we have $${{\mathcal{K}}}_\pi(Z_1, \dots, Z_{n - 1}, Z_nL_b) = {{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, Z_qR_b, Z_{q + 1}, \dots, Z_n)$$ for all $\pi \neq 1_\chi$, and hence $${{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{n - 1}, Z_nL_b) = {{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{q - 1}, Z_qR_b, Z_{q + 1}, \dots, Z_n)$$ by the same calculation as the $q = -\infty$ situation.
The verification for condition $(2)$ of Definition \[BiMulti\] follows from essentially the same induction arguments and casework as above with $p$ replacing $n$. The only difference is that if $q = -\infty$, then $p$ is the smallest element with respect to $\prec_\chi$, and hence necessarily belongs to an exterior block of $\pi$. Note this shows that the function ${{\mathcal{K}}}$ actually satisfies the additional properties that conditions $(1)$ and $(2)$ of Definition \[BiMulti\] hold for all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$.
Finally, for the modified condition $(4)$ as given in Definition \[CondBiMulti\], the result follows from the extended conditions $(1)$ and $(2)$ of Definition \[BiMulti\] as stated above along with the recursive definition in Definition \[CBFMomentPair\] and the fact that $\kappa$ is operator-valued bi-multiplicative.
Universal moment expressions for c-bi-free independence with amalgamation {#sec:moment-express}
=========================================================================
In this section, we will demonstrate that a family of pairs of ${{\mathcal{B}}}$-algebras is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if certain operator-valued moment expressions hold. To do so, we note that the shaded diagrams from Definition \[ShadedDiagrams\] and [@CNS2015-2]\*[Lemma 7.1.3]{} will be useful.
Let $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}_k^\circ, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$ be a family of ${{\mathcal{B}}}$-${{\mathcal{B}}}$-bimodules with pairs of specified $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued states, let $\lambda_k$ and $\rho_k$ be the left and right representations of ${{\mathcal{L}}}({{\mathcal{X}}}_k)$ on ${{\mathcal{L}}}({{\mathcal{X}}})$, and let ${{\mathcal{X}}}= (*_{{\mathcal{B}}})_{k \in K}{{\mathcal{X}}}_k$. Fix $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, $Z_k \in {{\mathcal{L}}}_{\chi(k)}({{\mathcal{X}}}_{\omega(k)})$, and let $\mu_k(Z_k) = \lambda_{\omega(k)}(Z_k)$ if $\chi(k) = \ell$ and $\mu_k(Z_k) = \rho_{\omega(k)}(Z_k)$ if $\chi(k) = r$.
For $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{\mathrm{lat}}(\chi, \omega)$, recursively define ${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$ as follows: If $D \in {{\mathcal{L}}}{{\mathcal{R}}}_0^{\mathrm{lat}}(\chi, \omega)$, then $${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = ({{\mathcal{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})})_\pi(\mu_1(Z_1), \dots, \mu_n(Z_n)) \in {{\mathcal{B}}},$$ where $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ is the bi-non-crossing partition corresponding to $D$. If every block of $D$ has a spine reaching the top, then enumerate the blocks from left to right according to their spines as $V_1, \dots, V_m$ with $V_j = \{k_{j, 1} < \cdots < k_{j, q_j}\}$, and set $${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = [(1 - \mathfrak{p}_{\omega(k_{1, 1})})Z_{k_{1, 1}}\cdots Z_{k_{1, q_1}}(1 \oplus 0)] \otimes \cdots \otimes [(1 - \mathfrak{p}_{\omega(k_{m, 1})})Z_{k_{m, 1}}\cdots Z_{k_{m, q_m}}(1 \oplus 0)],$$ which is an element of ${{\mathcal{X}}}^\circ$. Otherwise, apply the recursive process using ${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ as in Definition \[E-pi\] until every block of $D$ has a spine reaching the top.
Under the above assumptions and notation, it was demonstrated in [@CNS2015-2]\*[Lemma 7.1.3]{} that $$\label{MomentExpression}
\mu_1(Z_1)\cdots\mu_n(Z_n)(1 \oplus 0) = \sum_{k = 0}^n\sum_{D \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)}\left[\sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}_k(\chi, \omega)\\D' \geq_{\mathrm{lat}}D}}(-1)^{|D| - |D'|}\right]{{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$$ and, consequently, $$\label{E-Moment}
{{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n)) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)}\left[\sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \leq \sigma \leq \omega}}\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}(\pi, \sigma)\right]({{\mathcal{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})})_\pi(\mu_1(Z_1), \dots, \mu_n(Z_n)).$$
For $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$, we define ${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$ by exactly the same recursive process that used to define ${{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))$ for $D' \in {{\mathcal{L}}}{{\mathcal{R}}}^{\mathrm{lat}}(\chi, \omega)$. Note that, unlike ${{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))$, it is not necessarily true that ${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) \in {{\mathcal{X}}}$ for all $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ as such diagrams may have spines reaching the top which do not alternate in colour.
If ${{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = X_1 \otimes \cdots \otimes X_m$, let $$\mathfrak{q}({{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))) = \mathfrak{q}(X_1)\cdots\mathfrak{q}(X_m) \in {{\mathcal{D}}}.$$ Observe that although it is possible $X_1 \otimes \cdots \otimes X_m \notin {{\mathcal{X}}}^\circ$, it is still true that every $X_j$ belongs to some ${{\mathcal{X}}}_{k_j}^\circ$, and thus the above expression makes sense.
Finally for $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$, recursively define ${{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$ as follows: If $D \in {{\mathcal{L}}}{{\mathcal{R}}}_0^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$, then $${{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = {{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) \in {{\mathcal{B}}}\subset {{\mathcal{D}}}.$$ If every block of $D$ has a spine reaching the top, then enumerate the blocks from left to right according to their spines as $V_1, \dots, V_m$ with $V_j = \{k_{j, 1} < \cdots < k_{j, q_j}\}$, and set $${{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) = {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z_{k_{1, 1}}\cdots Z_{k_{1, q_1}})\cdots{{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z_{k_{m, 1}}\cdots Z_{k_{m, q_m}}) \in {{\mathcal{D}}}.$$ Otherwise, apply the recursive process using ${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ as in Definition \[E-pi\] until every block of $D$ has a spine reaching the top.
Note the values of ${{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))$ depend only on the values of ${{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}}_{k_j})}(Z_{k_{j, 1}}\cdots Z_{k_{j, q_j}})$ and the values of ${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}}_{k_j})}(Z_{k_{j, 1}}\cdots Z_{k_{j, q_j}})$ for some $k_j \in K$. Hence ${{\mathbb{F}}}_D$ makes sense in any ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-vector expectations and will not depend on the representation of the pairs of ${{\mathcal{B}}}$-algebras.
\[ChangeCoeff\] Under the above assumptions and notation, for all $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{\mathrm{lat}}(\chi, \omega)$ $$\sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)\\D' \leq_{\mathrm{cap}}D}}\mathfrak{q}\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right) = {{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)).$$
If $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{\mathrm{lat}}_0(\chi, \omega)$, then the only diagram $D' \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ such that $D' \leq_{\mathrm{cap}}D$ is $D$ itself. Thus the equation is trivially true by definition in this case.
For $D \in {{\mathcal{L}}}{{\mathcal{R}}}_m^{\mathrm{lat}}(\chi, \omega)$ with $0 < m \leq n$, it suffices to prove the following claim: Let $V_1, \dots, V_m$ be the blocks of $D$ with spines reaching the top, ordered from left to right according to their spines, let $V_1 = \{k_{1, 1} < \cdots < k_{1, q_1}\}$, and let $V_{1, 1}, \dots, V_{1, m_1}$ be the blocks of $D$ which reduce to appropriate $L_b$ or $R_b$ multiplied on the left and/or right of some $Z_{k_{1, j}}$ in the recursive process. Suppose $D', D'' \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ are such that $D' \leq_{{\mathrm{cap}}} D$, $D'' \leq_{{\mathrm{cap}}} D$, the spine of the block $V_1$ reaches the top in $D'$ but not in $D''$, and the spines of all other blocks in $D'$ and $D''$ agree. We claim that $$\begin{aligned}
&\mathfrak{q}\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right) + \mathfrak{q}\left({{\mathbb{E}}}_{D''}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)\\
&= {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}})\mathfrak{q}\left({{\mathbb{E}}}_{D' \setminus (V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1})}\left((\mu_1(Z_1), \dots, \mu_n(Z_n))|_{D' \setminus (V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1})}\right)\right),\end{aligned}$$ where $Z'_{k_{1, j}}$ is $Z_{k_{1, j}}$, potentially multiplied on the left and/or right by appropriate $L_b$ and $R_b$ such that the multiplications correspond to the blocks $V_{1, 1}, \dots, V_{1, m_1}$.
Indeed, if the claim is true, then for a given $D$ as above, the spine of $V_1$ reaches the top in exactly half of the cappings of $D$ and each such capping $D'$ can be paired with another capping $D''$ such that the only difference between $D'$ and $D''$ is that the spine of $V_1$ does not reach the top in $D''$. Adding up $\mathfrak{q}\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)$ and $\mathfrak{q}\left({{\mathbb{E}}}_{D''}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)$ for all pairs yield the result by induction.
To prove the claim, note if $m = 1$ (that is, the only spine that reaches the top is the spine of $V_1$), then $V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1} = D'$ and we have $$\begin{aligned}
\mathfrak{q}\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right) &= {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}}) - {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}})\\
&= {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}}) - \mathfrak{q}\left({{\mathbb{E}}}_{D''}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)\end{aligned}$$ as $D''$ has no spine reaching the top and $\mathfrak{q}(b) = b$. Thus the result follows when $m = 1$.
Otherwise, $m > 1$. Let $V = V_1 \cup V_{1, 1} \cup \cdots \cup V_{1, m_1}$. Since left ${{\mathcal{B}}}$-operators commute with elements of ${{\mathcal{L}}}_r({{\mathcal{X}}})$, right ${{\mathcal{B}}}$-operators commute with elements of ${{\mathcal{L}}}_\ell({{\mathcal{X}}})$, and by the properties of ${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ and ${{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}$ (i.e., there are bi-multiplicative-like properties implied by the recursive definition), it can be checked via casework that $$\begin{aligned}
\mathfrak{q}&\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right) \\
&= \left({{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}}) - {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}})\right)\mathfrak{q}\left({{\mathbb{E}}}_{D' \setminus V}\left((\mu_1(Z_1), \dots, \mu_n(Z_n))|_{D' \setminus V}\right)\right)\end{aligned}$$
and $${{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(Z'_{k_{1, 1}}\cdots Z'_{k_{1, q_1}})\mathfrak{q}\left({{\mathbb{E}}}_{D' \setminus V}\left((\mu_1(Z_1), \dots, \mu_n(Z_n))|_{D' \setminus V}\right)\right) = \mathfrak{q}\left({{\mathbb{E}}}_{D''}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)$$ for all $D'$ and $D''$. Thus the claim and proof follows.
To keep track of some coefficients that occur, we make the following definition.
\[CDandC’D\] For $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}_k(\chi, \omega)$, define $C'_D$ as follows: First define $$C_D = \begin{cases}
\displaystyle\sum_{\substack{D' \in \mathcal{L}\mathcal{R}_k(\chi, \omega)\\D' \geq_{\mathrm{lat}}D}}(-1)^{|D| - |D'|} &\text{if } D \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}}_k(\chi, \omega)\\
0 &\text{otherwise}
\end{cases}.$$ Recursively, starting with $k = n$, define $$C'_D = C_D - \sum^n_{m = k + 1}\sum_{\substack{D' \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}{\mathrm{cap}}}_m(\chi, \omega)\\D' \geq_{\mathrm{cap}}D}}C'_{D'}.$$
With Lemma \[ChangeCoeff\] complete, we obtain the following operator-valued analogue of [@GS2016]\*[Lemma 4.6]{}.
\[F-Moment\] Under the above assumptions and notation, $${{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n)) = \sum_{k = 0}^n\sum_{D \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}{\mathrm{cap}}}_k(\chi, \omega)}C'_D{{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)),$$ and $$C'_D = \sum_{\substack{ D' \in \mathcal{L}\mathcal{R}(\chi, \omega)\\D' \geq_{{\mathrm{lat}}{\mathrm{cap}}} D}}(-1)^{|D| - |D'|} = \sum^n_{m = k}\sum_{\substack{D' \in \mathcal{L}\mathcal{R}_m(\chi, \omega)\\D' \geq_{{\mathrm{lat}}{\mathrm{cap}}} D}}(-1)^{|D| - |D'|}$$ for $D \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}{\mathrm{cap}}}_k(\chi, \omega)$.
For $Z_1, \dots, Z_n$ as above, the expression ${{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n))$ is obtained by applying $\mathfrak{q}$ to the left-hand side of equation . Using Definition \[CDandC’D\], we have $$\begin{aligned}
&{{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n))\\
&= \mathfrak{q}\left(\mu_1(Z_1)\cdots\mu_n(Z_n)(1 \oplus 0)\right)\\
&= \sum_{k = 0}^n\sum_{D \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)}C_D\mathfrak{q}\left({{\mathbb{E}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)\\
&= \sum_{k = 0}^n\sum_{D \in {{\mathcal{L}}}{{\mathcal{R}}}_k^{\mathrm{lat}}(\chi, \omega)}C_D\left({{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)) - \sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)\\D' \leq_{\mathrm{cap}}D\\D' \neq D}}\mathfrak{q}\left({{\mathbb{E}}}_{D'}(\mu_1(Z_1), \dots, \mu_n(Z_n))\right)\right)\\
&= \sum_{k = 0}^n\sum_{D \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}{\mathrm{cap}}}_k(\chi, \omega)}C'_D{{\mathbb{F}}}_D(\mu_1(Z_1), \dots, \mu_n(Z_n)),\end{aligned}$$ where the third equality follows from Lemma \[ChangeCoeff\] and the fourth equality follows from Definition \[CDandC’D\] as the coefficient $C'_D$ for $D \in \mathcal{L}\mathcal{R}^{{\mathrm{lat}}{\mathrm{cap}}}_k(\chi, \omega)$ was specifically defined this way. The second result regarding $C'_D$ is exactly the content of [@GS2016]\*[Lemma 4.7]{}.
Combining these results, we have the following moment type characterization of c-bi-free independence with amalgamation.
\[MomentFormulae\] A family $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ of pairs of ${{\mathcal{B}}}$-algebras in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if $$\label{EA-Moment}
{{\mathbb{E}}}(Z_1\cdots Z_n) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)}\left[\sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \leq \sigma \leq \omega}}\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}(\pi, \sigma)\right]{{\mathcal{E}}}_\pi(Z_1, \dots, Z_n)$$ and $$\label{FA-Moment}
{{\mathbb{F}}}(Z_1\cdots Z_n) = \sum_{D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)}\left[\sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}(\chi, \omega)\\D' \geq_{{\mathrm{lat}}{\mathrm{cap}}} D}}(-1)^{|D| - |D'|}\right]{{\mathbb{F}}}_D(Z_1, \dots, Z_n)$$ for all $n \geq 1$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, and $Z_1, \dots, Z_n \in {{\mathcal{A}}}$ with $Z_k \in {{\mathcal{A}}}_{\omega(k), \chi(k)}$.
Under the above notation, if the family $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then there exists a family $\{({{\mathcal{X}}}_k, {{\mathcal{X}}}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)\}_{k \in K}$ such that $${{\mathbb{E}}}(Z_1\cdots Z_n) = {{\mathbb{E}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n)){\quad\text{and}\quad}{{\mathbb{F}}}(Z_1\cdots Z_n) = {{\mathbb{F}}}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\mu_1(Z_1)\cdots\mu_n(Z_n)),$$ where each $Z_k$ on the right-hand side of the above equations is identified as $\ell_k(Z_k)$ if $\chi(k) = \ell$ and $r_k(Z_k)$ if $\chi(k) = r$ acting on ${{\mathcal{X}}}_{\omega(k)}$. The fact that equation holds is part of [@CNS2015-2]\*[Theorem 7.1.4]{}, and the fact that equation holds follows from Lemma \[F-Moment\].
Conversely, suppose equations and hold. By Theorem \[Embedding\], there exist $(\mathcal{X}, \mathcal{X}^\circ, \mathfrak{p}, \mathfrak{q})$ and a unital homomorphism $\theta: {{\mathcal{A}}}\to {{\mathcal{L}}}({{\mathcal{X}}})$ such that $$\begin{gathered}
\theta(L_{b_1}R_{b_2}) = L_{b_1}R_{b_2},\quad\theta({{\mathcal{A}}}_\ell) \subset {{\mathcal{L}}}_\ell({{\mathcal{X}}}),\quad\theta({{\mathcal{A}}}_r) \subset {{\mathcal{L}}}_r({{\mathcal{X}}}),\\
\mathbb{E}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathbb{E}(Z),{\quad\text{and}\quad}\mathbb{F}_{{{\mathcal{L}}}({{\mathcal{X}}})}(\theta(Z)) = \mathbb{F}(Z)\end{gathered}$$ for all $b_1, b_2 \in {{\mathcal{B}}}$ and $Z \in {{\mathcal{A}}}$. For each $k \in K$, let $({{\mathcal{X}}}_k, {{\mathcal{X}}}^\circ_k, \mathfrak{p}_k, \mathfrak{q}_k)$ be a copy of $({{\mathcal{X}}}, {{\mathcal{X}}}^\circ, \mathfrak{p}, \mathfrak{q})$, and let $\ell_k$ and $r_k$ be copies of $\theta: {{\mathcal{A}}}\to {{\mathcal{L}}}({{\mathcal{X}}}_k)$. By [@CNS2015-2]\*[Lemma 7.1.3]{} and Lemma \[F-Moment\], we have $${{\mathbb{E}}}(Z_1\cdots Z_n) = {{\mathbb{E}}}_{{{\mathcal{L}}}((*_{{\mathcal{B}}})_{k \in K}{{\mathcal{X}}}_k)}(\mu_1(Z_1)\cdots\mu_n(Z_n)){\quad\text{and}\quad}{{\mathbb{F}}}(Z_1\cdots Z_n) = {{\mathbb{F}}}_{{{\mathcal{L}}}((*_{{\mathcal{B}}})_{k \in K}{{\mathcal{X}}}_k)}(\mu_1(Z_1)\cdots\mu_n(Z_n)),$$ where each $Z_k$ on the right-hand side of the above equations is identified as $\theta(Z_k)$ acting on ${{\mathcal{X}}}_{\omega(k)}$. Hence, the family $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ by definition.
As ${{\mathbb{F}}}_D(Z_1, \ldots, Z_n)$ and ${{\mathcal{E}}}_\pi(Z_1, \ldots, Z_n)$ depend only on the distributions of individual pairs $({{\mathcal{A}}}_{k,\ell}, {{\mathcal{A}}}_{k, r})$ inside our ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, we obtain that Definition \[defn:op-c-bi-free-definition\] is well-defined in that the joint distributions do not depend on the representations.
Additivity of operator-valued conditionally bi-free cumulant pairs {#sec:additivity}
==================================================================
The goal of this section is to prove the operator-valued analogue of [@GS2016]\*[Theorem 4.1]{}; namely that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. To establish the result, we will need a method, analogous to [@S2015]\*[Lemma 3.8]{} for constructing a pair of ${{\mathcal{B}}}$-algebras with any given operator-valued bi-free and conditionally bi-free cumulants. To this end, we discuss moment and cumulant series first.
Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, and let $({{\mathcal{C}}}_\ell, {{\mathcal{C}}}_r)$ be a pair of ${{\mathcal{B}}}$-algebras such that $${{\mathcal{C}}}_\ell = {\mathrm{alg}}(\{Z_i\}_{i \in I}, \varepsilon({{\mathcal{B}}}\otimes 1)) {\quad\text{and}\quad}{{\mathcal{C}}}_r = {\mathrm{alg}}(\{Z_j\}_{j \in J}, \varepsilon(1 \otimes {{\mathcal{B}}}^{\mathrm{op}}))$$ for some $\{Z_i\}_{i \in I} \subset {{\mathcal{A}}}_\ell$ and $\{Z_j\}_{j \in J} \subset {{\mathcal{A}}}_r$. By discussions in [@S2015]\*[Section 2]{} and by using the operator-valued conditionally bi-multiplicative properties, only certain operator-valued bi-free and conditionally bi-free moments/cumulants are required to study the joint distributions of elements in ${\mathrm{alg}}({{\mathcal{C}}}_\ell, {{\mathcal{C}}}_r)$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$. We make the following notation (in addition to [@S2015]\*[Notation 2.18]{} with some slight notational changes) and definition to describe the necessary moments and cumulants.
\[MCSeries\] Let ${{\mathcal{Z}}}= \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$ be as above, $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
- If $\omega(k) \in I$ for all $k$, define $$\begin{aligned}
\nu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathbb{E}}}\left(Z_{\omega(1)}L_{b_1}Z_{\omega(2)}\cdots L_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{B}}},\\
\mu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathbb{F}}}\left(Z_{\omega(1)}L_{b_1}Z_{\omega(2)}\cdots L_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{D}}},\\
\rho_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= \kappa_{1_{\chi_\omega}}\left(Z_{\omega(1)}, L_{b_1}Z_{\omega(2)}, \dots, L_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{B}}}, \text{ and}\\
\eta_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathcal{K}}}_{1_{\chi_\omega}}\left(Z_{\omega(1)}, L_{b_1}Z_{\omega(2)}, \dots, L_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{D}}}.\end{aligned}$$
- If $\omega(k) \in J$ for all $k$, define $$\begin{aligned}
\nu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathbb{E}}}\left(Z_{\omega(1)}R_{b_1}Z_{\omega(2)}\cdots R_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{B}}},\\
\mu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathbb{F}}}\left(Z_{\omega(1)}R_{b_1}Z_{\omega(2)}\cdots R_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{D}}},\\
\rho_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= \kappa_{1_{\chi_\omega}}\left(Z_{\omega(1)}, R_{b_1}Z_{\omega(2)}, \dots, R_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{B}}}, \text{ and}\\
\eta_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) &= {{\mathcal{K}}}_{1_{\chi_\omega}}\left(Z_{\omega(1)}, R_{b_1}Z_{\omega(2)}, \dots, R_{b_{n - 1}}Z_{\omega(n)}\right) \in {{\mathcal{D}}}.\end{aligned}$$
- Otherwise, let $k_\ell = \min\{k \, \mid \, \omega(k) \in I\}$ and $k_r = \min\{k \, \mid \, \omega(k) \in J\}$. Then $\{k_\ell, k_r\} = \{1, k_0\}$ for some $k_0$. Define $\nu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$ and $\mu_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$ to be $${{\mathbb{E}}}\left(Z_{\omega(1)}C_{b_1}^{\omega(2)}Z_{\omega(2)}\cdots C_{b_{k_0 - 2}}^{\omega(k_0 - 1)}Z_{\omega(k_0 - 1)}Z_{\omega(k_0)}C_{b_{k_0 - 1}}^{\omega(k_0 + 1)}Z_{\omega(k_0 + 1)}\cdots C_{b_{n - 3}}^{\omega(n - 1)}Z_{\omega(n - 1)}C_{b_{n - 2}}^{\omega(n)}Z_{\omega(n)}C_{b_{n - 1}}^{\omega(n)}\right) \in {{\mathcal{B}}}$$ and $${{\mathbb{F}}}\left(Z_{\omega(1)}C_{b_1}^{\omega(2)}Z_{\omega(2)}\cdots C_{b_{k_0 - 2}}^{\omega(k_0 - 1)}Z_{\omega(k_0 - 1)}Z_{\omega(k_0)}C_{b_{k_0 - 1}}^{\omega(k_0 + 1)}Z_{\omega(k_0 + 1)}\cdots C_{b_{n - 3}}^{\omega(n - 1)}Z_{\omega(n - 1)}C_{b_{n - 2}}^{\omega(n)}Z_{\omega(n)}C_{b_{n - 1}}^{\omega(n)}\right) \in {{\mathcal{D}}}$$ respectively, and define $\rho_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$ and $\eta_{\omega}^{\mathcal{Z}}(b_1, \dots, b_{n - 1})$ to be $$\kappa_{1_{\chi_\omega}}\left(Z_{\omega(1)}, C_{b_1}^{\omega(2)}Z_{\omega(2)}, \dots, C_{b_{k_0 - 2}}^{\omega(k_0 - 1)}Z_{\omega(k_0 - 1)}, Z_{\omega(k_0)}, C_{b_{k_0 - 1}}^{\omega(k_0 + 1)}Z_{\omega(k_0 + 1)}, \dots, C_{b_{n - 3}}^{\omega(n - 1)}Z_{\omega(n - 1)}, C_{b_{n - 2}}^{\omega(n)}Z_{\omega(n)}C_{b_{n - 1}}^{\omega(n)}\right) \in {{\mathcal{B}}}$$ and $${{\mathcal{K}}}_{1_{\chi_\omega}}\left(Z_{\omega(1)}, C_{b_1}^{\omega(2)}Z_{\omega(2)}, \dots, C_{b_{k_0 - 2}}^{\omega(k_0 - 1)}Z_{\omega(k_0 - 1)}, Z_{\omega(k_0)}, C_{b_{k_0 - 1}}^{\omega(k_0 + 1)}Z_{\omega(k_0 + 1)}, \dots, C_{b_{n - 3}}^{\omega(n - 1)}Z_{\omega(n - 1)}, C_{b_{n - 2}}^{\omega(n)}Z_{\omega(n)}C_{b_{n - 1}}^{\omega(n)}\right) \in {{\mathcal{D}}}$$ respectively, where $$C_b^{\omega(k)} = \begin{cases}
L_b & \text{if } \omega(k) \in I\\
R_b & \text{if } \omega(k) \in J
\end{cases}.$$
\[MCSeriesDefn\] Let ${{\mathcal{Z}}}= \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$ be as above. The *moment and cumulant series* of ${{\mathcal{Z}}}$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$ are the collections of maps $$\begin{aligned}
\nu^{{\mathcal{Z}}}&= \{\nu_\omega^{{{\mathcal{Z}}}}: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{B}}}\, \mid \, n \geq 1, \omega: \{1, \dots, n\} \to I \sqcup J\},\\
\mu^{{\mathcal{Z}}}&= \{\mu_\omega^{{{\mathcal{Z}}}}: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{D}}}\, \mid \, n \geq 1, \omega: \{1, \dots, n\} \to I \sqcup J\},\end{aligned}$$ and $$\begin{aligned}
\rho^{{\mathcal{Z}}}&= \{\rho_\omega^{{{\mathcal{Z}}}}: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{B}}}\, \mid \, n \geq 1, \omega: \{1, \dots, n\} \to I \sqcup J\},\\
\eta^{{\mathcal{Z}}}&= \{\eta_\omega^{{{\mathcal{Z}}}}: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{D}}}\, \mid \, n \geq 1, \omega: \{1, \dots, n\} \to I \sqcup J\},\end{aligned}$$ respectively. Note that if $n = 1$, then $\nu_\omega^{{\mathcal{Z}}}= \rho_\omega^{{\mathcal{Z}}}= {{\mathbb{E}}}(Z_{\omega(1)})$ and $\mu_\omega^{{\mathcal{Z}}}= \eta_\omega^{{\mathcal{Z}}}= {{\mathbb{F}}}(Z_{\omega(1)})$.
\[Existence\] Let $I$ and $J$ be non-empty disjoint index sets, and let ${{\mathcal{B}}}$ and ${{\mathcal{D}}}$ be unital algebras such that $1 := 1_{{\mathcal{D}}}\in {{\mathcal{B}}}\subset {{\mathcal{D}}}$. For every $n \geq 1$ and $\omega: \{1, \dots, n\} \to I \sqcup J$, let $\Theta_\omega: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{B}}}$ and $\Upsilon_\omega: {{\mathcal{B}}}^{n - 1} \to {{\mathcal{D}}}$ be linear in each coordinate. There exist a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ and elements $\{Z_i\}_{i \in I} \subset {{\mathcal{A}}}_\ell$ and $\{Z_j\}_{j \in J} \subset {{\mathcal{A}}}_r$ such that if $\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$, then $$\rho_\omega^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) = \Theta_\omega(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\eta_\omega^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) = \Upsilon_\omega(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
By the same construction presented in the proof of [@S2015]\*[Lemma 3.8]{}, there exist a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space $({{\mathcal{A}}}, {{\mathbb{E}}}, \varepsilon)$ and $\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$ with $\{Z_i\}_{i \in I} \subset {{\mathcal{A}}}_\ell$ and $\{Z_j\}_{j \in J} \subset {{\mathcal{A}}}_r$ such that $$\rho_\omega^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) = \Theta_\omega(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. Thus we need only define an expectation ${{\mathbb{F}}}$ to produce the correct operator-valued conditionally bi-free cumulants.
For $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n + 1} \in {{\mathcal{B}}}$, let $$C_b^{\omega(k)} = \begin{cases}
L_b & \text{if } \omega(k) \in I\\
R_b & \text{if } \omega(k) \in J
\end{cases},$$ and define $$\widehat{\Upsilon}_{1_{\chi_\omega}}\left(C_{b_1}^{\omega(1)}Z_{\omega(1)}, \dots, C_{b_{n - 1}}^{\omega(n - 1)}Z_{\omega(n - 1)}, C_{b_n}^{\omega(n)}Z_{\omega(n)}C_{b_{n + 1}}^{\omega(n)}\right) \in {{\mathcal{D}}}$$ like how $\widehat{\Theta}_{1_{\chi_\omega}}$ is defined in the proof of [@S2015]\*[Lemma 3.8]{} using $\Upsilon_\omega$ instead of $\Theta_\omega$. Subsequently, for $\omega: \{1, \dots, n\} \to I \sqcup J$ and $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$, define $$\widehat{\Upsilon}_\pi\left(C_{b_1}^{\omega(1)}Z_{\omega(1)}, \dots, C_{b_{n - 1}}^{\omega(n - 1)}Z_{\omega(n - 1)}, C_{b_n}^{\omega(n)}Z_{\omega(n)}C_{b_{n + 1}}^{\omega(n)}\right) \in {{\mathcal{D}}}$$ by selecting one of the many possible ways to reduce an operator-valued conditionally bi-multiplicative function where $\widehat{\Theta}_{1_\chi}$ is used for interior blocks and $\widehat{\Upsilon}_{1_\chi}$ is used for exterior blocks.
As seen in the proof of [@S2015]\*[Lemma 3.8]{}, every element in ${{\mathcal{A}}}$ is a linear combination of the form $$C_{b_1}^{\omega(1)}Z_{\omega(1)}\cdots C_{b_n}^{\omega(n)}Z_{\omega(n)}L_bR_{b'} + \mathcal{I},$$ where $n \geq 0$, $\omega: \{1, \dots, n\} \to I \sqcup J$ when $n \geq 1$, $b_1, \dots, b_n, b, b' \in {{\mathcal{B}}}$, and $\mathcal{I}$ is some two-sided ideal. Define ${{\mathbb{F}}}: {{\mathcal{A}}}\to {{\mathcal{D}}}$ by $${{\mathbb{F}}}(L_bR_{b'} + \mathcal{I}) = bb'$$ for all $b, b' \in {{\mathcal{B}}}$, and $${{\mathbb{F}}}\left(C_{b_1}^{\omega(1)}Z_{\omega(1)}\cdots C_{b_n}^{\omega(n)}Z_{\omega(n)}L_bR_{b'} + \mathcal{I}\right) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}\widehat{\Upsilon}_\pi\left(C_{b_1}^{\omega(1)}Z_{\omega(1)}, \dots, C_{b_{n - 1}}^{\omega(n - 1)}Z_{\omega(n - 1)}, C_{b_n}^{\omega(n)}Z_{\omega(n)}C_{bb'}\right)$$ for all $n \geq 1$ and $\omega: \{1, \dots, n\} \to I \sqcup J$, where $C_{bb'} = L_{bb'}$ if $\omega(n) \in I$ and $C_{bb'} = R_{bb'}$ if $\omega(n) \in J$, and extend ${{\mathbb{F}}}$ by linearity. By construction and commutation in ${{\mathcal{A}}}$, one can verify that ${{\mathbb{F}}}$ is well-defined and $${{\mathbb{F}}}(L_bR_{b'}Z + {{\mathcal{I}}}) = b{{\mathbb{F}}}(Z + {{\mathcal{I}}})b' {\quad\text{and}\quad}{{\mathbb{F}}}(ZL_b + {{\mathcal{I}}}) = {{\mathbb{F}}}(ZR_b + {{\mathcal{I}}})$$ for all $b, b' \in {{\mathcal{B}}}$ and $Z + {{\mathcal{I}}}\in {{\mathcal{A}}}$. Finally, since Definition \[OpVCBFCumulants\] completely determines the operator-valued conditionally bi-free cumulants and by our definition of $\hat{\Upsilon}$ via a choice of operator-valued conditionally bi-multiplicative reduction, [@S2015]\*[Lemma 3.8]{} with an induction argument together imply that if $\mathcal{Z} = \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$, then $$\eta_\omega^{\mathcal{Z}}(b_1, \dots, b_{n - 1}) = \Upsilon_\omega(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
We are now ready to prove the main result of this section.
\[VanishingEquiv\] A family $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ of pairs of ${{\mathcal{B}}}$-algebras in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if for all $n \geq 2$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, and $Z_k \in {{\mathcal{A}}}_{\omega(k), \chi(k)}$, we have $$\kappa_{1_\chi}(Z_1, \dots, Z_n) = {{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_n) = 0$$ whenever $\omega$ is not constant.
If all mixed cumulants vanish, then $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is bi-free over ${{\mathcal{B}}}$ so equation holds. To see that equation also holds, recall from [@GS2016]\*[Subsection 4.2]{} that ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$ denotes the set of all pairs $(\pi, \iota)$ where $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ is a bi-non-crossing partition and $\iota: \pi \to \{i, e\}$ is a function on the blocks of $\pi$. By Definitions \[CBFMomentPair\] and \[OpVCBFCumulants\], and the assumption that all mixed cumulants vanish, we have $${{\mathbb{F}}}(Z_1\cdots Z_n) = {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) = \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \leq \omega}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_n).$$ By applying Definition \[OpVCBFCumulants\] recursively, we obtain that $$\label{VanishingExpansion}
{{\mathbb{F}}}(Z_1\cdots Z_n) = \sum_{\substack{(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)\\\pi \leq \omega}}c(\chi, \omega; \pi, \iota)\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n),$$ where $c(\chi, \omega; \pi, \iota)\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n)$ for $(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$ is defined as follows: If there is an interior block $V$ of $\pi$ such that $\iota(V) = e$, then $c(\chi, \omega; \pi, \iota) = 0$. Otherwise, apply the recursive process using ${{\mathbb{E}}}$ as in Definition \[E-pi\] to the interior blocks of $\pi$, order the remaining $\chi$-intervals by $\prec_\chi$ as $V_1, \dots, V_m$, and define $$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n) = \Theta_{\pi|_{V_1}}((Z_1, \dots, Z_n)|_{V_1})\cdots\Theta_{\pi|_{V_m}}((Z_1, \dots, Z_n)|_{V_m}),$$ where $\Theta_{\pi|_{V_j}} = {{\mathbb{E}}}_{\pi|_{V_j}}$ if $\iota(V_j) = i$ and $\Theta_{\pi|_{V_j}} = {{\mathbb{F}}}_{\pi|_{V_j}}$ if $\iota(V_j) = e$.
Notice that, as with the scalar-valued case (see [@GS2016]\*[Remark 4.9]{}), $\Theta_{(\pi, \iota)}(Z_1, \dots, Z_n)$ and ${{\mathbb{F}}}_D(Z_1, \dots, Z_n)$ agree for certain $(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$ and $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$. Indeed, given $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$, defining $\pi$ via the blocks of $D$ and $\iota$ via $\iota(V) = e$ if the spine of $V$ reaches the top and $\iota(V) = i$ otherwise will produce such an equality.
If each $(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$ with $\pi \leq \omega$ and $c(\chi, \omega; \pi, \iota) \neq 0$ corresponds to some $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ in the sense as described above and $$c(\chi, \omega; \pi, \iota) = \sum_{\substack{D' \in {{\mathcal{L}}}{{\mathcal{R}}}(\chi, \omega)\\D' \geq_{{\mathrm{lat}}{\mathrm{cap}}} D}}(-1)^{|D| - |D'|}$$ for such $(\pi, \iota)$, then equations and coincide implying that $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ by Theorem \[MomentFormulae\]. Since the property that $(\pi, \iota)$ corresponds to a $D \in {{\mathcal{L}}}{{\mathcal{R}}}^{{\mathrm{lat}}{\mathrm{cap}}}(\chi, \omega)$ and the value of $c(\chi, \omega; \pi, \iota)$ do not depend on the algebras ${{\mathcal{B}}}$ and ${{\mathcal{D}}}$, the result follows from the ${{\mathcal{B}}}= {{\mathcal{D}}}= {{\mathbb{C}}}$ case by [@GS2016]\*[Lemma 4.13]{}.
Conversely, if $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then equations and hold by Theorem \[MomentFormulae\]. As shown in [@CNS2015-2]\*[Theorem 8.1.1]{}, equation is equivalent to the vanishing of mixed operator-valued bi-free cumulants. Thus we need only show that mixed operator-valued conditionally bi-free cumulants vanish. For fixed $n \geq 2$, $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\omega: \{1, \dots, n\} \to K$, and $Z_k \in {{\mathcal{A}}}_{\omega(k), \chi(k)}$, construct a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}', {{\mathbb{E}}}_{{{\mathcal{A}}}'}, {{\mathbb{F}}}_{{{\mathcal{A}}}'}, \varepsilon')$, pairs of ${{\mathcal{B}}}$-algebras $\{({{\mathcal{A}}}'_{k, \ell}, {{\mathcal{A}}}'_{k, r})\}_{k \in K}$, and elements $Z'_k \in {{\mathcal{A}}}'_{\omega(k), \chi(k)}$ such that
- for each $k \in \{1, \dots, n\}$, $\{Z'_j \, \mid \, \omega(j) = \omega(k), \chi(j) = \chi(k)\}$ generated ${{\mathcal{A}}}'_{\omega(k), \chi(k)}$,
- any joint operator-valued conditionally bi-free cumulant involving $Z'_1, \dots, Z'_n$ containing a pair $Z'_{k_1}, Z'_{k_2}$ with $\omega(k_1) \neq \omega(k_2)$ is zero, and
- for each $k \in \{1, \dots, n\}$, the joint distribution of $\{Z'_j \, \mid \, \omega(j) = \omega(k)\}$ with respect to $({{\mathbb{E}}}_{{{\mathcal{A}}}'}, {{\mathbb{F}}}_{{{\mathcal{A}}}'})$ equals the joint distribution of $\{Z_j \, \mid \, \omega(j) = \omega(k)\}$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$.
The above is possible via Lemma \[Existence\] by defining the operator-valued bi-free and conditionally bi-free cumulants appropriately.
By construction, $Z'_1, \dots, Z'_n$ have vanishing mixed cumulants and hence satisfy equations and by the first part of the proof. However, since for each $k \in \{1, \dots, n\}$, the joint distribution of $\{Z'_j \, \mid \, \omega(j) = \omega(k)\}$ with respect to $({{\mathbb{E}}}_{{{\mathcal{A}}}'}, {{\mathbb{F}}}_{{{\mathcal{A}}}'})$ equals the joint distribution of $\{Z_j \, \mid \, \omega(j) = \omega(k)\}$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$, equations and imply that the joint distribution of $Z_1, \dots, Z_n$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$ equals the joint distribution of $Z'_1, \dots, Z'_n$ with respect to $({{\mathbb{E}}}_{{{\mathcal{A}}}'}, {{\mathbb{F}}}_{{{\mathcal{A}}}'})$. Since the operator-valued bi-free and conditionally bi-free moments completely determine the operator-valued bi-free and conditionally bi-free cumulants, and since $Z'_1, \dots, Z'_n$ have vanishing mixed cumulants, the result follows.
Additional properties of c-bi-free independence with amalgamation {#sec:additional}
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In this section, we collect a list of additional properties of c-bi-free independence with amalgamation and operator-valued conditionally bi-free cumulants. All of the results below are analogues of known results in the current framework with essentially the same proofs. We begin by recalling the following notation from [@CNS2015-2]\*[Notation 6.3.1]{}.
Let $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $q \in \{1, \dots, n\}$. We denote by $\chi|_{\setminus q}$ the restriction of $\chi$ to the set $\{1, \dots, n\} \setminus q$. If $q \neq n$ and $\chi(q) = \chi(q + 1)$, define $\pi|_{q = q + 1} \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})$ to be the bi-non-crossing partition which results from identifying $q$ and $q + 1$ in $\pi$ (i.e., if $q$ and $q + 1$ are in the same block, then $\pi|_{q = q + 1}$ is obtained from $\pi$ by just removing $q$ from the block in which $q$ occurs, while if $q$ and $q + 1$ are in different blocks, then $\pi|_{q = q + 1}$ is obtained from $\pi$ by merging the two blocks and then removing $q$).
Vanishing of operator-valued cumulants
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The following demonstrates that, like with many other kinds of cumulants, the operator-valued conditionally bi-free cumulants of order at least two vanish if at least one input is a ${{\mathcal{B}}}$-operator.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, $\chi: \{1, \dots, n\} \to \{\ell, r\}$ with $n \geq 2$, and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$. If there exist $q \in \{1, \dots, n\}$ and $b \in {{\mathcal{B}}}$ such that $Z_q = L_b$ if $\chi(q) = \ell$ or $Z_q = R_b$ if $\chi(q) = r$, then $$\kappa_{1_\chi}(Z_1, \dots, Z_n) = \mathcal{K}_{1_\chi}(Z_1, \dots, Z_n) = 0.$$
The assertion that $\kappa_{1_\chi}(Z_1, \dots, Z_n) = 0$ was proved in [@CNS2015-2]\*[Proposition 6.4.1]{}, and the other assertion will be proved by induction with the base case easily verified by direct computations.
For the inductive step, suppose the assertion is true for all $\chi: \{1, \dots, m\} \to \{\ell, r\}$ with $2 \leq m \leq n - 1$. Fix $\chi: \{1, \dots, n\} \to \{\ell, r\}$ and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$. Suppose there exist $q \in \{1, \dots, n\}$ and $b \in {{\mathcal{B}}}$ such that $\chi(q) = \ell$ and $Z_q = L_b$ (the case $\chi(q) = r$ and $Z_q = R_b$ is similar). Let $$p = \max\{k \in \{1, \dots, n\} \, \mid \, \chi(k) = \ell, k < q\}.$$ There are two cases. If $p \neq -\infty$, then by the first assertion and the induction hypothesis, $$\begin{aligned}
{{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_n) &= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \neq 1_\chi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\{q\} \in \pi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, L_b, Z_{q + 1}, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_pL_b, Z_{p + 1}, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n)\end{aligned}$$ by properties of $(\kappa, {{\mathcal{K}}})$. On the other hand, we have $$\begin{aligned}
{{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) &= {{\mathbb{F}}}(Z_1\cdots Z_{q - 1}L_bZ_{q + 1}\cdots Z_n)\\
&= {{\mathbb{F}}}(Z_1\cdots Z_pL_bZ_{p + 1}\cdots Z_{q - 1}Z_{q + 1}\cdots Z_n)\\
&= {{\mathcal{F}}}_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_pL_b, Z_{p + 1}, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n)\\
&= \sum_{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_pL_b, Z_{p + 1}, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n),\end{aligned}$$ thus the assertion is true in this case. If $p = -\infty$, then by the first assertion and the induction hypothesis, $$\begin{aligned}
{{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_n) &= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\pi \neq 1_\chi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\{q\} \in \pi}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, L_b, Z_{q + 1}, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})}b{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n)\end{aligned}$$ by properties of $(\kappa, {{\mathcal{K}}})$ as $q = \min_{\prec_\chi}(\{1, \dots, n\})$ in this case. On the other hand, we have $$\begin{aligned}
{{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) &= {{\mathbb{F}}}(Z_1\cdots Z_{q - 1}L_bZ_{q + 1}\cdots Z_n)\\
&= {{\mathbb{F}}}(L_bZ_1\cdots Z_{q - 1}Z_{q + 1}\cdots Z_n)\\
&= b{{\mathcal{F}}}_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n)\\
&= \sum_{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})}b{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_{q - 1}, Z_{q + 1}, \dots, Z_n),\end{aligned}$$ thus the assertion is true in this case as well.
Operator-valued cumulants of products
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Next, we analyze operator-valued conditionally bi-free cumulants involving products of operators.
Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. If $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $Z_k \in {{\mathcal{A}}}_{\chi(k)}$, and $q \in \{1, \dots, n - 1\}$ with $\chi(q) = \chi(q + 1),$ then $${{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} = \pi}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n)$$ for all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})$.
We proceed by induction on $n$. If $n = 1$, there is nothing to check. If $n = 2$, then $${{\mathcal{K}}}_{0_\chi|_{1 = 2}}(Z_1Z_2) = {{\mathcal{K}}}_{1_\chi|_{1 = 2}}(Z_1Z_2) = {{\mathcal{F}}}_{1_\chi|_{1 = 2}}(Z_1Z_2) = {{\mathcal{F}}}_{1_\chi}(Z_1, Z_2) = {{\mathcal{K}}}_{0_\chi}(Z_1, Z_2) + {{\mathcal{K}}}_{1_\chi}(Z_1, Z_2)$$ as required. Suppose the assertion holds for $n - 1$, and note from [@CNS2015-2]\*[Theorem 6.3.5]{} that the analogous result also holds for the operator-valued bi-free cumulant function $\kappa$. Using the induction hypothesis and the operator-valued conditionally bi-multiplicativity of $(\kappa, {{\mathcal{K}}})$, we see for all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q}) \setminus \{1_{\chi|_{\setminus q}}\}$ that $${{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} = \pi}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n).$$ Hence, $$\begin{aligned}
&{{\mathcal{K}}}_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})\\\pi \neq 1_{\chi|_{\setminus q}}}}{{\mathcal{K}}}_\pi(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n)\\
&= {{\mathcal{F}}}_{1_\chi}(Z_1, \dots, Z_n) - \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})\\\pi \neq 1_{\chi|_{\setminus q}}}}\sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} = \pi}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n)\\
&= \sum_{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n) - \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} \neq 1_{\chi|_{\setminus q}}}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n)\\
&= \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} = 1_{\chi|_{\setminus q}}}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n),\end{aligned}$$ completing the inductive step.
Given two partitions $\pi, \sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, let $\pi \vee \sigma$ denote the smallest partition in ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ greater than $\pi$ and $\sigma$. Furthermore, suppose $m, n \geq 1$ with $m < n$ are fixed, and consider a sequence of integers $$0 = k(0) < k(1) < \cdots < k(m) = n.$$ For $\chi: \{1, \dots, m\} \to \{\ell, r\}$, define $\widehat{\chi}: \{1, \dots, n\} \to \{\ell, r\}$ by $$\widehat{\chi}(q) = \chi(p_q),$$ where $p_q$ is the unique number in $\{1, \dots, m\}$ such that $k(p_q - 1) < q \leq k(p_q)$. Let $\widehat{0_\chi}$ be the partition of $\{1, \dots, n\}$ with blocks $\{\{k(p - 1) + 1, \dots, k(p)\}\}_{p = 1}^m$. Recursively applying the previous lemma along with [@CNS2015-2]\*[Theorem 9.1.5]{} yields the following operator-valued analogue of [@GS2016]\*[Theorem 4.22]{}.
Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. Under the above notation, we have $${{\mathcal{K}}}_{1_\chi}\left(Z_1\cdots Z_{k(1)}, Z_{k(1) + 1}\cdots Z_{k(2)}, \dots, Z_{k(m - 1) + 1}\cdots Z_{k(m)}\right) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\widehat{\chi})\\\sigma \vee \widehat{0_\chi} = 1_{\widehat{\chi}}}}{{\mathcal{K}}}_\sigma(Z_1, \dots, Z_n)$$ for all $\chi: \{1, \dots, m\} \to \{\ell, r\}$ and $Z_k \in {{\mathcal{A}}}_{\widehat{\chi}(k)}$.
Operator-valued conditionally bi-moment and bi-cumulant pairs
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In [@S1998]\*[Subsection 3.2]{}, the classes of operator-valued moment and cumulant functions were introduced as a tool to calculate moment expressions of elements in amalgamated free products. The c-free extension (in the special case ${{\mathcal{B}}}= {{\mathbb{C}}}$) was achieved in [@M2002]\*[Section 3]{} and the bi-free analogue was obtained in [@CNS2015-2]\*[Subsection 6.3]{}. In this subsection, we extend the notions of operator-valued bi-moment and bi-cumulant functions from [@CNS2015-2]\*[Definition 6.3.2]{} to pairs of functions.
\[CondBiMC\] Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, and let $$\phi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ and $$\Phi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{D}}}$$ be an operator-valued conditionally bi-multiplicative pair.
1. We say that $(\phi, \Phi)$ is an *operator-valued conditionally bi-moment pair* if whenever $\chi: \{1, \dots, n\} \to \{\ell, r\}$ is such that there exists a $q \in \{1, \dots, n - 1\}$ with $\chi(q) = \chi(q + 1)$, then $$\phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \phi_{1_\chi}(Z_1, \dots, Z_n)$$ and $$\Phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \Phi_{1_\chi}(Z_1, \dots, Z_n)$$ for all $Z_k \in \mathcal{A}_{\chi(k)}$.
2. We say that $(\phi, \Phi)$ is an *operator-valued conditionally bi-cumulant pair* if whenever $\chi: \{1, \dots, n\} \to \{\ell, r\}$ is such that there exists a $q \in \{1, \dots, n - 1\}$ with $\chi(q) = \chi(q + 1)$, then $$\phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \phi_{1_\chi}(Z_1, \dots, Z_n) + \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\|\pi| = 2, q \not\sim_\pi q + 1}}\phi_\pi(Z_1, \dots, Z_n)$$ and $$\Phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \Phi_{1_\chi}(Z_1, \dots, Z_n) + \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\|\pi| = 2, q \not\sim_\pi q + 1}}\Phi_\pi(Z_1, \dots, Z_n)$$ for all $Z_k \in \mathcal{A}_{\chi(k)}$, where $\Phi_\pi(Z_1, \dots, Z_n)$ is defined by the operator-valued conditionally bi-multiplicativity of $(\phi, \Phi)$ using $\phi$ for an interior block and $\Phi$ for an exterior block.
The following demonstrates that the two notions of pairs of functions are naturally related by summing over bi-non-crossing partitions.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. If $$\phi, \psi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{B}}}$$ and $$\Phi, \Psi: \bigcup_{n \geq 1}\bigcup_{\chi: \{1, \dots, n\} \to \{\ell, r\}}{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi) \times {{\mathcal{A}}}_{\chi(1)} \times \cdots \times {{\mathcal{A}}}_{\chi(n)} \to {{\mathcal{D}}}$$ are such that $(\phi, \Phi)$ and $(\psi, \Psi)$ are operator-valued conditionally bi-multiplicative related by the formulae $$\phi_\pi(Z_1, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma \leq \pi}}\psi_\sigma(Z_1, \dots, Z_n)$$ and $$\Phi_\pi(Z_1, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma \leq \pi}}\Psi_\sigma(Z_1, \dots, Z_n)$$ for all $\chi: \{1, \dots, n\} \to \{\ell, r\}$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$, and $Z_k \in {{\mathcal{A}}}_{\chi(k)}$, then $(\phi, \Phi)$ is an operator-valued conditionally bi-moment pair if and only if $(\psi, \Psi)$ is an operator-valued conditionally bi-cumulant pair.
Let $\chi: \{1, \dots, n\} \to \{\ell, r\}$ be such that there exists a $q \in \{1, \dots, n - 1\}$ with $\chi(q) = \chi(q + 1)$. If $(\psi, \Psi)$ is an operator-valued conditionally bi-cumulant pair, then $$\phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \phi_{1_\chi}(Z_1, \dots, Z_n)$$ for all $Z_k \in {{\mathcal{A}}}_{\chi(k)}$ by [@CNS2015-2]\*[Theorem 6.3.5]{}. On the other hand, using the operator-valued conditionally bi-multiplicativity of $(\psi, \Psi)$ and part $(2)$ of Definition \[CondBiMC\], we have $$\Psi_\pi(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, \dots, Z_n) = \sum_{\substack{\sigma \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\\sigma|_{q = q + 1} = \pi}}\Psi_\sigma(Z_1, \dots, Z_n)$$ for all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi|_{\setminus q})$, and it follows from the same calculations as in the first part of the proof of [@CNS2015-2]\*[Theorem 6.3.5]{} that $$\Phi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \Phi_{1_\chi}(Z_1, \dots, Z_n)$$ for all $Z_k \in \mathcal{A}_{\chi(k)}$.
Conversely, if $(\phi, \Phi)$ is an operator-valued conditionally bi-moment pair, then $$\psi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \psi_{1_\chi}(Z_1, \dots, Z_n) + \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\|\pi| = 2, q \not\sim_\pi q + 1}}\psi_\pi(Z_1, \dots, Z_n)$$ for all $Z_k \in {{\mathcal{A}}}_{\chi(k)}$ by [@CNS2015-2]\*[Theorem 6.3.5]{}, and it follows from the same induction arguments as in the second part of the proof of [@CNS2015-2]\*[Theorem 6.3.5]{} that $$\Psi_{1_{\chi|_{\setminus q}}}(Z_1, \dots, Z_{q - 1}, Z_qZ_{q + 1}, Z_{q + 2}, \dots, Z_n) = \Psi_{1_\chi}(Z_1, \dots, Z_n) + \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)\\|\pi| = 2, q \not\sim_\pi q + 1}}\Psi_\pi(Z_1, \dots, Z_n)$$ for all $Z_k \in {{\mathcal{A}}}_{\chi(k)}$.
As an immediate corollary, we have the following expected result.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. The operator-valued conditionally bi-free moment pair $({{\mathcal{E}}}, {{\mathcal{F}}})$ is an operator-valued conditionally bi-moment pair and the operator-valued conditionally bi-free cumulant pair $(\kappa, {{\mathcal{K}}})$ is an operator-valued conditionally bi-cumulant pair.
Operations on operator-valued cumulants
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The following two results demonstrate how certain operations affect operator-valued conditionally bi-free cumulants under certain conditions. The same effects in the scalar-valued setting were observed in [@GS2016]\*[Lemmata 4.17 and 4.18]{}.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. Let $\chi: \{1, \dots, n\} \to \{\ell, r\}$ be such that $\chi(k_0) = \ell$ and $\chi(k_0 + 1) = r$ for some $k_0 \in \{1, \dots, n - 1\}$, and let $X \in {{\mathcal{A}}}_\ell$ and $Y \in {{\mathcal{A}}}_r$ be such that ${{\mathbb{E}}}(ZXYZ') = {{\mathbb{E}}}(ZYXZ')$ and ${{\mathbb{F}}}(ZXYZ') = {{\mathbb{F}}}(ZYXZ')$ for all $Z, Z' \in {{\mathcal{A}}}$. Define $\chi': \{1, \dots, n\} \to \{\ell, r\}$ by $$\chi'(k) = \begin{cases}
r &\text{if } k = k_0\\
\ell &\text{if } k = k_0 + 1\\
\chi(k) &\text{otherwise }
\end{cases}.$$ Then $${{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n) = {{\mathcal{K}}}_{1_{\chi'}}(Z_1, \dots, Z_{k_0 - 1}, Y, X, Z_{k_0 + 2}, \dots, Z_n)$$ for all $Z_1, \dots, Z_{k_0 - 1}, Z_{k_0 + 2}, \dots, Z_n \in {{\mathcal{A}}}$ with $Z_k \in {{\mathcal{A}}}_{\chi(k)}$.
By repeatedly applying Definition \[OpVCBFCumulants\] and using Definition \[MomentCumulant\] for interior blocks, we have $${{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n) = \sum_{(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)}d(\chi; \pi, \iota)\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n)$$ for some integer coefficients such that $d(\chi; \pi, \iota) = 0$ if there is an interior block $V$ of $\pi$ with $i(V) = e$, and $\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n)$ for non-zero $d(\chi; \pi, \iota)$ is defined as in the proof of Theorem \[VanishingEquiv\]. Similarly, we have $$\begin{aligned}
{{\mathcal{K}}}_{1_{\chi'}}(Z_1, \dots, Z_{k_0 - 1}, Y, & X, Z_{k_0 + 2}, \dots, Z_n) \\
& = \sum_{(\pi', \iota') \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi', ie)}d(\chi'; \pi', \iota')\Theta_{(\pi', \iota')}(Z_1, \dots, Z_{k_0 - 1}, Y, X, Z_{k_0 + 2}, \dots, Z_n).\end{aligned}$$ Note that there is a bijection from ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$ to ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi', ie)$ which sends a pair $(\pi, \iota)$ to the pair $(\pi', \iota')$ obtained by swapping $k_0$ and $k_0 + 1$. Furthermore, as only the lattice structure affects the expansions of the above formulae (alternatively, by appealing to the scalar-valued case in [@GS2016]\*[Subsection 4.2]{}), $d(\chi; \pi, \iota) = d(\chi'; \pi', \iota')$ under this bijection.
To complete the proof, it suffices to show that $$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n) = \Theta_{(\pi', \iota')}(Z_1, \dots, Z_{k_0 - 1}, Y, X, Z_{k_0 + 2}, \dots, Z_n)$$ for all $(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$. If $k_0$ and $k_0 + 1$ are in the same block of $\pi$, then one may reduce $$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n)$$ to an expression involving ${{\mathbb{E}}}(ZXYZ')$ or ${{\mathbb{F}}}(ZXYZ')$ for some $Z, Z' \in {{\mathcal{A}}}$, commute $X$ and $Y$ to get ${{\mathbb{E}}}(ZYXZ')$ or ${{\mathbb{F}}}(ZYXZ')$, and undo the reduction to obtain $$\Theta_{(\pi', \iota')}(Z_1, \dots, Z_{k_0 - 1}, Y, X, Z_{k_0 + 2}, \dots, Z_n).$$ On the other hand, if $k_0$ and $k_0 + 1$ are in different blocks of $\pi$, then the reductions of $$\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{k_0 - 1}, X, Y, Z_{k_0 + 2}, \dots, Z_n){\quad\text{and}\quad}\Theta_{(\pi', \iota')}(Z_1, \dots, Z_{k_0 - 1}, Y, X, Z_{k_0 + 2}, \dots, Z_n)$$ agree. Consequently, the proof is complete.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. Let $\chi: \{1, \dots, n\} \to \{\ell, r\}$ be such that $\chi(n) = \ell$, and let $X \in {{\mathcal{A}}}_\ell$ and $Y \in {{\mathcal{A}}}_r$ be such that ${{\mathbb{E}}}(ZX) = {{\mathbb{E}}}(ZY)$ and ${{\mathbb{F}}}(ZX) = {{\mathbb{F}}}(ZY)$ for all $Z \in {{\mathcal{A}}}$. Define $\chi': \{1, \dots, n\} \to \{\ell, r\}$ by $$\chi'(k) = \begin{cases}
r &\text{if } k = n\\
\chi(k) &\text{otherwise }
\end{cases}.$$ Then $${{\mathcal{K}}}_{1_\chi}(Z_1, \dots, Z_{n - 1}, X) = {{\mathcal{K}}}_{1_{\chi'}}(Z_1, \dots, Z_{n - 1}, Y)$$ for all $Z_1, \dots, Z_{n - 1} \in {{\mathcal{A}}}$ with $Z_k \in {{\mathcal{A}}}_{\chi(k)}$.
By the same arguments as the previous lemma, we have $$d(\chi; \pi, \iota)\Theta_{(\pi, \iota)}(Z_1, \dots, Z_{n - 1}, X) = d(\chi'; \pi', \iota')\Theta_{(\pi', \iota')}(Z_1, \dots, Z_{n - 1}, Y)$$ for all $(\pi, \iota) \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi, ie)$, where $(\pi', \iota')$ is obtained from $(\pi, \iota)$ by changing the last node from a left node to a right node. Consequently, the proof is complete.
In [@CNS2015-2]\*[Theorem 10.2.1]{}, it was demonstrated that for a family of ${{\mathcal{B}}}$-algebras with certain conditions, bi-free independence over ${{\mathcal{B}}}$ can be deduced from free independence over ${{\mathcal{B}}}$ of either the left ${{\mathcal{B}}}$-algebras or the right ${{\mathcal{B}}}$-algebras. The conditionally bi-free analogue in the scalar-valued setting was proved in [@GS2016]\*[Theorem 4.20]{}.
Let $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ be a $\mathcal{B}$-$\mathcal{B}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations. If $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is a family of pairs of ${{\mathcal{B}}}$-algebras in $({{\mathcal{A}}}, \mathbb{E}, \mathbb{F}, \varepsilon)$ such that
1. ${{\mathcal{A}}}_{m, \ell}$ and ${{\mathcal{A}}}_{n, r}$ commute for all $m, n \in K$,
2. for every $Y \in {{\mathcal{A}}}_{k, r}$, there exists an $X \in {{\mathcal{A}}}_{k, \ell}$ such that ${{\mathbb{E}}}(ZY) = {{\mathbb{E}}}(ZX)$ and ${{\mathbb{F}}}(ZY) = {{\mathbb{F}}}(ZX)$ for all $Z \in {{\mathcal{A}}}$,
then $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ if and only if $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$. Consequently, if $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then $\{{{\mathcal{A}}}_{k, r}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$.
If $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$ is c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then it is clear that $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ and $\{{{\mathcal{A}}}_{k, r}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$.
Suppose $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$ is c-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$. Given a mixed operator-valued bi-free or conditionally bi-free cumulant from $\{({{\mathcal{A}}}_{k, \ell}, {{\mathcal{A}}}_{k, r})\}_{k \in K}$, assumptions $(1)$ and $(2)$ imply that we may apply the previous two lemmata (or [@S2015]\*[Lemmata 2.16 and 2.17]{}) and reduce it to a mixed operator-valued free or conditionally free cumulant from $\{{{\mathcal{A}}}_{k, \ell}\}_{k \in K}$, which vanishes by c-free independence over $({{\mathcal{B}}}, {{\mathcal{D}}})$. Thus the result follows from Theorem \[VanishingEquiv\].
The operator-valued conditionally bi-free partial $\mathcal{R}$-transform {#sec:R-transform}
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In this section, we construct an operator-valued conditionally bi-free partial ${{\mathcal{R}}}$-transform generalizing [@GS2016]\*[Definition 5.3]{} and relate it to certain operator-valued moment transforms. As we will see in the proof, such transform is a function of three ${{\mathcal{B}}}$-variables instead of two by a similar reason as the operator-valued bi-free partial ${{\mathcal{R}}}$-transform developed in [@S2015]\*[Section 5]{}. As in [@S2015]\*[Section 5]{}, our proof will follow the combinatorial techniques used in [@S2016-2]\*[Section 7]{}. In that which follows, all algebras are assumed to be Banach algebras.
A *Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations* is a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ such that ${{\mathcal{A}}}$, ${{\mathcal{B}}}$, and ${{\mathcal{D}}}$ are Banach algebras, and $\varepsilon|_{{{\mathcal{B}}}\otimes 1}$, $\varepsilon_{1 \otimes {{\mathcal{B}}}^{\mathrm{op}}}$, ${{\mathbb{E}}}$, and ${{\mathbb{F}}}$ are bounded.
Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, let $Z_\ell \in {{\mathcal{A}}}_\ell$, $Z_r \in {{\mathcal{A}}}_r$, and let $b, d \in {{\mathcal{B}}}$. Consider the following series: $$\begin{aligned}
M_{Z_\ell}^\ell(b) &= 1 + \sum_{m \geq 1}{{\mathbb{E}}}((L_bZ_\ell)^m),\\
\mathbb{M}_{Z_\ell}^\ell(b) &= 1 + \sum_{m \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^m),\\
{{\mathcal{C}}}_{Z_\ell}^\ell(b) &= 1 + \sum_{m \geq 1}{{\mathcal{K}}}_{1_{\chi_{m, 0}}}(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}),\end{aligned}$$ and $$\begin{aligned}
M_{Z_r}^r(d) &= 1 + \sum_{n \geq 1}{{\mathbb{E}}}((R_dZ_r)^n),\\
\mathbb{M}_{Z_r}^r(d) &= 1 + \sum_{n \geq 1}{{\mathbb{F}}}((R_dZ_r)^n),\\
{{\mathcal{C}}}_{Z_r}^r(d) &= 1 + \sum_{n \geq 1}{{\mathcal{K}}}_{1_{\chi_{0, n}}}(\underbrace{R_dZ_r, \dots, R_dZ_r}_{n\,\text{entries}}).\end{aligned}$$
By similar arguments as in [@S2015]\*[Remark 5.2]{}, all of the series above converge absolutely for $b, d$ sufficiently small.
In the proof of Theorem \[CR-Transform\] below, the following relations will be used. Since the statements are slightly different than the ones in the literature (see, e.g., [@BPV2012]\*[equation (15)]{}), we will provide a proof.
\[CumulantTransform\] Under the above assumptions and notation, we have $${{\mathcal{C}}}_{Z_\ell}^\ell\left(M_{Z_\ell}^\ell(b)b\right) = 1 + M_{Z_\ell}^\ell(b) - M_{Z_\ell}^\ell(b)\mathbb{M}_{Z_\ell}^\ell(b)^{-1} {\quad\text{and}\quad}{{\mathcal{C}}}_{Z_r}^r\left(dM_{Z_r}^r(d)\right) = 1 + M_{Z_r}^r(d) - \mathbb{M}_{Z_r}^r(d)^{-1}M_{Z_r}^r(d)$$ for $b, d$ sufficiently small.
For $m \geq 1$, we have $${{\mathbb{F}}}((L_bZ_\ell)^m) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, 0})}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}).$$ For every partition $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, 0})$, let $W_\pi$ denote the block of $\pi$ containing $1$, which is necessarily an exterior block. Rearrange the above sum (which may be done as it converges absolutely) by first choosing $s \in \{1, \dots, m\}$, $W = \{1 = w_1 < \cdots < w_s\} \subset \{1, \dots, m\}$, and then summing over all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, 0})$ such that $W_\pi = W$, i.e., $${{\mathbb{F}}}((L_bZ_\ell)^m) = \sum_{s = 1}^m\sum_{\substack{W = \{1 = w_1 < \cdots < w_s\}\\W \subset \{1, \dots, m\}}}\sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, 0})\\W_\pi = W}}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}).$$ Furthermore, using operator-valued conditionally bi-multiplicative properties, the right-most sum in the above expression is $$b{{\mathcal{K}}}_{1_{\chi_{s, 0}}}(Z_\ell, L_{b_2}Z_\ell, \dots, L_{b_s}Z_\ell){{\mathbb{F}}}((L_bZ_\ell)^{m - w_s}),$$ where $b_k = {{\mathbb{E}}}((L_bZ_\ell)^{w_k - w_{k - 1} - 1})b$. Thus $${{\mathbb{F}}}((L_bZ_\ell)^m) = \sum_{\substack{1 \leq s \leq m\\0 \leq i_1, \dots, i_s \leq m\\i_1 + \cdots + i_s = m - s}}b{{\mathcal{K}}}_{1_{\chi_{s, 0}}}(Z_\ell, L_{f(i_1)}Z_\ell, \dots, L_{f(i_{s - 1})}Z_\ell){{\mathbb{F}}}((L_bZ_\ell)^{i_s}),$$ where $f(k) = {{\mathbb{E}}}((L_bZ_\ell)^k)b$. Note that $$\sum_{k \geq 0}f(k) = M_{Z_\ell}^\ell(b)b {\quad\text{and}\quad}\sum_{k \geq 0}{{\mathbb{F}}}((L_bZ_\ell)^k) = \mathbb{M}_{Z_\ell}^\ell(b).$$ Consequently, we obtain $$\sum_{m \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^m) = \sum_{s \geq 1}b{{\mathcal{K}}}_{1_{\chi_{s, 0}}}(Z_\ell, L_{M_{Z_\ell}^\ell(b)b}Z_\ell, \dots, L_{M_{Z_\ell}^\ell(b)b}Z_\ell)\mathbb{M}_{Z_\ell}^\ell(b),$$ therefore $$M_{Z_\ell}^\ell(b)\mathbb{M}_{Z_\ell}^\ell(b) = M_{Z_\ell}^\ell(b) + \sum_{s \geq 1}{{\mathcal{K}}}_{1_{\chi_{s, 0}}}(L_{M_{Z_\ell}^\ell(b)b}Z_\ell, L_{M_{Z_\ell}^\ell(b)b}Z_\ell, \dots, L_{M_{Z_\ell}^\ell(b)b}Z_\ell)\mathbb{M}_{Z_\ell}^\ell(b),$$ and hence $$M_{Z_\ell}^\ell(b) - M_{Z_\ell}^\ell(b)\mathbb{M}_{Z_\ell}^\ell(b)^{-1} = {{\mathcal{C}}}_{Z_\ell}^\ell\left(M_{Z_\ell}^\ell(b)b\right) - 1,$$ which proves the first equation. The proof for the second equation is nearly identical once one uses the fact that $d \mapsto R_d$ is an anti-homomorphism.
For $b, c, d \in {{\mathcal{B}}}$, $Z_\ell \in {{\mathcal{A}}}_\ell$, and $Z_r \in {{\mathcal{A}}}_r$, consider the following series of the pair $(Z_\ell, Z_r)$: $$\begin{aligned}
M_{(Z_\ell, Z_r)}(b, c, d) &= \sum_{m, n \geq 0}{{\mathbb{E}}}((L_bZ_\ell)^m(R_dZ_r)^nR_c),\\
\mathbb{M}_{(Z_\ell, Z_r)}(b, c, d) &= \sum_{m, n \geq 0}{{\mathbb{F}}}((L_bZ_\ell)^m(R_dZ_r)^nR_c),\\
{{\mathcal{C}}}_{(Z_\ell, Z_r)}(b, c, d) &= c + \sum_{m \geq 1}{{\mathcal{K}}}_{1_{\chi_{m, 0}}}(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m - 1\,\text{entries}}, L_bZ_\ell L_c)\\
&\quad + \sum_{\substack{m \geq 0\\n \geq 1}}{{\mathcal{K}}}_{1_{\chi_{m, n}}}(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}, \underbrace{R_dZ_r, \dots, R_dZ_r}_{n - 1\,\text{entries}}, R_dZ_rR_c),\end{aligned}$$ which converge absolutely for $b, c, d$ sufficiently small by similar arguments as in [@S2015]\*[Remarks 5.2 and 5.5]{}.
Notice if $(Z_{1, \ell}, Z_{1, r})$ and $(Z_{2, \ell}, Z_{2, r})$ are c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$, then $${{\mathcal{C}}}_{(Z_{1, \ell} + Z_{2, \ell}, Z_{1, r} + Z_{2, r})}(b, c, d) - c = ({{\mathcal{C}}}_{(Z_{1, \ell}, Z_{1, r})}(b, c, d) - c) + ({{\mathcal{C}}}_{(Z_{2, \ell}, Z_{2, r})}(b, c, d) - c)$$ by Theorem \[VanishingEquiv\]; that is, ${{\mathcal{C}}}_{(Z_\ell, Z_r)}(b, c, d) - c$ is an operator-valued conditionally bi-free partial ${{\mathcal{R}}}$-transform.
\[CR-Transform\] Let $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ be a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations, let $Z_\ell \in {{\mathcal{A}}}_\ell$, and let $Z_r \in {{\mathcal{A}}}_r$. Then $$\begin{aligned}
& {{\mathcal{C}}}_{(Z_\ell, Z_r)}(M_{Z_\ell}^\ell(b)b, M_{(Z_\ell, Z_r)}(b, c, d), dM_{Z_r}^r(d)) \\
& = M_{Z_\ell}^\ell(b)\mathbb{M}_{Z_\ell}^\ell(b)^{-1}\mathbb{M}_{(Z_\ell, Z_r)}(b, c, d)\mathbb{M}_{Z_r}^r(d)^{-1}M_{Z_r}^r(d) + M_{(Z_\ell, Z_r)}(b, c, d) \nonumber \\
& \quad + M_{Z_\ell}^\ell(b)(1 - \mathbb{M}_{Z_\ell}^\ell(b)^{-1})M_{(Z_\ell, Z_r)}(b, c, d) + M_{(Z_\ell, Z_r)}(b, c, d)(1 - \mathbb{M}_{Z_r}^r(d)^{-1})M_{Z_r}^r(d) - M_{Z_\ell}^\ell(b)cM_{Z_r}^r(d) \nonumber\end{aligned}$$ for $b, c, d \in {{\mathcal{B}}}$ sufficiently small.
Note that if ${{\mathcal{B}}}= {{\mathcal{D}}}= {{\mathbb{C}}}$, $b = z$, $d = w$, and $c = 1$, then Theorem \[CR-Transform\] produces exactly equation $(9)$ in [@GS2016]\*[Theorem 5.6]{} for the scalar-valued setting. On the other hand, if ${{\mathcal{B}}}= {{\mathcal{D}}}$ and ${{\mathbb{E}}}= {{\mathbb{F}}}$, then Theorem \[CR-Transform\] produces exactly equation $(10)$ in [@S2015]\*[Theorem 5.6]{} for the operator-valued bi-free setting.
For $m, n \geq 1$, let ${{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})$ denote the set of bi-non-crossing partitions where no block contains both left and right nodes. Using operator-valued conditionally bi-multiplicativity, we obtain $$\begin{aligned}
{{\mathbb{F}}}& ((L_bZ_\ell)^m(R_dZ_r)^nR_c)\\
&= \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}, \underbrace{R_dZ_r, \dots, R_dZ_r}_{n - 1\,\text{entries}}, R_dZ_rR_c)\\
&\quad + \sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, n})\\\pi \notin {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})}}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}, \underbrace{R_dZ_r, \dots, R_dZ_r}_{n - 1\,\text{entries}}, R_dZ_rR_c)\\
&= {{\mathbb{F}}}((L_bZ_\ell)^m)c{{\mathbb{F}}}((R_dZ_r)^n) + \Theta_{m, n}(b, c, d),\end{aligned}$$ where $\Theta_{m, n}(b, c, d)$ denotes the sum $$\sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, n})\\\pi \notin {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})}}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}, \underbrace{R_dZ_r, \dots, R_dZ_r}_{n - 1\,\text{entries}}, R_dZ_rR_c).$$
For every partition $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, n}) \setminus {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})$, let $V_\pi$ denote the block of $\pi$ with both left and right indices such that $\min(V_\pi)$ is the smallest among all blocks of $\pi$ with this property. Note that $V_\pi$ is necessarily an exterior block. Rearrange the sum in $\Theta_{m, n}(b, c, d)$ (which may be done as it converges absolutely) by first choosing $s \in \{1, \dots, m\}$, $t \in \{1, \dots, n\}$, $V \subset \{1, \dots, m + n\}$ such that $$V_\ell := V \cap \{1, \dots, m\} = \{u_1 < \cdots < u_s\} {\quad\text{and}\quad}V_r := V \cap \{m + 1, \dots, m + n\} = \{v_1 < \cdots < v_t\},$$ and then summing over all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, n}) \setminus {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})$ such that $V_\pi = V$. The result is $$\Theta_{m, n}(b, c, d) = \sum_{s = 1}^m\sum_{t = 1}^n\sum_{\substack{V\\V_\ell = \{u_1 < \cdots < u_s\}\\V_r = \{v_1 < \cdots < v_t\}}}\sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_{m, n})\\\pi \notin {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}_{\mathrm{vs}}(\chi_{m, n})\\V_\pi = V}}{{\mathcal{K}}}_\pi(\underbrace{L_bZ_\ell, \dots, L_bZ_\ell}_{m\,\text{entries}}, \underbrace{R_dZ_r, \dots, R_dZ_r}_{n - 1\,\text{entries}}, R_dZ_rR_c).$$
Using operator-valued conditionally bi-multiplicative properties, the right-most sum in the above expression is $$F_b{{\mathcal{K}}}_{1_{\chi_{s, t}}}(Z_\ell, L_{b_2}Z_\ell, \dots, L_{b_s}Z_\ell, Z_r, R_{d_2}Z_r, \dots, R_{d_{t - 1}}Z_r, R_{d_t}Z_rR_{{{\mathbb{E}}}((L_bZ_\ell)^{m - u_s}(R_dZ_r)^{n - v_t}R_c)})G_d,$$ where $$\begin{gathered}
b_k = {{\mathbb{E}}}((L_bZ_\ell)^{u_k - u_{k - 1} - 1})b, \quad d_k = d{{\mathbb{E}}}((R_dZ_r)^{v_k - v_{k - 1} - 1}), \\ F_b = {{\mathbb{F}}}((L_bZ_\ell)^{u_1 - 1})b, {\quad\text{and}\quad}G_d = d{{\mathbb{F}}}((R_dZ_r)^{v_1 - 1}). \end{gathered}$$ Consequently, we obtain that $\Theta_{m, n}(b, c, d)$ equals $$\begin{gathered}
\label{ThetaSum}
\sum_{\substack{1 \leq s \leq m\\0 \leq i_0, i_1, \dots, i_s \leq m\\i_0 + i_1 + \cdots + i_s = m - s}}\sum_{\substack{1 \leq t \leq n\\0 \leq j_0, j_1, \dots, j_t \leq n\\j_0 + j_1 + \cdots + j_t = n - t}}\\
F(i_1){{\mathcal{K}}}_{1_{\chi_{s, t}}}(Z_\ell, L_{f(i_2)}Z_\ell, \dots, L_{f(i_s)}Z_\ell, Z_r, R_{g(j_2)}Z_r, \dots, R_{g(j_t - 1)}Z_r, R_{g(j_t)}Z_rR_{{{\mathbb{E}}}((L_bZ_\ell)^{i_0}(R_dZ_r)^{j_0}R_c)})G(j_1),\end{gathered}$$ where $$f(k) = {{\mathbb{E}}}((L_bZ_\ell)^k)b, \quad g(k) = d{{\mathbb{E}}}((R_dZ_r)^k), \quad F(k) = {{\mathbb{F}}}((L_bZ_\ell)^k)b, {\quad\text{and}\quad}G(k) = d{{\mathbb{F}}}((R_dZ_r)^k).$$
Note that $$\sum_{k \geq 0}f(k) = M_{Z_\ell}^\ell(b)b, \quad \sum_{k \geq 0}g(k) = dM_{Z_r}^r(d), \quad \sum_{k \geq 0}F(k) = \mathbb{M}_{Z_\ell}^\ell(b)b, {\quad\text{and}\quad}\sum_{k \geq 0}G(k) = d\mathbb{M}_{Z_r}^r(d).$$ On the other hand, expanding $\mathbb{M}_{(Z_\ell, Z_r)}(b, c, d)$ using the fact everything converges absolutely produces $$\begin{aligned}
&\mathbb{M}_{(Z_\ell, Z_r)}(b, c, d)\\
&= c + \sum_{m \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^mR_c) + \sum_{n \geq 1}{{\mathbb{F}}}((R_dZ_r)^nR_c) + \sum_{m, n \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^m(R_dZ_r)^nR_c)\\
&= c + \sum_{m \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^mR_c) + \sum_{n \geq 1}{{\mathbb{F}}}((R_dZ_r)^nR_c) + \sum_{m, n \geq 1}{{\mathbb{F}}}((L_bZ_\ell)^m)c{{\mathbb{F}}}((R_dZ_r)^n) + \sum_{m, n \geq 1}\Theta_{m, n}(b, c, d)\\
&= \sum_{m, n \geq 0}{{\mathbb{F}}}((L_bZ_\ell)^m)c{{\mathbb{F}}}((R_dZ_r)^n) + \sum_{m, n \geq 1}\Theta_{m, n}(b, c, d)\\
&= \mathbb{M}_{Z_\ell}^\ell(b)c\mathbb{M}_{Z_r}^r(d) + \sum_{m, n \geq 1}\Theta_{m, n}(b, c, d).\end{aligned}$$ By rearranging the remaining sum involving $\Theta_{m, n}(b, c, d)$ to sum over all fixed $s, t$ in equation , and by choosing $b, d$ sufficiently small so that $M_{Z_\ell}^\ell(b)$, $M_{Z_r}^r(d)$, $\mathbb{M}_{Z_\ell}^\ell(b)$, and $\mathbb{M}_{Z_r}^r(d)$ are invertible, we obtain $$\begin{aligned}
&\sum_{m, n \geq 1}\Theta_{m, n}(b, c, d)\\
&= \sum_{s, t \geq 1}\mathbb{M}_{Z_\ell}^\ell(b)b\\
&\quad \times {{\mathcal{K}}}_{1_{\chi_{s, t}}}(\underbrace{Z_\ell, L_{M_{Z_\ell}^\ell(b)b}Z_\ell, \dots, L_{M_{Z_\ell}^\ell(b)b}Z_\ell}_{s\,\text{entries}}, \underbrace{Z_r, R_{dM_{Z_r}^r(d)}Z_r, \dots, R_{dM_{Z_r}^r(d)}Z_r}_{t - 1\,\text{entries}}, R_{dM_{Z_r}^r(d)}Z_rR_{M_{(Z_\ell, Z_r)}(b, c, d)}) \\
&\quad \times d\mathbb{M}_{Z_r}^r(d)\\
&= \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}\\
&\quad \times\sum_{s, t \geq 1}\left({{\mathcal{K}}}_{1_{\chi_{s, t}}}(\underbrace{L_{M_{Z_\ell}^\ell(b)b}Z_\ell, \dots, L_{M_{Z_\ell}^\ell(b)b}Z_\ell}_{s\,\text{entries}}, \underbrace{R_{dM_{Z_r}^r(d)}Z_r, \dots, R_{dM_{Z_r}^r(d)}Z_r}_{t - 1\,\text{entries}}, R_{dM_{Z_r}^r(d)}Z_rR_{M_{(Z_\ell, Z_r)}(b, c, d)})\right)\\
&\quad \times M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&= \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}[{{\mathcal{C}}}_{(Z_\ell, Z_r)}(M_{Z_\ell}^\ell(b)b, M_{(Z_\ell, Z_r)}(b, c, d), dM_{Z_r}^r(d)) - {{\mathcal{C}}}_{Z_\ell}^\ell(M_{Z_\ell}^\ell(b)b) M_{(Z_\ell, Z_r)}(b, c, d)\\
&\quad - M_{(Z_\ell, Z_r)}(b, c, d){{\mathcal{C}}}_{Z_r}^r(dM_{Z_r}^r(d)) + M_{(Z_\ell, Z_r)}(b, c, d)]M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&= \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}{{\mathcal{C}}}_{(Z_\ell, Z_r)}(M_{Z_\ell}^\ell(b)b, M_{(Z_\ell, Z_r)}(b, c, d), dM_{Z_r}^r(d))M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&\quad - \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}(1 + M_{Z_\ell}^\ell(b) - M_{Z_\ell}^\ell(b)\mathbb{M}_{Z_\ell}^\ell(b)^{-1})M_{(Z_\ell, Z_r)}(b, c, d)M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&\quad - \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}M_{(Z_\ell, Z_r)}(b, c, d)(1 + M_{Z_r}^r(d) - \mathbb{M}_{Z_r}^r(d)^{-1}M_{Z_r}^r(d))M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&\quad + \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}M_{(Z_\ell, Z_r)}(b, c, d)M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&= \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}\left({{\mathcal{C}}}_{(Z_\ell, Z_r)}(M_{Z_\ell}^\ell(b)b, M_{(Z_\ell, Z_r)}(b, c, d), dM_{Z_r}^r(d)) - M_{(Z_\ell, Z_r)}(b, c, d)\right)M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d)\\
&\quad - (\mathbb{M}_{Z_\ell}^\ell(b) - 1)M_{(Z_\ell, Z_r)}(b, c, d)M_{Z_r}^r(d)^{-1}\mathbb{M}_{Z_r}^r(d) - \mathbb{M}_{Z_\ell}^\ell(b)M_{Z_\ell}^\ell(b)^{-1}M_{(Z_\ell, Z_r)}(b, c, d)(\mathbb{M}_{Z_r}^r(d) - 1),\end{aligned}$$ where the fourth equality follows from Lemma \[CumulantTransform\]. The result now follows by combining these equations.
Operator-valued conditionally bi-free limit theorems {#sec:limit-thms}
====================================================
In this section, operator-valued conditionally bi-free limit theorems are studied. Recall first from Definition \[MCSeriesDefn\] that if ${{\mathcal{Z}}}= \{Z_i\}_{i \in I} \sqcup \{Z_j\}_{j \in J}$ is a two-faced family in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$, then the moment and cumulant series $(\nu^{{\mathcal{Z}}}, \mu^{{\mathcal{Z}}})$ and $(\rho^{{\mathcal{Z}}}, \eta^{{\mathcal{Z}}})$ completely describe the joint distribution of ${{\mathcal{Z}}}$ with respect to $({{\mathbb{E}}}, {{\mathbb{F}}})$. In that which follows, given a bi-non-crossing partition $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi)$ it is often convenient to define $$(\nu_\omega^{{\mathcal{Z}}})_\pi(b_1, \dots, b_{n - 1}), \quad (\mu_\omega^{{\mathcal{Z}}})_\pi(b_1, \dots, b_{n - 1}), \quad (\rho_\omega^{{\mathcal{Z}}})_\pi(b_1, \dots, b_{n - 1}), {\quad\text{and}\quad}(\eta_\omega^{{\mathcal{Z}}})_\pi(b_1, \dots, b_{n - 1})$$ by using operator-valued conditionally bi-multiplicativity and replacing $1_{\chi_\omega}$ with $\pi$ in Notation \[MCSeries\].
The operator-valued c-bi-free central limit theorem
---------------------------------------------------
Like any non-commutative probability theory, the first result is a central limit theorem in the operator-valued c-bi-free setting.
A two-faced family ${{\mathcal{Z}}}= ((Z_i)_{i \in I}, (Z_j)_{j \in J})$ in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ is said to have a *centred $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued c-bi-free Gaussian distribution* if $$\rho_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) = \eta_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) = 0$$ for all $n \geq 1$ with $n \neq 2$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in \mathcal{B}$.
In view of the definition above and the moment-cumulant formulae, it is enough to specify $\nu_\omega^{{\mathcal{Z}}}(b)$ and $\mu_\omega^{{\mathcal{Z}}}(b)$ for $\omega: \{1, 2\} \to I \sqcup J$ and $b \in {{\mathcal{B}}}$.
Let $I$ and $J$ be non-empty disjoint finite sets, let $M_{|I \sqcup J|}({{\mathcal{B}}})$ and $M_{|I \sqcup J|}({{\mathcal{D}}})$ denote the $|I \sqcup J|$ by $|I \sqcup J|$ matrices with entries in ${{\mathcal{B}}}$ and ${{\mathcal{D}}}$ respectively, and let $$\sigma: {{\mathcal{B}}}\to M_{|I \sqcup J|}({{\mathcal{B}}}), \quad b \mapsto (\sigma_{k, \ell}(b))_{k, \ell \in I \sqcup J} {\quad\text{and}\quad}\tau: {{\mathcal{B}}}\to M_{|I \sqcup J|}({{\mathcal{D}}}), \quad b \mapsto (\tau_{k, \ell}(b))_{k, \ell \in I \sqcup J}$$ be linear maps. A two-faced family ${{\mathcal{Z}}}= ((Z_i)_{i \in I}, (Z_j)_{j \in J})$ in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ is said to have a *centred $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued c-bi-free Gaussian distribution with covariance matrices $(\sigma, \tau)$* if, in addition to having a centred $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued c-bi-free Gaussian distribution, $$\nu_\omega^{{\mathcal{Z}}}(b) = \sigma_{\omega(1), \omega(2)}(b) \in {{\mathcal{B}}}{\quad\text{and}\quad}\mu_\omega^{{\mathcal{Z}}}(b) = \tau_{\omega(1), \omega(2)}(b) \in {{\mathcal{D}}}$$ for all $\omega: \{1, 2\} \to I \sqcup J$ and $b \in {{\mathcal{B}}}$.
Let $\{{{\mathcal{Z}}}_m = ((Z_{m; i})_{i \in I}, (Z_{m; j})_{j \in J})\}_{m = 1}^\infty$ be a sequence of two-faced families in a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ which are c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$. Moreover assume
1. ${{\mathbb{E}}}(Z_{m; k}) = {{\mathbb{F}}}(Z_{m; k}) = 0$ for all $m \geq 1$ and $k \in I \sqcup J$;
2. $\sup_{m \geq 1}\|\nu_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1})\| < \infty$ and $\sup_{m \geq 1}\|\mu_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1})\| < \infty$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$;
3. there are linear maps $\sigma: {{\mathcal{B}}}\to M_{|I \sqcup J|}({{\mathcal{B}}})$ and $\tau: {{\mathcal{B}}}\to M_{|I \sqcup J|}({{\mathcal{D}}})$ such that $$\lim_{N \to \infty}\frac{1}{N}\sum_{m = 1}^N\nu_\omega^{{{\mathcal{Z}}}_m}(b) = \sigma_{\omega(1), \omega(2)}(b) {\quad\text{and}\quad}\lim_{N \to \infty}\frac{1}{N}\sum_{m = 1}^N\mu_\omega^{{{\mathcal{Z}}}_m}(b) = \tau_{\omega(1), \omega(2)}(b)$$ for all $\omega: \{1, 2\} \to I \sqcup J$ and $b \in {{\mathcal{B}}}$.
Then the two-faced families $\{{{\mathcal{S}}}_N = ((S_{N ; i})_{i \in I}, (S_{N ; j})_{j \in J})\}_{N = 1}^\infty$, defined by $$S_{N ; k} = \frac{1}{\sqrt{N}}\sum_{m = 1}^NZ_{m; k}, \quad k \in I \sqcup J,$$ converges in distribution to a two-faced family ${{\mathcal{Y}}}= ((Y_i)_{i \in I}, (Y_j)_{j \in J})$ which has a centred $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued c-bi-free Gaussian distribution with covariance matrices $(\sigma, \tau)$.
Since the cumulant series uniquely determine the joint distributions, it suffices to show that $$\lim_{N \to \infty}\rho_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \rho_\omega^{{\mathcal{Y}}}(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\lim_{N \to \infty}\eta_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \eta_\omega^{{\mathcal{Y}}}(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. By definitions, this means $$\lim_{N \to \infty}\rho_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}\eta_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = 0$$ for all $\omega: \{1, \dots, n\} \to I \sqcup J$ such that $n \neq 2$, $$\lim_{N \to \infty}\rho_\omega^{{{\mathcal{S}}}_N}(b) = \sigma_{\omega(1), \omega(2)}(b) {\quad\text{and}\quad}\lim_{N \to \infty}\eta_\omega^{{{\mathcal{S}}}_N}(b) = \tau_{\omega(1), \omega(2)}(b)$$ for all $\omega: \{1, 2\} \to I \sqcup J$ and $b \in {{\mathcal{B}}}$.
For fixed $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, by the additive and multilinear properties of cumulants, we have $$\begin{aligned}
E &:= \lim_{N \to \infty}\rho_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}\frac{1}{N^{n/2}}\sum_{m = 1}^N\rho_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1}),\\
F &:= \lim_{N \to \infty}\eta_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}\frac{1}{N^{n/2}}\sum_{m = 1}^N\eta_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1}).\end{aligned}$$ If $n = 1$, then $$\begin{aligned}
E &= \lim_{N \to \infty}\frac{1}{\sqrt{N}}\sum_{m = 1}^N{{\mathbb{E}}}(Z_{m; \omega(1)}) = 0,\\
F &= \lim_{N \to \infty}\frac{1}{\sqrt{N}}\sum_{m = 1}^N{{\mathbb{F}}}(Z_{m; \omega(1)}) = 0\end{aligned}$$ by assumption $(1)$. If $n \geq 3$, then assumption $(2)$ and operator-valued conditionally bi-multiplicativity imply $$\sup_{m \geq 1}\|\rho_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1})\| := B < \infty {\quad\text{and}\quad}\sup_{m \geq 1}\|\eta_\omega^{{{\mathcal{Z}}}_m}(b_1, \dots, b_{n - 1})\| := D < \infty,$$ hence $$\|E\| \leq \lim_{N \to \infty}\frac{B}{N^{(n - 2)/2}} = 0 {\quad\text{and}\quad}\|F\| \leq \lim_{N \to \infty}\frac{D}{N^{(n - 2)/2}} = 0.$$ Otherwise $n = 2$ and $$E = \lim_{N \to \infty}\frac{1}{N}\sum_{m = 1}^N\rho_\omega^{{{\mathcal{Z}}}_m}(b) = \lim_{N \to \infty}\frac{1}{N}\sum_{m = 1}^N\nu_\omega^{{{\mathcal{Z}}}_m}(b) = \sigma_{\omega(1), \omega(2)}(b),$$ and similarly $F = \tau_{\omega(1), \omega(2)}(b)$, for all $\omega: \{1, 2\} \to I \sqcup J$ and $b \in {{\mathcal{B}}}$ by assumptions $(1)$ and $(3)$.
The operator-valued compound c-bi-free Poisson limit theorem
------------------------------------------------------------
The next result is a Poisson type limit theorem in the operator-valued c-bi-free setting. In what follows, all two-faced families are assumed to have non-empty disjoint left and right index sets $I$ and $J$, respectively. To formulate the statement, we introduce the following notation.
Let $(\nu_1, \mu_1)$ and $(\nu_2, \mu_2)$ be the moment series of two-faced families. For $\lambda \in {{\mathbb{R}}}$, denote by $$(\lambda\nu_1 + (1 - \lambda)\nu_2, \lambda\mu_1 + (1 - \lambda)\mu_2)$$ the moment series of some two-faced family such that $$(\lambda\nu_1 + (1 - \lambda)\nu_2)_\omega(b_1, \dots, b_{n - 1}) = \lambda(\nu_1)_\omega(b_1, \dots, b_{n - 1}) + (1 - \lambda)(\nu_2)_\omega(b_1, \dots, b_{n - 1})$$ and $$(\lambda\mu_1 + (1 - \lambda)\mu_2)_\omega(b_1, \dots, b_{n - 1}) = \lambda(\mu_1)_\omega(b_1, \dots, b_{n - 1}) + (1 - \lambda)(\mu_2)_\omega(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. Such a realization always exists by similar (and simpler) constructions as in the proofs of [@S2015]\*[Lemma 3.8]{} and Lemma \[Existence\]. Moreover, let $(\nu^\delta, \mu^\delta)$ be the special moment series such that $$\nu_\omega^\delta(b_1, \dots, b_{n - 1}) = \mu_\omega^\delta(b_1, \dots, b_{n - 1}) = 0$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
Let $(\nu, \mu)$ be the moment series of some two-faced family and let $\lambda \in {{\mathbb{R}}}$. A two-faced family ${{\mathcal{Z}}}= ((Z_i)_{i \in I}, (Z_j)_{j \in J})$ in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ is said to have a *$({{\mathcal{B}}}, {{\mathcal{D}}})$-valued compound c-bi-free Poisson distribution with rate $\lambda$ and jump distribution $(\nu, \mu)$* if $$\rho_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) = \lambda\nu_\omega(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\eta_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) = \lambda\mu_\omega(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
Let $(\nu, \mu)$ be the moment series of some two-faced family, let $\lambda \in {{\mathbb{R}}}$, and consider the sequence $\{(\nu_N, \mu_N)\}_{N = 1}^\infty$ of moment series defined by $$\nu_N = \left(1 - \frac{\lambda}{N}\right)\nu^\delta + \frac{\lambda}{N}\nu {\quad\text{and}\quad}\mu_N = \left(1 - \frac{\lambda}{N}\right)\mu^\delta + \frac{\lambda}{N}\mu.$$ If $\{{{\mathcal{Z}}}_{N; m} = ((Z_{N; m; i})_{i \in I}, (Z_{N; m; j})_{j \in J})\}_{m = 1}^N$ is a sequence of identically distributed two-faced families in a ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ which are c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$ with moment series $(\nu_N, \mu_N)$, then the two-faced families $\{{{\mathcal{S}}}_N = ((S_{N ; i})_{i \in I}, (S_{N ; j})_{j \in J})\}_{N = 1}^\infty$, defined by $$S_{N ; k} = \sum_{m = 1}^NZ_{N; m; k}, \quad k \in I \sqcup J,$$ converges in distribution to a two-faced family ${{\mathcal{Z}}}= ((Z_i)_{i \in I}, (Z_j)_{j \in J})$ which has a $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued compound c-bi-free Poisson distribution with rate $\lambda$ and jump distribution $(\nu, \mu)$.
For each $N \geq 1$, let $(\rho_N, \eta_N)$ be the cumulant series corresponding to $(\nu_N, \mu_N)$. For $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, we have $$\begin{aligned}
(\rho_N)_\omega(b_1, \dots, b_{n - 1}) &= \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}(\nu_N)_\pi(b_1, \dots, b_{n - 1})\mu_{{{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}}(\pi, 1_{\chi_\omega})\\
&= (\nu_N)_\omega(b_1, \dots, b_{n - 1}) + O(1/N^2)\\
&= \frac{\lambda}{N}\nu_\omega(b_1, \dots, b_{n - 1}) + O(1/N^2),\end{aligned}$$ and thus $$\begin{aligned}
\rho_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) &= \lim_{N \to \infty}\rho_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1})\\
&= \lim_{N \to \infty}\left(\lambda\nu_\omega(b_1, \dots, b_{n - 1}) + O(1/N)\right)\\
&= \lambda\nu_\omega(b_1, \dots, b_{n - 1}).\end{aligned}$$ Similarly, we have $(\eta_N)_\pi(b_1, \dots, b_{n - 1}) = O(1/N^2)$ for $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$ with at least two blocks, therefore $$(\eta_N)_\omega(b_1, \dots, b_{n - 1}) = (\mu_N)_\omega(b_1, \dots, b_{n - 1}) + O(1/N^2) = \frac{\lambda}{N}\mu_\omega(b_1, \dots, b_{n - 1}) + O(1/N^2),$$ and thus $$\eta_\omega^{{\mathcal{Z}}}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}\eta_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \lambda\mu_\omega(b_1, \dots, b_{n - 1})$$ as required.
A general operator-valued c-bi-free limit theorem
-------------------------------------------------
We finish this section with an operator-valued analogue of [@GS2016]\*[Theorem 6.8]{}.
\[LimitMC\] For every $N \in \mathbb{N}$, let ${{\mathcal{Z}}}_N = ((Z_{N; i})_{i \in I}, (Z_{N; j})_{j \in J})$ be a two-faced family in a Banach $\mathcal{B}$-$\mathcal{B}$-non-commutative-probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $(\mathcal{A}_N, {{\mathbb{E}}}_{{{\mathcal{A}}}_N}, {{\mathbb{F}}}_{{{\mathcal{A}}}_N}, \varepsilon_N)$. The following assertions are equivalent.
1. For all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in \mathcal{B}$, the limits $$\lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) \in {{\mathcal{B}}}{\quad\text{and}\quad}\lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) \in {{\mathcal{D}}}$$ exist.
2. For all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in \mathcal{B}$, the limits $$\lim_{N \to \infty}N\rho_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) \in {{\mathcal{B}}}{\quad\text{and}\quad}\lim_{N \to \infty}N\eta_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) \in {{\mathcal{D}}}$$ exist.
Moreover, if these assertions hold, then $$\begin{aligned}
&\lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N\rho_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\\
&\lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N\eta_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1})\end{aligned}$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in \mathcal{B}$.
Suppose assertion $(2)$ holds. Since $(\kappa_{{{\mathcal{A}}}_N}, {{\mathcal{K}}}_{{{\mathcal{A}}}_N})$ is operator-valued conditionally bi-multiplicative, we have $$(\rho_\omega^{{{\mathcal{Z}}}_n})_\pi(b_1, \dots, b_{n - 1}) = O\left(1/N^2\right) {\quad\text{and}\quad}(\eta_\omega^{{{\mathcal{Z}}}_n})_\pi(b_1, \dots, b_{n - 1}) = O\left(1/N^2\right)$$ for $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$ with at least two blocks. Hence $$\begin{aligned}
\lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) &= \lim_{N \to \infty}N\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}(\rho_\omega^{{{\mathcal{Z}}}_n})_\pi(b_1, \dots, b_{n - 1})\\
&= \lim_{N \to \infty}\left(N\rho_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) + O\left(1/N\right)\right),\end{aligned}$$ and similarly $$\lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}\left(N\eta_\omega^{{{\mathcal{Z}}}_n}(b_1, \dots, b_{n - 1}) + O\left(1/N\right)\right)$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in \mathcal{B}$.
The proof for the other direction is analogous by the operator-valued conditionally bi-multiplicativity of $({{\mathcal{E}}}_{{{\mathcal{A}}}_N}, {{\mathcal{F}}}_{{{\mathcal{A}}}_N})$ and the moment-cumulant formulae from Definitions \[MomentCumulant\] and \[OpVCBFCumulants\].
\[LimitThm\] For every $N \in {{\mathbb{N}}}$, let $({{\mathcal{A}}}_N, {{\mathbb{E}}}_N, {{\mathbb{F}}}_N, \varepsilon_N)$ be a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative-probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations and let $\{{{\mathcal{Z}}}_{N; m} = ((Z_{N; m; i})_{i \in I}, (Z_{N; m; j})_{j \in J})\}_{m = 1}^N$ be a sequence of identically distributed two-faced families in $({{\mathcal{A}}}_N, {{\mathbb{E}}}_N, {{\mathbb{F}}}_N, \varepsilon_N)$ which are c-bi-free over $({{\mathcal{B}}}, {{\mathcal{D}}})$. Furthermore, let ${{\mathcal{S}}}_N = ((S_{N; i})_{i \in I}, (S_{N; j})_{j \in J})$ be the two-faced family in $({{\mathcal{A}}}_N, {{\mathbb{E}}}_N, {{\mathbb{F}}}_N, \varepsilon_N)$ defined by $$S_{N; k} = \sum_{m = 1}^NZ_{N; m; k},\,\,\,\,\,k \in I \sqcup J.$$ The following assertions are equivalent.
1. There exists a two-faced family ${{\mathcal{Y}}}= ((Y_i)_{i \in I}, (Y_j)_{j \in J})$ in a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative-probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ such that ${{\mathcal{S}}}_N$ converges in distribution to ${{\mathcal{Y}}}$ as $N \to \infty$.
2. For all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, the limits $$\lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1})$$ exist and are independent of $m$.
Moreover, if these assertions hold, then the operator-valued bi-free and conditionally bi-free cumulants of ${{\mathcal{Y}}}$ are given by $$\rho_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\eta_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1})$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$.
Suppose assertion $(1)$ holds. For $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, we have $$\begin{aligned}
&\nu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1}) \\
&= \lim_{N \to \infty}\nu_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1})\\
&= \lim_{N \to \infty}({{\mathcal{E}}}_{{{\mathcal{A}}}_N})_{1_{\chi_\omega}}\left(S_{N; \omega(1)}, C^{\omega(2)}_{b_1}S_{N; \omega(2)}, \dots, C^{\omega(n)}_{b_{n - 2}}S_{N; \omega(n)}C^{\omega(n)}_{b_{n - 1}}\right)\\
&= \lim_{N \to \infty}\sum_{t(1), \dots, t(n) = 1}^N({{\mathcal{E}}}_{{{\mathcal{A}}}_N})_{1_{\chi_\omega}}\left(Z_{N; t(1); \omega(1)}, C^{\omega(2)}_{b_1}Z_{N; t(2); \omega(2)}, \dots, C^{\omega(n)}_{b_{n - 2}}Z_{N; t(n); \omega(n)}C^{\omega(n)}_{b_{n - 1}}\right)\\
&= \lim_{N \to \infty}\sum_{t(1), \dots, t(n) = 1}^N\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}(\kappa_{{{\mathcal{A}}}_N})_{\pi}\left(Z_{N; t(1); \omega(1)}, C^{\omega(2)}_{b_1}Z_{N; t(2); \omega(2)}, \dots, C^{\omega(n)}_{b_{n - 2}}Z_{N; t(n); \omega(n)}C^{\omega(n)}_{b_{n - 1}}\right)\\
&= \lim_{N \to \infty}\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}N^{|\pi|}(\kappa_{{{\mathcal{A}}}_N})_{\pi}\left(Z_{N; m; \omega(1)}, C^{\omega(2)}_{b_1}Z_{N; m; \omega(2)}, \dots, C^{\omega(n)}_{b_{n - 2}}Z_{N; m; \omega(n)}C^{\omega(n)}_{b_{n - 1}}\right)\\
&= \lim_{N \to \infty}\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}),\end{aligned}$$ where the second equality follows from Notation \[MCSeries\] by assuming $\{\omega(k)\}_{k = 1}^n$ intersects both $I$ and $J$ (the special cases that $\omega(k) \in I$ or $\omega(k) \in J$ for all $k$ can be checked similarly), and the fifth equality, which is independent of $m$, follows from the assumptions of c-bi-free independence over $({{\mathcal{B}}}, {{\mathcal{D}}})$ and identical distribution. Since $\nu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1})$ exist for all $n \geq 1$ and $\omega: \{1, \dots, n\} \to I \sqcup J$, it can be shown by induction on $n$ that the limits $$\lim_{N \to \infty}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})$$ exist for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. Indeed, the base case $n = 1$ follows from the assumption that $$\lim_{N \to \infty}N(\kappa_{{{\mathcal{A}}}_n})_{1_{\chi_\omega}}\left(Z_{N; m; \omega(1)}\right) = \lim_{N \to \infty}(\kappa_{{{\mathcal{A}}}_N})_{1_{\chi_\omega}}\left(S_{N; \omega(1)}\right) = \lim_{N \to \infty}({{\mathcal{E}}}_{{{\mathcal{A}}}_N})_{1_{\chi_\omega}}\left(S_{N; \omega(1)}\right) = {{\mathcal{E}}}_{1_{\chi_\omega}}(Y_{\omega(1)})$$ exist for all $\omega: \{1\} \to I \sqcup J$. For the inductive step, the limit $$\nu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1})$$ exists by assumption, and the limit $$\lim_{N \to \infty}\sum_{\substack{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)\\\pi \neq 1_{\chi_\omega}}}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})$$ exists by induction hypothesis with operator-valued conditionally bi-multiplicativity, thus the limit $$\lim_{N \to \infty}N\rho_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1}),$$ exists, and equals $$\lim_{N \to \infty}N\nu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1})$$ by Lemma \[LimitMC\]. On the other hand, it follows from a similar calculation as above that $$\begin{aligned}
\mu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1}) &= \lim_{N \to \infty}\mu_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1})\\
&= \lim_{N \to \infty}\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})\end{aligned}$$ for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, and a similar induction argument on $n$ shows that the limits $$\lim_{N \to \infty}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})$$ exist for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. In particular, choose $\pi = 1_{\chi_\omega}$ and apply Lemma \[LimitMC\], we obtain the existence of the limit $$\lim_{N \to \infty}N\mu_\omega^{{{\mathcal{Z}}}_{N; m}}(b_1, \dots, b_{n - 1}).$$
Conversely, suppose assertion $(2)$ holds. By Lemma \[LimitMC\] and operator-valued conditionally bi-multiplicativity, the limits $$\lim_{N \to \infty}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\lim_{N \to \infty}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})$$ exist for all $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$. Therefore, by the calculations above, $$\label{LimitingMomentsNu}
\lim_{N \to \infty}\nu_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}\lim_{N \to \infty}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})$$ and $$\label{LimitingMomentsMu}
\lim_{N \to \infty}\mu_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1}) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}\lim_{N \to \infty}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}),$$ and hence these limits exist. One concludes assertion $(1)$ by using Lemma \[Existence\] to construct a two-faced family ${{\mathcal{Y}}}= ((Y_i)_{i \in I}, (Y_j)_{j \in J})$ in a Banach ${{\mathcal{B}}}$-${{\mathcal{B}}}$-non-commutative-probability space with a pair of $({{\mathcal{B}}}, {{\mathcal{D}}})$-valued expectations $({{\mathcal{A}}}, {{\mathbb{E}}}, {{\mathbb{F}}}, \varepsilon)$ and define $\nu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1})$ and $\mu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1})$ to be the corresponding limit in equations and respectively.
Finally, for $n \geq 1$, $\omega: \{1, \dots, n\} \to I \sqcup J$, and $b_1, \dots, b_{n - 1} \in {{\mathcal{B}}}$, we have $$\begin{aligned}
\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}(\rho_\omega^{{{\mathcal{Y}}}})_\pi(b_1, \dots, b_{n - 1}) &= \nu_\omega^{{{\mathcal{Y}}}}(b_1, \dots, b_{n - 1})\\
&= \lim_{N \to \infty}\nu_\omega^{{{\mathcal{S}}}_N}(b_1, \dots, b_{n - 1})\\
&= \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}\lim_{N \to \infty}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}),\end{aligned}$$ and similarly $$\sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}(\eta_\omega^{{{\mathcal{Y}}}})_\pi(b_1, \dots, b_{n - 1}) = \sum_{\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)}\lim_{N \to \infty}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}).$$ A similar induction argument on $n$ shows that $$\begin{aligned}
&(\rho_\omega^{{{\mathcal{Y}}}})_\pi(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N^{|\pi|}(\rho_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1}) {\quad\text{and}\quad}\\
& (\eta_\omega^{{{\mathcal{Y}}}})_\pi(b_1, \dots, b_{n - 1}) = \lim_{N \to \infty}N^{|\pi|}(\eta_\omega^{{{\mathcal{Z}}}_{N; m}})_\pi(b_1, \dots, b_{n - 1})\end{aligned}$$ for all $\pi \in {{\mathcal{B}}}{{\mathcal{N}}}{{\mathcal{C}}}(\chi_\omega)$, from which the last claims follow from Lemma \[LimitMC\] applied to $\pi = 1_{\chi_\omega}$.
[13]{}
|
---
abstract: 'The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.'
address: 'Universitat Jaume I, Campus de Riu Sec, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, 12071 Castellón de la Plana, Spain'
author:
- 'Julio José Moyano-Fernández'
title: The universal zeta function for curve singularities and its relation with global zeta functions
---
[^1]
Introduction {#intro}
============
{#pa:I1}
Let $X$ be a complete, geometrically irreducible, singular algebraic curve defined over a perfect field $k$; from now on we will refer to such a curve simply as ‘algebraic curve over $k$’ . Let $K$ be the field of rational functions on $X$. Extending previous works of V. M. Galkin and B. Green—and based on the classical results of F. K. Schmidt [@Sch] for nonsingular curves—K.O. Stöhr (cf. [@St1], [@St2]) managed to attach a zeta function to $X$ for finite $k$ in the following manner: If $\mathcal{O}_X$ is the structure sheaf of $X$, he defined the Dirichlet series $$\zeta (\mathcal{O}_X,s) :=\sum_{\mathfrak{a} \succeq \mathcal{O}_X} q^{-s\deg{\mathfrak{a}}}, \ \ \ \ s \in \mathbb{C} \ \mbox{with} \ \mathrm{Re}(s) >0,$$ where the sum is taken over all positive divisors of $X$, and $\deg{(~\cdot~)}$ denotes the degree of those divisors. Observe that the change of variables $T=q^{-s}$ allows to consider the formal power series in $T$ $$Z(\mathcal{O}_X,T)=\sum_{n=0}^{\infty} \# (\{ \mbox{positive divisors of } X \mbox{ of degree } n\})\cdot T^{n}.$$ Moreover, Stöhr considered local zeta functions, i.e., zeta functions attached to every local ring $\mathcal{O}_P$ of points $P$ at $X$ of the form $$Z(\mathcal{O}_P,T):=\sum_{\mathfrak{a}\subseteq \mathcal{O}_P} T^{\deg{\mathfrak{a}}}=\sum_{n=0}^{\infty} \# (\{ \mathrm{positive~} \mathcal{O}_P\mathrm{-ideals~of~degree~} n\})\cdot T^{n}.$$ This series extends previous definitions by Galkin [@G] and Green [@Gr]. Furthermore, the Euler product formula for the formal power series yields the identity $$Z(\mathcal{O}_X,T)=\prod_{P\in X} Z(\mathcal{O}_P,T),$$ which actually establishes a link between the local and global theory. Every local factor $Z(\mathcal{O}_P,T)$ splits again into factors $$Z(\mathcal{O}_P, \mathcal{O}_P,T)=\sum_{n=0}^{\infty} \# (\{ \mathrm{principal~integral~} \mathcal{O}\mathrm{-ideals~of~codimension~} n\})\cdot T^{n}$$ which are determined by the value semigroup of $\mathcal{O}_P$ (see §2.1 for the definition of this semigroup) if the field is big enough, as Zúñiga showed in [@Z].
{#section}
On the other hand, when studying the Gorenstein property of one-dimensional local Cohen-Macaulay rings, Campillo, Delgado and Kiyek [@CDK (3.8)] observed the existence of a Laurent series—a polynomial in their situation—attached to those rings, and satisfying a functional equation in the case of Gorenstein rings. Further investigations by Campillo, Delgado and Gusein-Zade [@CDG1]–[@CDG9] led to the definition of a Poincaré series associated to a complex curve singularity as an integral with respect to the Euler characteristic (se also O. Viro [@V]). They even considered integration with respect to an Euler characteristic of motivic nature and so they introduced the notion of *generalized Poincaré series* of a complex curve singularity [@CDG10].
{#section-1}
In the spirit of the preceding paragraphs, the author showed in his thesis [@M] (see also the joint paper with his advisor Delgado [@DM]) that the factors $Z(\mathcal{O}_P, \mathcal{O}_P,T)$ coincide essentially with the generalized Poincaré series of Campillo, Delgado and Gusein-Zade, under a suitable specialization for finite fields (see §\[3.8\] below). These ideas have also provided some feedback: for instance Stöhr achieved a deeper insight into the nature of the local zeta functions (see [@St3], and [@St4] together with his student J.J. Mira).
{#section-2}
The key ingredient that allows to relate those different formal power series is the *universal zeta function* for a curve singularity defined by Zúñiga and the author in [@MZ]: for example, the local zeta functions and Poincaré series mentioned above are specializations of this universal zeta function. After some preliminaries, we devote Section \[section3\] to describe this series. Moreover, we claim that one may establish the local-global behaviour explained in §\[pa:I1\] for curves defined over non-finite fields. This conjectural behaviour has already shown some evidences in particular cases; see e.g. the theorem in Section \[section4\].
Preliminaries and notations
===========================
{#2.1}
Consider the normalization $\pi: \tilde{X} \to X$ of an algebraic curve $X$ over $k$, and let $\mathcal{O}=\mathcal{O}_P:=\mathcal{O}_{P,X}$ be the local ring of $X$ at $P$. For the sake of simplicity we will assume the ring $\mathcal{O}$ to be complete.
It is $\pi^{-1}(P)=\{Q_1,\ldots , Q_d\}$ and so the corresponding local rings $\mathcal{O}_{Q_i}$ are discrete valuation rings of $K$ over $\mathcal{O}$. The value semigroup associated to $\mathcal{O}$ is defined to be $$S(\mathcal{O}):=\{\underline{v}(\underline{z}) : \underline{z} \ \mbox{nonzero divisor in } \mathcal{O}\} \subseteq \mathbb{N}^d;$$ here $\underline{v}(\underline{z})=(v_1(z_1),\ldots , v_d(z_d))$, where each $v_i$ stands for the valuation associated with $\mathcal{O}_{Q_i}$; we write $S$ for this semigroup from now on. Let $c=c(S)$ denote the conductor of $S$, i.e. the smallest element $\underline{v}\in S$ such that $\underline{v}+\mathbb{N}^d\subseteq S$. Moreover, $\mathcal{O}^{\times}$ denotes the group of units of $\mathcal{O}$. Further details here and in the sequel can be checked in [@MZ] and the references therein.
{#section-3}
We say that the ring $\mathcal{O}$ is totally rational if all rings $\mathcal{O}_{Q_i}$, for $i=1,\ldots , d$ have $k$ as a residue field.
{#section-4}
The integral closure of $\mathcal{O}$ in $K/k$ is $\tilde{\mathcal{O}}=\tilde{\mathcal{O}}_P=\mathcal{O}_{Q_1}\cap \ldots \cap \mathcal{O}_{Q_d}$. We write $\tilde{\mathcal{O}}^{\times}$ for its group of units. The *singularity degree* $\delta_P$ of $\tilde{\mathcal{O}}$ is defined as $\delta_P=\delta:=\dim_k \tilde{\mathcal{O}}/\mathcal{O} <\infty$ (see e.g. [@Serre Chapter IV]).
{#section-5}
For $\underline{n}\in S$ we set $$\mathcal{I}_{\underline{n}}:=\left\{ I\subseteq\mathcal{O}\mid I=\underline{z}\mathcal{O}\text{, with }\underline{v}(\underline{z})=\underline{n}\right\} ,$$ and for $m\in\mathbb{N}$,$$\mathcal{I}_{m}:={\textstyle\bigcup\nolimits_{\substack{\underline{n}\in S\\\left\Vert
\underline{n}\right\Vert =m}}}
\mathcal{I}_{\underline{n}},$$ where $\left\Vert
\underline{n}\right\Vert$ denotes the sum of the components of the vector $\underline{n}=(n_1,\ldots , n_d)\in \mathbb{N}^d$.
{#section-6}
In the category $\mathrm{Var}_k$ of $k$-algebraic varieties, we define the Grothendieck ring $K_0(\mathrm{Var}_k)$, which is the ring generated by symbols $[V]$ for $V \in \mathrm{Var}_k$, with the relations $[V]=[W]$ if $V$ is isomorphic to $W$, $[V]=[V\setminus Z]+[Z]$ if $Z$ is closed in $V$, and $[V\times W]=[V][W]$. We write $\mathbb{L}:=[\mathbb{A}^1_{k}]$ for the class of the affine line, and $\mathcal{M}_k:=K_0(\mathrm{Var}_k)[\mathbb{L}^{-1}]$ for the ring obtained by localization with respect to the multiplicative set generated by $\mathbb{L}$.
{#2.7}
It is possible to associate to $\mathcal{I}_{\underline{n}}$ resp. $\mathcal{I}_m$ well-defined classes in the Grothendieck ring [@MZ Section 5]; those classes will be denoted by $[\mathcal{I}_{\underline{n}}] $ resp. $[\mathcal{I}_m]$. This allows to attach to the local ring $\mathcal{O}$ the zeta functions $$Z\left( T_{1},\ldots,T_{d},\mathcal{O}\right) :={\textstyle\sum\nolimits_{\underline{n}\in S}}
\left[ \mathcal{I}_{\underline{n}}\right] \mathbb{L}^{-\left\Vert
\underline{n}\right\Vert }T^{\underline{n}}\in\mathcal{M}_{k} [\![
T_{1},\ldots,T_{d} ]\!] , \nonumber$$ where $T^{\underline{n}}:=T_{1}^{n_{1}} \cdot\ldots\cdot T_{d}^{n_{d}}$, and $Z\left( T,\mathcal{O}\right) :=Z\left( T,\ldots,T,\mathcal{O}\right) $.
{#section-7}
**Definition.** Consider an algebraic curve $X$ over $k$. If $k$ has characteristic $p\geq0$, then we say that *$k$ is big enough for $X$* if for every singular point $P$ in $X$ the following two conditions hold: 1) the ring $\mathcal{O}$ is totally rational and 2) $\widetilde{\mathcal{O}}^{\times}/\mathcal{O}^{\times} \cong\left( G_{m}\right) ^{d-1}\times\left( G_{a}\right) ^{\delta-d+1}$, with $G_m=(k^{\times},\cdot)$ and $G_a=(k,+)$.
Note that the condition ‘$k$ is big enough for $X$’ is fulfilled when $p$ is big enough.
The universal zeta function for curve singularities {#section3}
===================================================
{#section-8}
For $k=\mathbb{C}$, we consider a semigroup $S\subseteq\mathbb{N}^{d}$ such that $S=S\left( \mathcal{O}\right) $. Moreover, for $\underline{n}\in S$ set $$\mathcal{I}_{\underline{n}}\left( U\right) :=\left( U-1\right)
^{-1}U^{\left\Vert \underline{n}\right\Vert +1}{\textstyle\sum\limits_{I\subseteq [d]}}
\left( -1\right) ^{\#(I)}U^{-\dim_k \big ( \mathcal{O}/\{\underline{z} \in \mathcal{O} : \underline{v}(\underline{z}) \geqslant \underline{n}+\underline{1}_{I} \} \big ) },$$ for an indeterminate $U$, and where $[d]:=\{1,2,\ldots ,d\}$, and $\underline{1}_{I}$ is the $d$-tuple with the components corresponding to the indices in $I$ equal to $1$, and the other components equal to $0$.
The notation $\mathcal{I}_{\underline{n}}\left( U\right)$ is appropriate, since that expression coincides with $\left[ \mathcal{I}_{\underline{n}}\right]$ when $U$ specializes to $\mathbb{L}$, cf. [@MZ Section 5].
{#section-9}
Let $\underline{c}=(c_1,\ldots, c_d)$ be the conductor of the semigroup $S$, cf. §\[2.1\]. Let $J:=\{1,\ldots , r\}\subseteq [d]$, and let $\underline{m}\in \mathbb{N}^d$ be such that $\underline{c}>\underline{m}$, i.e., $c_i>m_i$ for all $i\in [d]$. For a fixed $\emptyset \subsetneq J \subsetneq [d]$, set $r_J:=\# J$ and $$B_J:=\{\underline{m} \in \mathbb{N}^{r_J}: H_{J,\underline{m}} \neq \emptyset \},$$ where $H_{J,\underline{m}} :=\{\underline{n} \in S : n_j\geq c_j ~\mbox{if}~j\in J,~\mbox{and}~n_j=m_j~\mbox{otherwise}\}$.
**Definition** We define the *universal zeta function* $\mathcal{Z}\left( T_{1},\ldots,T_{d},U,S\right) $ associated with $S$ to be $$\begin{aligned}
{\textstyle\sum\limits_{\substack{\underline{n} \in S \\ \underline{0} \leqslant \underline{n} <
\underline{c}} }}
\mathcal{I}_{\underline{n}}\left( U\right) U^{-\left\Vert \underline
{n}\right\Vert }T^{\underline{n}}&+{\textstyle\sum\limits_{\emptyset\subsetneq J\subsetneq I_{0}}}
\text{ \ }{\textstyle\sum\limits_{\substack{\underline{m}\in B_{J}}}}
\left( U-1\right) U^{\left\Vert \underline{c}\right\Vert -\delta
-1}
\mathcal{I}_{\underline{f_J}(\underline{m})}\left( U\right)
U^{-\left\Vert
\underline{c}\right\Vert -\left\Vert \underline{f_{J}}(\underline{m})\right\Vert }\times \\
\times& \frac{T^{\underline{f_{J}}(\underline{m})}}{{\textstyle\prod\limits_{i=1}^{r_{J}}}
\left( 1-U^{-1}T_{i}\right) }
+\frac{\left( U-1\right) ^{d-1}U^{\delta-d+1}U^{-\left\Vert
\underline
{c}\right\Vert } T^{\underline{c}}}{{\textstyle\prod\limits_{i=1}^{d}}
\left( 1-U^{-1}T_{i}\right) },\end{aligned}$$ where $\underline{f_{J}}(\underline{m})=\left( c_{1},\ldots,c_{r_{J}},m_{r_{J}+1},\ldots,m_{d}\right) \in S$, with $m_{i}<c_{i}$, $r_{J}+1\leqslant i\leq d$, and $1\leqslant r_{J}<d$.
{#section-10}
Observe that this universal zeta function is completely determined by $S$. The adjective *universal* applied to this zeta function will be clear after the following paragraphs.
{#section-11}
The generalized Poincaré series $P_g(T_1,\ldots , T_d)$ of Campillo, Delgado and Gusein-Zade ([@CDG10]; see also [@CDK], [@DM]) as an integral with respect to an Euler characteristic of motivic nature is very close to the zeta function $Z(T_1,\ldots , T_d, \mathcal{O})$ of §\[2.7\], and therefore to the universal zeta function via the specialization $U=\mathbb{L}$:\
**Proposition.** *If $S=S(\mathcal{O})$ and $k$ is big enough for $Y$, then* $$Z(T_1,\ldots , T_d,\mathcal{O})=\mathbb{L}^{\delta + 1} P_g(T_1,\ldots , T_d) = \mathcal{Z}\left( T_{1},\ldots,T_{d},U,S\right) |_{U=\mathbb{L}}.$$
{#TheoremMONO}
\[TheoremC\] In addition, a certain specialization of the universal zeta function coincides with the zeta function of the monodromy transformation of a reduced plane curve singularity acting on its Milnor fibre, as we briefly explain now.\
**Definition.** Let $(X,0)\subseteq(\mathbb{C}^{2},0)$ be a reduced plane curve singularity defined by an equation $f=0$, with $f\in\mathcal{O}_{(\mathbb{C}^{2},0)}$ reduced. Let $h_{f}:V_{f}\rightarrow V_{f}$ be the monodromy transformation of the singularity $f$ acting on its Milnor fiber $V_{f}$. The zeta function of the monodromy $h_{f}$ is defined to be $$\varsigma_{f}\left( T\right):=\prod_{i\geqslant 0} \Big [ \mathrm{det}(\mathrm{id}-T\cdot (h_f)_{\ast} |_{H_i(V_f;\mathbb{C})}) \Big]^{(-1)^{i+1}}.$$
A result of Campillo, Delgado and Gusein-Zade ([@CDG1 Theorem 1]) allows us to prove:\
**Theorem.** *Let $k=\mathbb{C}$. Then for every $\mathcal{O}=\mathcal{O}_{\left( \mathbb{C}^{2},0\right) }/\left( f\right) $, with $f\in\mathcal{O}_{\left(
\mathbb{C}^{2},0\right) }$ reduced, and for every $S=S\left( \mathcal{O}\right) $, we have* $$\varsigma_{f}\left( T\right) =\mathcal{Z}\left( T_{1},\ldots,T_{d},U,S\right) \mid_{\begin{array}
[c]{l}T_{1}=\ldots=T_{d}=T\\
U=1
\end{array}
.}$$
{#section-12}
In [@Z] Zúñiga introduced a Dirichlet series $Z(\mathrm{Ca}(X),T)$ associated to the effective Cartier divisors on an algebraic curve $X$ defined over $k=\mathbb{F}_{q}$, which admits an Euler product of the form$$Z(\mathrm{Ca}(X),T)={\textstyle\prod\limits_{P\in X}}
Z_{\mathrm{Ca}(X)}(T,q,\mathcal{O}_{P,X}),$$ with $Z_{\mathrm{Ca}(X)}(T,q,\mathcal{O}_{P,X}):=
{\textstyle\sum\limits_{I\subseteq \mathcal{O}_{P,X}}}
T^{\dim_{k}\left(\mathcal{O}_{P,X}/I\right) }$, where $I$ runs through all the principal ideals of $\mathcal{O}_{P,X}$. In addition, $Z_{\mathrm{Ca}(X)}(T,q,\mathcal{O}_{P,X})=Z(T,\mathcal{O}_{P,X})$, cf. §\[2.7\].
Observe that this zeta function is nothing but the zeta function $Z(\mathcal{O},\mathcal{O},T)$ appearing as a local factor in the Stöhr zeta function, cf. Section \[intro\].
{#section-13}
**Remark.**\[3.8\] In the category of $\mathbb{F}_{q}$-algebraic varieties, $\left[
\cdot\right] $ specializes to the counting rational points additive invariant $\#\left( \cdot\right) $. We write $\#\left( Z\left( T_{1},\ldots
,T_{d},\mathcal{O}\right) \right) $ for the rational function obtained by specializing $\left[ \cdot\right] $ to $\#\left( \cdot\right) $. From a computational point of view, $\#\left( Z\left( T_{1},\ldots,T_{d},\mathcal{O}\right) \right) $ is obtained from $Z\left( T_{1},\ldots
,T_{d},\mathcal{O}\right) $ by replacing $\mathbb{L}$ by $q$.
{#TheoremD}
**Theorem.**
*Let $\ k=\mathbb{F}_{q}$ and let $\mathcal{Z}\left(
T_{1},\ldots,T_{d},U,S\right) $ be the universal zeta function for $S$. Moreover, let $X$ be an algebraic curve defined over $k$, and let $\mathcal{O}_{P,X}$ be the (complete) local ring of $X$ at a singular point $P$. Assume that $k$ is big enough for $X$ and that $S=S\big(\mathcal{O}_{P,X}\big) $.\
\
(1) For any $\mathcal{O}=\mathcal{O}_{\left( \mathbb{C}^{2},0\right) }/\left( f\right) $, with $f\in\mathcal{O}_{\left(
\mathbb{C}^{2},0\right) }$ reduced, and $S=S(\mathcal{O})$,$$Z_{\mathrm{Ca}(X)}\big( q^{-1}T,q,\mathcal{O}_{P,X}\big) =\#\left(
Z\big( T_{1},\ldots,T_{d},\mathcal{O}_{P,X}\big) \right)$$ $$=\mathcal{Z}\big( T_{1},\ldots,T_{d},U,S\big) \mid_{\begin{array}
[c]{l}{\small T_{1}=\ldots=T_{d}=T}\\
{\small U=q}
\end{array}
.}$$ In particular $Z_{\mathrm{Ca}(X)}\big( q^{-1}T,q,\mathcal{O}_{P,X}\big) $ depends only on $S$. Moreover, if $X$ is plane, then $Z_{\mathrm{Ca}(X)}\big( q^{-1}T,q,\mathcal{O}_{P,X}\big) $ is a complete invariant of the equisingularity class of $\mathcal{O}_{P,X}$.*
\(2) For any $\mathcal{O}=\mathcal{O}_{\left( \mathbb{C}^{2},0\right) }/\left( f\right) $, with $f\in\mathcal{O}_{\left(
\mathbb{C}^{2},0\right) }$, it holds that $$Z_{\mathrm{Ca}(X)}\big( q^{-1}T,q,\mathcal{O}_{P,X}\big) \mid
_{q \rightarrow 1}=\varsigma_{f}( T).$$
Some connections between local and global zeta functions {#section4}
========================================================
{#section-14}
For a smooth algebraic variety $Y$ defined over a field $k$, M. Kapranov defined a zeta function as the formal power series in an indeterminate $u$ $$\zeta_{\mathrm{mot},Y} (u)=\sum_{n=0}^{\infty} [Y^{(n)}] u^n \in K_0(\mathrm{Var}_k)[\![u]\!],$$ where $Y^{(n)}$ stands for the $n$-fold symmetric product of $Y$ (cf. [@K §1]). (For instance, if $k=\mathbb{F}_q$, then one obtains the usual Hasse-Weil zeta function of $Y$, cf. §\[3.8\]). When $Y$ is a curve, Baldassarri, Deninger and Naumann introduced in [@BDN] a two-variable version of the Kapranov zeta function, namely $$Z_{\mathrm{mot},Y}(t,u)=\sum_{n,\nu \geqslant 0} [\mathrm{Pic}_{\nu}^{n}] \frac{u^{\nu}-1}{u-1} t^n \in K_0(\mathrm{Var}_k)[\![u,t]\!],$$ where the algebraic $k$-scheme $\mathrm{Pic}_{\nu}^{n}=\mathrm{Pic}_{\geqslant \nu}^{n} \setminus \mathrm{Pic}_{\geqslant \nu+1}^{n}$ (with $\mathrm{Pic}_{\geqslant \nu}^{n} $ being the closed subvariety—in the Picard variety of degree $n$ line bundles on $Y$—of line bundles $\mathcal{L}$ with $h^0(\mathcal{L}) \geqslant \nu$) defines a class in $K_0(\mathrm{Var}_k)$.
{#section-15}
The connections between the universal zeta function and the motivic zeta functions of Kapranov and Baldassarri-Deninger-Naumann are being currently investigated by A. Melle, W. Zúñiga and the author; we believe that the zeta functions discussed in the previous sections are factors of motivic zeta functions of Baldassarri-Deninger-Naumann type for singular curves (as mentioned before, this is known when $k=\mathbb{F}_q$). In order to give some evidence supporting this belief, this note will be finished by stating the relation between local and global zeta functions in a particular situation.
The context will be the one of a *modulus*: Following Serre [@Serre], let $k$ be an algebraically closed field, and let $C$ be an irreducible, non-singular, complete algebraic curve defined over $k$. If $F$ is a finite subset of $C$, a modulus $\mathfrak{m}$ supported on $F$ is defined to be the assignment of an integer $n_{P} >0$ for each point $P \in F$; this is sometimes identified with the effective divisor $\sum_{P} n_{P} P$.
{#section-16}
It is possible to attach a curve to $\mathfrak{m}$ starting from $C$, essentially by “placing” the points in $F$ all together into one (see again [@Serre]). The resulting singular curve $C_{\mathfrak{m}}$ has then this point as its only singularity. It holds the following\
**Theorem.** [*Let $C_{\mathfrak{m}}$ be a curve arising from a modulus $\mathfrak{m}$ supported on a finite set of points of a curve $C$ as above, and let $P$ be its only singular point. Furthermore, let $\pi:\widetilde{C_{\mathfrak{m}}} \to C_{\mathfrak{m}}$ be the normalization morphism. Then $$Z_{\mathrm{mot},C_{\mathfrak{m}}}(\mathbb{L}^{-1}T,\mathbb{L})= Z_{\mathrm{mot},\widetilde{C_{\mathfrak{m}}}}(\mathbb{L}^{-1}T,\mathbb{L}) \prod_{i=1}^{\sharp(\pi^{-1}(P))} (1-\mathbb{L}^{-1} T)
\cdot Z (T, \mathcal{O}_{P}).$$* ]{} The proof of this statement will appear in a forthcoming paper.
[99]{}
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[^1]: The author was partially supported by the Spanish Government Ministerio de Economía, Industria y Competitividad (MINECO), grants MTM2012-36917-C03-03 and MTM2015-65764-C3-2-P, as well as by Universitat Jaume I, grant P1-1B2015-02.
|
---
abstract: 'For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.'
address:
- 'Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland.'
- 'Département d’informatique et de génie logiciel, Université Laval, Québec City (Québec), Canada G1V 0A6'
- 'Département de mathématiques et de statistique, Université Laval, Québec City (Québec), Canada G1V 0A6,.'
author:
- Łukasz Kosiński
- Étienne Martel
- Thomas Ransford
date: 7 August 2017
title: A uniform boundedness principle in pluripotential theory
---
[^1]
[^2]
[^3]
The uniform boundedness principle {#S:ubp}
=================================
Let $\Omega$ be an open subset of ${{\mathbb C}}^N$. A function $u:\Omega\to[-\infty,\infty)$ is called *plurisubharmonic* if:
1. $u$ is upper semicontinuous, and
2. $u|_{\Omega\cap L}$ is subharmonic, for each complex line $L$.
For background information on plurisubharmonic functions, we refer to the book of Klimek [@Kl91].
It is apparently an open problem whether in fact (2) implies (1) if $N\ge2$. In attacking this problem, we have repeatedly run up against an obstruction in the form of a uniform boundedness principle for plurisubharmonic functions. This principle, which we think is of interest in its own right, is the main subject of this note. Here is the formal statement.
\[T:ubp\] Let $D\subset{{\mathbb C}}^N$ and $G\subset{{\mathbb C}}^M$ be domains, where $N,M\ge1$, and let ${{\cal U}}$ be a family of continuous plurisubharmonic functions on $D\times G$. Suppose that:
1. ${{\cal U}}$ is locally uniformly bounded above on $D\times\{w\}$, for each $w\in G$;
2. ${{\cal U}}$ is locally uniformly bounded above on $\{z\}\times G$, for each $z\in D$.
Then:
1. ${{\cal U}}$ is locally uniformly bounded above on $D\times G$.
In other words, if there is an upper bound for ${{\cal U}}$ on each compact subset of $D\times G$ of the form $K\times\{w\}$ or $\{z\}\times L$, then there is an upper bound for ${{\cal U}}$ on every compact subset of $D\times G$. The point is that we have no *a priori* quantitative information about these upper bounds, merely that they exist. In this respect, the result resembles the classical Banach–Steinhaus theorem from functional analysis.
The proof of Theorem \[T:ubp\] is based on two well-known but non-trivial results from several complex variables: the equivalence (under appropriate assumptions) of plurisubharmonic hulls and polynomial hulls, and Hartogs’ theorem on separately holomorphic functions. The details of the proof are presented in §\[S:pfubp\].
The Banach–Steinhaus theorem is usually stated as saying that a family of bounded linear operators on a Banach space $X$ that is pointwise-bounded on $X$ is automatically norm-bounded. There is a stronger version of the result in which one assumes merely that the operators are pointwise-bounded on a non-meagre subset $Y$ of $X$, but with the same conclusion. This sharper form leads to new applications (for example, a nice one in the theory of Fourier series can be found in [@Ru87 Theorem 5.12]). Theorem \[T:ubp\] too possesses a sharper form, in which one of the conditions (i),(ii) is merely assumed to hold on a non-pluripolar set. This improved version of theorem is the subject of §\[S:ubpgen\].
We conclude the paper in §\[S:appl\] by considering applications of these results, and we also discuss the connection with the upper semicontinuity problem mentioned at the beginning of the section.
Proof of the uniform boundedness principle {#S:pfubp}
==========================================
We shall need two auxiliary results. The first one concerns hulls. Given a compact subset $K$ of ${{\mathbb C}}^N$, its *polynomial hull* is defined by $${\widehat}{K}:=\{z\in{{\mathbb C}}^N:|p(z)|\le\sup_K|p|\text{~for every polynomial~$p$ on ${{\mathbb C}}^N$}\}.$$ Further, given an open subset $\Omega$ of ${{\mathbb C}}^N$ containing $K$, the *plurisubharmonic hull* of $K$ with respect to $\Omega$ is defined by $${\widehat}{K}_{\operatorname{PSH}(\Omega)}:=\{z\in\Omega:u(z)\le\sup_Ku\text{~for every plurisubharmonic $u$ on $\Omega$}\}.$$ Since $|p|$ is plurisubharmonic on $\Omega$ for every polynomial $p$, it is evident that ${\widehat}{K}_{\operatorname{PSH}(\Omega)}\subset {\widehat}{K}$. In the other direction, we have the following result.
\[L:hulls\] Let $K$ be a compact subset of ${{\mathbb C}}^N$ and let $\Omega$ be an open subset of ${{\mathbb C}}^N$ such that ${\widehat}{K}\subset\Omega$. Then ${\widehat}{K}_{\operatorname{PSH}(\Omega)}={\widehat}{K}$.
The second result that we shall need is Hartogs’ theorem [@Ha06] that separately holomorphic functions are holomorphic.
\[L:Hartogs\] Let $D\subset{{\mathbb C}}^N$ and $G\subset{{\mathbb C}}^M$ be domains, and let $f:D\times G\to{{\mathbb C}}$ be a function such that:
- $z\mapsto f(z,w)$ is holomorphic on $D$, for each $w\in G$;
- $w\mapsto f(z,w)$ is holomorphic on $G$, for each $z\in D$.
Then $f$ is holomorphic on $D\times G$.
Suppose that the result is false. Then there exist sequences $(a_n)$ in $D\times G$ and $(u_n)$ in ${{\cal U}}$ such that $u_n(a_n)>n$ for all $n$ and $a_n\to a\in D\times G$. Let $P$ be a compact polydisk with centre $a$ such that $P\subset D\times G$. For each $n$, set $$P_n:=\{\zeta\in P:u_n(\zeta)\le n\}.$$ Then $P_n$ is compact, because the functions in ${{\cal U}}$ are assumed continuous. Further, since $P$ is convex, we have ${\widehat}{P_n}\subset P\subset D\times G$. By Lemma \[L:hulls\], we have ${\widehat}{P_n}={\widehat}{(P_n)}_{\operatorname{PSH}(D\times G)}$. As $a_n$ clearly lies outside this plurisubharmonic hull, it follows that $a_n$ also lies outside the polynomial hull of $P_n$. Thus there exists a polynomial $q_n$ such that $\sup_{P_n}|q_n|<1$ and $|q_n(a_n)|>1$. Let $r_n$ be a polynomial vanishing at $a_1,\dots,a_{n-1}$ but not at $a_n$, and set $p_n:=q_n^mr_n$, where $m$ is chosen large enough so that $$\sup_{P_n}|p_n|<2^{-n}
\quad\text{and}\quad
|p_n(a_n)|>n+\sum_{k=1}^{n-1}|p_k(a_n)|.$$ Let us write $P=Q\times R$, where $Q,R$ are compact polydisks such that $Q\subset D$ and $R\subset G$. Then, for each $w\in R$, the family ${{\cal U}}$ is uniformly bounded above on $Q\times\{w\}$, so eventually $u_n\le n$ on $Q\times\{w\}$. For these $n$, we then have $Q\times\{w\}\subset P_n$ and hence $|p_n|\le 2^{-n}$ on $Q\times\{w\}$. Thus the series $$f(z,w):=\sum_{n\ge1}p_n(z,w)$$ converges uniformly on $Q\times\{w\}$. Likewise, it converges uniformly on $\{z\}\times R$, for each $z\in D$. We deduce that:
- $z\mapsto f(z,w)$ is holomorphic on $\operatorname{int}(Q)$, for each $w\in \operatorname{int}(R)$;
- $w\mapsto f(z,w)$ is holomorphic on $\operatorname{int}(R)$, for each $z\in \operatorname{int}(Q)$.
By Lemma \[L:Hartogs\], $f$ is holomorphic on $\operatorname{int}(P)$. On the other hand, for each $n$, our construction gives $$|f(a_n)|
\ge|p_n(a_n)|-\sum_{k=1}^{n-1}|p_k(a_n)|-\sum_{k=n+1}^\infty| p_k(a_n)|>n.$$ Since $a_n\to a$, it follows that $f$ is discontinuous at $a$, the central point of $P$. We thus have arrived at a contradiction, and the proof is complete.
One might wonder if Theorem \[T:ubp\] remains true if we drop one of the assumptions (i) or (ii). Here is a simple example to show that it does not. For each $n\ge1$, set $$K_n:=\{z\in{{\mathbb C}}:|z|\le n,~1/n\le \arg(z)\le 2\pi\},$$ and let $(z_n)$ be a sequence such that $z_n\in{{\mathbb C}}\setminus K_n$ for all $n$ and $z_n\to0$. By Runge’s theorem, for each $n$ there exists a polynomial $p_n$ such that $\sup_{K_n}|p_n|\le 1$ and $|p_n(z_n)|>n$. The sequence $|p_n|$ is then pointwise bounded on ${{\mathbb C}}$, but not uniformly bounded in any neighborhood of $0$. Thus, if we define $u_n(z,w):=|p_n(z)|$, then we obtain a sequence of continuous plurisubharmonic functions on ${{\mathbb C}}\times{{\mathbb C}}$ satisfying (ii) but not (iii).
Although we cannot drop (i) or (ii) altogether, it *is* possible to weaken one of the conditions (i) or (ii) to hold merely on a set that is ‘not too small’, and still obtain the conclusion (iii). This is the subject of next section.
A stronger form of the uniform boundedness principle {#S:ubpgen}
====================================================
A subset $E$ of ${{\mathbb C}}^N$ is called *pluripolar* if there exists a plurisubharmonic function $u$ on ${{\mathbb C}}^N$ such that $u=-\infty$ on $E$ but $u\not\equiv-\infty$ on ${{\mathbb C}}^N$. Pluripolar sets have Lebesgue measure zero, and a countable union of pluripolar sets is again pluripolar. For further background on pluripolar sets, we again refer to Klimek’s book [@Kl91].
In this section we establish the following generalization of Theorem \[T:ubp\], in which we weaken one of the assumptions (i),(ii) to hold merely on a non-pluripolar set.
\[T:ubpgen\] Let $D\subset{{\mathbb C}}^N$ and $G\subset{{\mathbb C}}^M$ be domains, where $N,M\ge1$, and let ${{\cal U}}$ be a family of continuous plurisubharmonic functions on $D\times G$. Suppose that:
1. ${{\cal U}}$ is locally uniformly bounded above on $D\times\{w\}$, for each $w\in G$;
2. ${{\cal U}}$ is locally uniformly bounded above on $\{z\}\times G$, for each $z\in F$,
where $F$ is a non-pluripolar subset of $D$. Then:
1. ${{\cal U}}$ is locally uniformly bounded above on $D\times G$.
For the proof, we need the following generalization of Hartogs’ theorem, due to Terada [@Te67; @Te72].
\[L:Terada\] Let $D\subset{{\mathbb C}}^N$ and $G\subset{{\mathbb C}}^M$ be domains, and let $f:D\times G\to{{\mathbb C}}$ be a function such that:
- $z\mapsto f(z,w)$ is holomorphic on $D$, for each $w\in G$;
- $w\mapsto f(z,w)$ is holomorphic on $G$, for each $z\in F$,
where $F$ is a non-pluripolar subset of $D$. Then $f$ is holomorphic on $D\times G$.
We define two subsets $A,B$ of $D$ as follows. First, $z\in A$ if $w\mapsto\sup_{u\in{{\cal U}}}u(z,w)$ is locally bounded above on $G$. Second, $z\in B$ if there exists a neighborhood $V$ of $z$ in $D$ such that $(z,w)\mapsto\sup_{u\in{{\cal U}}}u(z,w)$ is locally bounded above on $V\times G$. Clearly $B$ is open in $D$ and $B\subset A$. Also $F\subset A$, so $A$ is non-pluripolar.
Let $z_0\in D\setminus B$. Then there exists $w_0\in G$ such that ${{\cal U}}$ is not uniformly bounded above on any neighborhood of $(z_0,w_0)$. The same argument as in the proof of Theorem \[T:ubp\] leads to the existence of a compact polydisk $P=Q\times R$ around $(z_0,w_0)$ and a function $f:Q\times R\to{{\mathbb C}}$ such that:
- $z\mapsto f(z,w)$ is holomorphic on $\operatorname{int}(Q)$, for each $w\in\operatorname{int}(R)$,
- $w\mapsto f(z,w)$ is holomorphic on $\operatorname{int}(R)$, for each $z\in \operatorname{int}(Q)\cap A$,
and at the same time $f$ is unbounded in each neighborhood of $(z_0,w_0)$. By Lemma \[L:Terada\], this is possible only if $\operatorname{int}(Q)\cap A$ is pluripolar.
Resuming what we have proved: if $z\in D$ and every neighborhood of $z$ meets $A$ in a non-pluripolar set, then $z\in B$.
We now conclude the proof with a connectedness argument. As $A$ is non-pluripolar, and a countable union of pluripolar sets is pluripolar, there exists $z_1\in D$ such that every neighborhood of $z_1$ meets $A$ in a non-pluripolar set, and consequently $z_1\in B$. Thus $B\ne\emptyset$. We have already remarked that $B$ is open in $D$. Finally, if $z\in D\setminus B$, then there is a an open neighborhood $W$ of $z$ that meets $A$ in a pluripolar set, hence $B\cap W$ is both pluripolar and open, and consequently empty. This shows that $D\setminus B$ is open in $D$. As $D$ is connected, we conclude that $B=D$, which proves the theorem.
We end the section with some remarks concerning the sharpness of Theorem \[T:ubpgen\].
Firstly, we cannot weaken both conditions (i) and (ii) simultaneously. Indeed, let ${{\mathbb D}}$ be the unit disk, and define a sequence $u_n:{{\mathbb D}}\times{{\mathbb D}}\to{{\mathbb R}}$ by $$u_n(z,w):=n(|z+w|-3/2).$$ Then
- $z\mapsto\sup_n u_n(z,w)$ is bounded above on ${{\mathbb D}}$ for all $|w|\le1/2$,
- $w\mapsto\sup_n u_n(z,w)$ is bounded above on ${{\mathbb D}}$ for all $|z|\le1/2$,
but the sequence $u_n(z,w)$ is not even pointwise bounded above at the point $(z,w):=(\frac{4}{5},\frac{4}{5})$.
Secondly, the condition in Theorem \[T:ubpgen\] that $F$ be non-pluripolar is sharp, at least for $F_\sigma$-sets. Indeed, let $F$ be an $F_\sigma$-pluripolar subset of $D$. Then there exists a plurisubharmonic function $v$ on ${{\mathbb C}}^N$ such that $v=-\infty$ on $F$ and $v(z_0)>-\infty$ for some $z_0\in D$. By convolving $v$ with suitable smoothing functions, we can construct a sequence $(v_n)$ of continuous plurisubharmonic functions on ${{\mathbb C}}^N$ decreasing to $v$ and such that the sets $\{v_n\le -n\}$ cover $F$. Let $(p_n)$ be a sequence of polynomials in one variable that is pointwise bounded in ${{\mathbb C}}$ but not uniformly bounded on any neighborhood of $0$ (such a sequence was constructed at the end of §\[S:pfubp\]). Choose positive integers $N_n$ such that $\sup_{|w|\le n}|p_n(w)|\le N_n$, and define $u_n:D\times{{\mathbb C}}\to{{\mathbb R}}$ by $$u_n(z,w):=v_{N_n}(z)+|p_n(w)|.$$ Then
- $z\mapsto\sup_n u_n(z,w)$ is locally bounded above on $D$ for all $w\in{{\mathbb C}}$,
- $w\mapsto\sup_n u_n(z,w)$ is locally bounded above on ${{\mathbb C}}$ for all $z\in F$,
but $\sup_nu_n(z,w)$ is not bounded above on any neighborhood of $(z_0,0)$.
Applications of the uniform boundedness principle {#S:appl}
=================================================
Our first application is to null sequences of plurisubharmonic functions.
\[T:null\] Let $D\subset{{\mathbb C}}^N$ and $G\subset{{\mathbb C}}^M$ be domains, and let $(u_n)$ be a sequence of positive continuous plurisubharmonic functions on $D\times G$. Suppose that:
- $u_n(\cdot,w)\to0$ locally uniformly on $D$ as $n\to\infty$, for each $w\in G$,
- $u_n(z,\cdot)\to0$ locally uniformly on $G$ as $n\to\infty$, for each $z\in F$,
where $F\subset D$ is non-pluripolar. Then $u_n\to0$ locally uniformly on $D\times G$.
Let $a\in D\times G$. Choose $r>0$ such that $\overline{B}(a,2r)\subset D\times G$. Writing $m$ for Lebesgue measure on ${{\mathbb C}}^N\times {{\mathbb C}}^M$, we have $$\begin{aligned}
\sup_{\zeta\in\overline{B}(a,r)}u_n(\zeta)
&\le \sup_{\zeta\in\overline{B}(a,r)}\frac{1}{m({B}(\zeta,r))}\int_{{B}(\zeta,r)}u_n\,dm\\
&\le \frac{1}{m({B}(0,r))}\int_{{B}(a,2r)}u_n\,dm.\end{aligned}$$ Clearly $u_n\to0$ pointwise on $B(a,2r)$. Also, by Theorem \[T:ubpgen\], the sequence $(u_n)$ is uniformly bounded on $B(a,2r)$. By the dominated convergence theorem, it follows that $\int_{B(a,2r)}u_n\,dm\to0$ as $n\to\infty$. Hence $\sup_{\zeta\in\overline{B}(a,r)}u_n(\zeta)\to0$ as $n\to\infty$.
Our second application relates to the problem mentioned at the beginning of §\[S:ubp\]. Recall that $u:\Omega\to[-\infty,\infty)$ is plurisubharmonic if
1. $u$ is upper semicontinuous, and
2. $u|_{\Omega\cap L}$ is subharmonic, for each complex line $L$,
and the problem is to determine whether in fact (2) implies (1). Here are some known partial results:
- Lelong [@Le45] showed that (2) implies (1) if, in addition, $u$ is locally bounded above.
- Arsove [@Ar66] generalized Lelong’s result by showing that, if $u$ if separately subharmonic and locally bounded above, then $u$ is upper semicontinuous. (Separately subharmonic means that (2) holds just with lines $L$ parallel to the coordinate axes.) Further results along these lines were obtained in [@AG93; @KT96; @Ri14].
- Wiegerinck [@Wi88] gave an example of a separately subharmonic function that is not upper semicontinuous. Thus Arsove’s result no longer holds without the assumption that $u$ be locally bounded above.
In seeking an example to show that (2) does not imply (1), it is natural to try to emulate Wiegerinck’s example, which was constructed as follows. Let $K_n, z_n$ and $p_n$ be defined as in the example at the end of §\[S:pfubp\]. For each $n$ define $v_n(z):=\max\{|p_n(z)|-1,\,0\}$. Then $v_n$ is a subharmonic function, $v_n=0$ on $K_n$ and $v_n(z_n)>n-1$. Set $$u(z,w):=\sum_kv_k(z)v_k(w).$$ If $w\in{{\mathbb C}}$, then $w\in K_n$ for all large enough $n$, so $v_n(w)=0$. Thus, for each fixed $w\in{{\mathbb C}}$, the function $z\mapsto u(z,w)$ is a finite sum of subharmonic functions, hence subharmonic. Evidently, the same is true with roles of $z$ and $w$ reversed. Thus $u$ is separately subharmonic. On the other hand, for each $n$ we have $$u(z_n,z_n)\ge v_n(z_n)v_n(z_n)>(n-1)^2,$$ so $u$ is not bounded above on any neighborhood of $(0,0)$.
This example does not answer the question of whether (2) implies (1) because the summands $v_k(z)v_k(w)$ are not plurisubharmonic as functions of $(z,w)\in{{\mathbb C}}^2$. It is tempting to try to modify the construction by replacing $v_k(z)v_k(w)$ by a positive plurisubharmonic sequence $v_k(z,w)$ such that the partial sums $\sum_{k=1}^nv_k$ are locally bounded above on each complex line, but not on any open neighborhood of $(0,0)$. However, Theorem \[T:ubp\] demonstrates immediately that this endeavor is doomed to failure, at least if we restrict ourselves to continuous plurisubharmonic functions.
This raises the following question, which, up till now, we have been unable to answer.
\[Q:cts\] Does Theorem \[T:ubp\] remain true without the assumption that the functions in ${{\cal U}}$ be continuous?
This is of interest because of the following result.
Assume that the answer to Question \[Q:cts\] is positive. Let $\Omega$ be an open subset of ${{\mathbb C}}^N$ and let $u:\Omega\to[-\infty,\infty)$ be a function such that $u|_{\Omega\cap L}$ is subharmonic for each complex line $L$. Define $$s(z):=\sup\{v(z):v \text{~plurisubharmonic on~}\Omega,~v\le u\}.$$ Then $s$ is plurisubharmonic on $\Omega$.
Let ${{\cal U}}$ be the family of plurisubharmonic functions $v$ on $\Omega$ such that $v\le u$. If the answer to Question \[Q:cts\] is positive, then ${{\cal U}}$ is locally uniformly bounded above on $\Omega$. Hence, by [@Kl91 Theorem 2.9.14], the upper semicontinuous regularization $s^*$ of $s$ is plurisubharmonic on $\Omega$, and, by [@Kl91 Proposition 2.6.2], $s^*=s$ Lebesgue-almost everywhere on $\Omega$. Fix $z\in\Omega$. Then there exists a complex line $L$ passing through $z$ such that $s^*=s$ almost everywhere on $\Omega\cap L$. Let $\mu_r$ be normalized Lebesgue measure on $B(z,r)\cap L$. Then $$s^*(z)\le\int_{B(z,r)\cap L} s^*\,d\mu_r=\int_{B(z,r)\cap L} s\,d\mu_r\le\int_{B(z,r)\cap L} u\,d\mu_r.$$ (Note that $u$ is Borel-measurable by [@Ar66 Lemma 1].) Since $u|_{\Omega\cap L}$ is upper semicontinuous, we can let $r\to0^+$ to deduce that $s^*(z)\le u(z)$. Thus $s^*$ is itself a member of ${{\cal U}}$, so $s^*\le s$, and thus finally $s=s^*$ is plurisubharmonic on $\Omega$.
Of course, $s=u$ if and only if $u$ is itself plurisubharmonic. Maybe this could provide a way of attacking the problem of showing that $u$ is plurisubharmonic?
[99]{}
D. H. Armitage, S. J. Gardiner, Conditions for separately subharmonic functions to be subharmonic, [*Potential Anal.*]{} 2 (1993), 255–261.
M. G. Arsove, On subharmonicity of doubly subharmonic functions, [*Proc. Amer. Math. Soc.*]{} 17 (1966), 622–626.
F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten. [*Math. Ann.*]{} 62 (1906), 1–88.
M. Klimek, [*Pluripotential Theory*]{}, Oxford University Press, New York, 1991.
S. Kołodziej, J. Thorbiörnson, Separately harmonic and subharmonic functions [*Potential Anal.*]{} 5 (1996), 463–466.
P. Lelong, Les fonctions plurisousharmoniques, [*Ann. Sci. Éc. Norm. Sup. (3)*]{} 62 (1945), 301–338.
J. Riihentaus, On separately subharmonic and harmonic functions. [*Complex Var. Elliptic Equ.*]{} 59 (2014), 149–161.
W. Rudin, [*Real and Complex Analysis*]{}, Third edition. McGraw–Hill, New York, 1987.
T. Terada, Sur une certaine condition sous laquelle une fonction de plusieurs variables complexes est holomorphe. Diminution de la condition dans le théorème de Hartogs. [*Research Ins. Math. Sci. Kyoto Univ.*]{} 2 (1967), 383–396.
T. Terada, Analyticités relatives à chaque variable. Analogies du théorème de Hartogs. [*J. Math. Kyoto Univ.*]{} 12 (1972), 263–296.
J. Wiegerinck, Separately subharmonic functions need not be subharmonic, [*Proc. Amer. Math. Soc.*]{} 104 (1988), 770–771.
[^1]: ŁK supported by the Ideas Plus grant 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education.
[^2]: EM supported by an NSERC undergraduate student research award
[^3]: TR supported by grants from NSERC and the Canada research chairs program
|
---
abstract: 'We propose an entanglement sharing protocol based on separable states. Initially, two parties, Alice and Bob, share a two-mode separable Gaussian state. Alice then splits her mode into two separable modes and distributes them between two players. Bob is separable from the players but he can create entanglement with either of the players if the other player moves to his location and collaborates with him. Any two parties are separable and the creation of entanglement is thus mediated by transmission of a mode which is separable from individual modes on Alice’s and Bob’s side. For the state shared by the players and Bob one cannot establish entanglement between any two modes even with the help of operation on the third mode provided that Bob is restricted to Gaussian measurements and the state thus carries a nontrivial signature of bound entanglement. The present protocol also demonstrates switching between different separability classes of tripartite systems by coherent operations on its bipartite parts and complements studies on protocols utilizing mixed partially entangled multipartite states.'
author:
- 'Ladislav Mišta Jr.'
title: Entanglement sharing with separable states
---
Introduction
============
Three correlated elementary quantum systems represent a basic primitive which already may exhibit genuine multipartite phenomena. The discovery that tripartite entanglement can provide a stronger violation of local realism [@Greenberger_89; @Pan_00] than bipartite one has triggered a large research activity with the aim to characterize it and find its applications. Early studies of tripartite entanglement focused on systems of three two-level particles (qubits) for which new forms of multipartite bound entanglement [@Bennett_99a; @Dur_00a] and inequivalent entanglement classes [@Dur_00b] have been found. Three qubits also proved to be a suitable platform for construction of a classical analog of bound entanglement know as bound information [@Acin_04], which so far has not been found in the bipartite scenario. In comparison with two qubits which can only be separable or entangled three-qubit states can be divided into five separability classes [@Dur_99] in dependence on their separability properties with respect to different qubits. Most of the applications of the three-qubit entanglement rely on the utilization of pure states from the class of fully inseparable states which are entangled with respect to all three qubits. They involve various protocols for information splitting ranging from secret sharing [@Hillery_99], telecloning [@Murao_99] and assisted teleportation [@Karlsson_98] as well as protocols for construction of quantum gates [@Gottesman_99] or controlled quantum cryptography [@Zukowski_98].
Tripartite entanglement has also been investigated within the framework of Gaussian states [@Braunstein_05] of infinitely-dimensional quantum systems. A convenient prototype of such a system is a system of three modes $A$, $A'$ and $B$ of an electromagnetic field which are characterized by position $x_{j}$ and momentum $p_{j}$ quadrature operators satisfying the canonical commutation rules $[x_j,p_k]=i\delta_{jk}$, $j,k=A,A',B$. Quantum states of three modes can be represented in phase space by a 6-variate Wigner quasiprobability distribution [@Wigner_32] and the set of Gaussian states comprises states with a Gaussian Wigner function. A three-mode Gaussian state $\rho_{AA'B}$ is therefore fully characterized by the vector of coherent displacements $d=\langle\xi\rangle=\mbox{Tr}(\rho_{AA'B}\xi)$, where we have introduced a column vector $\xi=(x_A,p_A,x_{A'},p_{A'},x_B,p_B)^{T}$, and by a $6\times6$ covariance matrix (CM) with elements $\gamma_{ij}=\langle\{\xi_i-d_i,\xi_j-d_j\}\rangle$, $i,j=1,\ldots,6$, where $\{A,B\}=AB+BA$ is the anticommutator.
Following the classification of Ref. [@Dur_99] we can divide three-mode states into five separability classes involving [@Giedke_01]:\
1. [*Fully inseparable states*]{} which are entangled with respect to all three bipartite splittings of modes $A,A'$ and $B$ into two groups. That is states entangled across $A-(A'B)$, $A'-(AB)$ as well as $B-(AA')$ splitting.\
2. [*One-mode biseparable states*]{} which are entangled with respect to two bipartite splittings, but separable with respect to the third one. Such a state exhibits entanglement across, e.g, $A-(A'B)$ and $A'-(AB)$ splitting but it is separable with respect to $B-(AA')$ splitting.\
3. [*Two-mode biseparable states*]{} which are entangled across one bipartite splitting, but separable with respect to the remaining two splittings. The state is therefore entangled, e.g., across $A-(A'B)$ splitting but separable with respect to $A'-(AB)$ and $B-(AA')$ splittings.\
4. [*Three-mode biseparable states*]{} which are separable across all three bipartite splittings but which cannot be written as a convex mixture of triple product states.\
5. [*Fully separable states*]{} which can be written as a convex mixture of triple product states. Like in the qubit case a genuine tripartite entanglement carried by fully inseparable three-mode states is practically exclusively used as a resource in quantum information protocols. It is due to a relative ease of its preparation [@Loock_00; @Aoki_03], detection [@Loock_03] and a number of quantum protocols which this type of entanglement offers [@Loock_01; @Yonezawa_04; @Tyc_02]. On the other hand, the other classes carry only partial or no entanglement, some exist just in the mixed-state scenario (two-mode and three-mode biseparable states [@Adesso_06]) and one might be then tempted to doubt about their practical utility. However, the astonishing protocol for entanglement distribution by a separable system [@Cubitt_03] teaches us about the opposite. Originally developed for qubits [@Cubitt_03], later extended to Gaussian states [@Mista_09], and experimentally demonstrated in [@Vollmer_13], it shows that also other, even mixed and just partially separable states may demonstrate new phenomena which are not encountered in the context of fully inseparable states. It demonstrates that two distant observers, Alice and Bob, can entangle modes $A$ and $B$ held by them by sending a third separable mode $A'$ between them. Initially, Alice holds modes $A$ and $A'$ whereas Bob holds mode $B$ of a suitable fully separable Gaussian state. By a beam splitter on modes $A$ and $A'$ Alice then transforms the state to the state entangled only across $A-(A'B)$ splitting (two-mode biseparable state) and transmits the separable mode $A'$ to Bob. He finally superimposes the received mode $A'$ with his mode $B$ on another beam splitter and thus he entangles modes $A$ and $B$ whereas mode $A'$ still remains separable from the two-mode subsystem $(AB)$ (one-mode biseparable state). The protocol thus shows that direct transmission of entanglement is not necessary to entangle two separate parties but only communication of a separable quantum system, local operations and a priori shared suitable fully separable state of three quantum systems suffice to accomplish the task.
In this paper we demonstrate the utility of mixed partially separable states from other separability classes for implementation of a certain entanglement sharing scheme. Specifically, we show that there is a correlated separable Gaussian state of two possibly distant modes $A$ and $B$ such that if mode $A$ is split into two modes $A$ and $A'$ this creates entanglement of each of the modes with the distant mode $B$ and the other split mode taken together. More precisely, the splitting entangles mode $A$ with the pair of modes $(A'B)$ and mode $A'$ with the pair of modes $(AB)$ at the same time. Moreover, any two modes are separable in the prepared three-mode state. By transmitting the mode $A'$ ($A$) which is [*separable*]{} from the mode $A$ ($A'$) to the location of mode $B$ the entanglement can be transformed by a simple beam splitter into entanglement of mode $A$ ($A'$) and the distant mode $B$. The protocol thus shares similarity with the quantum secret sharing protocol [@Hillery_99; @Cleve_99] for entanglement [@Choi_12] in which a dealer splits one part of a bipartite entanglement among several players in such a way that some collections of players can recover the entanglement with the dealer whereas the other collections cannot. If in the present case the modes $A$ and $B$ of the initial separable state are shared by two observers, called Alice and Bob, then entanglement created by splitting on Alice’s side of mode $A$ into two modes $A$ and $A'$ can be turned into the entanglement between Alice and Bob only if Bob has physically at his disposal also either of the split modes $A$ or $A'$. Remarkably, there is a certain interval of entanglement strengths for which Bob can establish entanglement with Alice by the coherent beam splitting operation on his mode and the received mode but he cannot establish it by any Gaussian measurement on his mode $B$ followed by displacement of modes $A$ and $A'$ which challenges the question about the presence of bound entanglement in the considered state. Like in the case of the entanglement distribution by a separable ancilla the protocol works only with mixed states and starts with a suitable fully separable three-mode Gaussian state. In contrast with a two-mode biseparable state which is created in the second step of the entanglement distribution protocol a one-mode biseparable state is created in the intermediate step of the present entanglement sharing protocol. Likewise, a one-mode biseparable state appears at the final step of the entanglement distribution protocol whereas the entanglement sharing scheme is crowned by a fully inseparable state.
The paper is structured as follows. In Sec \[sec\_1\] we explain the entanglement sharing protocol. In Sec. \[sec\_2\] we show the gap between unitary and measurement-based localizability of the entanglement for the state from the intermediate step of the protocol. Sec. \[sec\_3\] contains discussion and conclusion.
![Scheme of the entanglement sharing protocol. Mode $A$ in a position squeezed vacuum state and a vacuum mode $B$ are displaced as in Eq. (\[displacements\]). Mode $A$ is then split on a balanced beam splitter $BS_{AA'}$ into two modes $A$ and $A'$ which creates a state with no two-mode entanglement which possesses entanglement across $A-(A'B)$ and $A'-(AB)$ splittings and which is separable across $B-(AA')$ splitting. If modes $A'$ and $B$ are superimposed on a balanced beam splitter $BS_{A'B}$ and the dashed beam splitter $BS_{AB}$ is absent, entanglement is localized between modes $A$ and $B$ (solid lines with arrows). If instead the beam splitter $BS_{A'B}$ is removed and modes $A$ and $B$ are superimposed on a dashed balanced beam splitter $BS_{AB}$ entanglement is localized between modes $A'$ and $B$ (dashed lines with arrows). In both cases the resulting state is entangled across all three bipartite splittings and therefore the state is genuinely tripartite entangled. See text for details.[]{data-label="fig1"}](Secretsharing2.ps){width="5.5cm"}
Entanglement sharing protocol {#sec_1}
=============================
The scheme of the protocol is depicted in Fig. \[fig1\] The protocol starts with a Gaussian state of two modes $A$ and $B$ shared by a sender Alice and a receiver Bob which has the covariance matrix (CM) of the form: $$\begin{aligned}
\label{gammaAB}
\gamma_{AB}=\left(\begin{array}{cccc}
1+e^{-2r}(e^{2\varepsilon}-1) & 0 & e^{-2r}-1 & 0 \\
0 & e^{2r} & 0 & 0\\
e^{-2r}-1 & 0 & 2-e^{-2r} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}\right),\end{aligned}$$ where $r\geq0$ is the squeezing parameter and $\varepsilon\geq0$ is a noise parameter which will be specified later. The right upper $2\times2$ off-diagonal block is a diagonal matrix $\mbox{diag}(e^{-2r}-1,0)$ with zero determinant and hence the state is separable [@Simon_00]. The state can be easily prepared by displacing position quadratures of the state in mode $A$ with the CM $\gamma_{A}=\mbox{diag}(e^{-2(r-\varepsilon)},e^{2r})$ and the vacuum mode $B$ with CM $\gamma_{B}=\openone$ as $$\label{displacements}
x_{A}\rightarrow x_{A}+\bar{x},\quad x_{B}\rightarrow
x_{B}-\bar{x}.$$ Here, $\bar{x}$ is the classical Gaussian distributed displacement with variance satisfying $\langle\bar{x}^2\rangle=(1-e^{-2r})/2$ and $\openone$ is the $2\times2$ identity matrix.
Next, Alice splits her mode $A$ into two modes $A$ and $A'$ by superimposing mode $A$ with vacuum mode $A'$ on a balanced beam splitter. If we denote the vacuum CM of mode $A'$ as $\gamma_{A'}=\openone$ and describe the beam splitter by the orthogonal matrix $$\begin{aligned}
\label{Uij}
U_{ij}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
\openone & \openone \\
\openone & -\openone\\
\end{array}\right),\end{aligned}$$ where $i=A$, $j=A'$, we get the three-mode Gaussian state with the following CM $$\begin{aligned}
\label{gammaAAprB}
\gamma_{AA'B}=\left(\begin{array}{ccc}
\alpha & \delta & \tau \\
\delta & \alpha & \tau\\
\tau & \tau & \beta\\
\end{array}\right),\end{aligned}$$ with $\alpha,\beta,\tau$ and $\delta$ being diagonal matrices of the form $\alpha=\mbox{diag}(2+e^{-2r}(e^{2\varepsilon}-1),e^{2r}+1)/2$, $\beta=\mbox{diag}(2-e^{-2r},1)$, $\tau=\mbox{diag}(e^{-2r}-1,0)/\sqrt{2}$ and $\delta=\mbox{diag}(e^{-2r}(e^{2\varepsilon}-1),e^{2r}-1)/2$.
The performance of the proposed protocol is enabled by the remarkable separability properties of the state with CM (\[gammaAAprB\]). Let us investigate first the separability of the state with respect to the splitting of modes $A,A'$ and $B$ into two groups ($1\times2$-mode separability). Beam splitting transformation (\[Uij\]) on mode $A$ and mode $A'$ cannot create entanglement with mode $B$ and hence the state is separable across the $B-(AA')$ splitting. The state is, however, entangled with respect to the remaining $A-(A'B)$ and $A'-(AB)$ splittings as can be easily proven using the positive partial transposition criterion [@Peres_96; @Horodecki_96] expressed in terms of symplectic invariants [@Serafini_06]. Specifically, separability of mode $X$ from a pair of modes $(YZ)$ in a generic Gaussian state of three modes $X,Y$ and $Z$ with the CM $\gamma$ can be determined from the symplectic invariants of the matrix $\gamma^{(T_x)}\equiv\Lambda_{x}\gamma\Lambda_{x}^{T}$ associated with the partial transpose of the state with respect to mode $X$. Here, $\Lambda_{x}\equiv\sigma_{z}^{(X)}\oplus\openone^{(Y)}\oplus\openone^{(Z)}$, where $\sigma_{z}=\mbox{diag}(1,-1)$ is the Pauli diagonal matrix. The matrix $\gamma^{(T_x)}$ possesses three symplectic invariants denoted as $I_1,I_2$ and $I_3=\mbox{det}(\gamma)$ which coincide with the coefficients of the characteristic polynomial of the matrix $\Omega\gamma^{(T_{x})}$, i.e. $$\label{polynomial}
\mbox{det}(\Omega\gamma^{(T_{x})}-q\openone)=q^{6}+I_1q^{4}+I_2q^{2}+I_3,$$ where $$\begin{aligned}
\label{Omega}
\Omega=\bigoplus_{i=1}^{3}\left(\begin{array}{cc}
0 & 1 \\
-1 & 0\\
\end{array}\right).\end{aligned}$$ According to the criterion [@Serafini_06] mode $X$ is entangled with the two-mode subsystem $(YZ)$ if $$\label{Sigmadef}
\Sigma_{x}=I_3-I_2+I_1-1<0.$$ In the case of the CM (\[gammaAAprB\]) one gets explicitly after some algebra that $$\label{Sigma}
\Sigma_{A}=8e^{\varepsilon-r}\sinh(\varepsilon-r)\sinh^2(r),$$ which implies that if $r>\varepsilon$ it holds that $\Sigma_{A}<0$ and there is entanglement across $A-(A'B)$ splitting. Owing to the symmetry of the state under the exchange of modes $A$ and $A'$ (bisymmetric state [@Serafini_05]) it follows immediately that $\Sigma_{A}=\Sigma_{A'}$ and the state is therefore also entangled across $A'-(AB)$ splitting. Thus, the studied state is separable across one bipartite splitting and therefore it belongs to the class of one-mode biseparable Gaussian states.
Let us focus now on the separability of the two-mode reductions ($1\times1$-mode separability) of the state with the CM (\[gammaAAprB\]). Due to the separability of the $B-(AA')$ splitting mode $B$ is inevitably separable both from the mode $A$ as well as from the mode $A'$. The reduced state of modes $A$ and $A'$ was created by mixing of the state with CM $\gamma_{A}=
\mbox{diag}[1+e^{-2r}(e^{2\varepsilon}-1),e^{2r}]$ with the vacuum state on a beam splitter (\[Uij\]). Both the eigenvalues of the CM $\gamma_{A}$ are lower bounded by the unity and hence the corresponding normally ordered CM $\gamma_{A}^{(\mathcal{N})}\equiv\gamma_{A}-\openone$ is positive semidefinite. The normally ordered characteristic function of the state then possesses a Fourier transform which is not more singular than a Dirac delta function and the state with the CM $\gamma_{A}$ is thus classical. As mixing of such a state with a vacuum state on a beam splitter cannot create entanglement [@Kim_02] mode $A$ is therefore separable from mode $A'$ in the state with CM (\[gammaAAprB\]). In summary, for the state with the CM (\[gammaAAprB\]) any two modes are separable.
Note, that the aforementioned separability properties of the state with CM (\[gammaAAprB\]) can exist only in a mixed-state scenario. Indeed, for a pure state separability of mode $B$ from modes $(AA')$ implies the state to be a product state across the $B-(AA')$ splitting. Likewise, the separability of the mode $A$ from the mode $A'$ implies that the reduced state of the modes $A$ and $A'$ is also a product state. Consequently, a pure three-mode state where the mode $B$ is separable from modes $(AA')$ and mode $A$ is at the same time separable from mode $A'$ is therefore a triple product state which is fully separable. Such a state, however, cannot possess entanglement across, e.g., $A-(A'B)$ splitting as is the case of our state.
At the final stage of the protocol Alice keeps mode $A$ and sends the separable mode $A'$ to Bob. He superimposes the mode with his mode $B$ on a balanced beam splitter $U_{BA'}$ given in Eq. (\[Uij\]) where $i=B$ and $j=A'$ which creates a state with CM $$\begin{aligned}
\label{tildegammaAAprB}
\tilde{\gamma}_{AA'B}=\left(\begin{array}{ccc}
\alpha & \frac{\tau-\delta}{\sqrt{2}} & \frac{\tau+\delta}{\sqrt{2}} \\
\frac{\tau-\delta}{\sqrt{2}} & \frac{\alpha+\beta-2\tau}{2} & \frac{\beta-\alpha}{2}\\
\frac{\tau+\delta}{\sqrt{2}} & \frac{\beta-\alpha}{2} & \frac{\alpha+\beta+2\tau}{2}\\
\end{array}\right),\end{aligned}$$ where the $2\times 2$ submatrices $\alpha,\beta,\tau$ and $\delta$ are given below Eq. (\[gammaAAprB\]). If, on the other hand, Alice keeps mode $A'$ and sends the separable mode $A$ to Bob who mixes it with his mode on the beam splitter $$\begin{aligned}
\label{UAB}
U_{AB}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
\openone & -\openone \\
\openone & \openone\\
\end{array}\right),\end{aligned}$$ the CM of the resulting state reads $$\begin{aligned}
\label{tildetildegammaAAprB}
\tilde{\tilde{\gamma}}_{AA'B}=\left(\begin{array}{ccc}
\frac{\alpha+\beta-2\tau}{2} & \frac{\delta-\tau}{\sqrt{2}} & \frac{\alpha-\beta}{2}\\
\frac{\delta-\tau}{\sqrt{2}} & \alpha & \frac{\delta+\tau}{\sqrt{2}} \\
\frac{\alpha-\beta}{2} & \frac{\delta+\tau}{\sqrt{2}} & \frac{\alpha+\beta+2\tau}{2}\\
\end{array}\right).\end{aligned}$$ By retaining only the modes $A$ ($A'$) and $B$ it then follows that Alice and Bob are left with a reduced two-mode state with the CM $$\begin{aligned}
\label{tildegammaAB}
\tilde{\gamma}_{AB}=\tilde{\tilde{\gamma}}_{A'B}=\left(\begin{array}{cc}
\alpha & \frac{\delta+\tau}{\sqrt{2}} \\
\frac{\delta+\tau}{\sqrt{2}} & \frac{\alpha+\beta+2\tau}{2}\\
\end{array}\right),\end{aligned}$$ which describes an entangled state provided that the squeezing parameter $r$ is large enough. The threshold squeezing $r_{\rm e}$ above which the entanglement appears can be derived from the two-mode version of the sufficient condition for entanglement [@Serafini_06] $$\label{Sigma2}
\mbox{det}\tilde{\gamma}_{AB}-\Delta+1<0,$$ where $\Delta=\mbox{det}\alpha+\frac{1}{4}\mbox{det}\left(\alpha+\beta+2\tau\right)-\mbox{det}\left(\delta+\tau\right)$. Substituting here for the matrices $\alpha,\beta,\tau$ and $\delta$ defined below Eq. (\[gammaAAprB\]) one finds, that mode $A$ ($A'$) is entangled with mode $B$ if the squeezing $r$ satisfies $r>r_{\rm e}$, where
$$\begin{aligned}
\label{re}
r_{\rm e}=\frac{1}{2}\ln\left[\frac{11e^{2\varepsilon}+8\sqrt{2}-13+
\sqrt{(11e^{2\varepsilon}+8\sqrt{2}-13)^2+4e^{2\varepsilon}(8\sqrt{2}-1)}}{2(8\sqrt{2}-1)}\right].\end{aligned}$$
It is further of interest to look at the $1\times 2$-mode separability of the states with CMs (\[tildegammaAAprB\]) and (\[tildetildegammaAAprB\]) from the last step of the protocol. We have already seen that for the state with the CM (\[tildegammaAAprB\]) \[(\[tildetildegammaAAprB\])\] mode $A$ ($A'$) is entangled with mode $B$. This implies, that there is entanglement across $A-(A'B)$ $[A'-(AB)]$ as well as $B-(AA')$ $[B-(AA')]$ splitting. But what about entanglement across the remaining $A'-(AB)$ $[A-(A'B)]$ splitting? Analyzing the entanglement using again the criterion (\[Sigmadef\]) one gets for the CMs (\[tildegammaAAprB\]) and (\[tildetildegammaAAprB\]) the following expressions: $$\label{tildeSigma}
\tilde{\Sigma}_{A'}=\tilde{\tilde{\Sigma}}_{A}=\frac{\Sigma_{A}}{4},$$ where the quantity $\tilde{\Sigma}_{A'}$ ($\tilde{\tilde{\Sigma}}_{A}$) characterizes separability of the mode $A'$ ($A$) in the state with the CM (\[tildegammaAAprB\]) \[(\[tildetildegammaAAprB\])\] and $\Sigma_{A}$ is given in Eq. (\[Sigma\]). Hence, for the considered squeezing $r>\varepsilon$ the state in the last step of the present protocol is in both cases entangled across all three bipartite splittings and it therefore carries a genuine three-mode entanglement. The proposed sharing scheme thus also illustrates remarkable transformation properties of the three-mode fully separable state given by a product of a separable state of modes $A$ and $B$ with CM (\[gammaAB\]) and a vacuum state of mode $A'$. Namely, the beam splitting transformation (\[Uij\]) on a two-mode subsystem formed by modes $A$ and $A'$ transforms the state into the one-mode biseparable state which is separable with respect to $B-(AA')$ splitting and entangled across $A-(A'B)$ and $A'-(AB)$ splittings. Moreover, the second beam splitter on modes $A'$ ($A$) and $B$ preserves the latter entanglement and further creates entanglement also across $B-(AA')$ splitting, i.e., creates a genuine three-mode entanglement.
Unitary versus measurement-based localizability of the intermediate entanglement {#sec_2}
================================================================================
It can seem for the first sight that Bob may not need coherent beam splitting operation on his mode $B$ and the received mode $A'$ ($A$) to establish entanglement with Alice’s mode $A$ ($A'$). One can argue that the participants could first establish entanglement between the transmitted mode $A'$ ($A$) (held by Bob) and Alice’s mode $A$ ($A'$) simply by optimally measuring mode $B$ (which is separable from the pair of modes $(AA')$) followed by an optimal displacement of modes $A$ and $A'$. The entanglement thus obtained could be subsequently transformed into the entanglement of mode $B$ with Alice’s mode $A$ ($A'$) just by swapping the transmitted mode $A'$ ($A$) with the mode $B$. Now we show, that if Bob is restricted to Gaussian measurements there is a region of squeezing parameters $r$ for which he is unable to establish any entanglement with Alice by this measure-and-displace strategy. The task to be solved can be formulated as a task for finding maximum entanglement which can be localized between modes $A$ and $A'$ of the state with CM (\[gammaAAprB\]) by optimal Gaussian measurement on mode $B$ of the state. This problem, known as Gaussian localizable entanglement, has already been solved in the literature [@Fiurasek_07]. It can be conveniently analyzed using the concept of lower symplectic eigenvalue of the partially transposed state, which is for a generic CM $\sigma_{AB}$ of two modes $A$ and $B$ written in the $2\times2$-block form with respect to $A-B$ splitting $$\begin{aligned}
\label{sigmaAB}
\sigma_{AB}=\left(\begin{array}{cc}
A & C \\
C^{T} & B\\
\end{array}\right),\end{aligned}$$ given by [@Vidal_02] $$\begin{aligned}
\label{mu}
\mu=\sqrt{\frac{\Delta-\sqrt{\Delta^2-4\mbox{det}\sigma_{AB}}}{2}},\end{aligned}$$ where $\Delta=\mbox{det}A+\mbox{det}B-2\mbox{det}C$. The state with CM $\sigma_{AB}$ contains entanglement if and only if $\mu<1$. The symplectic eigenvalue $(\ref{mu})$ also characterizes the amount of entanglement in the state which can be quantified by the logarithmic negativity $E_{\mathcal{N}}(\sigma_{AB})=\mbox{max}(0,-\log_{2}\mu)
$[@Vidal_02]. As $E_{\mathcal{N}}$ is a monotonically decreasing function of the symplectic eigenvalue it follows that the smaller the value of $\mu$ the larger the entanglement. For certain families of three-mode Gaussian states one can find even analytically the symplectic eigenvalue (\[mu\]) for a two-mode state obtained by Gaussian measurement on the third mode minimized over all Gaussian measurements on the mode [@Fiurasek_07]. The present state with CM (\[gammaAAprB\]) belongs to the class of the bisymmetric states for which the problem of the Gaussian localizable entanglement can also be fully resolved using the analytical tools [@Mista_08]. It turns, that the localizable entanglement between modes $A$ and $A'$ is always achieved by homodyne detection of position quadrature $x_{B}$ on mode $B$, i.e., by projection of the mode onto an infinitely squeezed position eigenstate. This gives the minimal lower symplectic eigenvalue (\[mu\]) of the partial transpose of the conditional state of modes $A$ and $A'$ in the form:
$$\begin{aligned}
\label{mum}
\mu_{\rm m}= \left\{
\begin{array}{ll} e^{r} & \mbox{if } r<r_{\rm l}\equiv\frac{1}{2}\ln\left\{\frac{1}{3}+2\sqrt{-\frac{p}{3}}\cos\left[\frac{1}{3}\arccos\left(-\frac{q}{2}\sqrt{-\frac{27}{p^3}}\right)\right]\right\};\\
\sqrt{1+e^{-2r}(e^{2\varepsilon}-1)-\frac{(e^{-2r}-1)^2}{2-e^{-2r}}} & \mbox{otherwise},
\end{array}
\right.\nonumber\\\end{aligned}$$
where $$\label{pq}
p=\frac{1}{6}-e^{2\varepsilon},\quad q=\frac{5}{54}+\frac{e^{2\varepsilon}}{6}.$$ Clearly, the condition on the squeezing $r$ for which Bob can localize entanglement between modes $A$ and $A'$ can be derived by solving the inequality $\mu_{\rm m}<1$ with respect to $r$. This gives explicitly that the localizable entanglement is nonzero if $r>r_{\rm m}$, where $$\begin{aligned}
\label{rm}
r_{\rm m}=\frac{1}{2}\ln\left[e^{2\varepsilon}+\sqrt{e^{2\varepsilon}\left(e^{2\varepsilon}-1\right)}\right].\end{aligned}$$
![Lower symplectic eigenvalue $\mu_{AB}=\mu_{A'B}$ of the partial transpose of the states with CMs (\[tildegammaAB\]) (solid curve) and the symplectic eigenvalue $\mu_{\rm m}$, Eq. (\[mum\]), (dashed curve) versus the squeezing parameter $r$ for $\varepsilon=0.1$. The solid vertical lines correspond to the threshold squeezings $r_{\rm l}=0.079$, Eq. (\[mum\]), $r_{\rm e}$, Eq. (\[re\]), and $r_{\rm m}$, Eq. (\[rm\]). All the quantities plotted are dimensionless.[]{data-label="fig2"}](mufigure.eps){width="8.5cm"}
The performance of our protocol can be illustrated on a particular example when we set $\varepsilon=0.1$, which is depicted in Fig. \[fig2\]. It follows from Eq. (\[Sigma\]) that the state with the CM (\[gammaAAprB\]) contains entanglement across $A-(A'B)$ as well as $A'-(AB)$ splittings if $r>\varepsilon=0.1$. The entanglement can be transformed into two-mode entanglement of modes $A$ ($A'$) and $B$ by a balanced beam splitter on modes $A'$ ($A$) and $B$ if $r>r_{\rm
e}\doteq0.106$. In contrast, Bob can localize some entanglement between modes $A$ and $A'$ only if $r>r_m\doteq0.277$ according to Eq. (\[rm\]). Thus, in the interval $r_m\geq r>r_{\rm
e}$ Bob can get entanglement by a coherent operation on his mode and the received mode but cannot create entanglement between Alice’s mode and the received mode by any Gaussian measurement on his mode and optimal displacement of Alice’s and the received mode. Further analysis shows that the gap between the threshold squeezings (\[re\]) and (\[rm\]) exists also for the other values of the parameter $\varepsilon$. In Fig. \[fig3\] we plot the threshold squeezings as well as their difference $r_{\rm
m}-r_{\rm e}$ as a function of $\varepsilon$. The figure and further calculations reveal that the difference exists for any $\varepsilon>0$ and it is a monotonically increasing function of $\varepsilon$ which asymptotically approaches the limit value $(1/2)\ln[2(8\sqrt{2}-1)/11]\doteq0.314$. Thus, for the state with the CM (\[gammaAAprB\]) for any $\varepsilon>0$ there exists a region of squeezings $r_{\rm m}\geq r>r_{\rm e}$ for which entanglement between Alice and Bob cannot be established by performing any Gaussian measurement on mode $B$ and optimally displacing modes $A$ and $A'$ (where one of the modes is held by Bob), but it can be created by superimposing Bob’s mode $B$ and the received mode (either $A'$ or $A$) on a balanced beam splitter.
Note, that the gap exists also in the protocol for the Gaussian entanglement distribution by a separable ancilla [@Mista_09]. Here, the possibility to localize entanglement between mode $A$ and the transmitted mode $A'$ by measurement on Bob’s mode $B$ is prevented by separability of mode $A'$ from the pair of modes $(AB)$. Nevertheless, a balanced beam splitter on modes $A'$ and $B$ creates entanglement between modes $A$ and $B$. Passive coherent unitary operations exhibit superiority over the method based on the measurement and feed-forward also in the two-mode scenario [@Filip_10]. In this case, the coherent operations allow to extract squeezing from a squeezed signal classically correlated to a probe in cases when the measurement on the probe followed by a feed-forward correction on the signal fails.
![Squeezing thresholds $r_{\rm m}$, Eq. (\[rm\]), (solid curve), $r_{\rm e}$, Eq. (\[re\]), (dashed curve), and their difference $r_{\rm m}-r_{\rm e}$ (dotted curve). All the quantities plotted are dimensionless.[]{data-label="fig3"}](rmrefigure.eps){width="8.5cm"}
Discussion and conclusion {#sec_3}
=========================
The key state of our sharing protocol with the CM (\[gammaAAprB\]) is interesting from the point of view of the nondistillable (bound) entanglement [@MHorodecki_98]. A bipartite quantum state is nondistillable if it is impossible to transform many copies of the state by local operations and classical communication (LOCC) into fewer copies of nearly maximally entangled singlet state. The concept of bound entanglement can be easily generalized to three parties [@Dur_00a] in analogy with the concept of the classical multipartite bound information [@Acin_04]. We say that a tripartite quantum state is bound entangled if i) any two parties cannot distill singlet states by LOCC even with the help of the third party and ii) the state cannot be created by LOCC. Examples of the tripartite bound entanglement can be found both for two-level systems (qubits) [@Bennett_99a; @Dur_99] and Gaussian states [@Giedke_01]. They are given by the two-mode biseparable states which are separable across two bipartite splittings and entangled across the third one as well as three-mode biseparable states which are separable across all three bipartite splittings but they are not fully separable. This is because separability across at least two splittings immediately guarantees satisfaction of the condition i) whereas the presence of entanglement causes that also condition ii) holds. However, inseparability with respect to at least two bipartite splittings is currently known to be only a necessary condition for distillability but it is not known whether it is also sufficient. Thus although there are, for instance, distillable one-mode biseparable states both in the qubit [@Dur_99; @Cubitt_03] as well as Gaussian case [@Mista_08] one cannot rule out the possibility that there also exist bound entangled states belonging to this class [@Dur_99; @Giedke_01]. Our state with the CM (\[gammaAAprB\]) is entangled and hence fulfils the condition ii) but moreover it also possesses nontrivial properties which are necessary to satisfy the condition i). First, for any Gaussian state satisfying i) any two parties have to be separable. Namely, if there were entanglement between some pair of parties, then it would be distillable [@Giedke_01b]. For our state any two modes are separable and thus no entanglement can be distilled between them if the third party is totally ignored. What is more, not only entanglement cannot be distilled between $A$ and $B$ with the help of $A'$ as well as between $A'$ and $B$ with the help of $A$ because $B$ is separable from $(AA')$ but for certain squeezings entanglement also cannot be distilled between $A$ and $A'$ with the help of Bob provided that he is restricted to the Gaussian measurements on his mode, which is a nontrivial necessary condition for fulfillment of the requirement i).
In conclusion, we have constructed a two-mode separable Gaussian state which can be transformed by splitting of one of its modes on a beam splitter to a state without any two-mode entanglement but with entanglement across two bipartite splittings. If the initial two-mode state is shared by distant Alice and Bob, the beam splitter on Alice’s side creates entanglement between one of its outputs and a non-local composite system composed of the other output of the beam splitter and the distant Bob’s mode. Although the entanglement is between Alice’s mode and the system involving distant Bob’s mode it is not entanglement between Alice and Bob because Bob’s mode is separable at the same time. The entanglement can nevertheless be turned into entanglement between Alice and Bob by sending one output mode of Alice’s beam splitter (which is separable from the other output mode) to Bob and mixing it with Bob’s mode on another beam splitter. The protocol just described thus can be interpreted as an entanglement sharing scheme in which entanglement created by Alice’s beam splitter can be transformed into entanglement with Bob only if Bob has at his disposal physically also one output mode of the beam splitter. Besides, our analysis shows that for one copy of the key one-mode biseparable state of our protocol entanglement cannot be distilled between any two modes with the help of the third party if Bob is restricted to Gaussian measurements which is a nontrivial property necessary for the presence of the bound entanglement in the state. The question of whether the same holds true also for more generic even non-Gaussian operations on Bob’s mode and multiple copies is left for further research. We believe that our results contribute to the better understanding and utilization of entanglement as well as separable correlations in multipartite mixed quantum states.
Acknowledgment
==============
I would like to thank V. Chille, Ch. Peuntinger, Ch. Marquardt and N. Korolkova for fruitful discussions. The research has been supported by the GACR Project No. P205/12/0694.
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abstract: |
Kricker defined an invariant of knots in homology 3-spheres which is a rational lift of the Kontsevich integral and proved with Garoufalidis that this invariant satisfies splitting formulas with respect to a surgery move called null-move. We define a functorial extension of the Kricker invariant and prove splitting formulas for this functorial invariant with respect to null Lagrangian-preserving surgery, a generalization of the null-move. We apply these splitting formulas to the Kricker invariant.
**MSC: 57M27** 57M25 57N10
**Keywords:** 3-manifold, knot, homology sphere, cobordism category, Lagrangian cobordism, bottom-top tangle, beaded Jacobi diagram, Kontsevich integral, LMO invariant, Kricker invariant, Århus integral, Lagrangian-preserving surgery, finite type invariant, splitting formula.
author:
- Delphine Moussard
title: Splitting formulas for the rational lift of the Kontsevich integral
---
=1
Introduction
============
Context
-------
This paper presents the construction of a functorial extension of the Kricker rational lift of the Kontsevich integral, which aims at expliciting the properties of this invariant as a series of finite type invariants.
The notion of finite type invariants first appeared in independent works of Goussarov and Vassiliev involving invariants of knots in the 3–dimensional sphere $S^3$; in this case, finite type invariants are also called Vassiliev invariants. Finite type invariants of knots in $S^3$ are defined by their polynomial behaviour with respect to crossing changes. The discovery of the Kontsevich integral, which is a universal invariant among all finite type invariants of knots in $S^3$, revealed that this class of invariants is very prolific. It is known, for instance, that it dominates all Witten-Reshetikhin-Turaev’s quantum invariants. A theory of finite type invariants can be defined for any kind of topological objects provided that an elementary move on the set of these objects is fixed; the finite type invariants are defined by their polynomial behaviour with respect to this elementary move. For 3–dimensional manifolds, the notion of finite type invariants was introduced by Ohtsuki [@Oht4], who constructed the first examples for integral homology 3–spheres, and it has been widely developed and generalized since then. In particular, Goussarov and Habiro independently developed a theory which involves any 3–dimensional manifolds —and their knots— and which contains the Ohtsuki theory for ${\mathbb{Z}}$–spheres [@GGP; @Hab].
In [@Kri], Kricker constructed a rational lift of the Kontsevich integral of knots in integral homology 3–spheres (${\mathbb{Z}}$–spheres). In [@GK], he proved with Garoufalidis that his construction provides an invariant of knots in ${\mathbb{Z}}$–spheres. They also proved that the Kricker invariant satisfies some splitting formulas with respect to the so-called null-move. For knots in ${\mathbb{Z}}$–spheres with trivial Alexander polynomial, these formulas together with work of Garoufalidis and Rozansky [@GR] imply that the Kricker invariant is a universal finite type invariant with respect to the null-move.
Kricker’s construction easily generalizes to null-homologous knots in rational homology 3–spheres (${\mathbb{Q}}$–spheres); the main goal of this article is to prove splitting formulas for the Kricker invariant of these knots with respect to null Lagrangian-preserving surgery, a move which generalizes the null-move. For null-homologous knots in ${\mathbb{Q}}$–spheres with trivial Alexander polynomial, these formulas are used in [@M7] to prove that this extended Kricker invariant is a universal finite type invariant with respect to null Lagrangian-preserving surgeries.
Lescop defined in [@Les2] an invariant of null-homologous knots in ${\mathbb{Q}}$–spheres and proved in [@Les3] splitting formulas for this invariant with respect to null Lagrangian-preserving surgeries, similar to the ones proved in this paper for the Kricker invariant. Lescop conjectured in [@Les3] that her invariant is equivalent to the Kricker invariant. The mentioned results of Garoufalidis, Kricker, Lescop and Rozansky give such an equivalence for knots in ${\mathbb{Z}}$–spheres with trivial Alexander polynomial and the results of the present paper allow to generalize this equivalence to null-homologous knots in ${\mathbb{Q}}$–spheres with trivial Alexander polynomial [@M7 Theorem 2.11].
A similar situation arises in the study of finite type invariants of ${\mathbb{Q}}$–spheres with respect to Lagrangian-preserving surgeries. In this case, the Kontsevich–Kuperberg–Thurston invariant and the Le–Murakami–Ohtsuki invariant are both universal, up to degree 1 invariants deduced from the cardinality of the first homology group; this implies an equivalence result for these two invariants, see [@M2]. For the KKT invariant, splitting formulas with respect to Lagrangian-preserving surgeries were proved by Lescop [@Les]; for the LMO invariant, similar formulas were proved by Massuyeau [@Mas]. Massuyeau’s proof of his splitting formulas is based on an extension of the LMO invariant of ${\mathbb{Q}}$–spheres to a functor defined on a category of Lagrangian cobordisms that he constructed with Cheptea and Habiro [@CHM].
In this paper, we extend the LMO functorial invariant of Cheptea–Habiro–Massuyeau to a category of Lagrangian cobordisms with paths, inserting the Kricker’s idea in the construction. We obtain a functorial invariant from which the Kricker invariant of null-homologous knots in ${\mathbb{Q}}$–spheres is recovered. Following Massuyeau, we use the functoriality to obtain splitting formulas for our invariant and, as a consequence, for the Kricker invariant.
#### Notations and conventions.
For ${\mathbb{K}}={\mathbb{Z}},{\mathbb{Q}}$, a [*${\mathbb{K}}$–sphere*]{}, (resp. a [*${\mathbb{K}}$–cube*]{}) is a 3–manifold, compact and oriented, which has the same homology with coefficients in ${\mathbb{K}}$ as the standard 3–sphere (resp. 3–cube). The boundary of an oriented manifold with boundary is oriented with the “outward normal first” convention.
#### Acknowledgments.
I am supported by a Postdoctoral Fellowship of the Japan Society for the Promotion of Science. I am grateful to Tomotada Ohtsuki and the Research Institute for Mathematical Sciences for their support. I also wish to thank Gwénaël Massuyeau for interesting exchanges. Finally, I thank the referee whose useful comments helped to improve the paper.
Statement of the main result
----------------------------
We first give the definitions we need to state our main result.
#### Null LP–surgeries.
For $g\in \mathbb{N}$, a *genus $g$ rational homology handlebody (${\mathbb{Q}}$–handlebody)* is a 3–manifold which is compact, oriented, and which has the same homology with rational coefficients as the standard genus $g$ handlebody. Such a ${\mathbb{Q}}$–handlebody is connected, and its boundary is necessarily homeomorphic to the standard genus $g$ surface.
The *Lagrangian* $\mathcal{L}_C$ of a ${\mathbb{Q}}$–handlebody $C$ is the kernel of the map $i_*: H_1(\partial C;{\mathbb{Q}})\to H_1(C;{\mathbb{Q}})$ induced by the inclusion. The Lagrangian of a ${\mathbb{Q}}$–handlebody $C$ is indeed a Lagrangian subspace of $H_1(\partial C;{\mathbb{Q}})$ with respect to the intersection form. A [*Lagrangian-preserving pair*]{}, or [*LP–pair*]{}, is a pair ${\mathbf{C}}=\left(\frac{C'}{C}\right)$ of ${\mathbb{Q}}$–handlebodies equipped with a homeomorphism $h:\partial C\fl{\cong}\partial C'$ such that $h_*(\mathcal{L}_C)=\mathcal{L}_{C'}$.
Given a 3–manifold $M$, a *Lagrangian-preserving surgery*, or *LP–surgery*, on $M$ is a family ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ of LP–pairs such that the $C_i$ are embedded in $M$ and disjoint. The manifold obtained from $M$ by LP–surgery on ${\mathbf{C}}$ is defined as $$M({\mathbf{C}})=\left(M\setminus(\sqcup_{1\leq i \leq n} C_i)\right)\cup_{\partial}(\sqcup_{1\leq i \leq n} C_i').$$
Let $M$ be a 3–manifold such that $H_1(M;{\mathbb{Q}})=0$ and let $K$ be a disjoint union of knots or paths properly embedded in $M$. A *${\mathbb{Q}}$–handlebody null in $M\setminus K$* is a ${\mathbb{Q}}$–handlebody $C\subset M\setminus K$ such that the map $i_* : H_1(C;{\mathbb{Q}})\to H_1(M\setminus K;{\mathbb{Q}})$ induced by the inclusion has a trivial image. A *null LP–surgery* on $(M,K)$ is an LP–surgery ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ on $M\setminus K$ such that each $C_i$ is null in $M\setminus K$. The pair obtained by surgery is denoted by $(M,K)({\mathbf{C}})$.
#### The tensor $\mu({\mathbf{C}})$.
Given an LP–pair ${\mathbf{C}}=\left(\frac{C'}{C}\right)$, define the associated [*total manifold*]{} ${\mathcal{C}}=(-C)\cup C'$ and define $$\mu({\mathbf{C}})\in\hom(\Lambda^3 H^1({\mathcal{C}};{\mathbb{Q}}),{\mathbb{Q}})\cong\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})$$ by associating with a triple of cohomology classes the evaluation of their triple cup product on the fundamental class of ${\mathcal{C}}$. For a family ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ of LP–pairs, let ${\mathcal{C}}={\mathcal{C}}_1\sqcup\dots\sqcup{\mathcal{C}}_n$ and set: $$\mu({\mathbf{C}})=\mu({\mathbf{C}}_1)\otimes\dots\otimes\mu({\mathbf{C}}_n)\ \in\otimes_{i=1}^n\Lambda^3 H_1({\mathcal{C}}_i;{\mathbb{Q}}).$$ The natural identification $H_1({\mathcal{C}};{\mathbb{Q}})\cong\oplus_{i=1}^n H_1({\mathcal{C}}_i;{\mathbb{Q}})$ allows to see $\otimes_{i=1}^n\Lambda^3 H_1({\mathcal{C}}_i;{\mathbb{Q}})$ as a subspace of $\left(\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})\right)^{\otimes n}$. This subspace injects to $S^n\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})$ [*via*]{} the canonical surjection $\left(\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})\right)^{\otimes n}\twoheadrightarrow S^n\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})$. Hence we can view $\mu({\mathbf{C}})$ as an element of $S^n\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})$.
#### The bilinear form $\ell_{(S,\kappa)}({\mathbf{C}})$.
Let $(S,\kappa)$ be a [*${\mathbb{Q}}$SK–pair*]{}, [*i.e.*]{} a pair made of a ${\mathbb{Q}}$–sphere $S$ and a null-homologous knot $\kappa\subset S$. Let ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ be a null LP–surgery on $(S,\kappa)$. Let ${\mathcal{C}}={\mathcal{C}}_1\sqcup\dots\sqcup{\mathcal{C}}_n$ be the disjoint union of the associated total manifolds. Fix a lift $\tilde{C}_i$ of each $C_i$ in $\tilde{E}$. We will define a [*hermitian*]{} form: $$\ell_{(S,\kappa)}({\mathbf{C}}): H_1({\mathcal{C}};{\mathbb{Q}})\times H_1({\mathcal{C}};{\mathbb{Q}})\to{\mathbb{Q}(t)},$$ [*i.e.*]{} a ${\mathbb{Q}}$–bilinear form such that reversing the order of the arguments changes $t$ to $t^{-1}$. Let $a\in H_1({\mathcal{C}}_i;{\mathbb{Q}})$ and $b\in H_1({\mathcal{C}}_j;{\mathbb{Q}})$ be homology classes that can be represented by simple closed curves $\alpha\subset\partial C_i$ and $\beta\subset\partial C_j$, disjoint if $i=j$. Note that such homology classes generate $H_1({\mathcal{C}};{\mathbb{Q}})$ over ${\mathbb{Q}}$. Let $\tilde{\alpha}$ and $\tilde{\beta}$ be the copies of $\alpha$ and $\beta$ in $\tilde{C}_i$ and $\tilde{C}_j$. Set: $$\ell_{(S,\kappa)}({\mathbf{C}})(a,b)={\textrm{\textnormal{lk}}}_e(\tilde{\alpha},\tilde{\beta}),$$ where ${\textrm{\textnormal{lk}}}_e(\cdot,\cdot)$ stands for the equivariant linking number (see for instance [@M7 Section 2.1] for a definition). We get a well-defined hermitian form $\ell_{(S,\kappa)}({\mathbf{C}})$ associated with a choice of lifts of the $C_i$’s. We will keep this choice implicit; the statement of Theorem \[thmain\] is valid for any such choice.
#### Diagrammatic representations.
Let $V$ be a rational vector space. A [*$V$–colored Jacobi diagram*]{} is a unitrivalent graph whose trivalent vertices are oriented and whose univalent vertices are labelled by $V$, where an [*orientation*]{} of a trivalent vertex is a cyclic order of the three edges that meet at this vertex —fixed as in the pictures. Set: $${\mathcal{A}}_{\mathbb{Q}}(V)=\frac{{\mathbb{Q}}\langle V\textrm{--colored Jacobi diagrams}\rangle}{{\mathbb{Q}}\langle\textrm{AS, IHX, LV}\rangle},$$ where the relations are depicted in Figure \[figrelations1\].
\[scale=0.3\]
\[xshift=1cm\] (0,4) – (2,2); (2,2) – (4,4); (2,2) – (2,0); (5,2) node[$+$]{}; (8,2) .. controls +(2,0) and +(2.5,-1) .. (6,4); (8,2) .. controls +(-2,0) and +(-2.5,-1) .. (10,4); (8,2) .. controls +(-2,0) and +(-2.5,-1) .. (10,4); (8,0) – (8,2); (11,2) node[$=$]{}; (12.5,2) node[0]{}; (6,-1.5) node[AS]{};
(18,4) – (20,3) – (20,1) – (18,0); (20,1) – (22,0); (20,3) – (22,4); (23,2) node[$-$]{}; (24,4) – (25,2) – (27,2) – (28,4); (24,0) – (25,2); (27,2) – (28,0); (29,2) node[$+$]{}; (30,4) – (33,2) – (34,0); (31,2) – (34,4); (30,0) – (31,2) – (34,4); (31,2) – (33,2); (35,2) node[$=$]{}; (36.5,2) node[0]{}; (27,-1.5) node[IHX]{};
\[xscale=2.3,yscale=2,xshift=18.5cm\] in [0,2,4]{} [ (-0.5,0) – (,0.5) – (,2) (+0.5,0) – (,0.5);]{} (0,2.5) node [$v$]{} (1,1) node [$+$]{} (2,2.5) node [$w$]{} (3,1) node [$=$]{} (4,2.5) node [$v+w$]{}; (2,-0.7) node[LV]{};
A symmetric tensor in $S^n\Lambda^3 V$ can be represented by a Jacobi diagram [*via*]{} the following embedding. $$\begin{array}{c c c}
S^n\Lambda^3 V & \to & {\mathcal{A}}_{\mathbb{Q}}(V) \\
(u_1\wedge v_1\wedge w_1)\dots(u_n\wedge v_n\wedge w_n) & \mapsto & {
\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.7]
\draw[dashed] (-1,1) node[above] {$w_1$} -- (0,0) -- (0,1.1) (0,1) node[above] {$v_1$} (0,0) -- (1,1) node[above] {$u_1$};
\end{tikzpicture}}}\sqcup\dots\sqcup{
\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.7]
\draw[dashed] (-1,1) node[above] {$w_n$} -- (0,0) -- (0,1.1) (0,1) node[above] {$v_n$} (0,0) -- (1,1) node[above] {$u_n$};
\end{tikzpicture}}}
\end{array}$$ Now define a [*${\mathbb{Q}(t)}$–beaded Jacobi diagram*]{} as a trivalent graph whose vertices are oriented and whose edges are oriented and labelled by ${\mathbb{Q}(t)}$. Set: $${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(\varnothing)=\frac{{\mathbb{Q}}\langle {\mathbb{Q}(t)}\textrm{--beaded Jacobi diagrams}\rangle}{{\mathbb{Q}}\langle\textrm{AS, IHX, LE, Hol, OR}\rangle},$$ where the relations are depicted in Figures \[figrelations1\] and \[figrelations2\],
\[scale=0.3\]
(0,2) node[$x$]{}; (1,0) – (1,4); (1,0) – (1,3); (1.8,2.8) node[$P$]{}; (3.2,2) node[$+$]{}; (4.7,2) node[$y$]{}; (5.7,0) – (5.7,4); (5.7,0) – (5.7,3); (6.5,2.8) node[$Q$]{}; (7.9,2) node[$=$]{}; (9.4,0) – (9.4,4); (9.4,0) – (9.4,3); (12.4,2.8) node[$xP+yQ$]{}; (6.5,-1.5) node[LE]{};
\[xshift=22cm,yshift=1cm,scale=0.9\]
[ (0,0) – (0,3); (0,0) – (0,1.5);]{} (0,1.5) node\[right\] [$P$]{}; [ (0,0) – (0,3); (0,0) – (0,1.5);]{} (-1.5,-0.75) node\[above\] [$Q$]{}; [ (0,0) – (0,3); (0,0) – (0,1.5);]{} (1,-0.9) node\[above right\] [$R$]{}; (4,0) node[$=$]{};
\[xshift=8cm\] [ (0,0) – (0,3); (0,0) – (0,1.5);]{} (0,1.5) node\[right\] [$tP$]{}; [ (0,0) – (0,3); (0,0) – (0,1.5);]{} (-1.5,-0.75) node\[above\] [$tQ$]{}; [ (0,0) – (0,3); (0,0) – (0,1.5);]{} (1,-0.9) node\[above right\] [$tR$]{};
(4,-3.5) node[Hol]{};
\[xshift=38cm\] (0,0) – (0,4); (0,2.9) – (0,3) node\[right\] [$P(t)$]{}; (4,2) node[$=$]{}; (5.5,0) – (5.5,4); (5.5,3.1) – (5.5,3) node\[right\] [$P(t^{-1})$]{}; (4,-1.5) node[OR]{};
with the IHX relation defined with the central edge labelled by 1. Define the [*i–degree*]{}, or [*internal degree*]{}, of any Jacobi diagram as its number of trivalent vertices. Given a hermitian form $\ell:V\times V\to{\mathbb{Q}(t)}$, one can [*glue with $\ell$*]{} some legs of a $V$–colored Jacobi diagram as depicted in Figure \[figglue\].
\[yscale=0.7\] in [0,1.5]{} [ (-0.5,0) – (,0.5) – (,2) (+0.5,0) – (,0.5);]{} (0,2.5) node [$v$]{} (1.5,2.5) node [$w$]{} (3,1) node [$\rightsquigarrow$]{}; (4.5,0) – (4,0.5) (3.5,0) – (4,0.5) .. controls +(0,0.9) and +(-0.5,0) .. (4.75,2) .. controls +(0.5,0) and +(0,0.9) .. (5.5,0.5) – (5,0) (5.5,0.5) – (6,0); (4.72,2) – (4.78,2) node\[above\] [$\ell(v,w)$]{};
If $n$ is even, one can pairwise glue all legs of a $V$–colored Jacobi diagram of i–degree $n$ in order to get an element of ${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(\varnothing)$. This latter space is the target space of the Kricker invariant of ${\mathbb{Q}}$SK–pairs.
We can now state our main result, about the Kricker invariant ${\tilde{Z}}$, proved in Section \[secformules\]. Note that null LP–surgeries define a move on the set of ${\mathbb{Q}}$SK–pairs.
\[thmain\] Let $(S,\kappa)$ be a ${\mathbb{Q}}$SK–pair. Let ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ be a null LP–surgery on $(S,\kappa)$. Then: $$\sum_{I\subset\{1,\dots,n\}}(-1)^{|I|}{\tilde{Z}}\left((S,\kappa)({\mathbf{C}}_I)\right)\equiv_n
\left(\textrm{\begin{minipage}{6cm} \begin{center} sum of all ways of gluing all legs of $\mu({\mathbf{C}})$ with $\ell_{(S,\kappa)}({\mathbf{C}})/2$\end{center}\end{minipage}}\right),$$ where $\equiv_n$ means “equal up to terms of i–degree at least $n+1$”.
#### Example.
Let $(S^3,\mathcal{O})$ be the ${\mathbb{Q}}$SK–pair defined by the trivial knot $\mathcal{O}$ in the standard $3$–sphere. Let $C_1$ and $C_2$ be regular neighborhoods of the graphs $\Gamma_1$ and $\Gamma_2$ drawn in Figure \[figex\].
\[scale=0.8\]
(4,5) ellipse (4 and 1); (6,0.7) arc (90:180:0.7); in [5,...,1]{} [[(+0.7,0) arc (0:180:0.7); (+0.7,0) arc (0:180:0.7);]{}]{} in [1,...,6]{} [[(-0.7,0) arc (-180:0:0.7); (-0.7,0) arc (-180:0:0.7);]{}]{} in [1,...,6]{} [(-0.05,-0.7) – (,-0.7);]{} in [1,2,3]{} [(2\*-1,-0.7) node\[below\] [$\zeta_\x$]{};]{} in [1,2,3]{} [(2\*,-0.7) node\[below\] [$\xi_\x$]{};]{} [(2.6,4.2) .. controls +(0,0.7) and +(0,1.5) .. (3,3); (2.6,4.2) .. controls +(0,0.7) and +(0,1.5) .. (3,3);]{} (3,0.7) – (3,3) node [$\scriptstyle{\bullet}$]{}; (3,3) .. controls +(1,0) and +(0,1) .. (5,0.7); [(4,2) .. controls +(-1,0) and +(0,1) .. (2,0.7); (4,2) .. controls +(-1,0) and +(0,1) .. (2,0.7);]{} (4.1,6) – (4,6) node\[above\] [$\mathcal{O}$]{}; [(1,0.7) .. controls +(0,3) and +(0,1) .. (2.2,4.2); (1,0.7) .. controls +(0,3) and +(0,1) .. (2.2,4.2);]{} (2.2,4) – (2.2,1.5); [(2.2,1.2) .. controls +(0,-1) and +(0,-3) .. (2.6,3.95); (2.2,1.2) .. controls +(0,-1) and +(0,-3) .. (2.6,3.95);]{} in [1,...,5]{} [(+0.7,0) arc (0:100:0.7);]{} [(7.35,4.6) .. controls +(0.1,0.5) and +(0,0.5) .. (7.6,4); (7.35,4.6) .. controls +(0.1,0.5) and +(0,0.5) .. (7.6,4);]{} [(6.65,4.4) .. controls +(-0.3,1.5) and +(0.7,0) .. (6,0.7); (6.65,4.4) .. controls +(-0.3,1.5) and +(0.7,0) .. (6,0.7);]{} (7.6,4) .. controls +(0,-1) and +(0,1) .. (6.7,0); [(4,2) – (7,3); (4,2) – (7,3);]{} (4,0.7) – (4,2) node [$\scriptstyle{\bullet}$]{}; (7,3) .. controls +(-0.2,0) and +(0.1,-0.5) .. (6.7,4.15); (7,3) .. controls +(0.2,0) and +(-0.1,-0.5) .. (7.3,4.3); (0.7,2) node [$\Gamma_1$]{}; (7.6,2) node [$\Gamma_2$]{};
One can define an LP–surgery ${\mathbf{C}}=({\mathbf{C}}_1,{\mathbf{C}}_2)$ by associating with each $\Gamma_i$ a Borromean surgery, see for instance [@M7 Section 2.2]. The associated tensor is given by $\mu({\mathbf{C}}_1)=\zeta_1\wedge\zeta_2\wedge\zeta_3$ and $\mu({\mathbf{C}}_2)=\xi_1\wedge\xi_2\wedge\xi_3$. There are fifteen ways to glue all legs of $$\mu({\mathbf{C}})={
\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.7]
\draw[dashed] (-1,1) node[above] {$\zeta_3$} -- (0,0) -- (0,1.1) (0,1) node[above] {$\zeta_2$} (0,0) -- (1,1) node[above] {$\zeta_1$};
\end{tikzpicture}}}{
\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.7]
\draw[dashed] (-1,1) node[above] {$\xi_3$} -- (0,0) -- (0,1.1) (0,1) node[above] {$\xi_2$} (0,0) -- (1,1) node[above] {$\xi_1$};
\end{tikzpicture}}}$$ with $\frac12\ell_{(S^3,\mathcal{O})}({\mathbf{C}})$; all associated diagrams but one are trivial by the relation LE since they have a trivially labelled edge. Now ${\textrm{\textnormal{lk}}}_e(\zeta_1,\xi_1)=1$, ${\textrm{\textnormal{lk}}}_e(\zeta_2,\xi_2)=1$ and ${\textrm{\textnormal{lk}}}_e(\zeta_3,\xi_3)=t^{-1}$, so that we finally get: $${\tilde{Z}}\left((S^3,\mathcal{O})({\mathbf{C}})\right)+{\tilde{Z}}\left((S^3,\mathcal{O})\right)\equiv_2 -\raisebox{-0.8cm}{
\begin{tikzpicture} [scale=0.3]
\draw[dashed] (0,0) .. controls +(0,2) and +(0,2) .. (4,0);
\draw[dashed] (0,0) .. controls +(0,-2) and +(0,-2) .. (4,0);
\draw[dashed] (0,0) node {$\scriptscriptstyle{\bullet}$} -- (4,0) node {$\scriptscriptstyle{\bullet}$};
\draw[->] (1.95,-1.5) -- (2,-1.5) node[below] {$\scriptstyle{1}$};
\draw[->] (1.95,0) -- (2,0) node[below] {$\scriptstyle{1}$};
\draw[->] (1.95,1.5) -- (2,1.5) node[above] {$\scriptstyle{t}$};
\end{tikzpicture}},$$ where the vanishing of two terms in the left hand side is due to [@GGP Lemma 2.2].
Strategy
--------
In this Subsection, we give a rough overview of the strategy developed to prove Theorem \[thmain\].
The main object of this article is the construction of a functorial LMO invariant defined on a category of Lagrangian cobordisms with paths. The morphisms of this category are cobordisms between compact surfaces with one boundary component, satisfying a Lagrangian-preserving condition, with finitely many disjoint paths with fixed extremities which we think of as knots with a fixed part on the boundary. This category is equivalent to a category of bottom-top tangles in ${\mathbb{Q}}$–cubes, whose top part has a trivial linking matrix, with paths with fixed extremities. These bottom-top tangles can be viewed as morphisms in a category of (general) tangles with paths in ${\mathbb{Q}}$–cubes, with an important difference in the composition law. Now a tangle with paths in a ${\mathbb{Q}}$–cube can be expressed as the result of a surgery on a link in a tangle with trivial paths —segment lines— in $[-1,1]^3$. To sum up, with a Lagrangian cobordism with paths, we associate a tangle with disks —whose boundaries define the paths— in $[-1,1]^3$ with a surgery link. This is represented in the first line of the scheme in Figure \[figscheme\]. We initiate the construction of the invariant at the “tangle with disks” level.
(-0.3,5) node ; (2.2,5) – (2.8,5); (5,5) node ; (7.2,5) – (7.8,5); (10,5) node ; (0,3.7) – (0,1); (5,4.2) – (5,0.8); (10,4.2) – (10,0.8); (0,2.5) node\[right\] [${\tilde{Z}}$]{} (5,2.5) node\[right\] [$Z$]{} (10,2.5) node\[right\] [${Z^\bullet}$]{}; (0,0) node ; (5,0) node ; (10,0) node ; (3,2.6) – (2,2.6); (2.7,2.2) node [[*normalization*]{}]{}; (8,2.7) – (7,2.7); (7.7,2) node
-----------------------
[*formal Gaussian*]{}
[*integration*]{}
-----------------------
;
On the above mentioned categories, we define functorial invariants valued in categories of Jacobi diagrams with beads, [*i.e.*]{} unitrivalent graphs whose univalent vertices are labelled by some finite set or embedded in some 1–manifold —the skeleton— and whose edges are labelled (beaded) by powers of $t$, polynomials in ${\mathbb{Q}[t^{\pm1}]}$ or rational functions in ${\mathbb{Q}(t)}$. At the first step, we define a functor ${Z^\bullet}$ on the category of tangles with disks by applying the Kontsevich integral and adding a bead $t^{\pm1}$ on the skeleton when the corresponding component meets a disk of the tangle. At a second step, we apply the invariant ${Z^\bullet}$ to surgery presentations of tangles with paths in ${\mathbb{Q}}$–cubes. We use the formal Gaussian integration methods introduced by Bar-Natan, Garoufalidis, Rozansky and Thurston in [@AA1; @AA2] and adapted to the beaded setting in [@Kri; @GK]. We get a functor $Z$ on the category of tangles with paths in ${\mathbb{Q}}$–cubes. At the last step, given a Lagrangian cobordism with paths, we apply $Z$ to the associated bottom-top tangles with paths and normalize it following [@CHM] to obtain a functor ${\tilde{Z}}$ on the category of Lagrangian cobordisms with paths. Functoriality allows to prove splitting formulas for this invariant with respect to null Lagrangian-preserving surgeries.
Given a Lagrangian cobordism with one path between genus 0 surfaces, [*i.e.*]{} a ${\mathbb{Q}}$–cube with one path, one can glue a 3–ball to the boundary to get a ${\mathbb{Q}}$–sphere with a knot. In this way, the functor ${\tilde{Z}}$ provides an invariant of ${\mathbb{Q}}$SK–pairs which coincides with the Kricker invariant for knots in ${\mathbb{Z}}$–spheres. Splitting formulas for this invariant are deduced from the splitting formulas for the functor ${\tilde{Z}}$.
#### Plan of the paper.
We define the domain categories of cobordisms and tangles in Section \[secdomaincat\]. In Section \[sectargetcat\], we define the target categories of Jacobi diagrams and gives the tools of formal Gaussian integration. Section \[secwinding\] is devoted to the introduction of winding matrices, that will play the role of the linking matrices in the Cheptea–Habiro–Massuyeau construction of a functorial LMO invariant. The functors ${Z^\bullet}$, $Z$ and ${\tilde{Z}}$ are constructed in Section \[secfunctor\]. At the end of this section, from the functor ${\tilde{Z}}$, we deduce our version of the Kricker invariant for ${\mathbb{Q}}$SK–pairs; the behaviour of this invariant with respect to connected sum is stated. Finally, the splitting formulas are given in Section \[secformules\].
Domain categories: cobordisms and tangles {#secdomaincat}
=========================================
Cobordisms with paths {#subseccob}
---------------------
Given $g\in{\mathbb{N}}$, we fix a model surface $F_g$, compact, connected, oriented, of genus $g$, with one boundary component represented in Figure \[figFg\].
\[scale=0.5\] (0,0) – (8,4) – (30,4) – (22,0) – (0,0); (22,0) node [$*$]{}; (11.5,2) circle (1.2 and 0.6) (12,1.45) – (22,0); (12.7,1.95) – (12.7,2.05) node\[right\] [$\alpha_1$]{}; (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2) .. controls +(0,-1) and +(0,-1) .. (11.5,2) (10.2,1.3) – (22,0); (8,1.5) – (8.1,1.45) node\[below\] [$\beta_1$]{}; (15.5,2) node [$\dots$]{}; (22.5,2) circle (1.2 and 0.6) (22.6,1.4) – (22,0); (23.7,1.95) – (23.7,2.05) node\[right\] [$\alpha_g$]{}; (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (22.5,2) arc (0:180:2); (23,2) arc (0:180:2.5); (22,2) arc (0:180:1.5); (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (22.5,2) arc (0:180:2) .. controls +(0,-1) and +(0,-1) .. (22.5,2) (21.5,1.3) – (22,0); (19,1.5) – (19.1,1.45); (18.7,1) node [$\beta_g$]{}; (28,1) – (29,1) node\[right\] [$x$]{}; (28,1) – (29,1.5); (29,1.75) node\[right\] [$y$]{};
It is equipped with a fixed base point $*$ and a fixed basis $(\alpha_1,\dots,\alpha_g,\beta_1,\dots,\beta_g)$ of $\pi_1(F_g,*)$. Denote by ${C^{g^+}_{g^-}}$ the cube $[-1,1]^3$ with $g^+$ handles on the top boundary and $g^-$ tunnels in the bottom boundary. We have canonical embeddings $F_{g^+}\hookrightarrow\partial{C^{g^+}_{g^-}}$ and $F_{g^-}\hookrightarrow\partial{C^{g^+}_{g^-}}$. For $k\geq 0$ and $1\leq i\leq k$, set $h_i(k)=\frac{2i}{k+1}-1$. A [*cobordism with paths*]{} from $F_{g^+}$ to $F_{g^-}$ is an equivalence class of triples $(M,K,m)$ where:
- $M$ is a compact, connected, oriented 3–manifold,
- $m:\partial{C^{g^+}_{g^-}}\fl{\cong}\partial M$ is an orientation-preserving homeomorphism,
- $K=\sqcup_{i=1}^k K_i\subset M$ is a union of $k\geq 0$ oriented paths $K_i$ from $m(0,1,h_i(k))$ to $m(0,-1,h_i(k))$,
- ${\hat{K}}=\sqcup_{i=1}^k {\hat{K}}_i$ is an oriented [*boundary link*]{}, [*i.e.*]{} the ${\hat{K}}_i$ bound disjoint compact oriented surfaces in $M$, where ${\hat{K}}_i$ is the knot defined as the union of $K_i$ with the line segments $[(0,-1,h_i(k)),(1,-1,h_i(k))]$, $[(1,-1,h_i(k)),(1,1,h_i(k))]$ and $[(1,1,h_i(k)),(0,1,h_i(k))]$.
\[scale=0.2\]
in [30,52]{} [(,4) – (,19);]{}
\[xshift=22cm,yshift=0cm\] (0,0) – (8,4) – (30,4) – (22,0) – (0,0); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5); (15.5,2) node [$\dots$]{}; (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (22.5,2) arc (0:180:2); (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (23,2) arc (0:180:2.5); (22,2) arc (0:180:1.5);
in [5,10]{} [(52,+4) – (41,+4);]{} (33,5) .. controls +(2,1) and +(2,-1) .. (40,7) .. controls +(-2,1) and +(-1,2) .. (41,9); (33.9,12) – (33.8,11.9) node\[left\] [$K_2$]{}; (33,10) .. controls +(0,2) and +(0,2) .. (36,12) .. controls +(0,-2) and +(0,-2) .. (38,14) .. controls +(0,2) and +(0,2) .. (41,14); (36,5.72) – (35.9,5.7) node\[above\] [$K_1$]{}; (22,15) – (45,15);
\[xshift=22cm,yshift=15cm\] (0,0) – (8,4) – (30,4) – (22,0) – (0,0); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5); (15.5,2) node [$\dots$]{}; (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (22.5,2) arc (0:180:2); (22.5,2) circle (0.5 and 0.25) (18.5,2) circle (0.5 and 0.25); (23,2) arc (0:180:2.5); (22,2) arc (0:180:1.5);
in [22,44]{} [(,0.5) – (,14.6); (,0) – (,15);]{} in [5,10]{} [(33,) – (44,) – (52,+4);]{}
Two such triples are [*equivalent*]{} if they are related by an orientation-preserving homeomorphism which respects the boundary parametrizations and identifies the paths. We get embeddings $m_+:F_{g^+}\hookrightarrow\partial M$ and $m_-:F_{g^-}\hookrightarrow\partial M$.
Define a category ${\widetilde{\mathcal{C}ob}}$ of cobordisms with paths whose objects are non-negative integers and whose set of morphisms ${\widetilde{\mathcal{C}ob}}(g^+,g^-)$ is the set of cobordisms with paths from $F_{g^+}$ to $F_{g^-}$. The composition of a cobordism $(M,K,m)$ from $F_g$ to $F_f$ with a cobordism $(N,J,n)$ from $F_h$ to $F_g$ is given by gluing $N$ on the top of $M$, identifying $m_+(M)$ with $n_-(N)$ and reparametrizing the new manifold. The identity of $g\in{\mathbb{N}}$ is the cobordism $F_g\times[-1,1]$ with natural boundary parametrization and no path.
Forgetting the datum of the paths in the cobordisms, one gets the category ${\mathcal{C}ob}$ described in [@CHM], which we view as the subcategory of ${\widetilde{\mathcal{C}ob}}$ of cobordisms with no path. For a cobordism $(M,m)$ and a cobordism with paths $(N,J,n)$, define the tensor product $(M,m)\otimes(N,J,n)$ by horizontal juxtaposition in the $x$ direction.
We now define the subcategory of ${\widetilde{\mathcal{C}ob}}$ of Lagrangian cobordisms with paths. Set $A_g=\ker(\mathrm{incl}_*:H_1(F_g;{\mathbb{Q}})\to H_1(C^g_0;{\mathbb{Q}}))$ and $B_g=\ker(\mathrm{incl}_*:H_1(F_g;{\mathbb{Q}})\to H_1(C^0_g;{\mathbb{Q}}))$. These are Lagrangian subspaces of $H_1(F_g;{\mathbb{Q}})$ with respect to the intersection form and $A_g$ (resp. $B_g$) is generated by the homology classes of the curves $\alpha_i$ (resp. $\beta_i$). A cobordism with paths $(M,K,m)$ from $F_{g^+}$ to $F_{g^-}$ is [*Lagrangian(-preserving)*]{} if the following conditions are satisfied:
- $H_1(M;{\mathbb{Q}})=(m_-)_*(A_{g^-})\oplus(m_+)_*(B_{g^+})$,
- $(m_+)_*(A_{g^+})\subset(m_-)_*(A_{g^-})$ as subspaces of $H_1(M;{\mathbb{Q}})$.
The Lagrangian property is preserved by composition, and we denote by ${\widetilde{\mathcal{LC}ob}}$ the subcategory of ${\widetilde{\mathcal{C}ob}}$ of Lagrangian cobordisms with paths. The subcategory of Lagrangian cobordisms with no path is the category ${\mathcal{LC}ob}$ —denoted by ${\mathbb{Q}}{\mathcal{LC}ob}$ in [@Mas].
Define categories ${\mathcal{C}ob}_q$, ${\widetilde{\mathcal{C}ob}}_q$, ${\mathcal{LC}ob}_q$ and ${\widetilde{\mathcal{LC}ob}}_q$ of $q$–cobordisms with objects the non-commutative words in the single letter $\bullet$ and with set of morphisms from a word on $g^+$ letters to a word on $g^-$ letters the set of morphisms from $g^+$ to $g^-$ in ${\mathcal{C}ob}$, ${\widetilde{\mathcal{C}ob}}$, ${\mathcal{LC}ob}$ and ${\widetilde{\mathcal{LC}ob}}$ respectively.
Let $(M,K,m)$ be a Lagrangian cobordism with paths from $g^+$ to $g^-$. Since the space $H_1(M;{\mathbb{Q}})$ is non-trivial in general, we have to adapt the definition of null LP–surgeries given in the introduction. Let $N(K)$ be a tubular neighborhood of $K$ in $M$ and set $E=M\setminus\textrm{Int}(N(K))$. A standard homological computation gives $H_2(M;{\mathbb{Q}})=0$, so that the exact sequence in homology associated with the pair $(M,E)$ provides the following short exact sequence: $$0\to H_2(N(K),N(K)\cap E;{\mathbb{Q}})\to H_1(E;{\mathbb{Q}})\to H_1(M;{\mathbb{Q}})\to 0.$$ The image of $H_2(N(K),N(K)\cap E;{\mathbb{Q}})$ in $H_1(E;{\mathbb{Q}})$ is a subspace $H_K\cong{\mathbb{Q}}^k$ generated by meridians of the components of $K$, where $k$ is the number of these components. Now, the parametrizations $m_{\pm}$ of the top and bottom boundaries of $M$ can be decomposed into injective maps as follows.
(0,-0.2) node [$E$]{}; (-1,1) node [$F_{g^\pm}$]{}; (1,1) node [$M$]{}; (-0.5,1) – (0.5,1); (0,1) node\[above\] [$m_\pm$]{}; (-0.7,0.7) – (-0.2,0); (-0.4,0.3) node\[left\] [$e_\pm$]{}; (0.7,0.7) – (0.2,0);
Hence we have a canonical decomposition of $H_1(E;{\mathbb{Q}})$ as $$H_1(E;{\mathbb{Q}})=H_K\oplus(e_-)_*(A_{g^-})\oplus(e_+)_*(B_{g^+}),$$ where $(e_-)_*(A_{g^-})\oplus(e_+)_*(B_{g^+})\cong H_1(M;{\mathbb{Q}})$ [*via*]{} the inclusion map. We say that a ${\mathbb{Q}}$–handlebody $C\subset M\setminus K$ is *null with respect to $K$* if the composed map $$H_1(C;{\mathbb{Q}})\fl{incl_*} H_1(M\setminus K;{\mathbb{Q}})\longrightarrow H_K$$ has a trivial image, where the second map is the projection on the first factor in the above decomposition of $H_1(M\setminus K;{\mathbb{Q}})\cong H_1(E;{\mathbb{Q}})$. A *null LP–surgery* on $(M,K)$ is an LP–surgery ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ on $M\setminus K$ such that each $C_i$ is null with respect to $K$.
Bottom-top tangles with paths {#subsecbtt}
-----------------------------
Let us define the category of bottom-top tangles with paths. For a positive integer $g\geq0$, let $(p_1,q_1),\dots,(p_g,q_g)$ be $g$ pairs of points uniformly distributed on $[-1,1]\times\{0\}\subset[-1,1]^2\cong F_0$ as represented in Figure \[figpiqi\].
\[xscale=0.4,yscale=0.2\] (0,0) – (4,6) – (16,6) – (12,0) – (0,0); (3.5,3) node [$\scriptstyle{\bullet}$]{}; (3.5,3) node\[below\] [$p_1$]{}; (5,3) node [$\scriptstyle{\bullet}$]{}; (5,3) node\[below\] [$q_1$]{}; (8,3) node [$\dots$]{}; (11,3) node [$\scriptstyle{\bullet}$]{}; (11,3) node\[below\] [$p_g$]{}; (12.5,3) node [$\scriptstyle{\bullet}$]{}; (12.5,3) node\[above\] [$q_g$]{};
A [*bottom-top tangle with paths*]{} of type $(g^+,g^-)$ is an equivalence class of triples $(B,K,\gamma)$ where:
- $(B,K)=(B,K,b)$ is a cobordism with paths form $F_0$ to $F_0$,
- $\gamma=(\gamma^+,\gamma^-)$ is a framed oriented tangle in $B$ with $g^+$ components $\gamma_i^+$ from $b(\{p_i\}\times\{1\})$ to $b(\{q_i\}\times\{1\})$ and $g^-$ components $\gamma_i^-$ from $b(\{q_i\}\times\{-1\})$ to $b(\{p_i\}\times\{-1\})$,
- ${\hat{K}}$ is a boundary link in $B\setminus\gamma$.
Two such triples $(B,K,\gamma)$ and $(B',K',\gamma')$ are [*equivalent*]{} if $(B,K)$ and $(B',K')$ are related by an equivalence which identifies $\gamma$ and $\gamma'$.
In order to define the composition, we need the bottom-top tangle $([-1,1]^3,\varnothing,T_g)$ represented in Figure \[figTg\].
\[xscale=0.4,yscale=0.3\] (0,0) – (3,4) – (20,4) (20,14) – (3,14) (3,4) – (3,14) (20,4) – (20,14); (5.5,6.5) .. controls +(1,0) and +(0,-3) .. (7,12); (4,2) .. controls +(0,3) and +(-1,0) .. (5.5,7.5) .. controls +(1,0) and +(0,3) .. (7,2); (4,2) .. controls +(0,3) and +(-1,0) .. (5.5,7.5) .. controls +(1,0) and +(0,3) .. (7,2); (4,12) .. controls +(0,-3) and +(-1,0) .. (5.5,6.5); (4,12) .. controls +(0,-3) and +(-1,0) .. (5.5,6.5); (14.5,6.5) .. controls +(1,0) and +(0,-3) .. (16,12); (13,2) .. controls +(0,3) and +(-1,0) .. (14.5,7.5) .. controls +(1,0) and +(0,3) .. (16,2); (13,2) .. controls +(0,3) and +(-1,0) .. (14.5,7.5) .. controls +(1,0) and +(0,3) .. (16,2); (13,12) .. controls +(0,-3) and +(-1,0) .. (14.5,6.5); (13,12) .. controls +(0,-3) and +(-1,0) .. (14.5,6.5); (0,10) – (17,10) – (17,0); (0,10) – (17,10) – (17,0) (20,4) – (17,0) – (0,0) – (0,10) – (3,14) (17,10) – (20,14); (10,2) node [$\dots$]{} (10,12) node [$\dots$]{}; (5.5,2) node\[below\] [$1$]{} (14.5,2) node\[below\] [$g$]{}; in [4,7,13,16]{} in [2,12]{} (,) node [$\scriptstyle{\bullet}$]{}; (6.71,8.5) – (6.74,8.6); (15.71,8.5) – (15.74,8.6); (4.29,5.5) – (4.26,5.4); (13.29,5.5) – (13.26,5.4);
The composition of a bottom-top tangle $(B,K,\gamma)$ of type $(g,f)$ with a bottom-top tangle $(C,J,\upsilon)$ of type $(h,g)$ is given by first making the composition $(B,K)\circ([-1,1]^3,\varnothing)\circ(C,J)$ in the category ${\widetilde{\mathcal{C}ob}}$ and then perfoming the surgery on the $2g$ components link $\gamma^+\cup T_g\cup\upsilon^-$. We get a category ${{}_b^t\widetilde{\mathcal{T}}}$ whose objects are non-negative integers and whose set of morphisms ${{}_b^t\widetilde{\mathcal{T}}}(g^+,g^-)$ is the set of bottom-top tangles with paths of type $(g^+,g^-)$. The identity of $g\in{\mathbb{N}}$ is the bottom-top tangle in $[-1,1]^3$ with no path represented in Figure \[figtangleIdg\].
\[xscale=0.4,yscale=0.3\] (0,0) – (3,4) – (20,4) (20,14) – (3,14) (3,4) – (3,14) (20,4) – (20,14); (4,12) .. controls +(0,-3) and +(-1,0) .. (5.5,6.5); [(4,2) .. controls +(0,3) and +(-1,0) .. (5.5,7.5) .. controls +(1,0) and +(0,3) .. (7,2); (4,2) .. controls +(0,3) and +(-1,0) .. (5.5,7.5) .. controls +(1,0) and +(0,3) .. (7,2);]{} [(5.5,6.5) .. controls +(1,0) and +(0,-3) .. (7,12); (5.5,6.5) .. controls +(1,0) and +(0,-3) .. (7,12);]{} (13,12) .. controls +(0,-3) and +(-1,0) .. (14.5,6.5); [(13,2) .. controls +(0,3) and +(-1,0) .. (14.5,7.5) .. controls +(1,0) and +(0,3) .. (16,2); (13,2) .. controls +(0,3) and +(-1,0) .. (14.5,7.5) .. controls +(1,0) and +(0,3) .. (16,2);]{} [(14.5,6.5) .. controls +(1,0) and +(0,-3) .. (16,12); (14.5,6.5) .. controls +(1,0) and +(0,-3) .. (16,12);]{} (0,10) – (17,10) – (17,0); (0,10) – (17,10) – (17,0) (20,4) – (17,0) – (0,0) – (0,10) – (3,14) (17,10) – (20,14); (10,2) node [$\dots$]{} (10,12) node [$\dots$]{}; (5.5,2) node\[below\] [$1$]{} (14.5,2) node\[below\] [$g$]{}; in [4,7,13,16]{} in [2,12]{} (,) node [$\scriptstyle{\bullet}$]{}; (6.71,8.5) – (6.74,8.6); (15.71,8.5) – (15.74,8.6); (4.29,5.5) – (4.26,5.4); (13.29,5.5) – (13.26,5.4);
Forgetting the datum of the paths, one gets the category ${{}_b^t\mathcal{T}}$ of bottom-top tangles introduced in [@CHM], that we view as the subcategory of ${{}_b^t\widetilde{\mathcal{T}}}$ of bottom-top tangles with no path. For a bottom-top tangle $(B,\gamma)$ and a bottom-top tangle with paths $(C,J,\upsilon)$, define the tensor product $(B,\gamma)\otimes(C,J,\upsilon)$ by horizontal juxtaposition in the $x$ direction. Define categories ${{}_b^t\mathcal{T}}_q$ and ${{}_b^t\widetilde{\mathcal{T}}}_q$ of bottom-top $q$–tangles with objects the non-commutative words in the single letter $\bullet$.
The following result is a direct adaptation of [@CHM Theorem 2.10] which gives an isomorphism $D:{{}_b^t\mathcal{T}}\to{\mathcal{C}ob}$. The map $D$ is defined by digging tunnels around the components of the tangle.
\[propmapD\] There is an isomorphism $D:{{}_b^t\widetilde{\mathcal{T}}}\to{\widetilde{\mathcal{C}ob}}$ which identifies ${{}_b^t\mathcal{T}}$ with ${\mathcal{C}ob}$ and preserves the tensor product on ${{}_b^t\mathcal{T}}\otimes{{}_b^t\widetilde{\mathcal{T}}}$.
Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube. Let $\bar{\gamma}$ be the link obtained by closing the components of $\gamma$ with the line segments $[(p_i,\pm1),(q_i,\pm1)]$. Define the linking matrix ${\textrm{\textnormal{Lk}}}(\gamma)$ of $\gamma$ in $B$ with the linkings of the components of $\bar{\gamma}$. The characterization of the bottom-top tangles sent onto Lagrangian cobordisms by $D$ given in [@CHM Lemma 2.12] directly generalizes to:
Given a bottom-top tangle with paths $(B,K,\gamma)$, the cobordism with paths $D(B,K,\gamma)$ is Lagrangian if and only if $B$ is a ${\mathbb{Q}}$–cube and ${\textrm{\textnormal{Lk}}}(\gamma^+)$ is trivial.
Tangles with paths and tangles with disks {#subsectangles}
-----------------------------------------
Given a cobordism $(B,b)$ from $F_0$ to $F_0$, a [*tangle $\gamma$ in $B$*]{} is an isotopy (rel. $\partial B$) class of framed oriented tangles whose boundary points lie on the top and bottom surfaces and are uniformly distributed along the line segments $[-1,1]\times\{0\}\times\{1\}$ and $[-1,1]\times\{0\}\times\{-1\}$ in $\partial B=b(\partial[-1,1]^3)$. Associate with each boundary point of $\gamma$ the sign $+$ if $\gamma$ is oriented downwards at that point and the sign $-$ otherwise. This provides two words in the letters $+$ and $-$, one for the top surface and the other for the bottom surface. Lifting these two words into non-associative words $w_t(\gamma)$ and $w_b(\gamma)$ in the letters $(+,-)$, one gets a [*$q$–tangle*]{}. A $q$–tangle $\gamma$ in a cobordism with paths $(B,K,b)$ defines a [*$q$–tangle with paths*]{} $(B,K,\gamma)$ if ${\hat{K}}$ is a boundary link in $B\setminus\gamma$.
Define two categories ${\mathcal{T}_q\mathcal{C}ub}$ and ${\widetilde{\mathcal{T}}_q\mathcal{C}ub}$ with objects the non-associative words in the letters $(+,-)$ and morphisms the $q$–tangles in ${\mathbb{Q}}$–cubes for ${\mathcal{T}_q\mathcal{C}ub}$ and the $q$–tangles with paths in ${\mathbb{Q}}$–cubes for ${\widetilde{\mathcal{T}}_q\mathcal{C}ub}$, up to orientation-preserving homeomorphism respecting the boundary parametrization. Composition is given by vertical juxtaposition. Given a morphism $(C,\upsilon)$ in ${\mathcal{T}_q\mathcal{C}ub}$ and a morphism $(B,K,\gamma)$ in ${\widetilde{\mathcal{T}}_q\mathcal{C}ub}$, define the tensor product $(C,\upsilon)\otimes(B,K,\gamma)$ by horizontal juxtaposition in the $x$ direction.
\[lemmapreschir\] Let $(B,K,\gamma)$ be a $q$–tangle with paths in a ${\mathbb{Q}}$–cube. There exist a $q$–tangle with paths $([-1,1]^3,\Xi,\eta)$ and a framed link $L\subset[-1,1]^3\setminus(\Xi\cup\eta)$, with $\Xi$ a union of line segments and $L$ null-homotopic in $[-1,1]^3\setminus\Xi$, such that $(B,K,\gamma)$ is obtained from $([-1,1]^3,\Xi,\eta)$ by surgery on $L$. Moreover, two such surgery links are related by the following Kirby moves: the blow-up/blow-down move KI which adds or removes a split trivial component with framing $\pm1$ unknotted with $\eta$, and the handleslide move KII which adds a surgery component to another surgery component or to a component of the tangle $\eta$ (see Figure \[figK2\]).
\[scale=0.3\] (0,0) – (0,-6); (0,2) – (0,0) (0,-6) – (0,-8); (-4,-4.5) arc (-90:90:1.5); (-4,-1.5) arc (90:270:1.5); (-4,-0.5) node[$L_i$]{}; (0,-1) node\[right\] [$L_j$ or $\eta_j$]{}; (8,-3) – (9.5,-3);
\[xshift=19.5cm\] (0,0) – (0,-2.5) (0,-3.5) – (0,-6); (0,2) – (0,0) (0,-6) – (0,-8); (-4,-4.5) arc (-90:90:1.5); (-4,-1.5) arc (90:270:1.5); (-4,-4.8) arc (-90:90:1.8); (-4,-1.2) arc (90:270:1.8); (-2.26,-2.53) – (-2.26,-3.48); (-2.26,-2.5) – (0,-2.5) (-2.26,-3.5) – (0,-3.5);
Let $\Sigma$ be a Seifert surface of ${\hat{K}}$ which is the disjoint union of Seifert surfaces of the ${\hat{K}}_i$. Choose $\Sigma$ disjoint from $\gamma$. Take a link $J\subset B$ such that surgery on $J$ gives $[-1,1]^3$. Performing isotopies on $J$ if necessary, we can assume that $J$ does not meet $\Sigma$. The handles of (the image in $[-1,1]^3$ of) $\Sigma$ can be unlinked by adding surgery components as shown in Figure \[figundocrossing\].
\[htb\]
\[scale=0.3\]
(0.5,2) .. controls +(0,1) and +(0,1) .. (3.5,2); [(0,4) – (4,0); (0,4) – (4,0);]{} [(0,0) – (4,4); (0,0) – (4,4);]{} [(0.5,2) .. controls +(0,-1) and +(0,-1) .. (3.5,2); (0.5,2) .. controls +(0,-1) and +(0,-1) .. (3.5,2);]{} (-0.3,2) node [$\scriptstyle{+1}$]{};
(8,2) node [$\sim$]{};
\[xshift=12cm\] (0,0) – (4,4); [(0,4) – (4,0); (0,4) – (4,0);]{}
In this way, ${\hat{K}}$ can be turned into a trivial link. This provides a surgery link in $B\setminus(K\cup\gamma)$, disjoint from $\Sigma$, such that surgery on this link changes $(B,K,\gamma)$ to a $q$–tangle with paths $([-1,1]^3,\Xi,\eta)$ as required. Let $L$ be an inverse surgery link. It is null-homotopic in $[-1,1]^3\setminus\Xi$ since it is disjoint from $\Sigma$.
For the last assertion, apply [@HW Theorem 3.1] in $[-1,1]^3\setminus\Xi$. Note that a split trivial component with framing $\pm1$ can always be unknotted from $\eta$ using the KII move.
A family $(([-1,1]^3,\Xi,\eta),L)$ satisfying the conditions of the lemma with $L$ oriented is a [*surgery presentation*]{} of $(B,K,\gamma)$. When $\gamma$ (and thus $\eta$) is a bottom-top tangle, the components of $\eta$ can be closed by line segments in the top and bottom surfaces. The obtained curves are null-homotopic in $[-1,1]^3\setminus\Xi$ since ${\hat{\Xi}}$ is a boundary link in $[-1,1]^3\setminus\eta$.
The notion of a $q$–tangle in $[-1,1]^3$ with trivial paths, [*i.e.*]{} line segments, is equivalent to the following one. A [*$q$–tangle with disks*]{} is an equivalence class of pairs $(\gamma,k)$, where $\gamma$ is a $q$–tangle in $[-1,1]^3$, $k$ is a non-negative integer understood as the datum of $k$ disks $d_i=[0,1]\times[-1,1]\times\{h_i(k)\}$ and the link $\sqcup_{i=1}^k\partial d_i$ associated with the paths $d_i^\partial=\{0\}\times[-1,1]\times\{h_i(k)\}$ is a boundary link in $[-1,1]^3\setminus\gamma$. An example of such a tangle with disks is drawn in Figure \[figtang\], projected in the $y$ direction. Equivalence of such pairs is defined as isotopy relative to $(\partial[-1,1]^3)\cup(\cup_{i=1}^k d_i^\partial)$. Define two categories ${\mathcal{T}_q}$ and ${\widetilde{\mathcal{T}}_q}$ with objects the non-associative words in the letters $(+,-)$ and morphisms the $q$–tangles for ${\mathcal{T}_q}$ and the $q$–tangles with disks for ${\widetilde{\mathcal{T}}_q}$. Composition is given by vertical juxtaposition. Given a $q$–tangle $\gamma$ and a $q$–tangle with disks $(\upsilon,k)$, define the tensor product $\gamma\otimes(\upsilon,k)$ in ${\widetilde{\mathcal{T}}_q}((w_t(\gamma))(w_t(\upsilon)),(w_b(\gamma))(w_b(\upsilon)))$ by horizontal juxtaposition in the $x$ direction. Define similarly two categories $\mathcal{T}$ and $\widetilde{\mathcal{T}}$ of tangles and tangles with disks in $[-1,1]^3$ with objects the associative words in the letters $(+,-)$.
\[scale=0.4\] (6,8) .. controls +(0,-2) and +(1,0) .. (5.2,4.7); [(3.5,2) .. controls +(0,1) and +(0,-2) .. (6,4) .. controls +(0,0.7) and +(1,0) .. (5.2,5.3); (3.5,2) .. controls +(0,1) and +(0,-2) .. (6,4) .. controls +(0,0.7) and +(1,0) .. (5.2,5.3);]{} (5.2,5.3) .. controls +(-1,0) and +(1,0) .. (4.7,3); [(5.2,4.7) .. controls +(-0.8,0) and +(0,-1) .. (4.5,6) .. controls +(0,1) and +(0,-1) .. (4,8); (5.2,4.7) .. controls +(-0.8,0) and +(0,-1) .. (4.5,6) .. controls +(0,1) and +(0,-1) .. (4,8);]{} [(4,0) .. controls +(0,1) and +(0,-1) .. (5.5,2) .. controls +(0,0.5) and +(0.8,0) .. (4.7,3.5); (4,0) .. controls +(0,1) and +(0,-1) .. (5.5,2) .. controls +(0,0.5) and +(0.8,0) .. (4.7,3.5);]{} [((4.7,3.5) .. controls +(-0.8,0) and +(0,0.5) .. (4.5,2) .. controls +(0,-1) and +(0,1) .. (2,0); ((4.7,3.5) .. controls +(-0.8,0) and +(0,0.5) .. (4.5,2) .. controls +(0,-1) and +(0,1) .. (2,0);]{} [(4.7,3) .. controls +(-1,0) and +(0,-3) .. (2,8); (4.7,3) .. controls +(-1,0) and +(0,-3) .. (2,8);]{} [(6,0) .. controls +(0,2) and +(0,-1) .. (3.5,2); (6,0) .. controls +(0,2) and +(0,-1) .. (3.5,2);]{} (0,0) – (0,8) – (8,8) – (8,0) – (0,0); in [2,4,6]{} [(4,) node [$\scriptstyle{\bullet}$]{} – (8,);]{} (5.8,0.8) – (5.7,0.9); (4.5,0.83) – (4.6,0.9); (6.01,6.9) – (6.01,7);
Target categories: Jacobi diagrams with beads {#sectargetcat}
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Diagram spaces {#subsecdiagramspaces}
--------------
For a compact oriented 1–manifold $X$ and a finite set $C$, a [*Jacobi diagram on $(X,C)$*]{} is a unitrivalent graph whose trivalent vertices are oriented and whose univalent vertices are embedded in $X$ or labelled by $C$, where an orientation of a trivalent vertex is a cyclic order of the three edges that meet at this vertex —fixed as in the pictures. The manifold $X$ is called the [*skeleton*]{} of the diagram. Next, let $R$ be the ring ${\mathbb{Q}[t^{\pm1}]}$ or ${\mathbb{Q}(t)}$. An [*$R$–beaded Jacobi diagram on $(X,C)$*]{} is a Jacobi diagram on $(X,C)$ whose graph edges are oriented and labelled by $R$. Last, an [*$R$–winding Jacobi diagram on $(X,C)$*]{} is an $R$–beaded Jacobi diagram on $(X,C)$ whose skeleton is viewed as a union of edges —defined by the embedded vertices— that are labelled by powers of $t$, with the condition that the product of the labels on each component of $X$ is 1. As defined in the introduction, the i–degree of a trivalent diagram is its number of trivalent vertices. Set: $${\mathcal{A}}(X,*_C)=\frac{{\mathbb{Q}}\langle \textrm{Jacobi diagrams on }(X,C) \rangle}{{\mathbb{Q}}\langle \textrm{AS, IHX, STU} \rangle},$$ $${\widetilde{\mathcal{A}}}_R(X,*_C)=\frac{{\mathbb{Q}}\langle R\textrm{--beaded Jacobi diagrams on }(X,C) \rangle}{{\mathbb{Q}}\langle \textrm{AS, IHX, STU, LE, OR, Hol} \rangle},$$ $${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_C)=\frac{{\mathbb{Q}}\langle R\textrm{--winding Jacobi diagrams on }(X,C) \rangle}{{\mathbb{Q}}\langle \textrm{AS, IHX, STU, LE, OR, Hol, {Hol$_\textrm{w}$}} \rangle},$$ with the relations in Figures \[figrelations1\], \[figrelations2\] and \[figrelations3\], where the IHX relation for beaded and winding diagrams is defined with the central edge labelled by 1.
\[scale=0.3\]
(0,0) – (0,4); (0,2) – (0.8,2) node\[above\] [1]{}; (0.7,2) – (1.3,2) – (2.6,3) (1.3,2) – (2.6,1); (3.8,2) node[$=$]{}; (5,0) – (5,4); (5,2.6) – (7.6,2.6) (5,1.4) – (7.6,1.4); (8.8,2) node [$-$]{}; (10,0) – (10,4); (10,2.6) – (12.6,1.4) (10,1.4) – (12.6,2.6); (6.3,-1.5) node[STU]{};
\[xshift=20cm\] (0,0) – (0,4); (0,3) node [$\scriptscriptstyle{\bullet}$]{} node\[left\] [$t^i$]{}; (0,1) node [$\scriptscriptstyle{\bullet}$]{} node\[left\] [$t^j$]{}; (0,2) – (2.6,2); (0,2) – (1.5,2) node\[above\] [$P$]{}; (4,2) node [$=$]{}; (8,0) – (8,4); (8,3) node [$\scriptscriptstyle{\bullet}$]{} node\[left\] [$t^{i+1}$]{}; (8,1) node [$\scriptscriptstyle{\bullet}$]{} node\[left\] [$t^{j-1}$]{}; (8,2) – (10.6,2); (8,2) – (9.5,2) node\[above\] [$tP$]{}; (5,-1.5) node [[Hol$_\textrm{w}$]{}]{};
In the STU relation, the edges corresponding to each other have the same orientation and label. In the pictures, the skeleton is represented with full lines and the graph with dashed lines. We indeed consider the i–degree completion of these vector spaces, keeping the same notation.
For diagrams in ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_C)$, the condition on the labels on the skeleton implies that all labels can be pushed off each component of the skeleton using the [Hol$_\textrm{w}$]{} relation. When the component is an interval, there is a unique way to do so. Hence, when $X$ contains only intervals, ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_C)$ is isomorphic to ${\widetilde{\mathcal{A}}}_R(X,*_C)$.
For a finite set $S$, denote by ${\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S$ (resp. ${\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S$) the manifold made of $|S|$ intervals (resp. circles) indexed by the elements of $S$. In the following, ${\bar{\mathcal{A}}}$ stands for ${\mathcal{A}}$, ${\widetilde{\mathcal{A}}}_R$ or ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R$. In [@BN Theorem 8], Bar-Natan defines a formal PBW isomorphism: $$\chi_S:{\bar{\mathcal{A}}}(X,*_{C\cup S}){\fl{\scriptstyle{\cong}}}{\bar{\mathcal{A}}}(X\cup{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S,*_C).$$ For a Jacobi diagram $D$, the image $\chi_S(D)$ is the average of all possible ways to attach the $s$–colored vertices of $D$ on the corresponding $s$–indexed interval in ${\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S$ for each $s\in S$. The setting of [@BN] is not exactly the same, but the argument adapts directly. When $|S|=1$, closing the $S$–labelled component gives an isomorphism from ${\bar{\mathcal{A}}}(X\cup{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S,*_C)$ to ${\bar{\mathcal{A}}}(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C)$ [@BN Lemma 3.1]. However, this isomorphism does not hold for $|S|>1$. To recover an isomorphism onto ${\bar{\mathcal{A}}}(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C)$, some “link relations” were introduced in [@AA2 Section 5.2]. We recall these relations and introduce additional “winding relations”.
Given a (beaded, winding) Jacobi diagram $D$ on $(X,C\cup S)$ and a univalent vertex $*$ of $D$ labelled by $s\in S$, define the associated [*link relation*]{} as the vanishing of the sum of all diagrams obtained from $D$ by gluing the vertex $*$ on the edges adjacent to a univalent $s$–labelled vertex, as follows: , see Figure \[figlinkrel\] (we omit the orientation of the edges when it is not relevant thanks to the OR relation).
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(0,0) – (0,4); (0,2) – (1,2) – (2.5,1) (1.75,2.5) – (2.5,3); (1,2) – (1.75,2.5); (1.6,2.4) node\[above\] [$\scriptstyle{t}$]{}; (2.5,3) node\[right\] [$\scriptstyle{s}$]{} (2.5,1) node\[right\] [$\scriptstyle{s'}$]{}; (6,2) circle (1); (7,2) – (7,2.05); (6,3) – (6,4) (5.25,0.66) – (5,0.2) (6.5,1.12) – (7,0.2); (6,4) node [$*$]{}; (6,4) node\[right\] [$\scriptstyle{s}$]{} (5,0.2) node\[right\] [$\scriptstyle{s'}$]{} (7,0.2) node\[right\] [$\scriptstyle{s}$]{}; (5.5,1.12) – (5.25,0.66); (5.4,0.8) node\[left\] [$\scriptstyle{t^2}$]{};
(10,2) node [$\rightsquigarrow$]{};
\[xshift=12cm\] (0,0) – (0,4); (0,2) – (1,2) – (2.5,1) (1.5,2.33) – (2.5,3); (1,2) – (1.5,2.33); (1.25,2.6) node [$\scriptstyle{t}$]{}; (2.5,3) node\[right\] [$\scriptstyle{s}$]{} (2.5,1) node\[right\] [$\scriptstyle{s'}$]{}; (6,2) circle (1); (7,2) – (7,2.05); (5.25,0.66) – (5,0.2) (6.5,1.12) – (7,0.2); (6,3) .. controls +(0,2) and +(-1,1.5) .. (2,2.66); (5,0.2) node\[right\] [$\scriptstyle{s'}$]{} (7,0.2) node\[right\] [$\scriptstyle{s}$]{}; (5.5,1.12) – (5.25,0.66); (5.4,0.8) node\[left\] [$\scriptstyle{t^2}$]{};
(21,2) node [$+$]{};
\[xshift=23cm\] (0,0) – (0,4); (0,2) – (1,2) – (2.5,1) (1.75,2.5) – (2.5,3); (1,2) – (1.75,2.5); (1.6,2.4) node\[above\] [$\scriptstyle{t}$]{}; (2.5,3) node\[right\] [$\scriptstyle{s}$]{} (2.5,1) node\[right\] [$\scriptstyle{s'}$]{}; (6,2) circle (1); (7,2) – (7,2.05); (5.25,0.66) – (5,0.2) (6.5,1.12) – (7,0.2); (6,3) .. controls +(0,2) and +(1.84,1) .. (6.75,0.66); (5,0.2) node\[right\] [$\scriptstyle{s'}$]{} (7,0.2) node\[right\] [$\scriptstyle{s}$]{}; (5.5,1.12) – (5.25,0.66); (5.4,0.8) node\[left\] [$\scriptstyle{t^2}$]{};
(32,2) node [$=$]{} (33.5,2.06) node [$0$]{};
Given a winding Jacobi diagram $D$ on $(X,C\cup S)$, a label $s\in S$ and an integer $k$, the associated [*winding relation*]{} identifies $D$ with the diagram obtained from $D$ by [*pushing $t^k$ at each $s$–labelled vertex*]{}, [*i.e.*]{} by multiplying the label of each edge adjacent to a univalent $s$–labelled vertex by $t^k$ if the orientation of the edge goes backward the vertex and by $t^{-k}$ otherwise, see Figure \[figwindingrel\].
\[scale=0.45\]
(-2,1) – (-2,2) node\[right\] [$\scriptstyle{t^{-1}}$]{}; (-2,2) – (-2,3); (-1.8,0.8) node [$\scriptstyle{s}$]{} (-1.8,3.2) node [$\scriptstyle{s}$]{}; (0,0) – (0,4); (0,1) – (1,1) – (2.5,0) (1,1) – (1.75,1.5); (2.5,2) – (1.75,1.5); (1.6,1.4) node\[above\] [$\scriptstyle{t^3}$]{}; (2.7,2) node [$\scriptstyle{s}$]{} (2.7,0) node [$\scriptstyle{s'}$]{}; (0,3) – (1.4,3); (1.4,3) – (2.3,3); (1.4,3.5) node [$\scriptstyle{t^2}$]{}; (2.7,3) node [$\scriptstyle{s}$]{};
(5,2) node [$=$]{};
\[xshift=9cm\] (-2,1) – (-2,2) node\[right\] [$\scriptstyle{t^{-1}}$]{}; (-2,2) – (-2,3); (-1.8,0.8) node [$\scriptstyle{s}$]{} (-1.8,3.2) node [$\scriptstyle{s}$]{}; (0,0) – (0,4); (0,1) – (1,1) – (2.5,0) (1,1) – (1.75,1.5); (2.5,2) – (1.75,1.5); (1.6,1.4) node\[above\] [$\scriptstyle{t^4}$]{}; (2.7,2) node [$\scriptstyle{s}$]{} (2.7,0) node [$\scriptstyle{s'}$]{}; (0,3) – (1.4,3); (1.4,3) – (2.3,3); (1.4,3.5) node [$\scriptstyle{t}$]{}; (2.7,3) node [$\scriptstyle{s}$]{};
Denote ${\bar{\mathcal{A}}}(X,*_C,{\textrm{\footnotesize\textcircled{\raisebox{-0.9ex}{\normalsize*}}}}_S)$ (resp. ${\bar{\mathcal{A}}}(X,*_C,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_S)$) the quotient of ${\bar{\mathcal{A}}}(X,*_{C\cup S})$ by all link relations (resp. all link and winding relations) on $S$–labelled vertices. Note that if $X$ contains no closed component, then the spaces ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_C,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_S)$ and ${\widetilde{\mathcal{A}}}_R(X,*_C,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_S)$ are isomorphic. When some of the sets $X$, $C$, $S$ are empty, we simply drop the corresponding notation, mentionning $\varnothing$ only when they are all empty.
The isomorphisms $\chi_S:{\bar{\mathcal{A}}}(X,*_{C\cup S}){\fl{\scriptstyle{\cong}}}{\bar{\mathcal{A}}}(X\cup{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S,*_C)$ descend to isomorphisms: $$\chi_S:{\mathcal{A}}(X,*_C,{\textrm{\footnotesize\textcircled{\raisebox{-0.9ex}{\normalsize*}}}}_S){\fl{\scriptstyle{\cong}}}{\mathcal{A}}(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C),$$ $$\chi_S:{\widetilde{\mathcal{A}}}_R(X,*_C,{\textrm{\footnotesize\textcircled{\raisebox{-0.9ex}{\normalsize*}}}}_S){\fl{\scriptstyle{\cong}}}{\widetilde{\mathcal{A}}}_R(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C),$$ $$\chi_S:{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_C,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_S){\fl{\scriptstyle{\cong}}}{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C).$$
In the case ${\bar{\mathcal{A}}}={\mathcal{A}}\textrm{ or }{\widetilde{\mathcal{A}}}_R$, it is [@AA2 Theorem 3]. We recall briefly their argument in order to add the consideration of the winding relations when ${\bar{\mathcal{A}}}={\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R$.
The fact that the images by $\chi_S$ of the link relations map to 0 in ${\bar{\mathcal{A}}}(X\cup{\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,}_S,*_C)$ follows from the STU relation. For the winding relations, it follows from the [Hol$_\textrm{w}$]{} relation applied at each univalent vertex glued on the $s$–labelled component, where $s$ is the label involved in the relation. Now, take two diagrams in ${\bar{\mathcal{A}}}(X\cup{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_S,*_C)$ that are identified when closing an $S$–labelled component. We have to consider the two situations depicted in Figures \[figinverselinkrel\] and \[figinversewindingrel\], where the gray zone represents a hidden part of the diagram.
\[scale=0.4\]
(3,0) – (3,5); in [1,2,3,4]{} [(0,) – (3,);]{} (0,2.5) circle (1 and 2.5);
(5,2.5) node [$-$]{};
\[xshift=8cm\] (3,0) – (3,5); in [1,2,3]{} [(0,) – (3,);]{} (0.5,4) .. controls +(1.5,0) and +(-1.5,0) .. (3,0.3); (0,2.5) circle (1 and 2.5);
(13,2.5) node [$=$]{};
\[xshift=16cm\] (3,0) – (3,5); in [1,2,3]{} [(0,) – (3,);]{} (0.5,4) .. controls +(1,0) and +(0,0.5) .. (2,3); (0,2.5) circle (1 and 2.5);
(21,2.5) node [$+$]{};
\[xshift=24cm\] (3,0) – (3,5); in [1,2,3]{} [(0,) – (3,);]{} (0.5,4) .. controls +(1.3,0) and +(0,0.6) .. (2,2); (0,2.5) circle (1 and 2.5);
(29,2.5) node [$+$]{};
\[xshift=32cm\] (3,0) – (3,5); in [1,2,3]{} [(0,) – (3,);]{} (0.5,4) .. controls +(1.5,0) and +(0,0.8) .. (2,1); (0,2.5) circle (1 and 2.5);
Equalities are obtained by applying STU relations in the first case and [Hol$_\textrm{w}$]{} relations in the second case.
\[scale=0.35\]
(4,0) – (4,8); in [2,4,6]{} [(0,) – (4,);]{} in [3,5,7]{} [(4,) node [$\scriptscriptstyle{\bullet}$]{};]{} (4.7,3) node [$\scriptstyle{t^j}$]{} (4.7,5) node [$\scriptstyle{t^\ell}$]{} (4.7,7) node [$\scriptstyle{t^m}$]{}; (0,4) circle (1 and 3);
(7,4) node [$-$]{};
\[xshift=10cm\] (4,0) – (4,8); in [2,4,6]{} [(0,) – (4,);]{} in [1,3,5]{} [(4,) node [$\scriptscriptstyle{\bullet}$]{};]{} (4.7,3) node [$\scriptstyle{t^j}$]{} (4.7,5) node [$\scriptstyle{t^\ell}$]{} (4.7,1) node [$\scriptstyle{t^m}$]{}; (0,4) circle (1 and 3);
(17,4) node [$=$]{};
\[xshift=20cm\] (4,0) – (4,8); in [2,4,6]{} [(0,) – (4,);]{} in [4,6]{} [(2.5,) – (2.45,);]{} (2.7,4.6) node [$\scriptstyle{t^j}$]{} (2.9,6.6) node [$\scriptstyle{t^{j+\ell}}$]{}; (0,4) circle (1 and 3);
(27,4) node [$-$]{};
\[xshift=30cm\] (4,0) – (4,8); in [2,4,6]{} [(0,) – (4,);]{} in [2,4]{} [(2.5,) – (2.45,);]{} (2.7,2.6) node [$\scriptstyle{t^m}$]{} (2.9,4.6) node [$\scriptstyle{t^{j+m}}$]{}; (0,4) circle (1 and 3);
Application of $\chi^{-1}$ to the right members provides linear combinations of the same sum with the skeleton component dropped and possibly trees glued. Using the IHX relation in the first case (resp. the Hol relation in the second case), we obtain link relations (resp. winding relations).
Product and coproduct
---------------------
We first define a coproduct on the diagram spaces of the previous subsection. Given a (beaded, winding) Jacobi diagram $D$ on $(X,C)$, denote by $\dddot{D}$ its graph part, and by $\dddot{D}_i$, $i\in I$, the connected components of $\dddot{D}$. Set $D_J=D\setminus(\sqcup_{i\in I\setminus J}\dddot{D}_i)$. In the winding case, multiply the labels of the concatenated edges of the skeleton. Define the coproduct of a diagram $D$ by $$\Delta(D)=\sum_{J\subset I}D_J\otimes D_{I\setminus J}.$$ Note that the different relations on Jacobi diagrams respect the coproduct. This provides a notion of [*group-like*]{} elements, [*i.e.*]{} elements $G$ such that $\Delta(G)=G\otimes G$.
Set ${\bar{\mathcal{A}}}={\mathcal{A}}\textrm{ or }{\widetilde{\mathcal{A}}}_R$. We will define a Hopf algebra structure on ${\bar{\mathcal{A}}}(*_C)$. Define the product of two diagrams as the disjoint union. The unit $\epsilon:{\mathbb{Q}}\to{\bar{\mathcal{A}}}(*_C)$ is defined by $\epsilon(1)=\varnothing$ and the counit $\varepsilon:{\bar{\mathcal{A}}}(*_C)\to{\mathbb{Q}}$ is given by $\varepsilon(D)=0$ if $D\neq\varnothing$ and $\varepsilon(\varnothing)=1$. The antipode is given by $D\mapsto(-1)^{|I|}D$. We finally have a structure of a graded Hopf algebra on ${\bar{\mathcal{A}}}(*_C)$, where the grading is given by the i–degree. It is known that an element in a graded Hopf algebra is group-like if and only if it is the exponential of a [*primitive*]{} element, [*i.e.*]{} an element $G$ such that $\Delta(G)=1\otimes G+G\otimes 1$. Here, the primitive elements are the series of connected diagrams.
The isomorphisms $\chi$ of the previous subsection are not algebra morphisms, but they preserve the coproduct. For the spaces ${\bar{\mathcal{A}}}(*_C)$ with ${\bar{\mathcal{A}}}={\mathcal{A}}\textrm{ or }{\widetilde{\mathcal{A}}}_R$, we have an exponential map associated with the product. For general (beaded, winding) Jacobi diagrams, we will use the notation ${\textrm{\textnormal{exp}}_\sqcup}$, namely exponential with respect to the disjoint union, for linear combination of diagrams with no univalent vertex embedded in the skeleton, where the disjoint union applies only to the graph part.
Formal Gaussian integration
---------------------------
This part aims at defining a formal Gaussian integration along $S$ on ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R(X,*_{C\cup S})$.
A (beaded, winding) Jacobi diagram on $(X,C\cup S)$ is [*substantial*]{} if it has no [*strut*]{}, [*i.e.*]{} no isolated dashed edge. It is [*$S$–substantial*]{} if it has no [*$S$–strut*]{}, [*i.e.*]{} no strut with both vertices labelled in $S$.
Given two (beaded, winding) Jacobi diagrams $D$ and $E$ on $(X,C\cup S)$, one of whose is $S$–substantial, define $\langle D,E\rangle_S$ as the sum of all diagrams obtained by gluing all $s$–colored vertices of $D$ with all $s$–colored vertices of $E$ for all $s\in S$ —if the numbers of $s$–colored vertices in $D$ and $E$ do not match for some $s\in S$, then $\langle D,E\rangle_S=0$. In the beaded and winding cases, we must precise the orientation and label of the created edges. Such an edge is the gluing of two or three edges in the initial diagrams. Fix arbitrarily the orientation of the new edge. Let $P(t)$ (resp. $Q(t)$) be the product of the labels of the initial edges whose orientation coincides (resp. does not coincide). Define the label of the new edge as $P(t)Q(t^{-1})$, see Figure \[figbracket\].
\[htb\] $$\left\langle\raisebox{-1.2cm}{
\begin{tikzpicture} [scale=0.9]
\begin{scope}
\draw[dashed,->] (0,1) -- (0,0.5) node[right] {$t^2$}; \draw[dashed] (0,0.5) -- (0,0);
\draw[dashed,->] (0,1) arc (-90:90:0.5); \draw[dashed] (0,2) arc (90:270:0.5);
\draw (0,0) node[below] {$s'$}; \draw (0,2) node[above] {$t$};
\end{scope}
\begin{scope} [xshift=1.3cm]
\draw[dashed,->] (0,0) -- (0,1) node[right] {$t$}; \draw[dashed] (0,1) -- (0,2);
\draw (0,0) node[below] {$s$}; \draw (0,2) node[above] {$s$};
\end{scope}
\end{tikzpicture}}
,\raisebox{-1.2cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,0) -- (0,4);
\draw[dashed] (0,1) -- (1,1) (1.75,0.5) -- (2.5,0) node[right] {$s'$} (1.75,1.5) -- (2.5,2) node[right] {$s$};
\draw[->,dashed] (1,1) -- (1.75,1.5) node[above] {$t^4$};
\draw[->,dashed] (1,1) -- (1.75,0.5) node[below] {$t$};
\draw[dashed,->] (0,3) -- (1.4,3);
\draw[dashed] (1.4,3) -- (2.3,3);
\draw (1.4,3.5) node {$t^2$};
\draw (2.7,3) node {$s$};
\end{tikzpicture}}
\right\rangle\raisebox{-2ex}{${}_{\{s,s'\}}$}\ =\raisebox{-1cm}{
\begin{tikzpicture}
\draw[->] (0,0) -- (0,2.4);
\begin{scope} [xscale=0.4,yscale=0.6]
\draw[dashed,->] (0,3) -- (1,2.5) (0,1) -- (2,2) -- (1,2.5) node[above right] {$t^3$};
\draw[dashed,->] (2,2) -- (3,2) (4,2) -- (3,2) node[below] {$t$};
\end{scope}
\draw[dashed,->] (2.5,1.2) arc (0:180:0.45) arc (180:360:0.45) node[right] {$t$};
\end{tikzpicture}}
\ +\ \raisebox{-1cm}{
\begin{tikzpicture}
\draw[->] (0,0) -- (0,2.4);
\begin{scope} [xscale=0.4,yscale=0.6]
\draw[dashed,->] (0,3) -- (1,2.5) (0,1) -- (2,2) -- (1,2.5) node[above right] {$t$};
\draw[dashed,->] (2,2) -- (3,2) (4,2) -- (3,2) node[below] {$t$};
\end{scope}
\draw[dashed,->] (2.5,1.2) arc (0:180:0.45) arc (180:360:0.45) node[right] {$t$};
\end{tikzpicture}}$$
We have the following immediate lemma.
\[lemmawdrel\] If $D'$ and $E'$ are obtained from $D$ and $E$ by applying the same winding relation on $s$–labelled vertices for some $s\in S$, then $\langle D,E\rangle_S=\langle D',E'\rangle_S$.
This bracketting defines a ${\mathbb{Q}}$–bilinear operator on the diagram spaces ${\bar{\mathcal{A}}}(X,*_{C\cup S})$ for ${\bar{\mathcal{A}}}={\mathcal{A}},\ {\widetilde{\mathcal{A}}}_R,\textrm{ or }{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R$.
\[thJMM\] Assume the $1$–manifold $X$ is a disjoint union of intervals. If $G$ and $H$ are group-like in ${\bar{\mathcal{A}}}(X,*_{C\cup S})$, then $\langle G,H\rangle_S$ is also group-like.
When $X=\varnothing$, it is [@JMM Theorem 2.4]. The case of a non-empty $X$ follows since the isomorphism $\chi_{\pi_0(X)}:{\bar{\mathcal{A}}}(*_{\pi_0(X)\cup C\cup S}){\fl{\scriptstyle{\cong}}}{\bar{\mathcal{A}}}(X,*_{C\cup S})$ preserves the coproduct and the bracketting $\langle\cdot,\cdot\rangle_S$.
If $W=(W_{ij}(t))_{i,j\in S}$ is an $(S,S)$–matrix with coefficients in ${\mathbb{Q}(t)}$, we also denote $W=\sum_{i,j\in S}$.
An element $G\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(X,*_{C\cup S})$ is [*Gaussian*]{} if $G={\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W(t))\sqcup H$ where $W(t)$ is an $(S,S)$–matrix with coefficients in ${\mathbb{Q}[t^{\pm1}]}$ and $H$ is $S$–substantial. If $\det(W(t))\neq0$, $G$ is [*non-degenerate*]{} and we set: $$\int_S G=\langle\,{\textrm{\textnormal{exp}}_\sqcup}(-\frac{1}{2}W^{-1}(t)),H\,\rangle_S\ \in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}(X,*_C).$$
\[lemmaFGI\] Let $G={\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W(t))\sqcup H$ be a non-degenerate Gaussian in ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(X,*_{C\cup S})$.
- If a non-degenerate Gaussian ${\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W(t))\sqcup H'$ is equal to $G$ in ${\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(X,*_C,{\textrm{\footnotesize\textcircled{\raisebox{-0.9ex}{\normalsize*}}}}_S)$, then $\int_S({\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W(t))\sqcup H')=\int_S G$.
- If $G'={\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W'(t))\sqcup H'$ is obtained from $G={\textrm{\textnormal{exp}}_\sqcup}(\frac{1}{2}W(t))\sqcup H$ by applying a winding relation, then $\int_S G'=\int_S G$.
The first point is essentially given by the proof of Bar-Natan and Lawrence [@BNL Proposition 2.2] in the non-beaded case. Here, multiplication by does not preserve the link relations, but the supplementary terms vanish thanks to the AS relation when applying $\langle\,\cdot\,,{\textrm{\textnormal{exp}}_\sqcup}(-\frac{1}{2}
\raisebox{-0.64cm}{
\begin{tikzpicture} [scale=0.4]
\draw[dashed,->] (0,0) -- (0,1) node[right] {$\scriptstyle{W_{ss}(t)}$}; \draw[dashed] (0,1) -- (0,2);
\draw (0.2,-0.2) node {$\scriptstyle{s}$}; \draw (0.2,2.1) node {$\scriptstyle{s}$};
\end{tikzpicture}})\rangle$.
The second point follows from Lemma \[lemmawdrel\].
Categories of diagrams
----------------------
For ${\bar{\mathcal{A}}}={\mathcal{A}},\ {\widetilde{\mathcal{A}}}_R,\textrm{ or }{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_R$, define a category ${\bar{\mathcal{A}}}$ whose objects are associative words in the letters $(+,-)$ and whose set of morphisms are ${\bar{\mathcal{A}}}(v,u)=\oplus_X{\bar{\mathcal{A}}}(X)$, where $X$ runs over all compact oriented 1–manifolds with boundary identified with the set of letters of $u$ and $v$, with the following sign convention: for $u$, a $+$ when the orientation of $X$ goes towards the boundary point and a $-$ when it goes backward, and the converse for $v$. Composition is given by vertical juxtaposition, where the label of the created edges in the case of beaded or winding diagrams is defined with the same rule as in the definition of $\langle D,E\rangle$. The tensor product given by disjoint union defines a strict monoidal structure on ${\bar{\mathcal{A}}}$.
We finally define the target category of our extended Kricker invariant.
Given a positive integer $g$ and a symbol $\natural$, set $\lfloor g\rceil^\natural=\{1^\natural,\dots,g^\natural\}$. Set $\lfloor 0\rceil^\natural=\varnothing$.
Fix non-negative integers $f$ and $g$. An $R$–beaded Jacobi diagram on $(\varnothing,\lfloor g\rceil^+\cup\lfloor f\rceil^-)$ is [*top–substantial*]{} if it is $\lfloor g\rceil^+$–substantial.
Given two such diagrams $D$ and $E$, define their composition $D\circ E$ as the sum of all ways of gluing all $i^+$–labelled vertices of $D$ with all $i^-$–labelled vertices of $E$, fixing the orientations and labels of the created edges as in the definition of $\langle D,E\rangle_S$. We get a category ${{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}}$ whose objects are non-negative integers, with set of morphisms from $g$ to $f$ the subspace of ${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}({*_{\lfloor g\rceil^+\cup\lfloor f\rceil^-}})$ generated by top-substantial diagrams. The identity of $g$ is ${\textrm{\textnormal{exp}}_\sqcup}(\sum_{i=1}^g\raisebox{-2.3ex}{
\begin{tikzpicture} [scale=0.3]
\draw[dashed] (0,0) -- (0,2);
\draw (0,0) node[right] {$\scriptstyle{i^-}$} (0,2) node[right] {$\scriptstyle{i^+}$};
\end{tikzpicture}})$ —the sum is trivial if $g=0$. The tensor product defined by disjoint union of diagrams provides ${{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}}$ a strict monoidal structure.
Winding matrices {#secwinding}
================
In this section, we define winding matrices associated with tangles with disks and bottom-top tangles with paths and interpret them as equivariant linking matrices in the case of bottom-top tangles. They will be useful in expressions of our invariant and splitting formulas.
First definition
----------------
We first define winding matrices in tangles with disks. Let $(\gamma,k)$ be a tangle with disks $d_\ell$. Write $\gamma$ as the disjoint union of components $\gamma_i$ for $i=1,\dots,n$. Fix a diagram of $(\gamma,k)$ and a base point $\star_i$ for each closed component $\gamma_i$ far from the crossings and the disks. Define the associated [*winding*]{} $w(\gamma_i,\gamma_j)\in{\mathbb{Z}[t^{\pm1}]}$ of $\gamma_i$ and $\gamma_j$ in the following way. For a crossing $c$ between $\gamma_i$ and $\gamma_j$, denote $\varepsilon_{ij}(c)$ the algebraic intersection number of the union of the disks $d_\ell$ with the path that goes from $\star_i$, or the origin of $\gamma_i$, to $c$ along $\gamma_i$ and then from $c$ to $\star_j$, or the end-point of $\gamma_j$, along $\gamma_j$. If $i=j$, change component at the first occurence of $c$. Set $$w(\gamma_i,\gamma_j)=\left\lbrace\begin{array}{l l}
\displaystyle\frac{1}{2}\sum_c \textrm{sg}(c) t^{\varepsilon_{ij}(c)} & \textrm{ if } i\neq j \\ & \\
\displaystyle\frac{1}{2}\sum_c \textrm{sg}(c) (t^{\varepsilon_{ii}(c)}+t^{-\varepsilon_{ii}(c)}) & \textrm{ if } i=j
\end{array}\right.$$ where the sums are over all crossings between $\gamma_i$ and $\gamma_j$. Note that $w(\gamma_j,\gamma_i)(t)=w(\gamma_i,\gamma_j)(t^{-1})$. Now let $I$ and $J$ be two subsets of $\{1,\dots,n\}$ and denote by $\gamma_I$ and $\gamma_J$ the corresponding subtangles of $\gamma$. The [*winding matrix*]{} $W_{\gamma_I\gamma_J}$ associated with the fixed diagram and base points is the matrix whose coefficients are the windings $w(\gamma_i,\gamma_j)$ for $i\in I$ and $j\in J$ —denote it $W_{\gamma_I}$ when $I=J$. In this latter case, note that $W_{\gamma_I}$ is hermitian.
\[lemmawdmatrix1\] The winding matrix is invariant by isotopies which do not allow the base points to pass through the disks of the tangle. In particular, when $\gamma$ contains no closed components, it is an isotopy invariant.
First note that the winding matrix is preserved when a crossing passes through a disk $d_\ell$. It is also preserved when a base point of a closed component passes through a crossing since the algebraic intersection number of this component with the union of the disks $d_\ell$ is trivial. Hence it only remains to check invariance with respect to framed Reidemeister moves performed far from the base points and the disks, which is direct.
To completely understand the effect of an isotopy on the winding matrix $W_{\gamma_I,\gamma_J}$, we shall describe its modification when a base point passes through a disk of the tangle. Fix a closed component $\gamma_i$. Fix a diagram of $(\gamma,k)$ with the base point of $\gamma_i$ located “just before” a disk $d_\ell$ of the tangle, as shown in the first part of Figure \[figbasepoint\].
\[scale=0.5\]
(0.5,0) – (4,0) node\[right\] [$d_\ell$]{}; (3,2) .. controls +(0,-1) and +(1,1) .. (1,-2); (1.44,-1.5) – (1.36,-1.6) node\[right\] [$\gamma_i$]{}; (2.87,0.8) node [$\star_i$]{};
(7,0) node [$\rightsquigarrow$]{};
\[xshift=9cm\] (0.5,0) – (4,0) node\[right\] [$d_\ell$]{}; (3,2) .. controls +(0,-1) and +(1,1) .. (1,-2); (1.44,-1.5) – (1.36,-1.6) node\[right\] [$\gamma_i$]{}; (2.09,-0.8) node [$\star_i$]{};
Consider another diagram of $(\gamma,k)$ which differs from the previous one only by the position of the base point $\star_i$, which is as shown on the second part of Figure \[figbasepoint\]. Let $\varepsilon=\pm1$ give the sign of the intersection of $d_\ell$ and $\gamma_i$ which the base point passes through. It is easily seen that the winding matrix of the latter diagram is obtained from the winding matrix of the previous one by multiplication on the left by $T_i(t^{-\varepsilon})$ if $i\in I$ and on the right by $T_i(t^{\varepsilon})$ if $i\in J$, where $T_i(t)$ is the diagonal matrix whose diagonal coefficients are all 1 except a $t$ at the $i^{th}$ position.
We now define winding matrices for bottom-top tangles in ${\mathbb{Q}}$–cubes. Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube. Let $(([-1,1]^3,\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. Denote $(\eta,k)$ the associated tangle with disks. Fix a diagram of $([-1,1]^3,\Xi,\eta)$ and a base point $\star_i$ for each component $L_i$ of $L$. Define the [*winding matrix of $(B,K,\gamma)$*]{}, with coefficients in ${\mathbb{Q}(t)}$, as: $$W_\gamma=W_\eta-W_{\eta L}W_L^{-1}W_{L\eta}.$$ Note that $W_L(1)$ is the linking matrix of $L$, thus $W_L(1)\neq0$ since $B$ is a homology cube. Hence $W_L$ is invertible over ${\mathbb{Q}(t)}$.
The winding matrix $W_\gamma$ is an isotopy invariant of $(B,K,\gamma)$.
First, when the surgery presentation is fixed, the discussion of the previous subsection implies that the winding matrix does not depend on the choice of diagram and base points. Then independance with respect to Kirby moves is easily checked. We detail the less direct, which is the case when a surgery link component is added to a tangle component. Denote $\eta'$ the tangle obtained from $\eta$ by adding the surgery component $L_j$ to $\eta_i$. We have: $$W_{\eta'}=W_\eta+{}^tPW_{L\eta}+W_{\eta L}P+{}^tPW_LP \quad \textrm{and} \quad W_{L\eta'}=W_{L\eta}+W_LP,$$ where $P$ is the $|L|\times|\eta|$ matrix whose only non-trivial term is a 1 at the $(j,i)$ position.
Topological interpretation
--------------------------
Given a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube, or a tangle with disks defined from a bottom-top tangle with paths in $[-1,1]^3$ with a surgery link, close the components of the tangle, say $\gamma$, by line segments on the top and bottom surfaces to get a link $\bar{\gamma}$. This provides well-defined linking numbers for the tangle components. If there is no path or disk, the winding matrix is the linking matrix. It is clear when working in $[-1,1]^3$. For bottom-top tangles in ${\mathbb{Q}}$–cubes, apply the following easy fact, which can be proved by adapting the proof of Proposition \[propintertopo\].
\[fact3.18\] Let $L$ be an oriented framed link in $[-1,1]^3$ whose linking matrix ${\textrm{\textnormal{Lk}}}(L)$ is non-degenerate. Let $\xi$ and $\zeta$ be disjoint oriented knots in $[-1,1]^3\setminus L$ and denote $\xi',\zeta'$ the copies of $\xi,\zeta$ in the ${\mathbb{Q}}$–cube obtained from $[-1,1]^3$ by surgery on $L$. Then: $${\textrm{\textnormal{lk}}}(\xi',\zeta')={\textrm{\textnormal{lk}}}(\xi,\zeta)-{\textrm{\textnormal{Lk}}}(\xi,L).{\textrm{\textnormal{Lk}}}(L)^{-1}.{\textrm{\textnormal{Lk}}}(L,\zeta).$$
More generally, when there are paths or disks, the winding matrix evaluated at $t=1$ is the linking matrix. We shall give a similar interpretation for the winding matrix at a generic $t$.
Let $(B,K,\gamma)$ be a bottom-top tangle in a ${\mathbb{Q}}$–cube. Let $E$ be the exterior of $K$ in $B$ and let $\tilde{E}$ be its covering associated with the kernel of the map $\pi_1(E)\to{\mathbb{Z}}=\langle t\rangle$ which sends the positive meridians of the components of $K$ to $t$. The automorphism group of the covering is isomorphic to ${\mathbb{Z}}$; let $\tau$ be the generator associated with the action of the positive meridians. Let $\zeta$ be a knot in $\tilde{E}$ such that there are a rational 2–chain $\Sigma$ in $\tilde{E}$ and $P\in{\mathbb{Q}[t^{\pm1}]}$ which satisfy $\partial\Sigma=P(\tau)(\zeta)$. Let $\xi$ be another knot in $\tilde{E}$ such that the projections of $\zeta$ and $\xi$ in $E$ are disjoint. Define the [*equivariant linking number*]{} of $\zeta$ and $\xi$ as: $${\textrm{\textnormal{lk}}}_e(\zeta,\xi)=\frac{1}{P(t)}\sum_{j\in{\mathbb{Z}}}\langle\Sigma,\tau^j(\xi)\rangle\,t^j\ \in{\mathbb{Q}(t)}.$$ The equivariant linking number is well-defined since $H_2(\tilde{E},{\mathbb{Q}})=0$ (see for instance [@M4 Lemma 2.1]) and satisfies ${\textrm{\textnormal{lk}}}_e(\tau(\zeta),\xi)=t\,{\textrm{\textnormal{lk}}}_e(\zeta,\xi)$.
First consider a tangle with disks $(\gamma,k)$, with disks $d_\ell$ associated to the integer $k$, defined from a surgery presentation of a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube, so that the closure $\bar{\gamma}_i$ of each component $\gamma_i$ is well-defined. Fix a diagram of $(\gamma,k)$ and base points $\star_i$ of its components. For an interval component, choose the base point to be its origin. Set $d=\cup_{\ell=1}^kd_\ell$. Let $E$ be the exterior of $\partial d$ in $[-1,1]^3$ and let $\tilde{E}$ be the infinite cyclic covering defined above. Let $\tilde{E}_0\subset\tilde{E}$ be a copy of the exterior of $d$ in $[-1,1]^3$. Define the lift $\tilde{\gamma_i}$ of $\bar{\gamma}_i$ in $\tilde{E}$ by lifting $\star_i$ in $\tilde{E}_0$. Given two subtangles $\gamma_I$ and $\gamma_J$ of $\gamma$, define the [*equivariant linking matrix*]{} ${\textrm{\textnormal{Lk}}}_e(\tilde{\gamma}_I,\tilde{\gamma}_J)$ of their lifts with the equivariant linking numbers of the $\tilde{\gamma_i}$.
$W_{\gamma_I\gamma_J}={\textrm{\textnormal{Lk}}}_e(\tilde{\gamma}_I,\tilde{\gamma}_J)$
Set $\tilde{E}_\ell=\tau^\ell(\tilde{E}_0)$, where $\tau$ is the generator of the automorphism group of the covering $\tilde{E}$ which corresponds to the action of the positive meridians of the $\partial d_\ell$. Fix $i\in I$ and $j\in J$. Since $\bar{\gamma}_i$ is null-homotopic in $[-1,1]^3\setminus\partial d$, it bounds a disk $D$ immersed in $[-1,1]^3\setminus\partial d$. Let $\tilde{D}$ be the lift of $D$ obtained by lifting $\star_i$ in $\tilde{E}_0$. Set $\tilde{D}_\ell=\tilde{D}\cap\tilde{E}_\ell$ and let $D_\ell$ be the image of $\tilde{D}_\ell$ in $E$. Set $c_\ell=\partial D_\ell$ and $\tilde{c}_\ell=\partial\tilde{D}_\ell$. Similarly, define the $c'_\ell$ and $\tilde{c}'_\ell$ from $\bar{\gamma}_j$. Assume the $c'_\ell$ do not meet the $D_\ell$ along the disks of the tangle.
Thanks to Lemma \[lemmawdmatrix1\], we have $w(\gamma_i,\gamma_j)=\sum_{\ell,\ell'\in{\mathbb{Z}}}w(c_\ell,c'_{\ell'})t^{\ell-\ell'}$ for any choice of base points of the $c_\ell$ and $c'_{\ell'}$. Since these latter curves do not cross the disks of the tangle, we have $w(c_\ell,c'_{\ell'})={\textrm{\textnormal{lk}}}(c_\ell,c'_{\ell'})=\langle D_\ell,c'_{\ell'}\rangle$, thus $w(\gamma_i,\gamma_j)=\sum_{\ell,\ell'\in{\mathbb{Z}}}\langle D_\ell,c'_{\ell'}\rangle t^{\ell-\ell'}$. Lifting both $D_\ell$ and $c'_{\ell'}$ in $\tilde{E}_\ell$ does not change their algebraic intersection number, hence $$\begin{aligned}
w(\gamma_i,\gamma_j) & = & \sum_{\ell,\ell'\in{\mathbb{Z}}}\langle \tilde{D}_\ell,\tau^{\ell-\ell'}(\tilde{c}'_{\ell'})\rangle t^{\ell-\ell'} \\
& = & \sum_{\ell,\ell'\in{\mathbb{Z}}}\langle \tilde{D}_\ell,\tau^{\ell'}(\tilde{c}'_{\ell-\ell'})\rangle t^{\ell'} \\
& = & \sum_{\ell'\in{\mathbb{Z}}}\langle \tilde{D},\tau^{\ell'}(\tilde{\gamma_j})\rangle t^{\ell'} \\
& = & {\textrm{\textnormal{lk}}}_e(\tilde{\gamma}_i,\tilde{\gamma_j})
\end{aligned}$$ where the third equality holds since $\tau^{\ell'}(\tilde{c}'_{\ell-\ell'})=\tau^{\ell'}(\tilde{\gamma_j})\cap\tilde{E}_\ell$.
Now consider a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube $(B,K,\gamma)$. Since $\gamma$ is null-homotopic in $B\setminus K$, we have a well-defined [*equivariant linking matrix*]{} ${\textrm{\textnormal{Lk}}}_e(\tilde{\gamma})$. Here, all components are intervals, so we have a canonical choice of base points.
\[propintertopo\] Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube. Then $W_\gamma={\textrm{\textnormal{Lk}}}_e(\tilde{\gamma})$.
Let $(([-1,1]^3,\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. Fix a diagram of $([-1,1]^3,\Xi,\eta\cup L)$ and base points for the components of $L=\sqcup_{1\leq i\leq n}L_i$. Let $\tilde{E}$ be the infinite cyclic covering of the exterior of $\Xi$ in $[-1,1]^3$. Let $d$ be the disjoint union of disks in $[-1,1]^3$ bounded by ${\hat{\Xi}}$. Let $\tilde{L}=\sqcup_{1\leq i\leq n}\tilde{L}_i$, $\tilde{\gamma}$ and $\tilde{\eta}$ be the lifts of $L$, $\gamma$ and $\eta$ in $\tilde{E}$ with all base points in the same copy in $\tilde{E}$ of the exterior of $d$ in $[-1,1]^3$. We have to prove that: $${\textrm{\textnormal{Lk}}}_e(\tilde{\gamma})={\textrm{\textnormal{Lk}}}_e(\tilde{\eta})-{\textrm{\textnormal{Lk}}}_e(\tilde{\eta},\tilde{L}){\textrm{\textnormal{Lk}}}_e(\tilde{L})^{-1}{\textrm{\textnormal{Lk}}}_e(\tilde{L},\tilde{\eta}).$$ The infinite cyclic covering $\tilde{E}'$ of the exterior of $K$ in $B$ is obtained from $\tilde{E}$ by surgery on $\cup_{\ell\in{\mathbb{Z}}}\tau^\ell(\tilde{L})$. Let $\tilde{\eta}_x$ and $\tilde{\eta}_y$ be components of $\tilde{\eta}$, and let $\tilde{\gamma}_x$ and $\tilde{\gamma}_y$ be the corresponding components of $\tilde{\gamma}$. For any knot $\lambda$ in $\tilde{E}$ or $\tilde{E}'$, denote $m(\lambda)$ a positive meridian. For $1\leq i\leq n$, let $c_i$ be the parallel of $\tilde{L}_i$ which bounds a disk after surgery. In the group $H_1\left(\tilde{E}\setminus\cup_{\ell\in{\mathbb{Z}}}\left(\tau^\ell(\tilde{L})\cup\tau^\ell(\tilde{\eta}_y)\right);{\mathbb{Z}}\right)$, we have $$\tilde{\eta}_x={\textrm{\textnormal{lk}}}_e(\tilde{\eta}_x,\tilde{\eta}_y)m(\tilde{\eta}_y)+\sum_{i=1}^n{\textrm{\textnormal{lk}}}_e(\tilde{\eta}_x,\tilde{L}_i)m(\tilde{L}_i)$$ and $$c_i={\textrm{\textnormal{lk}}}_e(\tilde{L}_i,\tilde{\eta}_y)m(\tilde{\eta}_y)+\sum_{j=1}^n{\textrm{\textnormal{lk}}}_e(\tilde{L}_i,\tilde{L}_j)m(\tilde{L}_j),$$ where multiplication by $t$ is given by the action of $\tau$ in homology. In $H_1(\tilde{E}'\setminus\cup_{\ell\in{\mathbb{Z}}}\tau^\ell(\tilde{\gamma}_y);{\mathbb{Z}})$, this gives $\tilde{\gamma}_x={\textrm{\textnormal{lk}}}_e(\tilde{\eta}_x,\tilde{\eta}_y)m(\tilde{\gamma}_y)-{\textrm{\textnormal{Lk}}}_e(\tilde{\eta}_x,\tilde{L}){\textrm{\textnormal{Lk}}}_e(\tilde{L})^{-1}{\textrm{\textnormal{Lk}}}_e(\tilde{L},\tilde{\eta}_y)m(\tilde{\gamma}_y).$
It is easily checked that the null LP–surgery defined in the introduction provides a move on the set of bottom-top tangles with paths in ${\mathbb{Q}}$–cubes, which corresponds to the move of null LP–surgery on Lagrangian cobordisms defined at the end of Subsection \[subseccob\] [*via*]{} the map $D$ of Proposition \[propmapD\]. Moreover:
\[propinvariancewind\] The winding matrix of a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube is invariant under null LP–surgeries.
Let $(B,K,\gamma)$ be a bottom-top tangle with paths in a ${\mathbb{Q}}$–cube associated with a Lagrangian cobordism with paths $(M,K)$. Let ${\mathbf{C}}$ be a null LP–surgery on $(M,K)$ made of a single ${\mathbb{Q}}$SK–pair. Let $(M',K')$ be the Lagrangian cobordism with paths obtained by surgery and let $(B',K',\gamma')$ be the associated bottom-top tangle with paths in a ${\mathbb{Q}}$–cube. Let $\tilde{E}$ be the infinite cyclic covering of the exterior of $K$ in $B$. The nullity condition implies that the preimage of the ${\mathbb{Q}}$–handlebody $C$ is the disjoint union of ${\mathbb{Q}}$–handlebodies $C_\ell$ homeomorphic to $C$. The infinite cyclic covering $\tilde{E}'$ of the exterior of $K'$ in $B'$ is obtained from $\tilde{E}$ by null LP–surgeries on all the $C_\ell$. This concludes since LP–surgeries preserve linking numbers (see for instance [@M3 Lemma 2.1]).
Construction of an LMO functor on ${\widetilde{\mathcal{LC}ob}}$ {#secfunctor}
================================================================
The functor ${Z^\bullet}:{\widetilde{\mathcal{T}}_q}\to{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}$
----------------------------------------------------------------------------------------------------------------------------------
The definition of the functor ${Z^\bullet}:{\widetilde{\mathcal{T}}_q}\to{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}$ is based on the functor $Z:{\mathcal{T}_q}\to{\mathcal{A}}$ of [@CHM], which is a renormalization of the Le–Murakami functor [@LM1; @LM2]. We recall in Figure \[figfunctorZ\] the definition of $Z$ on the elementary $q$–tangles, where $\nu\in{\mathcal{A}}({\raisebox{-0.2ex}{\begin{tikzpicture} \draw (0,0) circle (0.16); \draw[->] (0.16,0.03) -- (0.16,0.04);\end{tikzpicture}}\,})\cong{\mathcal{A}}({\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[->] (0,0) -- (0,0.4);\end{tikzpicture}}\,})$ is the value of the Kontsevich integral on the zero framed unknot, $\Phi\in{\mathcal{A}}({\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,})$ is a Drinfeld associator with rational coefficients and $\Delta^{+++}_{u_1,u_2,u_3}:{\mathcal{A}}({\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}{\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,})\to{\mathcal{A}}({\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_{u_1u_2u_3})$ is obtained by applying $(|u_i|-1)$ times $\Delta$ on the $i$-th factor.
$$Z\left(\hspace{-1ex}\raisebox{-0.8cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,1) node[above] {$\scriptstyle{(+}$} -- (1,0) node[below] {$\scriptstyle{+)}$};
{\draw[white,line width=5pt] [->] (1,1) node[above] {$\scriptstyle{+)}$} -- (0,0) node[below] {$\scriptstyle{(+}$}; \draw [->] (1,1) node[above] {$\scriptstyle{+)}$} -- (0,0) node[below] {$\scriptstyle{(+}$};}
\end{tikzpicture}}
\right)=\exp\left(\frac{1}{2}\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,2) -- (0,1) -- (1,0);
\draw[->] (1,2) -- (1,1) -- (0,0);
\draw[dashed] (0,1.5) -- (1,1.5);
\end{tikzpicture}}\,\right)\in{\mathcal{A}}\left(\hspace{-0.7ex}\raisebox{-0.2cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,1) -- (1,0);
\draw[->] (1,1) -- (0,0);
\end{tikzpicture}}\right)
\qquad
Z\left(\hspace{-1ex}\raisebox{-0.8cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (1,1) node[above] {$\scriptstyle{+)}$} -- (0,0) node[below] {$\scriptstyle{(+}$};
{\draw[white,line width=5pt] [->] (0,1) node[above] {$\scriptstyle{(+}$} -- (1,0) node[below] {$\scriptstyle{+)}$}; \draw [->] (0,1) node[above] {$\scriptstyle{(+}$} -- (1,0) node[below] {$\scriptstyle{+)}$};}
\end{tikzpicture}}
\right)=\exp\left(-\frac{1}{2}\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,2) -- (0,1) -- (1,0);
\draw[->] (1,2) -- (1,1) -- (0,0);
\draw[dashed] (0,1.5) -- (1,1.5);
\end{tikzpicture}}\,\right)\in{\mathcal{A}}\left(\hspace{-0.7ex}\raisebox{-0.2cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,1) -- (1,0);
\draw[->] (1,1) -- (0,0);
\end{tikzpicture}}\right)$$ $$Z\left(\hspace{-1ex}\raisebox{-0.4cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (1,0) node[below] {$\scriptstyle{-)}$} .. controls +(0,1) and +(0,1) .. (0,0) node[below] {$\scriptstyle{(+}$};
\end{tikzpicture}}
\right)=\raisebox{-0.1cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (2,0) .. controls +(0,0.5) and +(0.5,0) .. (1.5,1) -- (0.5,1) .. controls +(-0.5,0) and +(0,0.5) .. (0,0);
\draw[fill=white] (1,1) circle (0.3);
\draw (1,1) node {$\nu$};
\end{tikzpicture}}\,\in{\mathcal{A}}\left(\hspace{-0.7ex}\raisebox{-0.05cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (1,0) .. controls +(0,1) and +(0,1) .. (0,0);
\end{tikzpicture}}\right)
\qquad
Z\left(\hspace{-1ex}\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,0) node[above] {$\scriptstyle{(+}$} .. controls +(0,-1) and +(0,-1) .. (1,0) node[above] {$\scriptstyle{-)}$};
\end{tikzpicture}}
\right)=\raisebox{-0.3cm}{
\begin{tikzpicture} [scale=0.7]
\draw[->] (0,0) .. controls +(0,-1) and +(0,-1) .. (1,0);
\end{tikzpicture}}\,\in{\mathcal{A}}\left(\hspace{-0.7ex}\raisebox{-0.25cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,0) .. controls +(0,-1) and +(0,-1) .. (1,0);
\end{tikzpicture}}\right)$$ $$Z\left(\hspace{-1ex}\raisebox{-0.7cm}{
\begin{tikzpicture} [scale=0.6]
\draw[->] (0,1) node[above] {$\scriptstyle{(u}$} -- (0,0) node[below] {$\scriptstyle{((u}\ $};
\draw[->] (1.5,1) node[above] {$\scriptstyle{(v}\,$} -- (0.5,0) node[below] {$\,\scriptstyle{v)}$};
\draw[->] (2,1) node[above] {$\ \scriptstyle{w))}$} -- (2,0) node[below] {$\scriptstyle{w)}$};
\end{tikzpicture}}\right)=\Delta^{+++}_{u,v,w}(\Phi)\,\in{\mathcal{A}}({\,\raisebox{-0.5ex}{\begin{tikzpicture} \draw[<-] (0,0) -- (0,0.4);\end{tikzpicture}}\,}_{uvw})$$
Let $(\gamma,k)$ be a $q$–tangle with disks. Assume $\gamma$ is transverse to $[-1,1]^2\times\{h_i(k)\}$ for all $i\in\{1,\dots,k\}$, and write $\gamma$ as a composition of $q$–tangles $\gamma_i$ by cutting along these levels, see Figure \[figdecouptangle\].
\[xscale=1.5,yscale=0.9\] (0,0) – (2,0) – (2,4) – (0,4) – (0,0); in [1,2,3]{} [(0,) – (1,); (1,) – (2,); (2,) node\[right\] [$(v_\x)(w_\x)$]{};]{} in [0,1,2,3]{} [(1,+0.5) node [$\gamma_\x$]{};]{}
Write the bottom word of $\gamma_i$ as $w_b(\gamma_i)=(v_i)(w_i)$, where $w_i$ corresponds to the part of the tangle which meets the disk $d_i$. Set: $${Z^\bullet}(\gamma,k)=Z(\gamma_0)\circ(I_{v_1}\otimes G_{w_1})\circ Z(\gamma_1)\circ\dots\circ(I_{v_k}\otimes G_{w_k})\circ Z(\gamma_k)\ \in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\gamma),$$ where $I_v$ is the identity on the word $v$ and $G_v$ is obtained from $I_v$ by adding a label $t$ (resp. $t^{-1}$) on skeleton components associated with a $-$ sign (resp. a $+$ sign), see Figure \[figIvGv\].
\[xscale=0.6,yscale=0.5\] (0,1) node [$I_{\scriptscriptstyle--+-}=$]{}; (2,0) – (2,2); (3,0) – (3,2); (4,0) – (4,2); (5,0) – (5,2);
\[xshift=10cm\] (0,1) node [$G_{\scriptscriptstyle--+-}=$]{}; (2,0) – (2,2); (3,0) – (3,2); (4,0) – (4,2); (5,0) – (5,2); in [2,3,4,5]{} [(,1) node [$\scriptscriptstyle{\bullet}$]{};]{} in [2,3,5]{} [(,1) node\[right\] [$\scriptstyle{t}$]{};]{} (3.9,1.1) node\[right\] [$\scriptstyle{t^{-1}}$]{};
At the level of objects, ${Z^\bullet}$ forgets the parentheses. Invariance with respect to isotopy and to the cutting of $\gamma$ is due to invariance of the functor $Z$ and the following observation of Kricker [@Kri Lemma 3.2.4].
For a winding Jacobi diagram $D\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(w,v)$, we have $G_v\circ D=D\circ G_w$.
Apply the relations Hol and [Hol$_\textrm{w}$]{} at all vertices of the diagram.
Furthermore, ${Z^\bullet}$ is a clearly a functor and it preserves the tensor product on ${\mathcal{T}_q}\otimes{\widetilde{\mathcal{T}}_q}$ since $Z$ is tensor-preserving.
\[lemmagrouplike\] For any $q$–tangle with disks $(\gamma,k)$, ${Z^\bullet}(\gamma,k)$ is group-like.
The fact that $Z(\gamma)$ is group-like for a $q$–tangle $\gamma$ follows from [@LM3 Theorem 5.1]. This concludes since the $G_v$ are obviously group-like and the coproduct commutes with the composition.
The functor $Z:{\widetilde{\mathcal{T}}_q\mathcal{C}ub}\to{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}$
------------------------------------------------------------------------------------------------------------------------------
The next step is to evaluate ${Z^\bullet}$ on the surgery presentation of a $q$–tangle with paths in a ${\mathbb{Q}}$–cube. Let $(B,K,\gamma)\in{\widetilde{\mathcal{T}}_q\mathcal{C}ub}(w,v)$. Let $(([-1,1]^3,\Xi,\eta),L)$ be a surgery presentation of $(B,K,\gamma)$. The trivial link $\hat{\Xi}$ is the union of the boundaries of the disks $d_i=[0,1]\times[-1,1]\times\{h_i(k)\}$, where $k$ is the number of components of $\Xi$. Hence we have a $q$–tangle with disks $(\eta\cup L,k)$ and ${Z^\bullet}(\eta\cup L,k)\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\eta\cup L)$. Set: $${Z^\circ}((\Xi,\eta),L)=\chi_{\pi_0(L)}^{-1}(\nu^{\otimes\pi_0(L)}\sharp_{\pi_0(L)}{Z^\bullet}(\eta\cup L,k))\ \in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\eta,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_{\pi_0(L)})$$ where the connected sum means that a copy of $\nu$ is summed to each component of $L$. Note that ${Z^\circ}((\Xi,\eta),L)$ is group-like since ${Z^\bullet}(\eta\cup L,k)$ and $\nu$ are group-like and $\chi_{\pi_0(L)}$ preserves the coproduct.
We want to apply formal Gaussian integration to ${Z^\circ}((\Xi,\eta),L)$. We work with a lift $\overline{{Z^\circ}((\Xi,\eta),L)}\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\eta,*_{\pi_0(L)})$. Fix a diagram of the $q$–tangle with disks $(\eta\cup L,k)$ transverse to the levels $\{h_i(k)\}$, and fix base points $\star_i$ on each component $L_i$ of $L$. Construct $\overline{{Z^\circ}((\Xi,\eta),L)}$ following the construction from the beginning of Section \[secfunctor\] for this diagram, with the skeleton components corresponding to the components of $L$ defined as intervals by cutting each component $L_i$ at the base point $\star_i$.
\[lemmawdmatrix\] The lift $\overline{{Z^\circ}((\Xi,\eta),L)}$ is group-like, and we have: $$\overline{{Z^\circ}((\Xi,\eta),L)}={\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_L\right)\sqcup H,$$ where $W_L$ is the winding matrix associated with our choice of diagram and base points and $H$ is $\pi_0(L)$–substantial.
Check as in Lemma \[lemmagrouplike\] that $\overline{{Z^\circ}((\Xi,\eta),L)}$ is group-like. We have to compute the part of $\overline{{Z^\circ}((\Xi,\eta),L)}$ made of $\pi_0(L)$–struts. We work with $\chi_{\pi_0(\eta)}^{-1}(\overline{{Z^\circ}((\Xi,\eta),L)})\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}[t^{\pm1}]}}(*_{\pi_0(\eta)\cup\pi_0(L)})$, which is also group-like, in order to have a Hopf algebra structure on our diagram space. In particular, the group-like property implies that $\chi_{\pi_0(\eta)}^{-1}(\overline{{Z^\circ}((\Xi,\eta),L)})$ is the exponential of a series of connected diagrams. Since $\nu$ and the associator $\Phi$ have no terms with exactly two vertices, the only contributions to the $\pi_0(L)$–struts part come from the crossings between components of $L$. For $i\neq j$, the definition of $Z$ and the [Hol$_\textrm{w}$]{} relation show that the contribution of a crossing $c$ between $L_i$ and $L_j$ is $\chi_{\pi_0(\eta)}^{-1}\raisebox{-0.8ex}{{\textrm{\Huge(}}}\frac{1}{2}\textrm{sg}(c)
\raisebox{-1cm}{
\begin{tikzpicture} [xscale=0.6,yscale=0.5]
\draw[->] (0,0) -- (0,2); \draw (0,0) node[below] {$\scriptstyle{L_i}$}; \draw[->] (2,0) -- (2,2); \draw (2,0) node[below] {$\scriptstyle{L_j}$};
\draw[dashed,->] (0,1) -- (1,1); \draw[dashed] (1,1) -- (2,1); \draw (1,1) node[above] {$\scriptstyle{t^{\varepsilon_{ij}(c)}}$};
\end{tikzpicture}}$. Hence the contribution of all crossings between $L_i$ and $L_j$ is . For $i=j$, the contribution of a self-crossing of $L_i$ is: $$\chi_{\pi_0(\eta)}^{-1}\left(\frac{1}{2}\textrm{sg}(c)
\raisebox{-0.9cm}{
\begin{tikzpicture} [xscale=0.7,yscale=0.7]
\draw[->] (0,0) -- (0,2); \draw (0,0) node[below] {$\scriptstyle{L_i}$};
\draw[dashed,->] (0,0.5) arc (-90:0:0.5); \draw[dashed] (0.5,1) arc (0:90:0.5); \draw (0.5,1) node[right] {$\scriptstyle{t^{\varepsilon_{ii}(c)}}$};
\end{tikzpicture}}\right)
=\textrm{sg}(c)\left(\raisebox{-0.9cm}{
\begin{tikzpicture} [xscale=0.7,yscale=0.5]
\draw[dashed,->] (0,0) -- (0,1); \draw[dashed] (0,1) -- (0,2);
\draw (0,0) node[below] {$\scriptstyle{L_i}$}; \draw (0,2) node[above] {$\scriptstyle{L_i}$}; \draw (0,1) node[right] {$\scriptstyle{t^{\varepsilon_{ii}(c)}}$};
\end{tikzpicture}}
+\frac{1}{2}\raisebox{-0.9cm}{
\begin{tikzpicture} [xscale=0.5,yscale=0.5]
\draw[dashed] (0,0) -- (0,1); \draw[dashed,->] (0,1) arc (-90:90:0.5); \draw[dashed] (0,2) arc (90:270:0.5);
\draw (0,0) node[below] {$\scriptstyle{L_i}$}; \draw (0,2) node[above] {$\scriptstyle{t^{\varepsilon_{ii}(c)}}$};
\end{tikzpicture}}\right).$$ Summed over all self-crossings of $L_i$, we get as strut part: $$\sum_c\frac{1}{2}\textrm{sg}(c)\raisebox{-0.9cm}{
\begin{tikzpicture} [xscale=0.7,yscale=0.5]
\draw[dashed,->] (0,0) -- (0,1); \draw[dashed] (0,1) -- (0,2);
\draw (0,0) node[below] {$\scriptstyle{L_i}$}; \draw (0,2) node[above] {$\scriptstyle{L_i}$}; \draw (0,1) node[right] {$\scriptstyle{t^{\varepsilon_{ii}(c)}}$};
\end{tikzpicture}}
=\frac{1}{2}\raisebox{-0.9cm}{
\begin{tikzpicture} [xscale=0.7,yscale=0.5]
\draw[dashed,->] (0,0) -- (0,1); \draw[dashed] (0,1) -- (0,2);
\draw (0,0) node[below] {$\scriptstyle{L_i}$}; \draw (0,2) node[above] {$\scriptstyle{L_i}$}; \draw (0,1) node[right] {$\scriptstyle{(W_L)_{ii}}$};
\end{tikzpicture}}.$$ Hence $\chi_{\pi_0(\eta)}^{-1}(\overline{{Z^\circ}((\Xi,\eta),L)})={\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_L\right)\sqcup H'$ where $H'\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}[t^{\pm1}]}}(*_{\pi_0(\eta)\cup\pi_0(L)})$ is $\pi_0(L)$–substantial. Set $H=\chi_{\pi_0(\eta)}(H')$.
The matrix $W_L(1)$ is the linking matrix of the link $L$, hence it is the presentation matrix of the first homology group of a ${\mathbb{Q}}$–cube. Thus $\det(W_L(1))\neq0$ and $\overline{{Z^\circ}((\Xi,\eta),L)}$ is a non-degenerate Gaussian. Lemma \[lemmaFGI\] implies:
The formal Gaussian integral $\int_{\pi_0(L)}\overline{{Z^\circ}((\Xi,\eta),L)}$ does not depend on the lift $\overline{{Z^\circ}((\Xi,\eta),L)}\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\eta,*_{\pi_0(L)})$ of ${Z^\circ}((\Xi,\eta),L)\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}[t^{\pm1}]}}(\eta,{\textrm{\textcircled{\scriptsize\raisebox{0.2ex}{\textcircled{\raisebox{-1.3ex}{\normalsize*}}}}}}_{\pi_0(L)})$.
This allows to set: $$\int_{\pi_0(L)}{Z^\circ}((\Xi,\eta),L)=\int_{\pi_0(L)}\overline{{Z^\circ}((\Xi,\eta),L)}\quad\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}(\gamma).$$
Let $(B,K,\gamma)$ be a $q$–tangle with paths in a ${\mathbb{Q}}$–cube. Fix a surgery presentation $(([-1,1]^3,\Xi,\eta),L)$ of $(B,K,\gamma)$. Then: $$Z(B,K,\gamma)=U_+^{-\sigma_+(L)}\sqcup U_-^{-\sigma_-(L)}\sqcup \int_{\pi_0(L)}{Z^\circ}((\Xi,\eta),L)\quad\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}(\gamma),$$ where $U_{\pm}={Z^\circ}((\varnothing,\varnothing),\raisebox{-0.1cm}{
\begin{tikzpicture}
\draw[->] (0,0) arc (0:360:0.2) node[right] {$\scriptstyle{\pm1}$};
\end{tikzpicture}})$, defines a functor $Z:{\widetilde{\mathcal{T}}_q\mathcal{C}ub}\to{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}$ which preserves the tensor product on ${\mathcal{T}_q\mathcal{C}ub}\otimes{\widetilde{\mathcal{T}}_q\mathcal{C}ub}$.
We have to check that $Z(B,K,\gamma)$ does not depend on the surgery presentation. Independance with respect to the orientation of the components of $L$ follows from the argument of [@AA2 Proposition 3.1]. The normalization term $U_+^{-\sigma_+(L)}\sqcup U_-^{-\sigma_-(L)}$ ensures independance with respect to the KI move as usual. Independance with respect to the KII move mainly follows from [@GK Section 5.4]. More precisely, the argument of [@GK Theorem 4] adapts [@LMMO Proposition 1] to relate the values of $\overline{{Z^\circ}((\Xi,\eta),L)}$ for surgery links that differ from each other by a KII move. Then [@GK Lemma 5.6] shows that this implies the invariance of the formal Gaussian integral. As noted in [@AA2 Section 5.1], the argument remains valid when a surgery component is added to a tangle component since [@GK Lemma 5.6] uses integration along the surgery component.
Restricting the functor $Z:{\widetilde{\mathcal{T}}_q\mathcal{C}ub}\to{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}$ to $q$–tangles in ${\mathbb{Q}}$–cubes with no path, one recovers the functor $Z:{\mathcal{T}_q\mathcal{C}ub}\to{\mathcal{A}}$ of [@CHM Definition 3.16]. When $\gamma$ is a bottom-top tangle and $K=\varnothing$, $\chi_{\pi_0(\gamma)}^{-1}(Z(B,\varnothing,\gamma))$ is group-like and $\chi_{\pi_0(\gamma)}^{-1}(Z(B,\varnothing,\gamma))={\textrm{\textnormal{exp}}_\sqcup}({\textrm{\textnormal{Lk}}}(\gamma))\sqcup H$ for some substantial and group-like $H$ [@CHM Lemma 3.17]. We generalize this in the next result.
\[lemmagrouplikeZ\] For any bottom-top $q$–tangle with paths $(B,K,\gamma)$ where $B$ is a ${\mathbb{Q}}$–cube, $\chi_{\pi_0(\gamma)}^{-1}(Z(B,K,\gamma))$ is group-like and: $$\chi_{\pi_0(\gamma)}^{-1}(Z(B,K,\gamma))={\textrm{\textnormal{exp}}_\sqcup}(W_\gamma)\sqcup H\quad\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\pi_0(\gamma)}),$$ for some substantial and group-like $H$.
The fact that $Z(B,K,\gamma)$ is group-like follows from the same property for $U_+$, $U_-$ and $\overline{{Z^\circ}((\Xi,\eta),L)}$, and Theorem \[thJMM\]. It implies that $\chi_{\pi_0(\gamma)}^{-1}(Z(B,K,\gamma))$ is group-like since $\chi_{\pi_0(\gamma)}^{\phantom{-1}}$ preserves the coproduct. The same computation as in the proof of Lemma \[lemmawdmatrix\] gives: $$\chi_{\pi_0(\eta)}^{-1}\left(\overline{{Z^\circ}((\Xi,\eta),L)}\right)={\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_L+\frac{1}{2}W_{\eta}+W_{L\eta}\right)\sqcup H',$$ where $H'$ is substantial —note that $\chi_{\pi_0(\gamma)}$ and $\chi_{\pi_0(\eta)}$ are essentially the same before and after surgery on $L$. Integrate along $\pi_0(L)$: $$\begin{aligned}
\chi_{\pi_0(\gamma)}^{-1}\left(\int_{\pi_0(L)}\overline{{Z^\circ}((\Xi,\eta),L)}\right)&&\\
&\hspace{-6cm}=&\hspace{-3cm}
\left\langle{\textrm{\textnormal{exp}}_\sqcup}\left(-\frac{1}{2}W_L^{-1}\right),{\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_{\eta}+W_{L\eta}\right)\sqcup H'\right\rangle_{\pi_0(L)}\\
&\hspace{-6cm}=&\hspace{-3cm}{\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_{\eta}-\frac{1}{2}{}^tW_{L\eta}(t^{-1})W_L^{-1}W_{L\eta}\right)\sqcup H\\
&\hspace{-6cm}=&\hspace{-3cm}{\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_{\gamma}\right)\sqcup H.
\end{aligned}$$
The functor ${\tilde{Z}}:{\widetilde{\mathcal{LC}ob}}_q\to{{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}}$
--------------------------------------------------------------------------------------------------------
In this section, we define a functor on Lagrangian $q$–cobordisms with paths by applying the invariant $Z$ on bottom-top $q$–tangles with paths in ${\mathbb{Q}}$–cubes. The invariant $Z$ is functorial on $q$–tangles but not on bottom-top $q$–tangles, due to the different composition laws. To deal with this, we introduce some specific elements ${\boldsymbol{\top}}_g\in{{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}(*_{\lfloor g\rceil^+\cup\lfloor g\rceil^-})}$ following [@CHM Sec. 4]. Set: $$\lambda(x,y;r)=\chi_{\{r\}}^{-1}\left(\left(\sum_{n\geq0}\frac 1{n!}\raisebox{-6ex}{
\begin{tikzpicture} [scale=0.35]
\draw[->] (0,0) node[below] {$\scriptstyle{r}$} -- (0,5);
\foreach \y in {1,3,4} {
\draw[dashed] (0,\y) -- (2,\y) node[right] {$\scriptstyle{x}$};}
\draw (1,2) node {$\vdots$};
\draw (3.8,2.5) node {$\left.\resizebox{0cm}{0.75cm}{\phantom{rien}}\right\rbrace \scriptstyle{n}$};
\end{tikzpicture}}\right)\circ\left(\sum_{n\geq0}\frac 1{n!}\raisebox{-6ex}{
\begin{tikzpicture} [scale=0.35]
\draw[->] (0,0) node[below] {$\scriptstyle{r}$} -- (0,5);
\foreach \y in {1,3,4} {
\draw[dashed] (0,\y) -- (2,\y) node[right] {$\scriptstyle{y}$};}
\draw (1,2) node {$\vdots$};
\draw (3.8,2.5) node {$\left.\resizebox{0cm}{0.75cm}{\phantom{rien}}\right\rbrace \scriptstyle{n}$};
\end{tikzpicture}}\right)\right)
\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\{x,y,r\}}),$$ $${\boldsymbol{\top}}(x^+,x^-)=U_+^{-1}\sqcup U_-^{-1}\sqcup\int_{\{r^+,r^-\}}\langle\lambda(1^-,x^-;r^-)\sqcup\lambda(x^+,1^+;r^+),\chi^{-1}(Z(T_1))\rangle_{\{1^+,1^-\}},$$ $${\boldsymbol{\top}}_g={\boldsymbol{\top}}(1^+,1^-)\sqcup\dots\sqcup{\boldsymbol{\top}}(g^+,g^-)\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}({*_{\lfloor g\rceil^+\cup\lfloor g\rceil^-}}),$$ where the bottom-top tangle $T_1$ is drawn in Figure \[figTg\]. As proven in [@CHM Lemma 4.9]:
\[lemmagrouplikeTg\] ${\boldsymbol{\top}}_g$ is a group-like element of ${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}({*_{\lfloor g\rceil^+\cup\lfloor g\rceil^-}})$ and ${\boldsymbol{\top}}_g=Id_g\sqcup H$ for some substantial and group-like $H$. In particular, ${\boldsymbol{\top}}_g$ is top-substantial and $\lfloor g\rceil^-$–substantial.
Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and denote $(B,K,\gamma)$ the associated bottom-top $q$–tangle with paths, of type $(g,f)$. We have $Z(B,K,\gamma)\in{\widetilde{\mathcal{A}}^\textrm{\textnormal{w}}}_{{\mathbb{Q}(t)}}(\gamma)\cong{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(\gamma)$ and we consider $\chi^{-1}(Z(B,K,\gamma))\in{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}({*_{\lfloor g\rceil^+\cup\lfloor f\rceil^-}})$. It may not be top-substantial, but since ${\boldsymbol{\top}}_g$ is $\lfloor g\rceil^-$–substantial, we can set: $${\tilde{Z}}(M,K)=\chi^{-1}(Z(B,K,\gamma))\circ{\boldsymbol{\top}}_g.$$ At the level of objects, ${\tilde{Z}}$ sends a word on its number of letters. Direct adaptation of the proof of [@CHM Lemma 4.10] implies that ${\tilde{Z}}$ preserves the composition and the next result follows, see [@CHM Theorem 4.13].
${\tilde{Z}}:{\widetilde{\mathcal{LC}ob}}_q\to{{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}}$ is a functor which preserves the tensor product on ${\mathcal{LC}ob}_q\otimes{\widetilde{\mathcal{LC}ob}}_q$.
Restricting the functor ${\tilde{Z}}:{\widetilde{\mathcal{LC}ob}}_q\to{{}^{ts}\hspace{-4pt}\widetilde{\mathcal{A}}}$ to Lagrangian $q$–cobordisms with no path, one recovers the functor ${\tilde{Z}}$ defined on ${\mathcal{LC}ob}_q$ in [@CHM Theorem 4.13].
Lemmas \[lemmagrouplikeZ\] and \[lemmagrouplikeTg\] imply:
Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and let $(B,K,\gamma)$ be the associated bottom-top $q$–tangle with paths. Then ${\tilde{Z}}(M,K)$ is group-like and ${\tilde{Z}}(M,K)={\textrm{\textnormal{exp}}_\sqcup}(W_\gamma)\sqcup H$ for some substantial and group-like $H$.
Application to ${\mathbb{Q}}$SK–pairs
-------------------------------------
Let $(S,\kappa)$ be a ${\mathbb{Q}}$SK–pair. Let $M$ be the ${\mathbb{Q}}$–cube obtained from $S$ by removing the interior of a ball $B^3$ disjoint from $\kappa$. Isotoping $\kappa$ in $M$ and fixing a boundary parametrization $m$ of $M$, one can view $\kappa$ as the knot ${\hat{K}}$ associated with a Lagrangian cobordism with one path $(M,K)$. Since the top and bottom words are empty, we get a Lagrangian $q$–cobordism with one path.
\[propinvariantQSK\] Let $(S,\kappa)$ be a ${\mathbb{Q}}$SK–pair. Define as above an associated Lagrangian $q$–cobordism with one path $(M,K)$. Then ${\tilde{Z}}(S,\kappa)={\tilde{Z}}(M,K)$ defines an invariant of ${\mathbb{Q}}$SK–pairs, which coincides with the Kricker invariant $Z^\mathrm{rat}$ for knots in ${\mathbb{Z}}$–spheres.
When associating a cobordism with one path with a ${\mathbb{Q}}$SK–pair, we make a choice in the way we isotope the knot to the closure of a path. Once we work with a surgery presentation of our cobordism, this choice corresponds to the sweeping move represented in Figure \[figsweepingmove\].
\[scale=0.3\] (0,0) – (10,0) – (10,10) – (0,10) – (0,0); in [1,7.5]{} [(1,) – (9,) – (9,+1.5) – (1,+1.5) – (1,);]{} (5,1.75) node [$T'$]{} (5,8.25) node [$T$]{}; in [2,3,4,7,8]{} [(,2.5) – (,7.5);]{} (2.5,5) node [$\scriptstyle{\dots}$]{} (7.5,3.5) node [$\scriptstyle{\dots}$]{} (7.5,6.5) node [$\scriptstyle{\dots}$]{}; (5,5) node [$\bullet$]{} – (10,5);
(4,1.5) node [$\longleftrightarrow$]{};
\[xshift=5cm,scale=0.3\] (0,0) – (10,0) – (10,10) – (0,10) – (0,0); (8.5,4) .. controls +(0,1) and +(0,-1) .. (4,5) .. controls +(0,1) and +(0,-1) .. (8.5,6); in [2,3,7,8]{} [[(,2.5) – (,7.5); (,2.5) – (,7.5);]{}]{} [(4,2.5) .. controls +(0,2) and +(0,-1) .. (8.5,4) (8.5,6) .. controls +(0,1) and +(0,-2) .. (4,7.5); (4,2.5) .. controls +(0,2) and +(0,-1) .. (8.5,4) (8.5,6) .. controls +(0,1) and +(0,-2) .. (4,7.5);]{} (2.5,5) node [$\scriptstyle{\dots}$]{} (7.5,3) node [$\scriptstyle{\dots}$]{} (7.5,7) node [$\scriptstyle{\dots}$]{}; in [1,7.5]{} [(1,) – (9,) – (9,+1.5) – (1,+1.5) – (1,);]{} (5,1.75) node [$T'$]{} (5,8.25) node [$T$]{}; (5,5) node [$\bullet$]{} – (10,5);
(9,1.5) node [$\sim$]{};
\[xshift=10cm,scale=0.3\] (0,0) – (10,0) – (10,10) – (0,10) – (0,0); (1.5,4) .. controls +(0,1) and +(0,-1) .. (4,5) .. controls +(0,1) and +(0,-1) .. (1.5,6); in [2,3,7,8]{} [[(,2.5) – (,7.5); (,2.5) – (,7.5);]{}]{} [(4,2.5) .. controls +(0,2) and +(0,-1) .. (1.5,4) (1.5,6) .. controls +(0,1) and +(0,-2) .. (4,7.5); (4,2.5) .. controls +(0,2) and +(0,-1) .. (1.5,4) (1.5,6) .. controls +(0,1) and +(0,-2) .. (4,7.5);]{} in [1,7.5]{} [(1,) – (9,) – (9,+1.5) – (1,+1.5) – (1,);]{} (5,1.75) node [$T'$]{} (5,8.25) node [$T$]{}; (2.5,5) node [$\scriptstyle{\dots}$]{} (7.5,3.5) node [$\scriptstyle{\dots}$]{} (7.5,6.5) node [$\scriptstyle{\dots}$]{}; (5,5) node [$\bullet$]{} – (10,5);
But the right hand side diagram of this figure shows this move is trivial —as noted in [@GK Lemma 3.26].
Coincidence with $Z^\mathrm{rat}$ is direct by construction.
The above proof does not work for a cobordism with more than one path, so we do not get an invariant of boundary links in ${\mathbb{Q}}$–spheres. One may obtain such an invariant by quotienting out the target space by suitable relations, see [@GK] for a construction of this kind.
Let $(S_1,\kappa_1)$ and $(S_2,\kappa_2)$ be ${\mathbb{Q}}$SK–pairs. The invariant ${\tilde{Z}}$ is given on their connected sum by: $${\tilde{Z}}((S_1,\kappa_1)\sharp(S_2,\kappa_2))={\tilde{Z}}(S_1,\kappa_1)\sqcup{\tilde{Z}}(S_2,\kappa_2).$$
As previously, associate Lagrangian $q$–cobordisms with one path $(M_1,K_1)$ and $(M_2,K_2)$ with $(S_1,\kappa_1)$ and $(S_2,\kappa_2)$ respectively. Construct a Lagrangian $q$–cobordism with one path $(M,K)$ associated with $(S,\kappa)=(S_1,\kappa_1)\sharp(S_2,\kappa_2)$ by stacking $(M_1,K_1)$ and $(M_2,K_2)$ together in the $y$ direction. Now $(M_1,K_1)$ and $(M_2,K_2)$ are obtained from the cube $[-1,1]^3$ with one disk by surgery on links $L_1$ and $L_2$ respectively. We obtain a surgery diagram for $(M,K)$ by drawing $L_1$ “in front” of $L_2$, or equivalently “around” $L_2$, see Figure \[figstack\]. The result follows from this latter diagram since there is no crossing between $L_1$ and $L_2$.
\[scale=0.5\]
(0,7.5) .. controls +(1,0) and +(0,1) .. (1,6) .. controls +(0,-2) and +(0,1) .. (-1,3); (0,6) .. controls +(-1,0) and +(0,1) .. (-2,4) .. controls +(0,-4) and +(0,-3) .. (1,3); [(0,7.5) .. controls +(-1,0) and +(0,1) .. (-1,6) .. controls +(0,-2) and +(0,1) .. (1,3); (0,7.5) .. controls +(-1,0) and +(0,1) .. (-1,6) .. controls +(0,-2) and +(0,1) .. (1,3);]{} [(0,6) .. controls +(1,0) and +(0,1) .. (2,4) .. controls +(0,-4) and +(0,-3) .. (-1,3); (0,6) .. controls +(1,0) and +(0,1) .. (2,4) .. controls +(0,-4) and +(0,-3) .. (-1,3);]{} [(0,3) arc (-90:90:2); (0,3) arc (-90:90:2);]{} [(0,2) arc (-90:90:3); (0,2) arc (-90:90:3);]{} [(0,8) .. controls +(-4,0) and +(-1,0) .. (-2.5,4.5); (0,8) .. controls +(-4,0) and +(-1,0) .. (-2.5,4.5);]{} [(0,3) .. controls +(-3,0) and +(1,0) .. (-2.5,5.5); (0,3) .. controls +(-3,0) and +(1,0) .. (-2.5,5.5);]{} [(0,2) .. controls +(-4,0) and +(-1,0) .. (-2.5,5.5); (0,2) .. controls +(-4,0) and +(-1,0) .. (-2.5,5.5);]{} [(0,7) .. controls +(-3,0) and +(1,0) .. (-2.5,4.5); (0,7) .. controls +(-3,0) and +(1,0) .. (-2.5,4.5);]{} (-4,0.5) rectangle (4,9.5); (0,5) node [$\bullet$]{} – (4,5); (2.24,7) – (2.14,7.1) node\[right\] [$L_1$]{}; (1.4,1.5) – (1.45,1.6) node\[right\] [$L_2$]{};
(7,5) node [$\sim$]{};
\[xshift=14cm\]
\[scale=0.6,yshift=3.3cm\] (0,7.5) .. controls +(1,0) and +(0,1) .. (1,6) .. controls +(0,-2) and +(0,1) .. (-1,3); (0,6) .. controls +(-1,0) and +(0,1) .. (-2,4) .. controls +(0,-4) and +(0,-3) .. (1,3); [(0,7.5) .. controls +(-1,0) and +(0,1) .. (-1,6) .. controls +(0,-2) and +(0,1) .. (1,3); (0,7.5) .. controls +(-1,0) and +(0,1) .. (-1,6) .. controls +(0,-2) and +(0,1) .. (1,3);]{} [(0,6) .. controls +(1,0) and +(0,1) .. (2,4) .. controls +(0,-4) and +(0,-3) .. (-1,3); (0,6) .. controls +(1,0) and +(0,1) .. (2,4) .. controls +(0,-4) and +(0,-3) .. (-1,3);]{}
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(-4,0.5) rectangle (4,9.5); (0,5) node [$\bullet$]{} – (4,5); (2.5,7.5) – (2.4,7.6) node\[right\] [$L_1$]{}; (1.2,4.2) – (1.2,4.3); (1.05,4.3) node\[right\] [$L_2$]{};
Splitting formulas {#secformules}
==================
We first mention useful lemmas, namely [@Mas Lemma 4.3] and [@Mas Lemma 4.4]. Recall the tensor $\mu({\mathbf{C}})$ was defined in the introduction.
\[lemmaboundpara\] For a ${\mathbb{Q}}$–handlebody $C$ of genus $g$, there exists a boundary parametrization $c:\partial C_0^g\to C$ such that $(C,c)\in{\mathcal{LC}ob}(g,0)$.
Let ${\mathbf{C}}=\left(\frac{C'}{C}\right)$ be an LP–pair of genus $g$. Take boundary parametrizations $c:\partial C_0^g\to C$ and $c':\partial C_0^g\to C'$ compatible with the fixed identification $\partial C\cong\partial C'$ such that $(C,c)\in{\mathcal{LC}ob}(g,0)$ and $(C',c')\in{\mathcal{LC}ob}(g,0)$. Then: $$\mu({\mathbf{C}})={\tilde{Z}}_1(C,c)-{\tilde{Z}}_1(C',c'),$$ where ${\tilde{Z}}_1$ is the i–degree 1 part of ${\tilde{Z}}$ and $\mu({\mathbf{C}})$ is considered as an element of ${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor g\rceil^+})$ [*via*]{} the inclusion $\Lambda^3 H_1({\mathcal{C}};{\mathbb{Q}})\hookrightarrow{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor g\rceil^+})$ defined by: $$[c_+(\beta_i)]\wedge[c_+(\beta_j)]\wedge[c_+(\beta_k)]\mapsto{
\raisebox{-0.5cm}{
\begin{tikzpicture} [scale=0.7]
\draw[dashed] (-1,1) node[above] {$k^+$} -- (0,0) -- (0,1.1) (0,1) node[above] {$j^+$} (0,0) -- (1,1) node[above] {$i^+$};
\end{tikzpicture}}}.$$
Let $(M,K)\in{\widetilde{\mathcal{LC}ob}}_q(w,v)$. Let ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ be a null LP–surgery on $(M,K)$. Let $e_i$ be the genus of $C_i$. For $1\leq i \leq n$, take boundary parametrizations $c_i:\partial C_0^{e_i}\to C_i$ and $c_i':\partial C_0^{e_i}\to C_i'$ compatible with the fixed identification $\partial C_i\cong\partial C_i'$ such that $(C_i,c_i)\in{\mathcal{LC}ob}(e_i,0)$ and $(C_i',c_i')\in{\mathcal{LC}ob}(e_i,0)$. Set $e=\sum_{i=1}^n e_i$. Take a collar neighborhood $m_-(F_f)\times[-1,\varepsilon-1]$ of the bottom surface $m_-(F_f)$. Take pairwise disjoint solid tubes $T_i$, $i=1,\dots,n$, such that $T_i$ connects $(c_i)_-(F_0)$ to a disk in $m_-(F_f)\times\{\varepsilon-1\}$ in the complement of the $C_j$’s, the collar neighborhood and $K$. This provides a decomposition of the cobordism $(M,K)$ as: $$(M,K)=((C_1,\varnothing)\otimes\dots\otimes(C_n,\varnothing)\otimes Id_f)\circ(N,J),$$ where $f$ is the number of letters of $v$ (see Figure \[figdecompositionCob\]). It is proved in [@Mas Section 4.4] that $N$ is a Lagrangian cobordism. The nullity condition on the surgery ensures that $\hat{J}$ is a boundary link. Thus $(N,J)$ is a Lagrangian cobordism with paths.
\[scale=0.25\]
in [8,30]{} [(,4) – (,19);]{} (30,4.5) – (8,4.5) – (0,0.5) (2.5,0.5) – (2.5,0) – (10.5,4) – (10.5,4.5);
(0,0) – (8,4) – (30,4) – (22,0) – (0,0);
\[xshift=4cm,yshift=0cm\] (11.5,2.5) arc (0:180:2); (11.5,2.5) circle (0.8 and 0.4) (7.5,2.5) circle (0.8 and 0.4); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5); (12.3,2.5) .. controls +(0,3) and +(0,3) .. (6.7,2.5); (10.7,2.5) .. controls +(0,1) and +(0,1) .. (8.3,2.5); (10.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (12.3,2.5) (6.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (8.3,2.5); (10.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (12.3,2.5) (6.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (8.3,2.5);
\[xshift=11cm,yshift=0cm\] (11.5,2.5) arc (0:180:2); (11.5,2.5) circle (0.8 and 0.4) (7.5,2.5) circle (0.8 and 0.4); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5); (12.3,2.5) .. controls +(0,3) and +(0,3) .. (6.7,2.5); (10.7,2.5) .. controls +(0,1) and +(0,1) .. (8.3,2.5); (10.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (12.3,2.5) (6.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (8.3,2.5); (10.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (12.3,2.5) (6.7,2.5) .. controls +(0,-0.5) and +(0,-0.5) .. (8.3,2.5);
in [5,10]{} [(30,+4) – (19,+4);]{} (11,5) .. controls +(2,1) and +(2,-1) .. (18,7) .. controls +(-2,1) and +(-2,1) .. (19,9); (11.9,12) – (11.8,11.9); (14,12) .. controls +(0,-2) and +(0,-2) .. (16,14) .. controls +(0,2) and +(0,2) .. (19,14); (14,5.72) – (13.9,5.7); (8,15) – (22,15);
(0,0) – (8,4) – (30,4) – (22,0) – (0,0);
\[xshift=-0.5cm,yshift=15cm\] (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5);
\[xshift=5.5cm,yshift=15cm\] (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5);
\[xshift=11.5cm,yshift=15cm\] (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (11.5,2) arc (0:180:2); (11.5,2) circle (0.5 and 0.25) (7.5,2) circle (0.5 and 0.25); (12,2) arc (0:180:2.5); (11,2) arc (0:180:1.5);
in [0,22]{} [(,0.5) – (,14.6); (,0) – (,15);]{}
\[xshift=9cm,yshift=8.7cm,scale=0.7\] (0,0) ..controls +(0,1) and +(-2,1) .. (4,2); (4,2) ..controls +(2,-1) and +(-2,-1) .. (8,2); (8,2) ..controls +(2,1) and +(-1.2,0) .. (12,1.3); (0,0) ..controls +(0,-1) and +(-2,-1) .. (4,-2); (4,-2) ..controls +(2,1) and +(-2,1) .. (8,-2); (8,-2) ..controls +(2,-1) and +(-1.2,0) .. (12,-1.3); (8,-2) ..controls +(2,-1) and +(-1.2,0) .. (12,-1.3); (14,2) ..controls +(2,1) and +(0,1) .. (18,0); (14,2) ..controls +(2,1) and +(0,1) .. (18,0); (14,-2) ..controls +(2,-1) and +(0,-1) .. (18,0); (12,1.3) ..controls +(0.5,0) and +(-1,-0.5) .. (14,2); (12,1.3) ..controls +(0.5,0) and +(-1,-0.5) .. (14,2); (12,-1.3) ..controls +(0.5,0) and +(-1,0.5) .. (14,-2); [ (2,0) ..controls +(0.5,-0.25) and +(-0.5,-0.25) .. (4,0) (2.3,-0.1) ..controls +(0.6,0.2) and +(-0.6,0.2) .. (3.7,-0.1)]{}; [ (2,0) ..controls +(0.5,-0.25) and +(-0.5,-0.25) .. (4,0) (2.3,-0.1) ..controls +(0.6,0.2) and +(-0.6,0.2) .. (3.7,-0.1)]{}; [ (2,0) ..controls +(0.5,-0.25) and +(-0.5,-0.25) .. (4,0) (2.3,-0.1) ..controls +(0.6,0.2) and +(-0.6,0.2) .. (3.7,-0.1)]{};
(11,5) – (22,5) – (30,9); (11,11) – (11,10) – (21,10); (11,10) .. controls +(0,2) and +(0,2) .. (14,12); (11,10) – (22,10) – (30,14); (5.7,2.5) circle (0.3 and 0.15); (5.7,2.5) .. controls +(0,2) and +(-1,-1) .. (9.7,7.7); (5.4,2.5) .. controls +(0,2) and +(-1,-1) .. (9.4,7.9); (6,2.5) .. controls +(0,2) and +(-1,-1) .. (9.95,7.45); (10,4.25) – (2.5,0.5); (10.5,4.5) – (2.5,0.5); (0.5,0.5) – (2,0.5) (3,0.5) – (6,0.5) (28,3.5) – (29.5,4.25); (0,0.5) – (22,0.5) – (30,4.5) (4,2.5) – (0,0.5); (-5,8) node [$(M,K)=$]{}; (33,11.9) node [$\left.\resizebox{0cm}{1.96cm}{\phantom{rien}}\right\rbrace (N,J)$]{}; (1,-1.5) node [$C$]{}; (12,-1.5) node [$Id_2$]{};
With the surgery ${\mathbf{C}}$ is associated the tensor $\mu({\mathbf{C}})\in{\mathcal{A}}_{\mathbb{Q}}(H_1({\mathcal{C}};{\mathbb{Q}}))$. Let $W$ be a square matrix of size $e$ with coefficients in ${\mathbb{Q}(t)}$. Interpret $W$ as a hermitian form on $H_1({\mathcal{C}};{\mathbb{Q}})$ written in the basis $(([(c_i)_+(\beta_j)])_{1\leq j\leq e_i})_{1\leq i\leq n}$. Given an $H_1({\mathcal{C}};{\mathbb{Q}})$–colored Jacobi diagram, one can glue some legs of the diagram with $W$, see Figure \[figglue\]. Changing the labels of the univalent vertices [*via*]{} the bijection $$[(c_i)_+(\beta_j)]\mapsto\sum_{\ell=1}^{i-1}e_\ell+j$$ onto $\{1,\dots,e\}$, this provides a diagram in ${\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor e\rceil^+})$.
The following result is a direct adaptation of [@Mas Section 4.4], with the winding matrices playing the role of the linking matrices.
Let $(M,K)$ be a Lagrangian $q$–cobordism with paths and let $(B,K,\gamma)$ be the associated bottom-top $q$–tangle with paths. Let ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ be a null LP–surgery on $(M,K)$. Define as above a decomposition of the cobordism $(M,K)$. Choose top and bottom words for $(N,J)$ and the $(C_i,\varnothing)$ in order to get a decomposition of the Lagrangian $q$–cobordism $(M,K)$ as $(M,K)=((C_1,\varnothing)\otimes\dots\otimes(C_n,\varnothing)\otimes Id_v)\circ(N,J)$. Let $(D,J,\varsigma)$ be the bottom-top $q$–tangle with paths associated with $(N,J)$. Let $\varsigma^{\mathbf{c}}$ be the subtangle of $\varsigma^-$ corresponding to the $C_i$’s. Let $e$ be the number of components of $\varsigma^{\mathbf{c}}$. Let $\tilde{\rho}_{\mathbf{c}}:{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor e\rceil^+})\to{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor g\rceil^+\cup\lfloor f\rceil^-})$ be the linear form which changes the labels of the univalent vertices as follows: $$\tilde{\rho}_{\mathbf{c}}(\ell^+)=\sum_{j=1}^g W_{\varsigma^{\mathbf{c}}}(\varsigma_j^+,\varsigma_\ell)\cdot j^+ +\sum_{i=1}^f W_{\varsigma^{\mathbf{c}}}(\varsigma_{e+i}^-,\varsigma_\ell)\cdot i^-.$$ Then: $$\sum_{I\subset\{1,\dots,n\}}(-1)^{|I|}{\tilde{Z}}\left((M,K)({\mathbf{C}}_I)\right)\equiv_n
{\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_\gamma\right)\sqcup
\tilde{\rho}_{\mathbf{c}}\left(\textrm{\begin{minipage}{4.5cm} \begin{center} sum of all ways of gluing some legs of $\mu({\mathbf{C}})$ with $W_{\varsigma^{\mathbf{c}}}/2$\end{center}\end{minipage}}\right),$$ where ${\mathbf{C}}_I=(({\mathbf{C}}_i)_{i\in I})$ and $\equiv_n$ means “equal up to i–degree at least $n+1$ terms”.
Note that $W_\gamma=W_{\varsigma\setminus\varsigma^{\mathbf{c}}}$ by Proposition \[propinvariancewind\].
For a cobordism with one path, the next result gives a more intrinsic version of these formulas, which does not refer to a decomposition of the cobordism. A similar result is given by in [@Mas Lemma 4.1] for a cobordism with no path.
Given a null LP–surgery ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ on a Lagrangian cobordism with paths $(M,K)$, define a hermitian form $\ell_{(M,K)}({\mathbf{C}}):H_1({\mathcal{C}};{\mathbb{Q}})\times H_1({\mathcal{C}};{\mathbb{Q}})\to{\mathbb{Q}(t)}$ in the same way as $\ell_{(S,\kappa)}({\mathbf{C}})$ was defined in the introduction. Also define a map $\rho_{\mathbf{c}}:{\mathcal{A}}_{\mathbb{Q}}(H_1({\mathcal{C}};{\mathbb{Q}}))\to{\widetilde{\mathcal{A}}}_{{\mathbb{Q}(t)}}(*_{\lfloor g\rceil^+\cup\lfloor f\rceil^-})$ which changes the labels of the univalent vertices by first sending them in $H_1(M;{\mathbb{Q}})$ [*via*]{} $H_1({\mathcal{C}};{\mathbb{Q}})\cong\oplus_{i=1}^nH_1(C_i;{\mathbb{Q}})\to H_1(M;{\mathbb{Q}})$, and then writing them in terms of the $[m_+(\beta_i)]$ and $[m_-(\alpha_i)]$.
A direct adaptation of (the end of) [@Mas Section 4.4] gives:
\[propformulas1\] Let $(M,K)\in{\widetilde{\mathcal{LC}ob}}_q(w,v)$ be a Lagrangian $q$–cobordism with one path. Let ${\mathbf{C}}=({\mathbf{C}}_1,\dots,{\mathbf{C}}_n)$ be a null LP–surgery on $(M,K)$. Let $(B,K,\gamma)$ be the bottom-top tangle with one path associated with $(M,K)$. Then: $$\sum_{I\subset\{1,\dots,n\}}(-1)^{|I|}{\tilde{Z}}\left((M,K)({\mathbf{C}}_I)\right)\equiv_n
{\textrm{\textnormal{exp}}_\sqcup}\left(\frac{1}{2}W_\gamma\right)\sqcup
\rho_{\mathbf{c}}\left(\textrm{\begin{minipage}{4.5cm} \begin{center} sum of all ways of gluing some legs of $\mu({\mathbf{C}})$ with $\ell_{(M,K)}({\mathbf{C}})/2$\end{center}\end{minipage}}\right).$$
Use Propositions \[propinvariantQSK\] and \[propformulas1\]. The strut part disappears since we deal with a cobordism from $F_0$ to $F_0$. The map $\rho_{\mathbf{c}}$ kills all terms with at least one univalent vertex since a Lagrangian cobordism from $F_0$ to $F_0$ has trivial first homology group over ${\mathbb{Q}}$.
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---
abstract: 'We discuss the Ping-Pong protocol which was proposed by Boström and Felbinger. We derive a simple trade-off inequality between distinguishability of messages for Eve and detectability of Eve for legitimate users. Our inequality holds for arbitrary initial states. That is, even if Eve prepares an initial state, she cannot distinguish messages without being detected. We show that the same inequality holds also on another protocol in which Alice and Bob use one-way quantum communication channel twice.'
author:
- 'Takayuki Miyadera$\ ^{1}$'
- 'Masakazu Yoshida$\ ^{2}$'
- 'Hideki Imai$\ ^{3}$'
title: ' On Ping-Pong protocol and its variant '
---
Introduction
============
In 2002, Boström and Felbinger [@BF02] proposed a quantum protocol which is called Ping-Pong protocol. Being different from other protocols such as BB84 or E91, this protocol uses two-way quantum communication. They showed a trade-off inequality between information gain by Eve and the error probability detected by Alice and Bob on the ideal setting of the protocol. That is, information gain by Eve is inevitably detected by Alice and Bob. While they insist that this protocol works as a secure direct communication as well as a key distribution protocol, there have been several discussions on its security from various points of view [@QCai03; @Woj03; @QCai06; @BF08]. The purpose of the present paper is not to discuss the security issue of the protocol but to give a simple derivation of another trade-off inequality between distinguishability of messages for Eve and detectability of Eve for legitimate users. The inequality holds for arbitrary initial states. Thus even if an initial state is prepared by Eve, she cannot distinguish the messages without being detected. As a byproduct, we show that the same inequality holds on a variant of the original protocol in which Bob sends his quantum system twice to Alice.
This paper is organized as follows. In the next section, we give a short description of the original Ping-Pong protocol. In section \[analysis\], a trade-off inequality is derived in a simple manner. In section \[variant\], a variant of the original protocol is given. It is shown that our trade-off inequality still holds on this variant.
Protocol
========
In this section we give a brief explanation on the simplest version of the protocols for Alice to send Bob one-bit message (or secret key). Bob first prepares a maximally entangled state $|\phi_0\rangle :=\frac{1}{\sqrt{2}}(|11\rangle +|00\rangle)$. He sends one of the bipartite systems which is called system A. It is described by a Hilbert space ${\cal H}_A (\simeq {\bf C}^2)$. Another system possessed by Bob is called system B with its Hilbert space ${\cal H}_B (\simeq {\bf C}^2)$. Bob confirms Alice’s receipt of the system A [@confirm]. Alice randomly chooses one from $\{\mbox{Control},\ \mbox{Message}\}$. If she chose “Control", she lets Bob know it and they both make measurements of $\sigma_z(A)$ and $\sigma_z(B)$ on their own systems respectively [@sigma]. If their outcomes disagree, they know existence of Eve and abort the protocol. On the other hand, if Alice chose “Message", she encodes her one-bit message to her system A. She does nothing on system A for the message $0$. She operates $\sigma_z(A)$ on it for the message $1$, which changes the phase with respect to $|0\rangle$. Alice sends back the system A to Bob. Bob makes a Bell measurement on the composite system A and B to know the encoded message. As pointed out in [@QCai03; @BF08], this naive protocol yields a simple “attack" that disturbs the message without being detected. That is, just an attack only on the second quantum communication from Alice to Bob does not affect the error probability in the control mode but can change the message while Eve cannot obtain any information. As claimed in [@QCai03; @BF08], this disadvantage may be avoided by introducing authentication phase after the protocol or slightly modifying the protocol itself. We, however, do not treat this problem here. What we are interested in is whether Eve can distinguish the messages $0$ and $1$ without being detected.
Analysis
========
Let us see what Eve can do in this protocol. Eve prepares her own system E which is described by a Hilbert space ${\cal H}_E$. We write the initial state of system E as $|\Omega\rangle$. She interacts it with system A when system A is sent between Alice and Bob. That is, she has two chances to obtain the information. Let us denote the first interaction by a unitary map $W :{\cal H}_A \otimes {\cal H}_E \to {\cal H}_A \otimes {\cal H}_E$ and the second interaction by $V: {\cal H}_A \otimes {\cal H}_E \to {\cal H}_A \otimes {\cal H}_E$. The state after the first attack is described by $|\Psi\rangle:=W |\phi \otimes \Omega\rangle$. The final state over the tripartite system A, B and E in a message mode becomes $V|\Psi\rangle$ when Alice’s message is $0$ and becomes $V\sigma_z(A)|\Psi\rangle$ when Alice’s message is $1$. Eve’s purpose is to distinguish them. The states to be distinguished by Eve are $$\begin{aligned}
\rho_0&:=& \mbox{tr}_{AB}\left(V|\Psi \rangle
\langle \Psi| V^* \right)
\\
\rho_1&:=& \mbox{tr}_{AB}\left(V\sigma_z(A)|\Psi \rangle
\langle \Psi | \sigma_z(A) V^*
\right) . \end{aligned}$$ We employ fidelity [@Uhlmann; @Jozsa] as a measure for (in)distinguishability of states. The fidelity between two states $\rho$ and $\sigma$ is defined by $F(\rho,\sigma):=\mbox{tr}\sqrt{\rho^{1/2} \sigma \rho^{1/2}}$. It takes $1$ if and only if $\rho=\sigma$ and takes a nonnegative value less than $1$ in general. The key lemma is the following which played an important role in [@MIWig] to derive a version of Wigner-Araki-Yanase theorem.
\[lemma1\] Suppose that we have two systems that are described by Hilbert spaces ${\cal H}_1$ and ${\cal H}_2$, and a pair of pure states $|\phi_0 \rangle, |\phi_1 \rangle \in {\cal H}_1
\otimes {\cal H}_2$. If we put states on ${\cal H}_2$ as $$\begin{aligned}
\rho_j :=\mbox{tr}_1 \left(|\phi_j \rangle
\langle \phi_j |
\right),\end{aligned}$$ for $j=0,1$, then for an arbitrary operator $X$ on ${\cal H}_1$, $$\begin{aligned}
|\langle \phi_0 |X|\phi_1 \rangle |
\leq \Vert X\Vert F(\rho_0,\rho_1)\end{aligned}$$ holds, where $\Vert \cdot\Vert$ is an operator norm defined by $\Vert X\Vert:=\sup_{|\phi\rangle \neq 0}
\frac{\Vert X|\phi\rangle \Vert}{\Vert |\phi\rangle\Vert}$.
[**Proof:**]{}\
We consider an arbitrary positive-operator-valued measure (POVM) $\{E_{\alpha}\}$ on ${\cal H}_2$, that is, every positive operator $E_{\alpha}$ acts only on ${\cal H}_2$ and satisfies $\sum_{\alpha}E_{\alpha}={\bf 1}$. We obtain $$\begin{aligned}
|\langle \phi_0|X|\phi_1\rangle |
=|\sum_{\alpha} \langle \phi_0| E_{\alpha} X|\phi_1\rangle |
=|\sum_{\alpha} \langle \phi_0| E_{\alpha}^{1/2} X E_{\alpha}^{1/2} |
\phi_1 \rangle |,
%\label{eq1}\end{aligned}$$ where we used the commutativity between $E_{\alpha}^{1/2}$ and $X$. We further obtain $$\begin{aligned}
|\langle \phi_0|X|\phi_1\rangle |
&\leq & \sum_{\alpha}|\langle \phi_0|E_{\alpha}^{1/2}
X E_{\alpha}^{1/2} |\phi_1\rangle | \\
&\leq& \sum_{\alpha} \langle \phi_0 |E_{\alpha} |\phi_0 \rangle^{1/2}
\langle \phi_1 |E_{\alpha}^{1/2} X^* X E_{\alpha}^{1/2} |\phi_1
\rangle^{1/2} \\
&\leq & \sum_{\alpha} \langle \phi_0 |E_{\alpha} |\phi_0 \rangle^{1/2}
\langle \phi_1 |E_{\alpha} |\phi_1
\rangle^{1/2} \Vert X\Vert,\end{aligned}$$ where we used the Cauchy-Schwarz inequality to derive the second line and the definition of the operator norm to derive the third line. By using a property $F(\rho,\sigma)=
\inf_{E:POVM} \sum_{\alpha}
\sqrt{
\mbox{tr}(\rho E_{\alpha})\mbox{tr}(\sigma E_{\alpha})}$ which was shown in [@FC95; @BCFJS], we take the infimum of the above inequality over all the POVMs to obtain $$\begin{aligned}
|\langle \phi_0|X|\phi_1\rangle |
\leq \Vert X \Vert F(\rho_0, \rho_1).\end{aligned}$$ It ends the proof.
In applying this lemma to Wigner-Araki-Yanase theorem, it was important to have a conserved quantity. Also in the Ping-Pong protocol, we have a conserved quantity. In fact, since the system B is kept by Bob during whole the protocol, the attack does not give any effect on the operator on system B. That is, for any operator $X$ on ${\cal H}_B$, $W^* V^* X V W =X$ and $V^* XV=X$ hold. We take the second equation and operate $\langle \Psi| $ and $\sigma_z(A)|\Psi \rangle$ to it. We obtain $$\begin{aligned}
\langle \Psi | V^*X V \sigma_z(A)|\Psi
\rangle
&=&\langle \Psi | X \sigma_z(A)| \Psi
\rangle .
%\\
%&=& \langle \Psi | X W \sigma_z(A)|\Psi
%\rangle,\end{aligned}$$ Taking the absolute value of the above equation, we apply the lemma with ${\cal H}_1={\cal H}_A\otimes {\cal H}_B$, ${\cal H}_2={\cal H}_E$, $|\phi_0\rangle=V |\Psi \rangle$ and $|\phi_1 \rangle=V \sigma_z(A) |\Psi \rangle$ to obtain, $$\begin{aligned}
\Vert X \Vert F\left(\rho_0,\rho_1
\right)
\geq
|\langle \Psi| X \sigma_z(A)|\Psi
\rangle |.
%=|\langle \phi_0 \otimes \Omega|W^* V^* X V \sigma_z(A)W|\phi_0
%\otimes \Omega \rangle |
%\leq \end{aligned}$$ Thus the indistinguishability of the messages for Eve is bounded from below by a correlation function after the first attack. If we put $X=\sigma_z(B)$, this correlation function becomes $$\begin{aligned}
\langle \Psi| \sigma_z(B) \sigma_z(A)
|\Psi \rangle
=p(0,0)+p(1,1)-p(0,1)-p(1,0)=1-2p(\sigma_z(A)\neq \sigma_z(B)),
\end{aligned}$$ where $p(i,j)$ is probability for Alice and Bob to obtain $\sigma_z(A)=i$ and $\sigma_z(B)=j$ respectively in $|\Psi\rangle$. That is, this is a probability distribution of the outcomes in the control mode. Thus we obtain $$\begin{aligned}
\left| 1-2p(\sigma_z(A)\neq \sigma_z(B))
\right|
\leq F(\rho_0,\rho_1).
\end{aligned}$$ Note that this inequality holds for an arbitrary state $|\Psi\rangle$ over the tripartite state since we did not use its concrete form. Thus we proved the following theorem.
In the Ping-Pong protocol, Eve cannot distinguish the messages $0$ and $1$ without being detected. In fact, if we put $p(\sigma_z(A)\neq \sigma_z(B))$ probability for Alice and Bob to obtain different outcomes in the control mode, indistinguishability measured by the fidelity is bounded as $$\begin{aligned}
\left| 1-2p(\sigma_z(A)\neq \sigma_z(B))\right|
\leq F(\rho_0,\rho_1). \end{aligned}$$ Here the initial state can be arbitrary. Even if it was prepared by Eve, the above trade-off inequality still holds.
It should be remarked that although the above trade-off inequality holds for arbitrarily prepared states, it does not mean that the protocol works in such cases. In fact, in such cases Alice and Bob cannot share the messages even if they do not detect Eve. That is, success in message sharing and information gain by Eve are different matters in this protocol.
A variant of the protocol {#variant}
=========================
In the original Ping-Pong protocol Bob first sends a qubit to Alice and receives it in the end of the protocol. In this section, we consider its variant. After the confirmation of Alice’s receipt of a qubit, Bob, instead of Alice, sends a message to Alice. For the definiteness, we describe the whole protocol in the following. Bob first prepares a maximally entangled state $|\phi_0\rangle :=\frac{1}{\sqrt{2}}(|11\rangle +|00\rangle)$. He sends one of the bipartite systems which is called system A. It is described by a Hilbert space ${\cal H}_A$. Another system possessed by Bob is called system B with its Hilbert space ${\cal H}_B$. Bob confirms Alice’s receipt of the system A. Bob randomly chooses one from $\{\mbox{Control},\ \mbox{Message}\}$. If he chose “Control", he lets Alice know it and they both make measurements of $\sigma_z(A)$ and $\sigma_z(B)$ on their own systems respectively. If their outcomes disagree, they know existence of Eve and abort the protocol. On the other hand, if Bob chose “Message", he encodes his one-bit message to his system B. He does nothing on system B for the message $0$. He operates $\sigma_z(B)$ on it for the message $1$, which changes the phase with respect to $|0\rangle$. Bob sends the system B to Alice. Alice makes a Bell measurement to the composite system A and B to know the encoded message.
We can prove again the following theorem.
In the above variant of the Ping-Pong protocol, Eve cannot distinguish the message $0$ and $1$ without being detected. Let us denote by $\mu_0$ Eve’s final state corresponding to the message $0$ and $\mu_1$ one corresponding to the message $1$. If we put $p(\sigma_z(A)\neq \sigma_z(B))$ probability for Alice and Bob to obtain different outcomes in the control mode, indistinguishability between $\mu_0$ and $\mu_1$ is bounded as $$\begin{aligned}
\left| 1-2p(\sigma_z(A)\neq \sigma_z(B))\right|
\leq F(\mu_0,\mu_1). \end{aligned}$$ Here the initial state can be arbitrary. Even if it was prepared by Eve, the above trade-off inequality still holds.
[**Proof:**]{}
The proof runs in the same manner with the previous theorem. Eve, with her own system E, interacts system A and system B when they are sent from Bob to Alice. We denote by $|\Psi\rangle$ the state over system A, B and E after the first attack and denote the second attack by a unitary map $U: {\cal H}_B \otimes {\cal H}_E
\to {\cal H}_B \otimes {\cal H}_E$. In the message mode, the states Eve wants to distinguish are $\mu_0:=\mbox{tr}_{AB}(U|\Psi\rangle
\langle \Psi |U^*)$ and $\mu_1:=\mbox{tr}_{AB}
(U \sigma_z(B)|\Psi\rangle \langle \Psi|\sigma_z(B)U^*)$ Since the second attack does not change the operator on ${\cal H}_A$, $U^* \sigma_z(A) U =\sigma_z(A)$ holds. We operate $\langle \Psi|\cdot \sigma_z(A)|\Psi\rangle$ on this equation to obtain, $$\begin{aligned}
\langle \Psi|U^* \sigma_z(A) U \sigma_z(B)|\Psi\rangle
=\langle \Psi|\sigma_z(A) \sigma_z(B)|\Psi\rangle.\end{aligned}$$ Applying Lemma \[lemma1\] to the absolute value of the left hand side with ${\cal H}_1={\cal H}_A\otimes {\cal H}_B$, ${\cal H}_2 ={\cal H}_E$, $X=\sigma_z(A)$, $|\phi_0\rangle
=U|\Psi\rangle$ and $|\phi_1\rangle=U\sigma_z(B)|\Psi\rangle$, we obtain, $$\begin{aligned}
\left| 1-2p(\sigma_z(A)\neq \sigma_z(B))\right| \leq F(\mu_0,\mu_1).\end{aligned}$$ It ends the proof. Q.E.D.
discussions
===========
In this paper, we treated the Ping-Pong protocol and derived a trade-off inequality between distinguishability of states for Eve and detectability for legitimate users. The inequality holds for arbitrary states that may be prepared even by Eve. We showed that the same inequality holds in a slightly different protocol in which the quantum communication is one-way. It, however, should be remarked that this trade-off relation does not directly mean the security of the protocols. For instance, Eve can change the message without being detected by making an attack only on the second communication phase. Furthermore, if Alice and Bob intend to use the protocols for direct communication, they need to confirm sufficiently many times the cleanness of the line before sending a message. In fact, otherwise Eve may obtain the message with non-negligible probability. Thus further investigation on definition and analysis of the security should be needed.
[9]{} K. Boström, T. Felbinger, Phys. Rev. Lett. [**89**]{}, 187902 (2002). Q.-Y Cai, Phys. Rev. Lett. [**91**]{}, 109801 (2003). A. Wojcik, Phys. Rev. Lett. [**90**]{}, 157901 (2003). Q.-Y. Cai, Phys. Lett. A [**351**]{}, 23 (2006). K. Boström, T. Felbinger, Phys. Lett. A [**372**]{}, 3953 (2008). Alice, for instance, can confirm her receipt of a particle by measuring number operator which does not destroy internal degrees of freedom. She then announces the receipt to Bob. In this paper, we use $\sigma_z=|1\rangle \langle 1|
-|0\rangle \langle 0|$. A. Uhlmann, Rep. Math. Phys. [**9**]{}, 273 (1976). R. Jozsa, J. Mod. Opt. [**41**]{}, 2315 (1994). T. Miyadera, H. Imai, Phys. Rev. A [**74**]{}, 024101 (2006). C. A. Fuchs, C. M. Caves, Open Sys. Info. Dyn. [**3**]{}, 1 (1995). H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, B. Schumacher, Phys. Rev. Lett. [**76**]{}, 2818 (1996).
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---
abstract: 'Robust object skeleton detection requires to explore rich representative visual features and effective feature fusion strategies. In this paper, we first re-visit the implementation of HED, the essential principle of which can be ideally described with a linear reconstruction model. Hinted by this, we formalize a Linear Span framework, and propose Linear Span Network (LSN) which introduces Linear Span Units (LSUs) to minimizes the reconstruction error. LSN further utilizes subspace linear span besides the feature linear span to increase the independence of convolutional features and the efficiency of feature integration, which enhances the capability of fitting complex ground-truth. As a result, LSN can effectively suppress the cluttered backgrounds and reconstruct object skeletons. Experimental results validate the state-of-the-art performance of the proposed LSN.'
author:
- Chang Liu
- Wei Ke
- Fei Qin
- 'Qixiang Ye ^()^'
bibliography:
- '1576.bib'
title: Linear Span Network for Object Skeleton Detection
---
Introduction
============
Skeleton is one of the most representative visual properties, which describes objects with compact but informative curves. Such curves constitute a continuous decomposition of object shapes [@ref31], providing valuable cues for both object representation and recognition. Object skeletons can be converted into descriptive features and spatial constraints, which enforce human pose estimation [@ref4], semantic segmentation [@ref16], and object localization [@ref32].
Researchers have been exploiting the representative CNNs for skeleton detection and extraction [@ref1; @ref6; @ref7; @ref33] for years. State-of-the-art approaches root in effective multi-layer feature fusion, with the motivation that low-level features focus on detailed structures while high-level features are rich in semantics [@ref1]. As a pioneer work, the holistically-nested edge detection (HED) [@ref6] is computed as a pixel-wise classification problem, without considering the complementary among multi-layer features. Other state-of-the-art approaches, $e.g.$, fusing scale-associated deep side-outputs (FSDS) [@ref7; @ref33] and side-output residual network (SRN) [@ref1] investigates the multi-layer association problem. FSDS requires intensive annotations of the scales for each skeleton point, while SRN struggles to pursuits the complementary between adjacent layers without complete mathematical explanation. The problem of how to principally explore and fuse more representative features remains to be further elaborated.
![A comparison of holistically-nested edge detection (HED) network [@ref6] and linear span network (LSN). HED uses convolutional features without considering their complementary. The union of the output spaces of HED is small, denoted as the pink area. As an improved solution, LSN spans a large output space. []{data-label="fig:shot"}](fig1){width="70.00000%"}
Through the analysis, it is revealed that HED treats the skeleton detection as a pixel-wise classification problem with the side-output from convolutional network. Mathematically, this architecture can be equalized with a linear reconstruction model, by treating the convolutional feature maps as linear bases and the $1\times1$ convolutional kernel values as weights. Under the guidance of the linear span theory [@ref30], we formalize a linear span framework for object skeleton detection. With this framework, the output spaces of HED could have intersections since it fails to optimize the subspace constrained by each other, Fig. 1. To ease this problem, we design Linear Span Unit (LSU) according to this framework, which will be utilized to modify convolutional network. The obtained network is named as Linear Span Network (LSN), which consists feature linear span, resolution alignment, and subspace linear span. This architecture will increase the independence of convolutional features and the efficiency of feature integration, which is shown as the smaller intersections and the larger union set, Fig. 1. Consequently, the capability of fitting complex ground-truth could be enhanced. By stacking multiple LSUs in a deep-to-shallow manner, LSN captures both rich object context and high-resolution details to suppress the cluttered backgrounds and reconstruct object skeletons. The contributions of the paper include:
- A linear span framework that reveals the essential nature of object skeleton detection problem, and proposes that the potential performance gain could be achieved with both the increased independence of spanning sets and the enlarged spanned output space.
- A Linear Span Network (LSN) can evolve toward the optimized architecture for object skeleton detection under the guidance of linear span framework.
Related work
============
Early skeleton extraction methods treat skeleton detection as morphological operations [@ref9; @ref10; @ref11; @ref12; @ref13; @ref14; @ref37]. One hypothesis is that object skeletons are the subsets of lines connecting center points of super-pixels [@ref12]. Such line subsets could be explored from super-pixels using a sequence of deformable discs to extract the skeleton path [@ref13]. In [@ref14], The consistence and smoothness of skeleton are modeled with spatial filters, $e.g.$, a particle filter, which links local skeleton segments into continuous curves. Recently, learning based methods are utilized for skeleton detection. It is solved with a multiple instance learning approach [@ref15], which picks up a true skeleton pixel from a bag of pixels. The structured random forest is employed to capture diversity of skeleton patterns [@ref16], which can be also modeled with a subspace multiple instance learning method [@ref17].
With the rise of deep learning, researchers have recently formulated skeleton detection as image-to-mask classification problem by using learned weights to fuse the multi-layer convolutional features in an end-to-end manner. HED [@ref6] learns a pixel-wise classifier to produce edges, which can be also used for skeleton detection. Fusing scale-associated deep side-outputs (FSDS) [@ref7] learns multi-scale skeleton representation given scale-associated ground-truth. Side-output residual network (SRN) [@ref1] leverages the output residual units to fit the errors between the object symmetry/skeleton ground-truth and the side-outputs of multiple convolutional layers.
The problem about how to fuse multi-layer convolutional features to generate an output mask, $e.g.$, object skeleton, has been extensively explored. Nevertheless, existing approaches barely investigate the problem about the linear independence of multi-layer features, which limits their representative capacity. Our approach targets at exploring this problem from the perspective of linear span theory by feature linear span of multi-layer features and subspace linear span of the spanned subspaces.
Problem Formulation
===================
Re-thinking HED
---------------
In this paper, we re-visit the implementation of HED, and reveal that HED as well as its variations can be all formulated by the linear span theory [@ref30].
HED utilizes fully convolutional network with deep supervision for edge detection, which is one of the typical low-level image-to-mask task. Denoting the convolutional feature as $C$ with $m$ maps and the classifier as $w$, HED is computed as a pixel-wise classification problem, as $${\hat {y_j}} = \sum\limits_{k = 1}^m {{w_k} \cdot {c_{k,j}}} ,j = 1,2, \cdots ,|{\hat Y}|,
\label{hed::classifier}$$ where ${c_{k,j}}$ is the feature value of the $j$-th pixel on the $k$-th convolutional map and ${\hat y_j}$ is the classified label of the $j$-th pixel in the output image ${\hat Y}$.
Not surprisingly, this can be equalized as a linear reconstruction problem, as $${ Y} = \sum\limits_{k = 1}^m {{\lambda _k}{v_k}},
\label{hed::reconstruction}$$ where ${\lambda _k}$ is linear reconstruction weight and $v_k$ is the $k$-th feature map in $C$.
We treat each side-output of HED as a feature vector in the linear spanned subspace $V_i=span({v_1^i,v_2^i,\cdots,v_m^i})$, in which $i$ is the index of convolutional stages. Then HED forces each subspace $V_i$ to approximate the ground-truth space $\mathcal{Y}$. We use three convolutional layers as an example, which generate subspaces $V_1$, $V_2$, and $V_3$. Then the relationship between the subspaces and the ground-truth space can be illustrated as lines in a 3-dimension space in Fig. 2(a).
As HED does not optimize the subspaces constrained by each other, it fails to explore the complementary of each subspace to make them decorrelated. The reconstructions can be formulated as $$\left\{\begin{array}{l}
V_1 \approx \mathcal{Y}\\
V_2 \approx \mathcal{Y}\\
V_3 \approx \mathcal{Y}
\end{array}.\right.
\label{hed:span}$$ When $v_1$, $v_2$, and $v_3$ are linearly dependent, they only have the capability to reconstruct vectors in a plane. That is to say, when the point $Y$ is out of the plane, the reconstruction error is hardly eliminated, Fig. 2(a).
Obviously, if $v_1$, $v_2$, and $v_3$ are linearly independent, $i.e.$, not in the same plane, Fig. 2(b), the reconstruction could be significantly eased. To achieve this target, we can iteratively formulate the reconstruction as $$\left\{\begin{array}{l}
V_1 \approx \mathcal{Y}\\
V_1 + V_2 \approx \mathcal{Y} \\
V_1 + V_2 + V_3 \approx \mathcal{Y}
\end{array}.\right.
\label{lsn::span}$$ It s observed that $V_2$ is refined with the constraint of $V_1$. And $V_3$ is optimized in the similar way, which aims for vector decorrelation. The sum of subspaces, $i.e.$, $V_1+V_2$ is denoted with the dark blue plane, and $V_1+V_2+V_3$ is denoted with the light blue sphere, Fig. 2(b).
Now, it is very straightforward to generalize Eq. (\[lsn::span\]) to $$\sum\limits_{k = 1}^l {{V_k}} \approx \mathcal{Y}, l=1,2,\cdots,n.
\label{genera::span}$$ One of the variations of HED, $i.e.$, SRN, which can be understand as a special case of Eq. (5) with $\sum\nolimits_{k = l - 1}^l {{V_k}} \approx \mathcal{Y}$, has already shown the effectiveness.
Linear Span View
----------------
Based on the discussion of last section, we can now strictly formulate a mathematical framework based on linear span theory [@ref30], which can be utilized to guide the design of Linear Span Network (LSN) toward the optimized architecture.
In linear algebra, linear span is defined as a procedure to construct a linear space by a set of vectors or a set of subspaces.
***Definition 1.*** $\mathcal{Y}$ is a linear space over ${ \mathbb{R}}$. The set $\left\{ {{v_1},{v_2},...,{v_m}} \right\} \subset \mathcal{Y}$ is a spanning set for $\mathcal{Y}$ if every $y$ in $\mathcal{Y}$ can be expressed as a linear combination of ${v_1},{v_2},...,{v_m}$, as $$y = \sum\limits_{k=1}^{m} {{\lambda_k}{v_k}}, \ \lambda_1,...,\lambda_m \in \mathbb{R},
\label{ls:def1}$$ and $\mathcal{Y}=span(\left\{ {{v_1},{v_2},...,{v_m}}\right\})$.
***Theorem 1.*** Let ${v_1},{v_2},...,{v_m}$ be vectors in $\mathcal{Y}$. Then $\left\{ {{v_1},{v_2},...,{v_m}} \right\}$ spans $\mathcal{Y}$ if and only if, for the matrix $F = \left[ {{v_1}{\rm{ }}{v_2}{\rm{ }}...{\rm{ }}{v_m}} \right]$, the linear system $F\lambda = y$ is consistent for every $y$ in $\mathcal{Y}$.
***Remark 1.*** According to *Theorem* 1, if the linear system is consistent for almost every vector in a linear space, the space can be approximated by the linear spanned space. This theorem uncovers the principle of LSN, which pursues a linear system as mentioned above setting up for as many as ground-truth.
***Definition 2.*** A finite set of vectors, which span $\mathcal{Y}$ and are linearly independent, is called a basis for $\mathcal{Y}$.
***Theorem 2.*** Every linearly independent set of vectors $\left\{ {{v_1},{v_2},...,{v_m}} \right\}$ in a finite dimensional linear space $\mathcal{Y}$ can be completed to a basis of $\mathcal{Y}$.
***Theorem 3.*** Every subspace $U$ has a complement in $\mathcal{Y}$, that is, another subspace $V$ such that vector $y$ in $\mathcal{Y}$ can be decomposed uniquely as $$y = u + v, u \ in \ U, v \ in \ V.
\label{ls::theo3}$$
***Definition 3.*** $\mathcal{Y}$ is said to be the sum of its subspaces $V_1,...,V_m$ if every $y$ in $Y$ can be expressed as $$y = v_1+...+v_m, v_j \ in \ V_j.
\label{ls::def3}$$
***Remark 2.*** We call the spanning of feature maps to a subspace as feature linear span, and the sum of subspaces as subspace linear span. From *Theorem 2* and *Theorem* 3, it is declared that the union of the spanning sets of subspaces is the spanning set of the sum of the subspaces. That is to say, in the subspace linear span we can merge the spanning sets of subspaces step by step to construct a larger space.
***Theorem 4.*** Supposing $\mathcal{Y}$ is a finite dimensional linear space, $U$ and $V$ are two subspaces of $\mathcal{Y}$ such that $\mathcal{Y} = U + V $, and $W$ is the intersection of $U$ and $V$, $i.e.$, $W = U \cap V $. Then $$\dim \mathcal{Y} = \dim U + \dim V - \dim W.$$
***Remark 3.*** From *Theorem* 4, the smaller the dimension of the intersection of two subspaces is, the bigger the dimension of the sum of two subspaces is. Then, successively spanning the subspaces from deep to shallow with supervision increases independence of spanning sets and enlarges the sum of subspaces. It enfores the representative capacity of convolutional features and integrates them in a more effective way.
Linear Span Network
===================
With the help of the proposed framework, the Linear Span Network (LSN) is designed for the same targets with HED and SRN, $i.e.$, the object skeleton detection problem. Following the linear reconstruction theory, a novel architecture named Linear Span Unit(LSU) has been defined first. Then, LSN is updated from VGG-16 [@ref33] with LSU and hints from *Remark* 1-3. VGG-16 has been chosen for the purpose of fair comparison with HED and SRN. In what follows, the implementation of LSU and LSN are introduced.
Linear Span Unit
----------------
![Linear Span Unit, which is used in both feature linear span and subspace linear span. In LSU, the operation of linear reconstruction is implemented by a concatenation layer and a $1\times1$ convolutional layer. []{data-label="fig:lsu"}](fig3){width="80.00000%"}
The architecture of Linear Span Unit (LSU) is shown in Fig. \[fig:lsu\], where each feature map is regarded as a feature vector. The input feature vectors are unified with a concatenation (concat for short) operation, as
$${C} = \mathop {concat}\limits_{k = 1}^{m} {\rm{ }}(c_{k}),
\label{lsu::concat}$$
where $c_k$ is the $k$-th feature vector. In order to compute the linear combination of the feature vectors, a convolution operation with $1\times1\times m$ convolutional kernels is employed: $${{s_i}} = \sum\limits_{k = 1}^m {{\lambda_{k,i}} \cdot {c_{k}}} ,i = 1,2, \cdots ,n,
\label{lsu::reconstructor}$$ where ${\lambda_{k,i}}$ is the convolutional parameter with $k$ elements for the $i$-th reconstruction output. The LSU will generate $n$ feature vectors in the subspace spanned by the input feature vectors. A slice layer is further utilized to separate them for different connections, which is denoted as $$\bigcup\limits_{i=1}^{n} s_i = slice({{\rm{S}}}).
\label{lsu::slice}$$
Linear Span Network Architecture
--------------------------------
![The architecture of the proposed Linear Span Network (LSN), which leverages Linear Span Units (LSUs) to implement three components of the feature linear span, the resolution alignment, and the subspace linear span. The feature linear span uses convolutional features to build subspaces. The LSU is re-used to unify the resolution among multi-stages in resolution alignment. The subspace linear span summarizes the subspaces to fit the ground-truth space. []{data-label="fig:LSN"}](fig4){width="98.00000%"}
The architecture of LSN is shown in Fig. \[fig:LSN\], which is consisted of three components, $i.e.$, feature linear span, resolution alignment, and subspace linear span are illustrated. The VGG-16 network with 5 convolutional stages [@ref34] is used as the backbone network.
In feature linear span, LSU is used to span the convolutional feature of the last layer of each stage according to Eq. \[lsu::reconstructor\]. The supervision is added to the output of LSU so that the spanned subspace approximates the ground-truth space, following *Remark* 1. If only feature linear span is utilized, the LSN is degraded to HED [@ref6]. Nevertheless, the subspaces in HED separately fit the ground-truth space, and thus fail to decorrelate spanning sets among subspaces. According to *Remark* 2 and 3, we propose to further employ subspace linear span to enlarge the sum of subspaces and deal with the decorrelation problem.
As the resolution of the vectors in different subspaces is with large variation, simple up-sampling operation will cause the Mosaic effect, which generates noise in subspace linear span. Without any doubt, the resolution alignment is necessary for LSN. Thus, in Fig. \[fig:LSN\], LSUs have been laid between any two adjacent layers with supervision. As a pre-processing component to subspace linear span, it outputs feature vectors with same resolution.
The subspace linear span is also implemented by LSUs, which further concatenates feature vectors from deep to shallow layers and spans the subspaces with Eq. (\[genera::span\]). According to *Remark* 3, a step-by-step strategy is utilized to explore the complementary of subspaces. With the loss layers attached on LSUs, it not only enlarges the sum of subspaces spanned by different convolutional layers, but also decorrelates the union of spanning sets of different subspaces. With this architecture, LSN enforces the representative capacity of convolutional features to fit complex ground-truth.
Experiments
===========
Experimental setting
--------------------
**Datasets:** We evaluate the proposed LSN on pubic skeleton datasets including SYMMAX [@ref15], WH-SYMMAX [@ref17], SK-SMALL [@ref7], SK-LARGE [@ref33], and Sym-PASCAL[@ref1]. We also evaluate LSN to edge detection on the BSDS500 dataset [@ref35] to validate its generality.
SYMMAX is derived from BSDS300 [@ref35], which contains 200/100 training and testing images. It is annotated with local skeleton on both foreground and background. WH-SYMMAX is developed for object skeleton detection, but contains only cropped horse images, which are not comprehensive for general object skeleton. SK-SMALL involves skeletons about 16 classes of objects with 300/206 training and testing images. Based on SK-SMALL, SK-LARGE is extended to 746/745 training and testing images. Sym-PASCAL is derived from the PASCAL-VOC-2011 segmentation dataset [@ref36] which contains 14 object classes with 648/787 images for training and testing.
The BSDS500 [@ref35] dataset is used to evaluate LSN’s performance on edge detection. This dataset is composed of 200 training images, 100 validation images, and 200 testing images. Each image is manually annotated by five persons on average. For training images, we preserve their positive labels annotated by at least three human annotators.
**Evaluation protocol:** Precision recall curve (PR-curve) is use to evaluate the performance of the detection methods. With different threshold values, the output skeleton/edge masks are binarized. By comparing the masks with the ground-truth, the precision and recall are computed. For skeleton detection, the F-measure is used to evaluate the performance of the different detection approaches, which is achieved with the optimal threshold values over the whole dataset, as $$F = \frac{{2PR}}{{P + R}}.$$ To evaluate edge detection performance, we utilize three standard measures [@ref35]: F-measures when choosing an optimal scale for the entire dataset (ODS) or per image (OIS), and the average precision (AP).
**Hyper-parameters:** For both skeleton and edge detection, we use VGG16 [@ref34] as the backbone network. During learning we set the mini-batch size to 1, the loss-weight to 1 for each output layer, the momentum to 0.9, the weight decay to 0.002, and the initial learning rate to 1e-6, which decreases one magnitude for every 10,000 iterations.
LSN Implementation
------------------
We evaluate four LSN architectures for subspace linear span and validate the iterative training strategy.
**LSN architectures.** If there is no subspace linear span, Fig. \[fig:LSN\], LSN is simplified to HED [@ref6], which is denoted as LSN\_1. The F-measure of LSN\_1 is 49.53%. When the adjacent two subspaces are spanned, it is denoted as LSN\_2, which is the same as SRN [@ref1]. LSN\_2 achieve significant performance improvement over HED which has feature linear span but no subspace span. We compare LSNs with different number of subspaces to be spanned, and achieve the best F-measure of 66.82%. When the subspace number is increased to 4, the skeleton detection performance drops. The followings explained why LSN\_3 is the best choice.
If the subspaces to be spanned are not enough, the complementary of convolutional features from different layers could not be effectively explored. On the contrary, if a LSU fuses feature layers that have large-scale resolution difference, it requires to use multiple up-sampling operations, which deteriorate the features. Although resolution alignment significantly eases the problem, the number of adjacent feature layers to be fused in LSU remains a practical choice. LSN\_3 reported the best performance by fusing a adjacent layer of higher resolution and a adjacent layer of lower resolution.On one hand, the group of subspaces in LSN\_3 uses more feature integration. On the other hand, there is not so much information loss after an $2\times$ up-sampling operation.
-------------------------------------------------- --------------
Architecture F-measure(%)
LSN\_1 (HED, feature linear span only) 49.53
LSN\_2 (SRN, feature and 2-subspace linear span) 65.88
LSN\_3 (LSN, feature and 3-subspace linear span) **66.15**
LSN\_4 (LSN, feature and 4-subspace linear span) 65.89
\[LSU\_Table\]
-------------------------------------------------- --------------
: The performance of different LSN implementations on the SK-LARGE dataset. LSN\_3 that fuses an adjacent layer of higher resolution and an adjacent layer of lower resolution reported the best performance.
------------------------------- -------- ------------ ----------- ------- -------
w/o RA end-to-end iter1 iter2 iter3
F-measure(%) 66.15 66.63 **66.82** 66.74 66.68
\[Training\_strategy\_table\]
------------------------------- -------- ------------ ----------- ------- -------
: The performance for different training strategies.
**Training strategy.** With three feature layers spanned, LSN needs up-sampling the side-output feature layers from the deepest to the shallowest ones. We use the supervised up-sampling to unify the resolution of feature layers.
During training, the resolution alignment is also achieved by stacking LSUs. We propose a strategy that train the two kinds of linear span, $i.e.$, feature linear span with resolution alignment and subspace linear span, iteratively. In the first iteration, we tune the LSU parameters for feature linear span and resolution alignment using the pre-trained VGG model on ImageNet, as well as update the convolutional parameters. Keeping the LSU parameters for resolution alignment unchanged, we tune LSU parameters for feature linear span and subspace linear span using the new model. In other iteration, the model is fine-tuned on the snap-shot of the previous iteration. With this training strategy, the skeleton detection performance is improved from 66.15% to 66.82%, Table \[Training\_strategy\_table\]. The detection performance changes marginally when more iterations are used. We therefore use the single iteration (iter1) in all experiments.
**LSU effect.** In Fig. \[Bases\], we use a giraffe’s skeleton from SK-LARGE as an example to compare and analyze the learned feature vectors (bases) by HED [@ref6], SRN [@ref6], and LSN. In Fig. \[Bases\](a) and (c), we respectively visualize the feature vectors learned by HED [@ref6] and the proposed LSN. It can be seen in the first column that the HED’s results incorporate more background noise and mosaic effects. This shows that the proposed LSN can better span an output feature space. In Fig. \[Bases\](b) and (d), we respectively visualize the subspace vectors learned by SRN [@ref1] and the proposed LSN. It can be seen in the first column that the SRN’s results incorporate more background noises. It requires to depress such noises by using a residual reconstruction procedure. In contrast, the subspace vectors of LSN is much clearer and compacter. This fully demonstrates that LSN can better span the output space and enforce the representative capacity of convolutional features, which will ease the problems of fitting complex outputs with limited convolutional layers.
\
Performance and Comparison
--------------------------
{width="2.3in"} captype[figure]{}
captype[table]{}
---------------------- ----------- --------------------
Mehods F-measure Runtime/s
Lindeberg [@ref37] 0.270 4.05
Levinshtein [@ref12] 0.243 146.21
Lee [@ref13] 0.255 609.10
MIL [@ref15] 0.293 42.40
HED [@ref6] 0.495 **0.05** $\dagger$
SRN [@ref1] 0.655 0.08 $\dagger$
LMSDS [@ref33] 0.649 **0.05** $\dagger$
LSN (ours) **0.668** 0.09 $\dagger$
\[tab:sk\_large\]
---------------------- ----------- --------------------
\[fig:sklarge\]
**Skeleton detection.**
The proposed LSN is evaluated and compared with the state-of-the-art approaches, and the performance is shown in Fig. \[fig:sklarge\] and Table \[tab:sk\_large\]. The result of SRN [@ref1] is obtained by running authors’ source code on a Tesla K80 GPU, and the other results are provided by [@ref33].
The conventional approaches including Lindeberg [@ref37], Levinshtein [@ref12], and Lee [@ref13], produce the skeleton masks without using any learning strategy. They are time consuming and achieve very low F-measure of 27.0%, 24.3%, and 25.5%, respectively. The typical learning approach, $i.e.$, multiple instance learning (MIL) [@ref15], achieves F-measure of 29.3%. It extractes pixel-wised feature with multi-orientation and multi-scale, and averagely uses 42.40 seconds to distinguish skeleton pixels from the backgrounds in a single image.
The CNN based approaches achieve huge performance gain compared with the conventional approaches. HED [@ref6] achieves the F-measure of 49.5% and uses 0.05 seconds to process an images, while SRN [@ref1] achieves 64.9% and uses 0.08 seconds. The scale-associated multi-task method, LMSDS [@ref33], achieves the performance of 64.9%, which is built on HED with the pixel-level scale annotations. Our proposed LSN reportes the best detection performance of 66.8% with a little more runtime cost compared with HED and SRN.
The results show that feature linear span is efficient for skeleton detection. As discussed above, HED and SRN are two special case of LSN. LSN that used three spanned layers in each span unit is a better choice than the state-of-the art SRN. Some skeleton detection results are shown in Fig. \[fig::skeleton\_examples\]. It is illustrated that HED produces lots of noise while the FSDS is not smooth. Comparing SRN with LSN, one can see that LSN rectifies some false positives as shown in column one and column three and reconstruct the dismiss as shown in column six.
![Skeleton detection examples by state-of-the-art approaches including HED [@ref6], FSDS [@ref7], SRN [@ref1], and LSN. The red boxes are false positive or dismiss in SRN, while the blue ones are correct reconstruction skeletons in LSN at the same position. (Best viewed in color with zoon-in.)[]{data-label="fig::skeleton_examples"}](fig7){width="0.95\linewidth"}
-------------------------- ----------- ----------- ----------- ------------
WH-SYMMAX SK-SMALL SYMMAX Sym-PASCAL
Levinshtein [@ref12] 0.174 0.217 – 0.134
Lee [@ref13] 0.223 0.252 – 0.135
Lindeberg [@ref37] 0.277 0.227 0.360 0.138
Particle Filter [@ref14] 0.334 0.226 – 0.129
MIL [@ref15] 0.365 0.392 0.362 0.174
HED [@ref6] 0.743 0.542 0.427 0.369
FSDS [@ref7] 0.769 0.623 0.467 0.418
SRN [@ref1] 0.780 0.632 0.446 **0.443**
LSN (ours) **0.797** **0.633** **0.480** 0.425
\[tab::four\_datasets\]
-------------------------- ----------- ----------- ----------- ------------
: Performance comparison of the state-of-the-art approaches on the public WH-SYMMAX [@ref17], SK-SMALL [@ref7], SYMMAX [@ref15], and Sym-PASCAL [@ref1] datasets.
The proposed LSN is also evaluated on other four commonly used datasets, including WH-SYMMAX [@ref17], SK-SMALL [@ref7], SYMMAX [@ref15], and Sym-PASCAL [@ref1]. The F-measure are shown in Table \[tab::four\_datasets\]. Similar with SK-LARGE, LSN achieves the best detection performance on WH-SYMMAX, SK-SMALL, and SYMMAX, with the F-measure 79.7%, 63.3% and 48.0%. It achieves 5.4%, 8.1%, and 5.3% performance gain compared with HED, and 1.7%, 0.1%, and 2.4% gain compared with SRN. On Sym-PASCAL, LSN achieves comparable performance of 42.5% vs. 44.3% with the state-of-the art SRN.
**Edge detection.** Edge detection task has similar implementation with skeleton that discriminate whether a pixel belongs to an edge. It also can be reconstructed by the convolutional feature maps. In this section, we compare the edge detection result of the proposed LSN with some other state-of-the-art methods, such as Canny [@ref40], Sketech Tokens [@ref39], Structured Edge (SE) [@ref38], gPb [@ref35], DeepContour [@ref18], HED [@ref6], and SRN [@ref1], Fig. 8 and Table 5.
In Fig. 8, it is illustrated that the best conventional approach is SE with F-measure (ODS) of 0.739 and all the CNN based approaches achieve much better detection performance. HED is one of the baseline deep learning method, which achieved 0.780. The proposed LSN reportes the highest F-measure of 0.790, which has a very small gap (0.01) to human performance. The F-measure with an optimal scale for the per image (OIS) was 0.806, which was even higher than human performance, Table 5. The good performance of the proposed LSN demonstrates its general applicability to image-to-mask tasks.
{width="2.2in"} captype[figure]{}
captype[table]{}
---------------- ----------- ----------- ------- ----------------
Mehods ODS OIS AP FPS
Canny [@ref40] 0.590 0.620 00578 15
ST [@ref39] 0.721 0.739 0.768 1
gPb [@ref35] 0.726 0.760 0.727 1/240
SE [@ref18] 0.739 0.759 0.792 2.5
DC [@ref18] 0.757 0.776 0.790 1/30 $\dagger$
HED [@ref6] 0.780 0.797 0.814 2.5 $\dagger$
SRN [@ref1] 0.782 0.800 0.779 2.3 $\dagger$
LSN (ours) **0.790** **0.806** 0.618 2.0 $\dagger$
Human 0.800 0.800 – –
---------------- ----------- ----------- ------- ----------------
\[fig:sklarge\]
Conclusion
==========
Skeleton is one of the most representative visual properties, which describes objects with compact but informative curves. In this paper, the skeleton detection problem is formulated as a linear reconstruction problem. Consequently, a generalized linear span framework for skeleton detection has been presented with formal mathematical definition. We explore the Linear Span Units (LSUs) to learn a CNN based mask reconstruction model. With LSUs we implement three components including feature linear span, resolution alignment, and subspace linear span, and update the Holistically-nested Edge Detection (HED) network to Linear Span Network (LSN). With feature linear span, the ground truth space can be approximated by the linear spanned output space. With subspace linear span, not only the independence among spanning sets of subspaces can be increased, but also the spanned output space can be enlarged. As a result, LSN will have better capability to approximate the ground truth space, $i.e.$, against the cluttered background and scales. Experimental results validate the state-of-the-art performance of the proposed LSN, while we provide a principled way to learn more representative convolutional features.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was partially supported by the National Nature Science Foundation of China under Grant 61671427 and Grant 61771447, and Beijing Municipal Science & Technology Commission under Grant Z181100008918014.
|
---
abstract: |
Let $M_c=M(2,0,c)$ be the moduli space of ${\mathcal{O} }(1)$-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where ${\mathcal{O} }(1)$ is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [@ogrady], O’Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\ge
3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This obviously implies that there is no symplectic desingularization.
address: 'Dept of Mathematics, Seoul National University, Seoul 151-747, Korea'
author:
- 'Jaeyoo Choy and Young-Hoon Kiem'
title: Nonexistence of a crepant resolution of some moduli spaces of sheaves on a K3 surface
---
[^1]
Introduction
============
Let $X$ be a complex projective K3 surface with polarization $H={\mathcal{O} }_X(1)$ generic in the sense of [@ogrady] §0. Let $M(r,c_1,c_2)$ be the moduli space of rank $r$ $H$-semistable torsion-free sheaves on $X$ with Chern classes $(c_1,c_2)$ in $H^*(X,{\mathbb{Z}})$. Let $M^s(r,c_1,c_2)$ be the open subscheme of $H$-stable sheaves in $M(r,c_1,c_2)$. In [@Muk84], Mukai shows that $M^s(r,c_1,c_2)$ is smooth and has a holomorphic symplectic structure. By [@Gi77], if either $(c_1.H)$ or $c_2$ is an odd number, then $M(2,c_1,c_2)$ is equal to $M^s(2,c_1,c_2) $ and thus $M(2,c_1,c_2)$ is a smooth projective irreducible symplectic variety. However if both $(c_1.H)$ and $c_2$ are even numbers then generally $M(2,c_1,c_2)$ admits singularities. We restrict our interest to the trivial determinant case i.e. $c_1=0$ and let $M_c=M(2,0,c)$ where $c=2n$ ($n\geq2$). It is well-known that $M_{2n}$ is an irreducible, normal ([@Yo01] Theorem 3.18) and projective variety ([@HL97] Theorem 4.3.4) of dimension $8n-6$ ([@Muk84] Theorem 0.1) with only Gorenstein singularities ([@HL97] Theorem 4.5.8, [@Ei95] Corollary 21.19). Since $M_{2n}$ contains the smooth open subset $M^s_{2n}$, there arises a natural question: does there exist a resolution of $M_{2n}$ such that the Mukai form on $M^s_{2n}$ extends to the resolution without degeneration? When $c=4$, O’Grady successfully extends the Mukai form on $M^s_{2n}$ to some resolution without degeneration ([@og97; @ogrady]). At the same time, he conjectures nonexistence of a symplectic desingularization of $M_{2n}$ for $n\ge 3$ ([@ogrady], (0.1)). Our main result in this paper is the following.
\[thm:main result\] If $n\geq3$, there is no crepant resolution of $M_{2n}$.
The highest exterior power of a symplectic form gives a non-vanishing section of the canonical sheaf on $M_{2n}$. Likewise any symplectic desingularization of $M_{2n}$ has trivial canonical divisor and hence it must be a crepant resolution. Therefore, O’Grady’s conjecture is a consequence of Theorem \[thm:main result\].
\[cor:O’Grady’s conjecture\] If $n\geq3$, there is no symplectic desingularization of $M_{2n}$.
The idea of the proof of Theorem \[thm:main result\] is to use a new invariant called the stringy E-function [@Bat98; @DL99]. Since $M_{2n}$ is normal irreducible variety with log terminal singularities ([@ogrady], 6.1), the stringy E-function of $M_{2n}$ is a well-defined rational function. If there is a crepant resolution ${\widetilde{M} }_{2n}$ of $M_{2n}$, then the stringy E-function of $M_{2n}$ is equal to the Hodge-Deligne polynomial (E-polynomial) of ${\widetilde{M} }_{2n}$ (Theorem \[thm:Batyrev’s result\]). In particular, we deduce that the stringy E-function $E_{st}(M_{2n};u,v)$ must be a polynomial. Therefore, Theorem \[thm:main result\] is a consequence of the following.
\[prop:stringy E-function test\] The stringy E-function $E_{st}(M_{2n};u,v)$ is not a polynomial for $n\geq3$.
To prove that $E_{st}(M_{2n};u,v)$ is not a polynomial for $n\geq 3$, we show that $E_{st}(M_{2n};z,z)$ is not a polynomial in $z$. Thanks to the detailed analysis of Kirwan’s desingularization in [@og97] and [@ogrady] which is reviewed in section 4, we can find an expression for $E_{st}(M_{2n};z,z)$ and then with some efforts on the combinatorics of rational functions we show that $E_{st}(M_{2n};z,z)$ is not a polynomial in section 3. In section 2, we recall basic facts on stringy E-function and in section 5 we prove a lemma which computes the E-polynomial of a divisor.
In [@ogrady], O’Grady gets a symplectic desingularization ${\widetilde{M} }_{2n}$ of $M_{2n}$ in the case when $n=2$. This turns out to be a new irreducible symplectic variety, which means that it does not come from a generalized Kummer variety nor from a Hilbert scheme parameterizing 0-dimensional subschemes on a K3 surface [@og98; @Bea83]. Corollary \[cor:O’Grady’s conjecture\] shows that unfortunately we cannot find any more irreducible symplectic variety in this way.
After we finished the first draft of this paper, we learned that Kaledin and Lehn [@KL04] proved Corollary \[cor:O’Grady’s conjecture\] in a completely different way. We are grateful to D. Kaledin for informing us of their approach. The second named author thanks Professor Jun Li for useful discussions concerning the article [@VW94]. Finally we would like to express our gratitude to the referee for careful reading and challenging us for many details which led us to improve the manuscript and correct an error in Proposition 3.2.
Preliminaries
=============
In this section we collect some facts that we shall use later.
For a topological space $V$, the Poincaré polynomial of $V$ is defined as $$\label{eqn:Poincare polynomial}
P(V;z)=\sum_{i}(-1)^ib_i(V)z^i$$ where $b_i(V)$ is the $i$-th Betti number of $V$. It is well-known from [@Go90] that the Betti numbers of the Hilbert scheme of points $X^{[n]}$ in $X$ are given by the following: $$\label{eqn:Betti for X[n]} \sum_{n\geq
0}P(X^{[n]};z)t^n=\prod_{k\geq1}
\prod_{i=0}^{4}(1-z^{2k-2+i}t^k)^{(-1)^{i+1}b_i(X)}.$$ Next we recall the definition and basic facts about stringy E-functions from [@Bat98; @DL99]. Let $W$ be a normal irreducible variety with at worst log-terminal singularities, i.e.
1. W is ${\mathbb{Q}}$-Gorenstein;
2. for a resolution of singularities $\rho: V\to W$ such that the exceptional locus of $\rho$ is a divisor $D$ whose irreducible components $D_1,\cdots,D_r$ are smooth divisors with only normal crossings, we have $$K_V=\rho^*K_W+\sum^r_{i=1} a_iD_i$$ with $a_i>-1$ for all $i$, where $D_i$ runs over all irreducible components of $D$. The divisor $\sum^r_{i=1}a_iD_i$ is called the *discrepancy divisor*.
For each subset $J\subset I=\{1,2,\cdots,r\}$, define $D_J=\cap_{j\in J}D_j$, $D_\emptyset=V$ and $D^0_J=D_J-\cup_{i\in
I-J}D_i$. Then the stringy E-function of $W$ is defined by $$\label{eqn:stringy E-function}
E_{st}(W;u,v)=\sum_{J\subset I}E(D^0_J;u,v)\prod_{j\in
J}\frac{uv-1}{(uv)^{a_j+1}-1}$$ where $$E(Z;u,v) =
\sum_{p,q}\sum_{k\geq 0} (-1)^kh^{p,q}(H^k_c(Z;{\mathbb{C}}))u^pv^q$$ is the Hodge-Deligne polynomial for a variety $Z$. Note that the Hodge-Deligne polynomials have
1. the additive property: $E(Z;u,v)=E(U;u,v)+E(Z-U;u,v)$ if $U$ is a smooth open subvariety of $Z$;
2. the multiplicative property: $E(Z;u,v)=E(B;u,v)E(F;u,v)$ if $Z$ is a Zariski locally trivial $F$-bundle over $B$.
By [@Bat98] Theorem 6.27, the function $E_{st}$ is independent of the choice of a resolution (Theorem 3.4 in [@Bat98]) and the following holds.
\[thm:Batyrev’s result\] ([@Bat98] Theorem 3.12) Suppose $W$ is a ${\mathbb{Q}}$-Gorenstein algebraic variety with at worst log-terminal singularities. If $\rho:V\to W$ is a crepant desingularization (i.e. $\rho^*K_W=K_V$) then $E_{st}(W;u,v)=E(V;u,v)$. In particular, $E_{st}(W;u,v)$ is a polynomial.
Proof of Proposition \[prop:stringy E-function test\]
=====================================================
In this section we first find an expression for the stringy E-function of the moduli space $M_{2n}$ for $n\geq 3$ by using the detailed analysis of Kirwan’s desingularization in [@og97; @ogrady]. Then we show that it cannot be a polynomial, which proves Proposition \[prop:stringy E-function test\].
We fix a generic polarization of $X$ as in [@ogrady]. The moduli space $M_{2n}$ has a stratification $$M_{2n}=M^s_{2n}\sqcup
(\Sigma-\Omega)\sqcup \Omega$$ where $M^s_{2n}$ is the locus of stable sheaves and $\Sigma\simeq(X^{[n]}\times X^{[n]})/{\rm
involution}$ is the locus of sheaves of the form $I_Z\oplus
I_{Z'}$ ($[Z],[Z']\in X^{[n]}$) while $\Omega\simeq X^{[n]}$ is the locus of sheaves $I_Z\oplus I_{Z}$. For $n\geq 3$, Kirwan’s desingularization $\rho:{\widehat{M} }_{2n}\to M_{2n}$ is obtained by blowing up $M_{2n}$ first along $\Omega$, next along the proper transform of $\Sigma$ and finally along the proper transform of a subvariety $\Delta$ in the exceptional divisor of the first blow-up. This is indeed a desingularization by [@ogrady] Proposition 1.8.3.
Let $D_1=\hat{\Omega}$, $D_2=\hat{\Sigma}$ and $D_3=\hat{\Delta}$ be the (proper transforms of the) exceptional divisors of the three blow-ups. Then they are smooth divisors with only normal crossings as we will see in Proposition \[prop:analysis on exc\] and the discrepancy divisor of $\rho:{\widehat{M} }_{2n}\to M_{2n}$ is ([@ogrady], 6.1) $$(6n-7)D_1+(2n-4)D_2+(4n-6)D_3.$$ Therefore the singularities are log-terminal for $n\geq 2$, and from (\[eqn:stringy E-function\]) the stringy E-function of $M_{2n}$ is given by $$\begin{aligned}
\label{eqn:stringy E-function
of M_c}
E(M^s_{2n};u,v)+E(D^0_1;u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}}
+E(D^0_2;u,v){\textstyle\frac{1-uv}{1-(uv)^{2n-3}}}\nonumber \\
+E(D^0_3;u,v) {\textstyle\frac{1-uv}{1-(uv)^{4n-5}}}
+E(D^0_{12};u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}\frac{1-uv}{1-(uv)^{2n-3}}} \\
+E(D^0_{23};u,v){\textstyle\frac{1-uv}{1-(uv)^{2n-3}}\frac{1-uv}{1-(uv)^{4n-5}}}
+E(D^0_{13};u,v){\textstyle\frac{1-uv}{1-(uv)^{4n-5}}\frac{1-uv}{1-(uv)^{6n-6}}}
\nonumber \\
+E(D^0_{123};u,v){\textstyle\frac{1-uv}{1-(uv)^{6n-6}}
\frac{1-uv}{1-(uv)^{2n-3}}\frac{1-uv}{1-(uv)^{4n-5}}} .\nonumber\end{aligned}$$ We need to compute the Hodge-Deligne polynomials of $D^0_J$ for $J\subset \{1,2,3\}$. Let $({\mathbb{C}}^{2n},\omega)$ be a symplectic vector space. Let ${\mathrm{Gr}}^{\omega}(k,2n)$ be the Grassmannian of $k$-dimensional subspaces of ${\mathbb{C}}^{2n}$, isotropic with respect to the symplectic form $\omega$ (i.e. the restriction of $\omega$ to the subspace is zero).
\[lem:Hodge poly of Gr\] For $k\leq n$, the Hodge-Deligne polynomial of ${\mathrm{Gr}}^\omega(k,2n)$ is $$\prod_{1\leq i\leq k} \frac{1-(uv)^{2n-2k+2i}}{1-(uv)^i}.$$
Consider the incidence variety $$Z= \{(a,b)\in
{\mathrm{Gr}}^\omega(k-1,2n)\times {\mathrm{Gr}}^\omega(k,2n)|a\subset b\}.$$ This is a ${\mathbb{P}}^{2n-2k+1}$-bundle over ${\mathrm{Gr}}^\omega(k-1,2n)$ and a ${\mathbb{P}}^{k-1}$-bundle over ${\mathrm{Gr}}^\omega(k,2n)$. We have the following equalities between Hodge-Deligne polynomials: $$\begin{aligned}
E(Z;u,v)&=&\frac{1-(uv)^{2n-2k+2}}{1-uv}
E({\mathrm{Gr}}^\omega(k-1,2n);u,v)\\ &=& \frac{1-(uv)^{k}}{1-uv}
E({\mathrm{Gr}}^\omega(k,2n);u,v). \end{aligned}$$ The desired formula follows recursively from ${\mathrm{Gr}}^\omega(1,2n)={\mathbb{P}}^{2n-1}$.\
Let $\hat{{\mathbb{P}}}^5$ be the blow-up of ${\mathbb{P}}^5$ (projectivization of the space of $3\times 3$ symmetric matrices) along ${\mathbb{P}}^2$ (the locus of rank 1 matrices). We have the following from [@og97] and [@ogrady]. The proof will be presented in §\[sec: Proof of Lemma\].
\[prop:analysis on exc\] Let $n\geq 3$.
\(1) $D_1$ is a $\hat{{\mathbb{P}}}^5$-bundle over a ${\mathrm{Gr}}^\omega(3,2n)$-bundle over $X^{[n]}$.
\(2) $D_2^0$ is a free ${\mathbb{Z}}_2$-quotient of a Zariski locally trivial $I_{2n-3}$-bundle over $ X^{[n]}\times
X^{[n]}-\mathbf{\Delta} $ where $\mathbf{\Delta}$ is the diagonal in $ X^{[n]}\times X^{[n]}$ and $I_{2n-3}$ is the incidence variety given by $$I_{2n-3}=\{(p,H)\in {\mathbb{P}}^{2n-3}\times \breve{{\mathbb{P}}}^{2n-3}| p\in
H\}.$$
\(3) $D_3$ is a ${\mathbb{P}}^{2n-4}$-bundle over a Zariski locally trivial $ {\mathbb{P}}^2$-bundle over a Zariski locally trivial ${\mathrm{Gr}}^\omega(2,2n)$-bundle over $X^{[n]}$.
\(4) $D_{12}$ is a ${\mathbb{P}}^2$-bundle over a ${\mathbb{P}}^2$-bundle over a ${\mathrm{Gr}}^\omega(3,2n)$-bundle over $X^{[n]}$.
\(5) $D_{23}$ is a ${\mathbb{P}}^{2n-4}$-bundle over a $ {\mathbb{P}}^1$-bundle over a ${\mathrm{Gr}}^\omega(2,2n)$-bundle over $X^{[n]}$.
\(6) $D_{13}$ is a $ {\mathbb{P}}^2$-bundle over a ${\mathbb{P}}^2$-bundle over a ${\mathrm{Gr}}^\omega(3,2n)$-bundle over $X^{[n]}$.
\(7) $D_{123}$ is a ${\mathbb{P}}^1$-bundle over a ${\mathbb{P}}^2$-bundle over a ${\mathrm{Gr}}^\omega(3,2n)$-bundle over $X^{[n]}$.\
All the above bundles except in (2) and (3) are Zariski locally trivial. Moreover, $D_i$ ($i=1,2,3$) are smooth divisors such that $D_1\cup D_2\cup D_3$ is normal crossing.
From Lemma \[lem:Hodge poly of Gr\] and Proposition \[prop:analysis on exc\], we have the following corollary by the additive and multiplicative properties of the Hodge-Deligne polynomial.
\[eqn:computation of stringy E-function\] $$E(D_1;u,v) = \Bigl({\textstyle
\frac{1-(uv)^6}{1-uv}-\!\frac{1-(uv)^3}{1-uv}+\!\bigl(\frac{1-(uv)^3}{1-uv}\bigr)^2}\Bigr)
\! \times\!\!\! \prod_{1\leq i\leq 3}\! \Bigl({\textstyle
\frac{1-(uv)^{2n-6+2i}}{1-(uv)^i}}\Bigr)\!\! \times\!
E(X^{[n]};u,v),$$
$$E(D_3;u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\cdot\frac{1-(uv)^3}{1-uv}} \times
\prod_{1\leq i\leq 2}\Bigl({\textstyle \frac{1-(uv)^{2n-4+2i}}
{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v),$$
$$E(D_{12};u,v) = \Bigl({\textstyle
\frac{1-(uv)^3}{1-uv}}\Bigr)^2\times \prod_{1\leq i\leq
3}\Bigl({\textstyle \frac{1-(uv)^{2n-6+2i}}{1-(uv)^i}}\Bigr)
\times E(X^{[n]};u,v),$$
$$E(D_{23};u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\cdot\frac{1-(uv)^2}{1-uv}}
\times\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v),$$
$$E(D_{13};u,v) = {\textstyle \frac{
1-(uv)^3}{1-uv}\cdot\frac{1-(uv)^{2n-4}}{1-uv}} \times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v),$$ $$E(D^0_{123};u,v) ={\textstyle
\frac{1-(uv)^2}{1-uv}\cdot\frac{1-(uv)^{2n-4}}{1-uv}}\times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-(uv)^{2n-4+2i}}{1-(uv)^i}}\Bigr)\times E(X^{[n]};u,v).$$
Perhaps the only part that requires proof is the equation for $E(D_3;u,v)$. From Proposition \[prop:analysis on exc\] (3), $D_3$ is a projective variety which is a ${\mathbb{P}}^{2n-4}$-bundle over a smooth projective variety, say $Y$, whose E-polynomial is $$E({\mathbb{P}}^2;u,v)\times E(\mathrm{Gr}^\omega (2,2n);u,v)\times
E(X^{[n]};u,v).$$ By the Leray-Hirsch theorem ([@V02I] p.182), we have $$\begin{aligned}
H^*(D_3;{\mathbb{C}})\cong H^*(Y;{\mathbb{C}})\otimes
H^*({\mathbb{P}}^{2n-4};{\mathbb{C}})\cong H^*(Y;{\mathbb{C}})\otimes {\mathbb{C}}[\lambda]/(\lambda^{2n-3})\\ \cong H^*(Y;{\mathbb{C}})\oplus
H^*(Y;{\mathbb{C}})\lambda\oplus \cdots \oplus
H^*(Y;{\mathbb{C}})\lambda^{2n-4}\end{aligned}$$ where $\lambda$ is a class of type $(1,1)$ which comes from the Kähler class. The above determines the Hodge structure of $D_3$ because the Hodge structure is compatible with the cup product. Therefore we deduce that $$E(D_3;u,v) = {\textstyle
\frac{1-(uv)^{2n-3}}{1-uv}\times E(Y;u,v)}.$$
For the E-polynomial of $D_2^0$ we have the following lemma whose proof is presented in section \[sec: Computation of E-poly of D\_0\^2\]. Recall that $$I_{2n-3}=\{((x_i),(y_j))\in {\mathbb{P}}^{2n-3}\times
{\mathbb{P}}^{2n-3}\,|\, \sum_{i=0}^{2n-3} x_iy_i=0\}$$ and there is an action of ${\mathbb{Z}}_2$ which interchanges $(x_i)$ and $(y_j)$. Let $H^r(I_{2n-3})^+$ denote the ${\mathbb{Z}}_2$-invariant subspace of $H^r(I_{2n-3})$ .
\[lem: Hodge Deligne poly of D02\] $$\begin{aligned}
\label{eqn: compute D02} \lefteqn{
E(D^0_2;z,z)=P(I_{2n-3};z) \Bigl(
\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2 \Bigr)} && \\ && +
P^+(I_{2n-3};z)\bigl(P(X^{[n]};z^2)-P(X^{[n]};z) \bigr)\nonumber
\end{aligned}$$ where $P^+(I_{2n-3};z)=\displaystyle \sum_{r\geq0}(-1)^rz^r\dim
H^r(I_{2n-3})^+$. Moreover $$\begin{aligned}
\label{eqn: E D02 is divisible by some Q}
E(D^0_2;z,z)=\frac{1-(z^2)^{2n-3}}{1-z^2} Q(z^2)\end{aligned}$$ for some polynomial $Q$.
*Proof of Proposition \[prop:stringy E-function test\].*
Let us prove that cannot be a polynomial. Let $$S(z)=E_{st}(M_{2n};z,z)-E(M^s_{2n};z,z).$$ It suffices to show that $S(z)$ is not a polynomial for all $n\geq3$ because $E(M^s_{2n};z,z)$ is a polynomial.
Note that given any $n\geq 3$, we can explicitly compute $E(X^{[n]};z,z)$ and $E(D^0_2;z,z)$ by (\[eqn:Betti for X\[n\]\]) and Lemma \[lem: Hodge Deligne poly of D02\]. If $n=3$, direct calculation shows that $S(z)$ is as follows: $$\begin{aligned}
S(z)& =&
1+46z^2+852z^4+12308z^6+111641z^8+886629z^{10}+4233151z^{12}\\
& & +4990239z^{14}+4999261z^{16}+4230852z^{18}+884441z^{20}+113877z^{22}\\
& & +12928z^{24}+3749z^{26}+3200z^{28}+2877z^{30}+299z^{32}+\cdots.\end{aligned}$$ It is easy to see from (\[eqn:stringy E-function of M\_c\]) and Corollary \[eqn:computation of stringy E-function\] that if $S(z)$ were a polynomial, it should be of degree $\le 30$. Since the series $S(z)$ has a nonzero coefficient of $z^{32}$, $S(z)$ cannot be a polynomial. So we assume from now on that $n\ge 4$.
Express the rational function $S(z)$ as $$\frac{N(z)}{(1-(z^2)^{2n-3})(1-(z^2)^{4n-5})(1-(z^2)^{6n-6})}.$$ All we need to show is that the numerator $N(z)$ is not divisible by the denominator $(1-(z^2)^{2n-3})(1-(z^2)^{4n-5})(1-(z^2)^{6n-6})$.
As $E(X^{[n]};z,z)$ and $E(D^0_2;z,z)$ do not have nonzero terms of odd degree by (\[eqn:Betti for X\[n\]\]) and Lemma \[lem: Hodge Deligne poly of D02\], all the nonzero terms in $S(z)$ are of even degree by (\[eqn:stringy E-function of M\_c\]) and Corollary \[eqn:computation of stringy E-function\]. Hence, we can write $S(z)=s(z^2)=s(t)$ for some rational function $s(t)$ in $t=z^2$. The numerator $N(z)=n(z^2)=n(t)$ is not divisible by $1-(z^2)^{2n-3}$ if and only if $n(t)$ is not divisible by $1-t^{2n-3}$. By direct computation using (\[eqn:stringy E-function of M\_c\]), Corollary \[eqn:computation of stringy E-function\] and Lemma \[lem: Hodge Deligne poly of D02\], $n(t)$ modulo $1-t^{2n-3}$ is congruent to $$\begin{aligned}
\label{eqn:denumerator modulo}
\shoveleft(1-t)^2(1-t^{4n-5})\times\Bigl({\textstyle
\frac{1-t^3}{1-t}}\Bigr)^2\times \prod_{1\leq i\leq
3}\Bigl({\textstyle \frac{1-t^{2n-6+2i}}{1-t^i}}\Bigr) \times
p(X^{[n]};t) \\ -(1-t)^2(1-t^{4n-5})\times {\textstyle
\frac{1-t^2}{1-t}\cdot\frac{1-t^{2n-4}}{1-t}}\times \prod_{1\leq
i\leq 2}\Bigl({\textstyle \frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr)
\times p(X^{[n]};t) \nonumber \\ -(1-t)^2(1-t^{6n-6})\times
{\textstyle \frac{1-t^2}{1-t}\cdot\frac{1-t^{2n-4}}{1-t}}\times
\prod_{1\leq i\leq 2}\Bigl({\textstyle
\frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr) \times p(X^{[n]};t) \nonumber
\\ + (1-t)^3\times{\textstyle \frac{1-t^2}{1-t}
\cdot\frac{1-t^{2n-4}}{1-t}}\times \prod_{1\leq i\leq
2}\Bigl({\textstyle \frac{1-t^{2n-4+2i}}{1-t^i}}\Bigr)\times
p(X^{[n]};t) \nonumber \end{aligned}$$ where $p(X^{[n]};t)=P(X^{[n]};z)$ with $t=z^2$. We write (\[eqn:denumerator modulo\]) as a product $\bar s(t)\cdot
p(X^{[n]};t)$ for some polynomial $\bar s(t)$. For the proof of our claim for $n\geq 4$, it suffices to prove the following:
1. if $n$ is not divisible by 3, then $1-t$ is the GCD of $1-t^{2n-3}$ and $\bar s(t)$, and $\frac{1-t^{2n-3}}{1-t}$ does not divide $p(X^{[n]};t)$;
2. if $n$ is divisible by 3, then $1-t^3$ is the GCD of $1-t^{2n-3}$ and $\bar s(t)$, and $\frac{1-t^{2n-3}}{1-t^3}$ does not divide $p(X^{[n]};t)$.
For (1), suppose $n$ is not divisible by 3. From (\[eqn:denumerator modulo\]), $\bar s(t)$ is divisible by $1-t$. We claim that $\bar s(t)$ is not divisible by any irreducible factor of $\frac{1-t^{2n-3}}{1-t}$, i.e. for any root $\alpha$ of $1-t^{2n-3}$ which is not 1, $\bar s(\alpha)\neq 0$. Using the relation $\alpha^{2n-3}=1$, we compute directly that $$\label{eqn:bar s} \bar s(\alpha)={\textstyle
-\frac{\alpha(1-\alpha^{-1}){(1-\alpha^3)}^2}{1+\alpha}},$$ which is not 0 because 3 does not divide $2n-3$.
Next we check that $\frac{1-t^{2n-3}}{1-t}$ does not divide $p(X^{[n]};t)$. We put $${\displaystyle p(X^{[n]};t)=\sum_{0\leq i\leq 2n} c_it^i}$$ and write $p(X^{[n]};t)$ as follows: $$\begin{aligned}
\label{eqn:p(t) when 3 NOT divides n} \lefteqn{
\sum_{0\leq i\leq 2n} c_it^i =
(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3
} \\ & & + \sum_{4\leq i\leq 2n-4} c_it^i + c_{2n-3}(t^{2n-3}-1)
+ c_{2n-2}t(t^{2n-3}-1)\nonumber \\ & & +
c_{2n-1}t^2(t^{2n-3}-1)+ c_{2n}t^3(t^{2n-3}-1).\nonumber\end{aligned}$$ Therefore, the divisibility of $p(X^{[n]};t)$ by $\frac{1-t^{2n-3}}{1-t}$ is that of $(c_0+c_{2n-3})+ (c_1+c_{2n-2})t+ (c_2+c_{2n-1})t^2
+(c_3+c_{2n})t^3 + {\displaystyle \sum_{4\leq i\leq 2n-4} c_it^i}$ by $\frac{1-t^{2n-3}}{1-t}$. Since $\frac{1-t^{2n-3}}{1-t}={\displaystyle \sum_{0\leq i\leq 2n-4}
t^i}$, the polynomial $(c_0+c_{2n-3})+ (c_1+c_{2n-2})t+
(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3 + {\displaystyle \sum_{4\leq
i\leq 2n-4} c_it^i}$ is divisible by $\frac{1-t^{2n-3}}{1-t}$ if and only if it is a scalar multiple of ${\displaystyle \sum_{0\leq
i\leq 2n-4} t^i}$, i.e. $c_0+c_{2n-3}=c_1+c_{2n-2}=
c_2+c_{2n-1}=c_3+c_{2n}= c_4=\cdots =c_{2n-4}$ ($n\geq 4$).
Table \[table:list of ci\] is the list of $c_i$ ($1\leq i\leq
4$) for $n\geq 3$, which comes from direct computation using the generating functions (\[eqn:Betti for X\[n\]\]) for the Betti numbers of $X^{[n]}$. By Table \[table:list of ci\], we can check that this is impossible. Indeed, for $n\geq 6$, $c_0=1$, $c_1=23$, $c_2=300$ and $c_3=2876$, which implies $c_{2n-3}=2876$, $c_{2n-2}=300$, $c_{2n-1}=23$ and $c_{2n-2}=1$ by Poincaré duality. Thus $c_0+c_{2n-3}=2877$ while $c_1+c_{2n-2}=323$. For $4\leq n\leq 5$, the proof is also direct computation using Table \[table:list of ci\].
$n=3$ $n=4$ $n=5$ $n=6$ $n=7$ $n\geq 8$
------- ------- ------- ------- ------- ------- -----------
$c_1$ 23 23 23 23 23 23
$c_2$ 299 300 300 300 300 300
$c_3$ 2554 2852 2875 2876 2876 2876
$c_4$ 299 19298 22127 22426 22449 22450
: list of $c_i$ \[table:list of ci\]
For (2), suppose 3 divides $n$ and $n\neq 3$. Then from (\[eqn:bar s\]), $(1-t^3)$ divides $\bar s(t)$. More precisely, for a third root of unity $\alpha$, $\bar s(\alpha)=0$. On the other hand, if $\alpha$ is a root of $1-t^{2n-3}$ but not a third root of unity then we can observe that $\bar s(\alpha)\neq 0$ by (\[eqn:bar s\]). Therefore, since every root of $1-t^{2n-3}$ is a simple root, any irreducible factor of $\frac{1-t^{2n-3}}{1-t^3}$ does not divide $\bar s(t)$.
We next check that the polynomial $\frac{1-t^{2n-3}}{1-t^3}$ does not divide $p(X^{[n]};t)$. Write $p(X^{[n]};t)=\displaystyle
\sum_{0\leq i\leq 2n} c_it^i$ as follows: $$\begin{aligned}
\label{eqn:p(t) when 3 divides n} \lefteqn{\sum_{0\leq i\leq 2n} c_it^i =
(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2 +(c_3+c_{2n})t^3}
\\ & & + \sum_{4\leq i\leq 2n-6} c_it^i - c_{2n-5}
\Bigl(\sum_{i=0}^{\frac{2n-9}3} t^{3i+1} \Bigr)-
c_{2n-4} \Bigl( \sum_{i=0}^{\frac{2n-9}3} t^{3i+2}\Bigr) \nonumber \\
& & + c_{2n-5}t\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} +
c_{2n-4}t^2\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} +
c_{2n-3}(t^{2n-3}-1) \nonumber \\ & & + c_{2n-2}t(t^{2n-3}-1) +
c_{2n-1}t^2(t^{2n-3}-1)+ c_{2n}t^3(t^{2n-3}-1) \nonumber\end{aligned}$$ where the equality comes from $$\begin{aligned}
t^{2n-5} = -\sum_{i=0}^{\frac{2n-9}3} t^{3i+1}
+t\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}}\ \ {\rm and}\ \
t^{2n-4} = -\sum_{i=0}^{\frac{2n-9}3} t^{3i+2} +
t^2\cdot{\textstyle \frac{1-t^{2n-3}}{1-t^3}} \end{aligned}$$ since $\frac{1-t^{2n-3}}{1-t^3}=\displaystyle
\sum_{i=0}^{\frac{2n-6}3}t^{3i}$. Therefore, $p(X^{[n]};t)$ modulo $\frac{1-t^{2n-3}}{1-t^3}$ is congruent to $$\begin{aligned}
\lefteqn{R(t)=(c_0+c_{2n-3})+(c_1+c_{2n-2})t+(c_2+c_{2n-1})t^2
+(c_3+c_{2n})t^3} \\ & & + \displaystyle\sum_{4\leq i\leq 2n-6}
c_it^i - c_{2n-5} \Bigl(\displaystyle\sum_{i=0}^{\frac{2n-9}3}
t^{3i+1} \Bigr)- c_{2n-4} \Bigl(
\displaystyle\sum_{i=0}^{\frac{2n-9}3}
t^{3i+2}\Bigr).\end{aligned}$$ Now $R(t)$ is divisible by $\frac{1-t^{2n-3}}{1-t^3}=\displaystyle
\sum_{i=0}^{\frac{2n-6}3}t^{3i}$ if and only if $R(t)$ is a scalar multiple of $\displaystyle \sum_{i=0}^{\frac{2n-6}3}t^{3i}$ because $R(t)$ is of degree $\le 2n-6$. Thus the coefficient of $R(t)$ with respect to $t^2$ should be 0 i.e. $c_2+c_{2n-1}-c_{2n-4}=0$. However, $c_2+c_{2n-1}-c_{2n-4}=c_2+c_1-c_4$ is not zero by Table \[table:list of ci\]. This proves Proposition \[prop:stringy E-function test\] for the case where 3 divides $n$ and $n\neq 3$. So the proof of Proposition \[prop:stringy E-function test\] is completed for any $n\geq 3$.\
In case of smooth projective curves, we remark that the stringy E-function of the moduli space of rank 2 bundles is explicitly computed ([@kiem] and [@KL]). We were not able to compute the stringy E-function of $M_{2n}$ precisely, because we do not know how to compute the Hodge-Deligne polynomial $E(M^s_{2n};u,v)$ of the locus $M^s_{2n}$ of stable sheaves.
Analysis of Kirwan’s desingularization {#sec: Proof of Lemma}
======================================
This section is devoted to the proof of Proposition \[prop:analysis on exc\]. All can be extracted from [@og97] but we spell out the details for reader’s convenience.
To begin with, note that for each $Z\in X^{[n]}$, the tangent space $T_{X^{[n]},Z}$ of the Hilbert scheme $X^{[n]}$ is canonically isomorphic to ${\mathrm{Ext}}^1(I_Z,I_Z)$ where $I_Z$ is the ideal sheaf of the 0-dimensional closed subscheme $Z$. By the Yoneda pairing map and Serre duality, we have a skew-symmetric pairing $\omega:{\mathrm{Ext}}^1(I_Z,I_Z)\otimes {\mathrm{Ext}}^1(I_Z,I_Z) \to
{\mathrm{Ext}}^2(I_Z,I_Z)\cong {\mathbb{C}}$, which gives us a symplectic form $\omega$ on the tangent bundle $T_{X^{[n]}}$ by [@Muk84] Theorem 0.1. Note that the Killing form on $sl(2)$ gives an isomorphism $sl(2)^\vee\cong sl(2)$. Let $W=sl(2)^\vee\cong sl(2)\cong {\mathbb{C}}^3$. The adjoint action of $PGL(2)$ on $W$ gives us an identification $SO(W)\cong PGL(2)$ ([@og97] §1.5). For a symplectic vector space $(V,\omega)$, let ${\mathrm{Hom}}^\omega(W,V)$ be the space of homomorphisms from $W$ to $V$ whose image is isotropic. Let ${\mathrm{Hom}}^\omega(W,T_{X^{[n]}}) $ be the bundle over $X^{[n]}$ whose fiber over $Z\in X^{[n]}$ is ${\mathrm{Hom}}^\omega(W,T_{X^{[n]},Z})$. Clearly ${\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$ is Zariski locally trivial over $X^{[n]}$. Let ${\mathrm{Hom}}_k^\omega(W,T_{X^{[n]}})$ be the subbundle of ${\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$ of rank $\leq k$ elements in ${\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$. Also let ${\mathrm{Gr}}^\omega(3,T_{X^{[n]}})$ be the relative Grassmannian of isotropic 3-dimensional subspaces in $T_{X^{[n]}}$ and let ${\mathcal{B} }$ denote the tautological rank 3 bundle on ${\mathrm{Gr}}^\omega(3,T_{X^{[n]}})$. Obviously these bundles are all Zariski locally trivial as well.
Let ${\mathbb{P}}{\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$ (resp. ${\mathbb{P}}{\mathrm{Hom}}_k^\omega(W,T_{X^{[n]}})$) be the projectivization of ${\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$ (resp. ${\mathrm{Hom}}_k^\omega(W,T_{X^{[n]}})$). Likewise, let ${\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{B} })$ and ${\mathbb{P}}{\mathrm{Hom}}_k(W,{\mathcal{B} })$ denote the projectivizations of the bundles ${\mathrm{Hom}}(W,{\mathcal{B} })$ and ${\mathrm{Hom}}_k(W,{\mathcal{B} })$. Note that there are obvious forgetful maps $$\begin{aligned}
f:{\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{B} })\to{\mathbb{P}}{\mathrm{Hom}}^\omega(W,T_{X^{[n]}})\ \mbox{\rm
and}\\
f_k:{\mathbb{P}}{\mathrm{Hom}}_k(W,{\mathcal{B} })\to{\mathbb{P}}{\mathrm{Hom}}_k^\omega(W,T_{X^{[n]}})\end{aligned}$$ Since the pull-back of the defining ideal of ${\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}})$ is the ideal of ${\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{B} })$ (both are actually given by the determinants of $2\times 2$ minor matrices), $f$ gives rise to a map between blow-ups $$\overline{f}:Bl_{{\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{B} })}{\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{B} })\to
Bl_{{\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}})}{\mathbb{P}}{\mathrm{Hom}}^\omega(W,T_{X^{[n]}}).$$ Let us denote $Bl_{{\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{B} })}{\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{B} })$ by $Bl^{\mathcal{B} }$ and $Bl_{{\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}})}{\mathbb{P}}{\mathrm{Hom}}^\omega(W,T_{X^{[n]}})$ by $Bl^T$. We denote the proper transform of ${\mathbb{P}}{\mathrm{Hom}}_2(W,{\mathcal{B} })$ in $Bl^{\mathcal{B} }$ by $Bl_2^{\mathcal{B} }$ and the proper transform of ${\mathbb{P}}{\mathrm{Hom}}_2^\omega(W,T_{X^{[n]}})$ by $Bl_2^T$. Since $Bl_2^{\mathcal{B} }$ is a Cartier divisor which is mapped onto $Bl_2^T$ and the pull-back of the defining ideal of $Bl_2^T$ is the ideal sheaf of $Bl_2^{\mathcal{B} }$, $\overline{f}$ lifts to $$\label{eq4.-2}
\hat{f}:Bl^{\mathcal{B} }\to
Bl_{Bl_2^T}Bl^T.$$ By [@og97] §3.1 IV, $\hat f$ is an isomorphism on each fiber over $X^{[n]}$, so in particular $\hat f$ is bijective. Therefore, $\hat f$ is an isomorphism by Zariski’s main theorem.
Note that ${\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{B} }){/\!\!/}SO(W)$ (resp. ${\mathbb{P}}{\mathrm{Hom}}_k(W,{\mathcal{B} }){/\!\!/}SO(W)$) is isomorphic to the space of conics ${\mathbb{P}}(S^2{\mathcal{B} })$ (resp. rank $\leq k$ conics ${\mathbb{P}}(S^2_k{\mathcal{B} })$) where the $SO(W)$-quotient map is given by $[\alpha]\mapsto[\alpha\circ\alpha^t]$ where $\alpha^t$ denotes the transpose of $\alpha\in {\mathrm{Hom}}(W,{\mathcal{B} })$ ([@og97] §3.1). Let $\hat {\mathbb{P}}(S^2{\mathcal{B} }) =
Bl_{{\mathbb{P}}(S^2_1{\mathcal{B} })}{\mathbb{P}}(S^2{\mathcal{B} })$ denote the blow-up along the locus of rank 1 conics. Then $Bl^{\mathcal{B} }{/\!\!/}SO(W)$ is canonically isomorphic to $\hat {\mathbb{P}}(S^2{\mathcal{B} })$ by [@k2] Lemma 3.11. Since ${\mathcal{B} }$ is Zariski locally trivial, so is $\hat {\mathbb{P}}(S^2{\mathcal{B} })$ over ${\mathrm{Gr}}^\omega(3,T_{X^{[n]}})$.
Now consider Simpson’s construction of the moduli space $M_{2n}$ ([@og97] §1.1). Let $Q$ be the closure of the set of semistable points $Q^{ss}$ in the Quot-scheme whose quotient by the natural $PGL(N)$ action is $M_{2n}$ for some even integer $N$. Then $Q^{ss}$ parameterizes semistable torsion-free sheaves $F$ together with surjective homomorphisms $h:{\mathcal{O} }^{\oplus N}\to F(k)$ which induces an isomorphism ${\mathbb{C}}^N\cong H^0(F(k))$ and $H^1(F(k))=0$. Let $\Omega_Q$ denote the subset of $Q^{ss}$ which parameterizes sheaves of the form $I_Z\oplus I_Z$ for some $Z\in
X^{[n]}$. This is precisely the locus of closed orbits with maximal dimensional stabilizers, isomorphic to $PGL(2)$ and the quotient of $\Omega_Q$ by $PGL(N)$ is $X^{[n]}$.
We can give a more precise description of $\Omega_Q$ as follows. Let ${\mathcal{L} }\to X^{[n]}\times X$ be the universal rank 1 sheaf such that ${\mathcal{L} }|_{Z\times X}$ is isomorphic to the ideal sheaf $I_Z$. By [@HL97] Theorem 10.2.1, the tangent bundle $T_{X^{[n]}}$ is in fact isomorphic to ${\mathcal{E} }xt^1_{X^{[n]}}({\mathcal{L} },{\mathcal{L} })$. Let $p:X^{[n]}\times X\to X^{[n]}$ be the projection onto the first component. For $k\gg 0$, $p_*{\mathcal{L} }(k)$ is a vector bundle of rank $N/2$. Let $$\label{eq4.-1}q:{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))\to X^{[n]}$$ be the $PGL(N)$-bundle over $X^{[n]}$ whose fiber over $Z$ is ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,
H^0(I_Z(k)\oplus I_Z(k)))$. Note that the standard action of $GL(N)$ on ${\mathbb{C}}^N$ and the obvious action of $GL(2)$ on $p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k)$ induce a $PGL(N)\times
PGL(2)$-action on ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))\to X^{[n]}$ .
\[4.1\] (1) $\Omega_Q\cong {\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k)){/\!\!/}SO(W).$\
(2) Via the above isomorphism, the normal cone of $\Omega_Q$ in $Q^{ss}$ is $$q^*\mathrm{Hom}^{\omega}(W,T_{X^{[n]}}){/\!\!/}SO(W)\to
{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k)){/\!\!/}SO(W)$$ whose fiber over a point lying over $Z\in X^{[n]}$ is $\mathrm{Hom}^{\omega}(W,T_{X^{[n]},Z})$.
\(1) Let $\hat p:{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))\times X\to {\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))$ be the obvious projection so that we have $q\circ \hat
p=p\circ (q\times 1_X)$. Let $H$ be the dual of the tautological line bundle over ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))$. There is a canonical isomorphism ${\mathcal{O} }^{\oplus N}\cong
q^*(p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k))\otimes H$. This induces a surjective homomorphism $$\begin{aligned}
{\mathcal{O} }^{\oplus N}\to \hat{p}^*q^*(p_*{\mathcal{L} }(k)\oplus
p_*{\mathcal{L} }(k))\otimes H =(q\times 1)^*(p^*p_*{\mathcal{L} }(k)\oplus
p^*p_*{\mathcal{L} }(k))\otimes H\\ \to (q\times 1)^*({\mathcal{L} }(k)\oplus
{\mathcal{L} }(k))\otimes H\end{aligned}$$ over ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,
p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k))\times X$. By the universal property of the Quot-scheme, we get a morphism ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,
p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k))\to Q^{ss}$ whose image is clearly contained in $\Omega_Q$. This map is $PGL(2)$-invariant and hence we get a morphism $$\label{eq4.00}\phi_\Omega:{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,
p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k)){/\!\!/}SO(W)\to \Omega_Q.$$ It is easy to check that $\phi_\Omega$ is bijective. Since $\Omega_Q$ is smooth ([@og97] (1.5.1)), $\phi_\Omega$ is an isomorphism by Zariski’s main theorem.
\(2) Let ${\mathcal{O} }^{\oplus N} \to {\mathcal{E} }(k)$ denote the universal quotient sheaf on $Q^{ss}\times X$ restricted to $\Omega_Q$ and let ${\mathcal{F} }$ be the kernel of the twisted homomorphism ${\mathcal{O} }^{\oplus N}(-k) \to
{\mathcal{E} }$ so that we have an exact sequence $$0\to {\mathcal{F} }\to {\mathcal{O} }^{\oplus N}(-k) \to {\mathcal{E} }\to 0$$ over $\Omega_Q\times X$. The induced long exact sequence gives us $$\label{eqn: rel long exact sequence}
{\mathscr H\!om}_{\Omega_Q}({\mathcal{O} }^{\oplus N}(-k),{\mathcal{E} })\to
{\mathscr H\!om}_{\Omega_Q}({\mathcal{F} },{\mathcal{E} })\to {\mathscr E\!xt}^1_{\Omega_Q}({\mathcal{E} },{\mathcal{E} })\to
{\mathscr E\!xt}^1_{\Omega_Q}({\mathcal{O} }^{\oplus N}(-k),{\mathcal{E} })$$ Let $\pi:\Omega_Q\times X\to \Omega_Q$ be the obvious projection. Note that ${\mathscr E\!xt}^1_{\Omega_Q}({\mathcal{O} }^{\oplus
N}(-k),{\mathcal{E} })=R^1\pi_*({\mathcal{E} }(k))^{\oplus N}=0$ and that ${\mathscr H\!om}_{\Omega_Q}({\mathcal{O} }^{\oplus N}(-k),{\mathcal{E} })\cong
{\mathscr H\!om}_{\Omega_Q}({\mathcal{O} }^{\oplus N},{\mathcal{E} }(k))$ is a vector bundle over $\Omega_Q$ whose fiber is $gl(N)$ because ${\mathcal{O} }_X^{\oplus N}\cong
H^0(E(k))$ for any $[{\mathcal{O} }_X^{\oplus N}\to E(k)]\in Q^{ss}$. Let $T^*_{Q^{ss}}, T^*_{\Omega_Q}$ be cotangent sheaves over $Q^{ss}$ and ${\Omega_Q}$ respectively. By a famous result of Grothendieck ([@Gr95] §5) we know $$(T^*_{Q^{ss}}|_{\Omega_Q})^\vee\cong
{\mathscr H\!om}_{\Omega_Q}({\mathcal{F} },{\mathcal{E} })$$ which contains the tangent bundle of $\Omega_Q$ as a subbundle. So the first homomorphism in is the tangent map of the group action of $PGL(N)$[^2] on $\Omega_Q$ and the second homomorphism is the Kodaira-Spencer map.
Via the isomorphism $\phi_\Omega$ , we have a map $$\delta:{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k))\to {\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N, p_*{\mathcal{L} }(k)\oplus p_*{\mathcal{L} }(k)){/\!\!/}SO(W)\cong
\Omega_Q.$$ From the proof of (1) above, the pull-back of ${\mathcal{E} }$ by $\delta\times 1$ is isomorphic to $(q\times 1)^*({\mathcal{L} }(k)\oplus
{\mathcal{L} }(k))\otimes H$ and thus the vector bundle $\delta^*{\mathcal{E} }xt^1_{\Omega_Q}({\mathcal{E} },{\mathcal{E} })$ is isomorphic to $$q^*{\mathcal{E} }xt^1_{X^{[n]}}({\mathcal{L} }, {\mathcal{L} })\otimes gl(2)\cong
q^*T_{X^{[n]}}\otimes gl(2).$$ The pull-back of the tangent sheaf of $X^{[n]}$ sits in $q^*T_{X^{[n]}}\otimes gl(2)$ as $q^*T_{X^{[n]}}\otimes \left(\begin{matrix}1&0\\
0&1\end{matrix}\right)$. Hence the pull-back by $\delta$ of the normal bundle of $\Omega_Q$ (in the sense of [@og97] §1.3) is isomorphic to $$q^*T_{X^{[n]}}\otimes sl(2)\cong
q^*\mathrm{Hom}(W,T_{X^{[n]}}).$$ By [@og97] (1.5.10), the normal cone is fiberwisely the same as the Hessian cone. (See [@og97] §1.3 for more details on the Hessian cone.) Since the normal cone is contained in the Hessian cone, the normal cone is equal to the Hessian cone which is the inverse image of zero by the Yoneda square map $\Upsilon:{\mathcal{E} }xt^1_{\Omega_Q}({\mathcal{E} },{\mathcal{E} })\to
{\mathcal{E} }xt^2_{\Omega_Q}({\mathcal{E} },{\mathcal{E} })$. It is elementary to see that $\delta^*\Upsilon^{-1}(0)$ is precisely $q^*\mathrm{Hom}^\omega(W,T_{X^{[n]}}).$ Since $SO(W)$ acts freely we obtain (2). See [@og97] (1.5.1) for a description of the normal cone at each point.
Let $\Sigma_Q$ denote the subset of $Q^{ss}$ whose sheaves are of the form $I_Z\oplus I_W$ for some $Z,W\in X^{[n]}$. Then $\Sigma_Q-\Omega_Q$ is precisely the locus of points in $Q^{ss}$ whose stabilizer is isomorphic to ${\mathbb{C}}^*$. Let $\pi_R:R\to Q^{ss}$ be the blow-up of $Q^{ss}$ along $\Omega_Q$ and let $\Omega_R$ denote the exceptional divisor. By the above lemma,we have $$\label{eq4.0}\Omega_R\cong q^*{\mathbb{P}}\mathrm{Hom}^{\omega}(W,T_{X^{[n]}}){/\!\!/}SO(W).$$ The following lemma is an easy exercise.
\[4.2\] (1) The locus of points in ${\mathbb{P}}\mathrm{Hom}^\omega(W,T_{X^{[n]},Z})^{ss}$ whose stabilizer is 1-dimensional by the action of $SO(W)$ is precisely ${\mathbb{P}}\mathrm{Hom}^\omega _1(W,T_{X^{[n]},Z})^{ss}$.\
(2) The locus of nontrivial stabilizers is ${\mathbb{P}}\mathrm{Hom}^\omega
_2(W,T_{X^{[n]},Z})^{ss}$.
Let $$\label{eq4.1}\Delta_R=q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}}){/\!\!/}SO(W).$$ Let $\Sigma_R$ be the proper transform of $\Sigma_Q$. Then $\Sigma_R^{ss}$ is precisely the locus of points in $R^{ss}$ with 1-dimensional stabilizers by [@k2]. Therefore we have the following from Lemma \[4.2\].
\[4.3\] $\Sigma_R^{ss}\cap
\Omega_R=q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})^{ss}{/\!\!/}SO(W).$
We have an explicit description of $\Sigma_R^{ss}$ from [@og97] §1.7 III as follows. Let $$\beta:\mathcal{X}^{[n]}\to X^{[n]}\times X^{[n]}$$ be the blow-up along the diagonal and let $\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$ where $\mathbf{\Delta}$ is the diagonal. Let ${\mathcal{L} }_1$ (resp. ${\mathcal{L} }_2$) be the pull-back to $\mathcal{X}^{[n]}\times X$ of the universal sheaf ${\mathcal{L} }\to X^{[n]}\times X$ by $p_{13}\circ (\beta\times 1)$ (resp. $p_{23}\circ (\beta\times 1)$) where $p_{ij}$ is the projection onto the first (resp. second) and third components. Let $p:\mathcal{X}^{[n]}\times X\to \mathcal{X}^{[n]}$ be the projection onto the first component. Then for $k\gg0$, $p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k)$ is a vector bundle of rank $N$. Let $$q:{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k))\to \mathcal{X}^{[n]}$$ be the $PGL(N)$-bundle. There is an action of $O(2)$ on ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k))$. We quote [@og97] (1.7.10) and (1.7.1).
\[4.4\] (1) $\Sigma_R^{ss}\cong{\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus
p_*{\mathcal{L} }_2(k)){/\!\!/}O(2)$\
(2) The normal cone of $\Sigma_R^{ss}$ in $R^{ss}$ is a locally trivial bundle over $\Sigma_R^{ss}$ with fiber the cone over a smooth quadric in ${\mathbb{P}}^{4n-5}$.
In fact we can give a more explicit description of the normal cone when restricted to $\Sigma_R^0:=\Sigma_R^{ss}-\Omega_R$. Similarly as in the proof of Lemma \[4.1\], the normal vector bundle to $\Sigma_R^0$ is isomorphic to the vector bundle (of rank $4n-4$) $$\label{eq4.2} q^*[{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)]{/\!\!/}O(2)$$ over ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k)){/\!\!/}O(2)$ where $O(2)$ acts as follows: if we realize $O(2)$ as the subgroup of $PGL(2)$ generated by $$SO(2)=\{\theta_\alpha=\left(\begin{matrix}\alpha&0\\
0&\alpha^{-1}\end{matrix}\right)\}/\{\pm Id\},\qquad
\tau=\left(\begin{matrix} 0&1\\1&0\end{matrix}\right)$$ $\theta_\alpha$ multiplies $\alpha$ (resp. $\alpha^{-1}$) to ${\mathcal{L} }_1$ (resp. ${\mathcal{L} }_2$) and $\tau$ interchanges ${\mathcal{L} }_1$ and ${\mathcal{L} }_2$ by the induced action on $\mathcal{X}^{[n]}$ of interchanging the first and second factors of $X^{[n]}\times
X^{[n]}$. The normal cone is the inverse image $q^*\Upsilon^{-1}(0)$ of zero in terms of the Yoneda pairing $$\label{eq4.3}\Upsilon:{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)\to {\mathcal{E} }xt^2_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_1).$$
Let $\pi_S:S\to R^{ss}$ denote the blow-up of $R^{ss}$ along $\Sigma_R^{ss}$ and let $\Sigma_S$ be the exceptional divisor of $\pi_S$ while $\Omega_S$ (resp. $\Delta_S$) denotes the proper transform of $\Omega_R$ (resp. $\Delta_R$). By , we have $$\label{eq4.4} {\Sigma_S}
|_{\pi_S^{-1}(\Sigma_R^0)}\cong q^*{\mathbb{P}}\Upsilon^{-1}(0){/\!\!/}O(2)\subset q^*{\mathbb{P}}[{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)]{/\!\!/}O(2).$$
By [@og97] (1.8.10), $S^s=S^{ss}$ and $S^s$ is smooth. The quotient $S{/\!\!/}PGL(N)$ has only ${\mathbb{Z}}_2$-quotient singularities along $\Delta_S{/\!\!/}PGL(N)$. Let $\pi_T:T\to S^{s}$ be the blow-up of $S^{s}$ along $\Delta_S^{s}$. Then $T{/\!\!/}PGL(N)$ is nonsingular and this is Kirwan’s desingularization $\rho:{\widehat{M} }_{2n}\to M_{2n}$.
Let $\Omega_T$ and $\Sigma_T$ denote the proper transforms of $\Omega_S$ and $\Sigma_S$ respectively. Let $\Delta_T$ be the exceptional divisor of $\pi_T$. Their quotients $\Omega_T{/\!\!/}PGL(N)$, $\Sigma_T{/\!\!/}PGL(N)$ and $\Delta_T{/\!\!/}PGL(N)$ are denoted by $D_1=\hat\Omega$, $D_2=\hat\Sigma$ and $D_3=\hat\Delta$ respectively.
With this preparation, we now embark on the proof of Proposition \[prop:analysis on exc\].
**Proof of (1).** This is just [@og97] (3.0.1). More precisely, by and Corollary \[4.3\], $\Omega_S$ is the blow-up of $$q^*{\mathbb{P}}\mathrm{Hom}^{\omega}(W,T_{X^{[n]}}){/\!\!/}SO(W)\text{ along
}q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}}){/\!\!/}SO(W).$$ By , $\Omega_T$ is the blow-up of $\Omega_S$ along the proper transform of $$q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}}){/\!\!/}SO(W)$$ and $D_1=\hat\Omega$ is the quotient of $\Omega_T$ by the action of $PGL(N)$. Since the action of $PGL(N)$ commutes with the action of $SO(W)$, $D_1$ is in fact the quotient by $SO(W)\times PGL(N)$ of the variety obtained from $q^*{\mathbb{P}}\mathrm{Hom}^{\omega}(W,T_{X^{[n]}})$ by two blow-ups. So $D_1$ is also the consequence of taking the quotient by $PGL(N)$ first and then the quotient by $SO(W)$ second. Since $q$ is a principal $PGL(N)$ bundle, the result of the first quotient is just $Bl_{Bl_2^T}Bl^T$ in which is isomorphic to $Bl^{\mathcal{B} }$. If we take further the quotient by $SO(W)$, then as discussed above the result is $D_1=\hat{\mathbb{P}}(S^2{\mathcal{B} })$.
**Proof of (2).** We use Lemma \[4.4\], , and . Note that $\Sigma_R^0$ does not intersect with $\Omega_R$ and $\Delta_R$. Hence $D_2^0$ is the quotient of $q^*{\mathbb{P}}\Upsilon^{-1}(0){/\!\!/}O(2)$ which is a subset of $q^*{\mathbb{P}}[{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)]{/\!\!/}O(2)$, by the action of $PGL(N)$. The above are bundles over the restriction of $${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k)){/\!\!/}O(2)$$ to the complement $\mathcal{X}^{[n]}_0$ of the diagonal $\mathbf{\Delta}$ in $X^{[n]}\times X^{[n]}$. As in the proof of (1), observe that $D_2^0$ is in fact the quotient of $q^*{\mathbb{P}}\Upsilon^{-1}(0)$ by the action of $PGL(N)\times O(2)$ since the actions commute. So we can first take the quotient by the action of $PGL(N)$, then by the action of $SO(2)$, and finally by the action of ${\mathbb{Z}}_2=O(2)/SO(2)$. Since ${\mathbb{P}}\mathrm{Isom}({\mathbb{C}}^N,p_*{\mathcal{L} }_1(k)\oplus p_*{\mathcal{L} }_2(k))$ is a principal $PGL(N)$-bundle, the quotient by $PGL(N)$ gives us $${\mathbb{P}}\Upsilon^{-1}(0)\subset {\mathbb{P}}[{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)]$$ over $\mathcal{X}^{[n]}_0$. The algebraic vector bundles ${\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)$ and ${\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)$ are certainly Zariski locally trivial and in fact these bundles are dual to each other by the Yoneda pairing $\Upsilon$ which is non-degenerate (possibly after tensoring with a line bundle). In particular, $\Upsilon^{-1}(0)$ is Zariski locally trivial.
Next we take the quotient by the action of $SO(2)\cong {\mathbb{C}}^*$. This action is trivial on the base $\mathcal{X}^{[n]}_0$ and $SO(2)$ acts on the fibers. Hence ${\mathbb{P}}\Upsilon^{-1}(0)/SO(2)$ is a Zariski locally trivial subbundle of $${\mathbb{P}}[{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\oplus {\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)]{/\!\!/}{\mathbb{C}}^*\cong {\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\times_{\mathcal{X}^{[n]}_0}
{\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)$$ over $\mathcal{X}^{[n]}_0$ given by the incidence relations in terms of the identification $${\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\cong {\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)^\vee.$$ Finally, $D_2^0$ is the ${\mathbb{Z}}_2$-quotient of ${\mathbb{P}}\Upsilon^{-1}(0)/SO(2)$.
**Proof of (3).** By [@og97] (1.7.10), the intersection of $\Sigma_R^{ss}$ and $\Omega_R$ is smooth. By Corollary \[4.3\], $\Delta_S$ is the blow-up of $q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}}){/\!\!/}SO(W)$ along $q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}}){/\!\!/}SO(W)$. Hence $\Delta_S{/\!\!/}PGL(N)$ is the quotient of $$Bl_{q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_1
(W,T_{X^{[n]}})}q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})$$ by the action of $SO(W)\times PGL(N)$. By taking the quotient by the action of $PGL(N)$ we get $$Bl_{ {\mathbb{P}}\mathrm{Hom}^{\omega}_1
(W,T_{X^{[n]}})} {\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}})$$since $q$ is a principal $PGL(N)$-bundle. Next we take the quotient by the action of $SO(W)$. Let $\mathrm{Gr}^\omega(2,T_{X^{[n]}})$ be the relative Grassmannian of isotropic 2-dimensional subspaces in $T_{X^{[n]}}$ and let $\mathcal A$ be the tautological rank 2 bundle on $\mathrm{Gr}^\omega(2,T_{X^{[n]}})$. We claim $$\label{}Bl_{{\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}})}
{{\mathbb{P}}{\mathrm{Hom}}_2^\omega(W,T_{X^{[n]}})}{/\!\!/}SO(W)\simeq {\mathbb{P}}(S^2{\mathcal{A} })$$ which is a ${\mathbb{P}}^{2}$-bundle over a ${\mathrm{Gr}}^\omega (2,2n)$-bundle over $X^{[n]}$. It is obvious that the bundles are Zariski locally trivial.
There are forgetful maps $$\begin{aligned}
f:{\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{A} })\to
{\mathbb{P}}{\mathrm{Hom}}_2^\omega(W,T_{X^{[n]}}) \\ f_1:{\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{A} })\to
{\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}}) \end{aligned}$$ where the subscript 1 denotes the locus of rank $\leq1$ homomorphisms. Because the ideal of ${\mathbb{P}}{\mathrm{Hom}}^\omega_1(W,T_{[n]})$ pulls back to the ideal of ${\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{A} })$, $f$ lifts to $$\hat
f:Bl_{{\mathbb{P}}{\mathrm{Hom}}_1(W,{\mathcal{A} })} {\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{A} })\to
Bl_{{\mathbb{P}}{\mathrm{Hom}}_1^\omega(W,T_{X^{[n]}})}
{{\mathbb{P}}{\mathrm{Hom}}_2^\omega(W,T_{X^{[n]}})}$$ This map is bijective ([@og97] (3.5.1)) and hence $\hat f$ is an isomorphism by Zariski’s main theorem because the varieties are smooth. Now observe that the quotient ${\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{A} }){/\!\!/}SO(W)$ is ${\mathbb{P}}(S^2{\mathcal{A} })$ where the quotient map is given by $[\alpha]\mapsto
[\alpha\circ\alpha^t]$. Hence $\Delta_S{/\!\!/}PGL(N) $ is the blow-up of ${\mathbb{P}}{\mathrm{Hom}}(W,{\mathcal{A} }){/\!\!/}SO(W)\cong{\mathbb{P}}(S^2{\mathcal{A} })$ along the locus of rank 1 quadratic forms ${\mathbb{P}}(S^2_1{\mathcal{A} })$ ([@k2] Lemma 3.11) which is a Cartier divisor. So we proved that $$\Delta_S{/\!\!/}PGL(N)\cong {\mathbb{P}}(S^2{\mathcal{A} }).$$
Finally $S{/\!\!/}PGL(N)$ is singular only along $\Delta_S{/\!\!/}PGL(N)$ and the singularities are ${\mathbb{C}}^{2n-3}/\{\pm
1\}$ by Luna’s slice theorem [@og97] (1.2.1). Since $D_3$ is the exceptional divisor of the blow-up of $S{/\!\!/}PGL(N)$ along $\Delta_S{/\!\!/}PGL(N)$, we conclude that $D_3$ is a ${\mathbb{P}}^{2n-4}$-bundle over ${\mathbb{P}}(S^2\mathcal A)$.
**Proof of (4).** By Corollary \[4.3\], $\Sigma_S^s\cap\Omega_S$ is the exceptional divisor of the blow-up $Bl_{q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_1(W,T_{X^{[n]}})}q^*{\mathbb{P}}\mathrm{Hom}^{\omega}(W,T_{X^{[n]}}){/\!\!/}SO(W) $ and $\Sigma_T^s\cap \Omega_T$ is now the blow-up of the exceptional divisor along the proper transform of $q^*{\mathbb{P}}\mathrm{Hom}^{\omega}_2(W,T_{X^{[n]}}){/\!\!/}SO(W)$. Using the isomorphism , this is the exceptional divisor of $$q^*Bl_{{\mathbb{P}}(S^2_1{\mathcal{B} })}{\mathbb{P}}(S^2{\mathcal{B} })\to q^*{\mathbb{P}}(S^2{\mathcal{B} })$$ over $\mathrm{Gr}^\omega(3,T_{X^{[n]}})$. Since $q$ is a principal $PGL(N)$-bundle, $D_1\cap D_2=\Sigma_T^s\cap \Omega_T{/\!\!/}PGL(N)$ is the exceptional divisor of the blow-up $Bl_{{\mathbb{P}}(S^2_1{\mathcal{B} })}{\mathbb{P}}(S^2{\mathcal{B} })$. Because the exceptional divisor is a Zariski locally trivial ${\mathbb{P}}^2 $-bundle over ${\mathbb{P}}(S^2_1{\mathcal{B} })$ and ${\mathbb{P}}(S^2_1{\mathcal{B} })$ itself is a Zariski locally trivial ${\mathbb{P}}^2$-bundle over $\mathrm{Gr}^\omega(3,T_{X^{[n]}})$, we proved (4).
**Proof of (5).** From the above proof of (3) it follows immediately that $\Sigma_S^s\cap \Delta_S{/\!\!/}PGL(N)$ is ${\mathbb{P}}(S^2_1{\mathcal{A} })$ and $D_2\cap D_3$ is a ${\mathbb{P}}^{2n-4}$ bundle over ${\mathbb{P}}(S^2_1{\mathcal{A} })$ which is Zariski locally trivial.
**Proof of (6).** As in the above proof of (4), we start with and use the isomorphism to see that $D_1\cap D_3$ is the proper transform of ${\mathbb{P}}(S^2_2{\mathcal{B} })$ in the blow-up $Bl_{{\mathbb{P}}(S^2_1{\mathcal{B} })}{\mathbb{P}}(S^2{\mathcal{B} })$. This is a Zariski locally trivial ${\mathbb{P}}^2$-bundle over a Zariski locally trivially ${\mathbb{P}}^2$-bundle over $\mathrm{Gr}^\omega(3,T_{X^{[n]}})$.
**Proof of (7).** This follows immediately from the proof of (4) and (6).
From the above descriptions, it is clear that $D_i$ ($i=1,2,3$) are normal crossing smooth divisors.
Hodge-Deligne polynomial of $D_2^0 $ {#sec: Computation of E-poly of D_0^2}
====================================
In this section we prove Lemma \[lem: Hodge Deligne poly of D02\]. Recall $$I_{2n-3}=\{((x_i),(y_j))\in {\mathbb{P}}^{2n-3}\times
{\mathbb{P}}^{2n-3}\,|\, \sum_{i=0}^{2n-3} x_iy_i=0\}.$$ It is elementary ([@GH78] p. 606) to see that $$H^*(I_{2n-3};{\mathbb{Q}})\cong {\mathbb{Q}}[a,b]/\langle a^{2n-2}, b^{2n-2},
a^{2n-3}+a^{2n-4}b+a^{2n-5}b^2+\cdots+b^{2n-3} \rangle$$ where $a$ (resp. $b$) is the pull-back of the first Chern class of the tautological line bundle of the first (resp. second) ${\mathbb{P}}^{2n-3}$. The ${\mathbb{Z}}_2$-action interchanges $a$ and $b$ and the invariant subspace of $H^*(I_{2n-3};{\mathbb{Q}})$ is generated by classes of the form $a^ib^j+a^jb^i$. As a vector space $H^*(I_{2n-3};{\mathbb{Q}})$ is $$\label{eq5.0}{\mathbb{Q}}\text{-span}\{a^ib^j\,|\, 0\le i\le 2n-3, 0\le j\le
2n-4\}$$ while the invariant subspace is $${\mathbb{Q}}\text{-span}\{a^ib^j+a^jb^i\,|\, 0\le i\le j\le 2n-4\}.$$ The index set $\{(i,j)\,|\, 0\le i\le j\le 2n-4\}$ is mapped to its complement in $\{(i,j)\,|\, 0\le i\le 2n-3, 0\le j\le 2n-4\}$ by the map $(i,j)\mapsto (j+1,i)$. This immediately implies that $$\label{eq5.3}
P(I_{2n-3};z)=(1+z^2)P^+(I_{2n-3};z)$$ By or the observation that $I_{2n-3}$ is the Zariski locally trivial ${\mathbb{P}}^{2n-4}$-bundle over ${\mathbb{P}}^{2n-3}$, we have $$\label{eq5.2}P(I_{2n-3};z)=\frac{1-(z^2)^{2n-2}}{1-z^2}\cdot
\frac{1-(z^2)^{2n-3}}{1-z^2}.$$ Because $1+z^2$ divides $\frac{1-(z^2)^{2n-2}}{1-z^2}$, $\frac{1-(z^2)^{2n-3}}{1-z^2}$ also divides $P^+(I_{2n-3};z)$. Therefore, (\[eqn: E D02 is divisible by some Q\]) is a direct consequence of since $P(X^{[n]};z)$ has no odd degree terms by .
Now let us prove . Let $$\psi:\widetilde{D}_2^0:={\mathbb{P}}\Upsilon^{-1}(0)/SO(2)\to
\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$$ be the Zariski locally trivial $I_{2n-3}$-bundle in the proof of Proposition \[prop:analysis on exc\] (2) in §4. Recall that $D_2^0=\widetilde{D}_2^0/{\mathbb{Z}}_2$. We have seen in the proof of Proposition \[prop:analysis on exc\] (2) in §4 that there is a ${\mathbb{Z}}_2$-equivariant embedding $$\imath:\widetilde{D}_2^0\hookrightarrow
{\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)\times_{\mathcal{X}^{[n]}_0}
{\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)$$ where the ${\mathbb{Z}}_2$-action interchanges ${\mathcal{L} }_1$ and ${\mathcal{L} }_2$.
Let $\lambda$ (resp. $\eta$) be the pull-back to $\widetilde{D}_2^0$ of the first Chern class of the tautological line bundle over ${\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_1,{\mathcal{L} }_2)$ (resp. ${\mathbb{P}}{\mathcal{E} }xt^1_{\mathcal{X}^{[n]}_0}({\mathcal{L} }_2,{\mathcal{L} }_1)$). By definition, $\lambda$ and $\eta$ restrict to $a$ and $b$ respectively. The ${\mathbb{Z}}_2$-action interchanges $\lambda$ and $\eta$. By the Leray-Hirsch theorem[^3] we have an isomorphism $$\label{eq5.5} H^*_c(\widetilde{D}_2^0)\ \
\cong \ H^*_c(\mathcal{X}^{[n]}_0)\otimes
H^*(I_{2n-3}).$$ As the pull-back and the cup product preserve mixed Hodge structure, determines the mixed Hodge structure of $H^*_c(\widetilde{D}_2^0)$. The ${\mathbb{Z}}_2$-invariant part is $$\label{eq5.6}
H^*_c(\widetilde{D}_2^0)^+\cong \left(
H^*_c(\mathcal{X}^{[n]}_0)^+\otimes H^*(I_{2n-3})^+\right) \oplus
\left( H^*_c(\mathcal{X}^{[n]}_0)^-\otimes H^*(I_{2n-3})^-\right)$$ where the superscript $\pm$ denotes the $\pm
1$-eigenspace of the ${\mathbb{Z}}_2$-action. Because $H^*_c(D_2^0)\cong
H^*_c(\widetilde{D}_2^0/{\mathbb{Z}}_2)\cong H^*_c(\widetilde{D}_2^0)^+$ ([@Gr57] Theorem 5.3.1 and Proposition 5.2.3), $E(D_2^0;u,v)$ is equal to $$\label{eq5.7}
E^+(\widetilde{D}_2^0;u,v)=E^+(\mathcal{X}^{[n]}_0;u,v)E^+(I_{2n-3};u,v)
+E^-(\mathcal{X}^{[n]}_0;u,v)E^-(I_{2n-3};u,v).$$ where $E^\pm(Y;u,v)=\sum_{p,q}\sum_{k\geq0} (-1)^k
h^{p,q}(H^k_c(Y)^\pm) u^pv^q$.
It is easy to see $$\begin{aligned}
P^+(X^{[n]}\times
X^{[n]};z)=\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2,\\
{P^-(X^{[n]}\times
X^{[n]};z)=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2}\end{aligned}$$ (Macdonald’s formula). Since $X^{[n]}\times X^{[n]}$ is smooth projective, we have $$\begin{aligned}
E^+(X^{[n]}\times
X^{[n]};z,z)=\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2\\
E^-(X^{[n]}\times
X^{[n]};z,z)=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2\end{aligned}$$ Now as $\mathcal{X}^{[n]}_0=X^{[n]}\times X^{[n]}-\mathbf{\Delta}$ and $\mathbf{\Delta}\cong X^{[n]}$ is ${\mathbb{Z}}_2$-invariant, by the additive property of the E-polynomial we have $$\begin{aligned}
E^+(\mathcal{X}^{[n]}_0;z,z)=E^+(X^{[n]}\times X^{[n]};z,z)
-E(X^{[n]};z,z)\\ =
\frac{P(X^{[n]};z)^2+P(X^{[n]};z^2)}2-P(X^{[n]};z),\end{aligned}$$ $$\begin{aligned}
E^-(\mathcal{X}^{[n]}_0;z,z)=E^-(X^{[n]}\times X^{[n]};z,z)\\
=\frac{P(X^{[n]};z)^2-P(X^{[n]};z^2)}2.\end{aligned}$$ The equation is an immediate consequence of the above equations and .
[10]{}
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[^1]: Young-Hoon Kiem was partially supported by KOSEF R01-2003-000-11634-0; Jaeyoo Choy was partially supported by KRF 2003-070-C00001
[^2]: In fact the term prior to the first term of is ${\mathscr H\!om}_{\Omega_Q}({\mathcal{E} },{\mathcal{E} })$ which contains ${\mathcal{O} }$ obviously and the quotient of ${\mathscr H\!om}_{\Omega_Q}({\mathcal{O} }^{\oplus N}(-k),{\mathcal{E} })$ by ${\mathcal{O} }$ is a vector bundle whose fiber is the Lie algebra of $PGL(N)$.
[^3]: The Leray-Hirsch theorem in [@V02I] p.182 is stated for ordinary cohomology but the statement holds also for compact support cohomology. See the proof in [@V02I] p.195
|
---
author:
- 'F. Aharonian, A. Akhperjanian, M. Beilicke, K. Bernlöhr, H. Börst, H. Bojahr, O. Bolz, T. Coarasa, J. Contreras, J. Cortina, L. Costamante, S. Denninghoff, V. Fonseca, M. Girma, N. Götting, G. Heinzelmann, G. Hermann, A. Heusler, W. Hofmann, D. Horns, I. Jung, R. Kankanyan, M. Kestel, J. Kettler, A. Kohnle, A. Konopelko, H. Kornmeyer, D. Kranich, H. Krawczynski, H. Lampeitl, M. Lopez, E. Lorenz, F. Lucarelli, O. Mang, H. Meyer, R. Mirzoyan, M. Milite, A. Moralejo, E. Ona, M. Panter, A. Plyasheshnikov, G. Pühlhofer, G. Rauterberg, R. Reyes, W. Rhode, J. Ripken, G. Rowell, V. Sahakian, M. Samorski, M. Schilling, M. Siems, D. Sobzynska, W. Stamm, M. Tluczykont, H.J. Völk, C. A. Wiedner, W. Wittek, and R. A. Remillard'
date:
- 'Received / Accepted'
- 'Received / Accepted'
title: Variations of the TeV energy spectrum at different flux levels of Mkn 421 observed with the HEGRA system of Cherenkov telescopes
---
Introduction
============
The nearby BL Lac object Mkn 421 ($z=0.031$) was the first extra-galactic source of very-high energy gamma-ray emission detected [@1992Natur.358..477P]. The source was monitored closely with the HEGRA imaging Cherenkov telescopes [@1999APh....10..275H] during the observational periods from 1997-1998 while the source remained in a low flux state with an average flux level of a third of the flux observed from the and a power law energy spectrum with a photon index of $3.09\pm0.07$ . It is well known that the source exhibits time variability on a sub-hour timescale at TeV-energies [@1996Natur.383..319G], which constrains the size of the emission region ($R<10^{15}\,\mbox{cm}
(\delta/10)\,(T_{\rm var}/1\,{\rm hr})$) and indicates a relativistic bulk motion of the emitting plasma with a Doppler factor $\delta=(\Gamma-\sqrt{\Gamma^2-1}\cos\theta)^{-1}>10$ at an angle $\theta$ to the line of sight to avoid severe internal absorption [@1998MNRAS.293..239C].
The observed variations of the source in the optical and radio energy region have been found to correlate only marginally with the TeV flux [@2001AAS...199.9817N; @ICRC2001...jordan]. However, the X-ray activity shows good correlation with the TeV flux level. During the observational periods December 1999 until May 2001, extended multi-wavelength observation campaigns were conducted by X-ray satellites (RXTE, BeppoSAX) and TeV observatories (HEGRA, CAT and VERITAS collaborations). During coordinated simultaneous observations in January/February 2001 with RXTE, Mkn 421 showed a soft X-ray spectrum consistent with a peak position of the synchrotron spectrum below $3$ keV. The X-ray flux integrated between 2-10 keV reached a value of $1.6\cdot10^{-9}$ erg/(cm$^2$s) on January 31, 2001 [@ICRC2001...horns] where Mkn 421 showed also a high TeV flux level. This is the highest X-ray state for Mkn 421 reported during a pointed observation. Interpretation of the broadband spectral energy distribution (SED) in the framework of synchrotron-self-Compton models is discussed in Sect. \[MWL\].
The TeV energy spectrum of Mkn 421 has been measured by several instruments (HEGRA, Whipple, CAT) at different flux levels of the source . The observed spectral shape shows tentative indications for variations and a possible correlation with the overall flux level. However, since the instruments have largely different energy thresholds and systematic uncertainties, a firm claim of spectral variations of the TeV emission from Mkn 421 could so far not be made. For Mkn 501, observations of a spectral hardening during the 1997 outbursts in TeV emission have been claimed , but could not be verified by other experiments [@2001ApJ...546..898A]. The TeV-spectrum of Mkn 501 softened during the subsequent years following the strong outburst while the mean flux has decreased by a factor of 10 to $1/3$ of the flux observed from the Crab-Nebula [@2001ApJ...546..898A].
Mkn 421 exhibited strong activity from January to May 2001 [@2001IAUC.7568....3B] with a peak diurnal flux level on March 23/24, 2001 (MJD 51991.9-51992.2) with a value of $(12.5\pm0.4)$ above 1 TeV corresponding to $7.4$ times the flux of the Crab-Nebula. Only 6 days before reaching the peak flux, the diurnal flux had been as low as $(0.5\pm0.1)$ , a factor of 25 lower.
We report on observations of Mkn 421 carried out with the HEGRA system of 5 imaging air Cherenkov telescopes. Sects. \[lightcurve\] & \[spectrum\] contain the results on flux and spectral shape of Mkn 421 during the observational periods from December 1999 until May 2001. We show clear evidence for variations of the spectral shape of a TeV $\gamma$-ray source, getting harder with increased flux level (Sect. \[sec:corr\]). Intranight flux and spectral variations are evident in at least two nights of observations (Sects. \[diurnal:flux\] & \[diurnal:spectrum\]). The difference between cut-off energies present in the TeV spectra of Mkn 421 and Mkn 501 is discussed in Sect. \[subsection:differences\]
------------------------- ------ ---------- -------------------------
Date Year Obs.Time $\langle \theta\rangle$
$[hrs]$ $[^\circ]$
December 3-11 1999 4.8 17.8
January 31-February 15 2000 58.4 20.9
March 3-March 12 2000 17.9 13.2
March 24-March 29 2000 28.4 19.8
April 23-May 5 2000 21.6 17.1
November 24-December 6 2000 20.8 30.5
December 28 2000 2.0 16.4
January 17-February 5 2001 94.2 21.2
February 13-February 28 2001 63.2 20.5
March 17-March 31 2001 51.6 17.8
April 14-April 26 2001 8.1 20.6
May 9- May 23 2001 14.9 20.7
Total 385.9 20.3
------------------------- ------ ---------- -------------------------
: \[table:obstime\] Individual observation times, and mean zenith angles $\langle \theta \rangle$ are listed for the different observational periods.
Data selection and analyses
===========================
{width="1.30\vsize"} {width="1.30\vsize"}
[lcccccc]{} Period
------------------------------------------------------------------------
& $T_{obs}$ & &$N_0$ & $\alpha$ & $E_0$ & $\chi^2_\mathrm{red}(d.o.f.)$[^1]\
& $[hrs]$ & $\left[\frac{10^{-11}}{\mbox{cm}^2\,\mbox{s}}\right]$& $\left[\frac{10^{-11}}{\mbox{cm}^2\,\mbox{s}\,\mbox{TeV}}\right]$ & & $[$TeV$]$ &\
& & &Fixed $E_0$&\
Dec 1999-May 2000 & 131.1 & $1.43\pm0.05$ & $4.3\pm0.3$ &$2.39\pm0.09$ &3.6 & $0.8(8)$\
Nov 2000-May 2001 & 254.8 & $4.19\pm0.04$ & $11.4\pm0.3$ &$2.19\pm0.02$ &3.6 & $1.2(8)$\
$0<F_{-11}<1$[^2] & 42.9 & $0.53\pm0.04$ & $2.2\pm0.3$ &$3.0\pm0.2$ &3.6 & $0.3(3)$\
$1<F_{-11}<2$ & 77.4 & $1.55\pm0.04$ & $5.2\pm0.2$ &$2.47\pm0.08$ &3.6 & $0.9(6)$\
$2<F_{-11}<4$ & 54.1 & $2.76\pm0.05$ & $8.8\pm0.3$ &$2.36\pm0.05$ &3.6 & $0.9(6)$\
$4<F_{-11}<8$ & 129.4 & $4.22\pm0.03$ & $12.1\pm0.2$ &$2.18\pm0.02$ &3.6 & $0.8(6)$\
$8<F_{-11}<16$& 17.8 & $10.3\pm0.1$ & $27.3\pm0.5$ &$2.06\pm0.03$ &3.6 & $0.5(6)$\
MJD51964/51965 & 3.6 & $10.6\pm0.4$& $32\pm1$ &$2.36\pm0.06$ &3.6 & $0.9(6)$\
MJD51991/51992 & 4.1 & $12.5\pm0.4$& $31\pm1$ &$2.04\pm0.05$ &3.6 & $1.2(6)$\
& & & *no cut-off* &\
Dec 1999-May 2000 & & & $3.8\pm0.2$ &$3.19\pm0.04$ & & $2.9(8)$\
Nov 2000-May 2001 & & & $10.1\pm0.1$ &$3.05\pm0.01$ & & $53(8)$\
MJD51989/51990,preflare & 1.9 & $3.6\pm0.3$ & $8.1\pm0.7$ & $2.9\pm0.1$ & &$0.5(6)$\
MJD51989/51990,flare & 1.1 & $7.54\pm0.05$ & $12.0\pm1.0$ & $2.2\pm0.1$ & &$1.0(5)$\
MJD51990/51991,preflare & 1.5 & $3.1\pm0.4$ & $8.3\pm0.8$ & $3.2\pm0.2$ & &$0.4(5)$\
MJD51990/51991,flare & 1.0 & $10.0\pm0.7$ & $15.2\pm0.9$ & $2.5\pm0.1$ & &$0.7(5)$\
The data used here were collected from December 1999 until May 2001 and comprises in total 385.9 hours of observations under stable weather conditions, good atmospheric transparency and good detector performance (see Table \[table:obstime\] for a breakdown of the observation time for individual months). Data taken up to zenith angles of $45^\circ$ are used which constitute more than 95% of the data taken in total.
Based upon comparing the observed rate with the expected cosmic ray event rate, individual runs were rejected if the relative deviation from the expectation exceeds 20%. In total 35hrs of data fail to pass this selection.\
A large fraction of data were taken with a 5-telescope setup. However, in January 2000 and April 2001, only 4 telescopes were operating. The collection areas for the spectral analysis of data taken with a 4 or 5 telescope setup are calculated separately to take the difference in acceptance between the two setups into account. Additionally, the degrading mirror reflectivity and the conversion factors relating the registered digitised Cherenkov amplitude values in the cameras to the number of photoelectrons detected have been determined for monthly observational periods and are applied to the Monte-Carlo simulation to calculate the collection areas for each period separately.
All observations were carried out in the so-called *wobble*-mode. During *wobble*-mode observations the source direction is positioned $\pm0\fdg5$ shifted in declination with respect to the centre of the field of view of the camera. This observation mode allows for simultaneous estimate of the background (OFF) rate induced by charged cosmic rays. The analysis uses an extended OFF-region to reduce the statistical error on the background estimate. A ring segment ($180^\circ$ opening angle), from $0\fdg3$ to $0\fdg7$ distance from the camera centre at the opposite side of the ON region has been chosen. The events suitable for spectral analysis are selected in a similar way as described in , applying only loose cuts on the image shape and the shower direction in order to minimise systematic effects from energy dependent cut efficiencies. The loose directional and shape cuts accept events with the reconstructed direction being within an angular distance of $0\fdg22$ to the source and the image shape parameter *mean scaled width* $mscw<1.2$. 80% of the registered $\gamma$-events pass both the directional and the shape cut. To ensure that the images are not truncated because of the limited field of view, images with a $distance>1\fdg7$ to the centre of the camera are rejected. At the same time the reconstructed core position is required to be within 200 m distance to the central telescope. The energy reconstruction requires two images with more than 40 photoelectrons after applying a two-stage tail-cut removing night sky background contamination of the image. The angle subtended between the major axes of at least two of the registered images is required to be larger than $20^\circ$. The number of excess events after applying all selection criteria used in this analysis amounts to $\approx$ 40000, constituting the largest photon sample collected from a single source with the HEGRA telescopes.
Spectral analysis method
------------------------
The energy reconstruction and the reconstruction of energy spectra have been described in detail elsewhere . For a large part of the data, contemporaneous observations of the Crab-Nebula[^3] are available, allowing to compare and to verify the expectation for detection rates and cut-efficiencies for $\gamma$-induced air showers derived from Monte-Carlo simulations with data.
In the possible presence of a deviation from a pure power law in the form of an exponential cut-off, the reconstruction of the spectral shape beyond this cut-off is influenced by the energy resolution. Even with an event-by-event energy reconstruction accuracy of $\Delta E/E\approx 20\,\%$ as achieved with the stereoscopic reconstruction technique of air showers, a considerable fraction of events with overestimated energies contribute to the flux of neighbouring bins.
This becomes increasingly important for energies well in excess of the cut-off energy. Different methods like deconvolution using suitable algorithms and forward-folding techniques are applicable and have been pursued. An advanced energy reconstruction technique with an improved energy resolution of $\approx 10\,\%$ [@2000APh....12..207H] has been proven to show similar results as obtained with the analysis technique used here . However, the results presented here are obtained with the conventional energy reconstruction method being less sensitive to changes in the detector performance and therefore less susceptible to systematic errors.
The influence of the energy resolution has been absorbed in the collection area $A(E_{\rm reco},\theta)$, $\theta$ indicating the zenith angle. The method applied here follows the same approach as described in , using a collection area $A=A(E_{\rm reco.},\theta)$ being a function of the reconstructed energy $E_{\rm reco}$. The collection area therefore depends upon the spectral shape assumed. An iterative method is applied to test a spectral hypothesis on the data. The procedure starts with an assumed arbitrary spectral shape. For the sake of simplicity, a Crab-like power law spectrum with a photon index of $2.6$ is assumed. If the reconstructed spectrum shows deviations from this shape, a new hypothesis can be tested by using a different spectral shape to derive the collection area and following the same procedure once more. The process is iterative and usually converges after two iterations.
The method has been carefully checked with simulated energy spectra and also against other methods and has proven to be less dependent upon the exact modelling of the detector response with Monte-Carlo methods than more sophisticated procedures as mentioned above.
Systematic uncertainties {#sect:systematics}
------------------------
The systematic errors include as most important contributions the uncertainties in the conversion factors used to calculate the number of photo-electrons from the digitised pulses of the photo multiplier tubes, non-linearities of the electronic chain and the conservatively estimated variations of the response of the telescopes in the threshold region. Additionally, the uncertainties in the spectral shape enter the calculation of the collection area and cause an additional systematic uncertainty especially at energies well above the exponential cut-off energy.
Besides the mentioned uncertainties, the overall flux determination is subject to a $15\,\%$ uncertainty in the calibration of the absolute energy scale of the telescopes. This uncertainty only affects the absolute flux calibration and does not influence the reconstructed shape of the spectrum. The stability of the calibration of the energy scale has been carefully tested with data taken on the Crab-Nebula constraining possible time dependent effects seen as variations on the reconstructed flux to be less than $10$% which translates into less than $6$% uncertainty on changes of the energy scale (assuming a power law energy spectrum with a photon index of $2.6$). A more in-depth discussion of the systematic errors is given in .
Results
=======
Light curve {#lightcurve}
-----------
The observed integral flux above an energy of 1 TeV for individual nights is shown in Fig. \[plot:lcurve\]. In the year 2000, the highest diurnal flux level reached $(6.4\pm0.3)$ (MJD 51662/51663). Note the smooth increase and decrease of the flux level in the course of a few days observed from MJD 51659 until MJD 51665 (May 2000). The flux averaged from December 1999 to May 2000 is $(1.43\pm0.04_{\rm stat}\pm0.2_{\rm sys})$ . In the observational period from November 2000 until May 2001 the average flux and the amplitude of variations increased. The maximum diurnally averaged flux was observed in MJD 51991/51992 (March 23/24 2002) with $(12.5\pm0.4)$ (see also Sect. \[diurnal:spectrum\]). The time averaged flux for the 2000/2001 measurements is $(4.19\pm0.04_{\rm stat}\pm0.4_{\rm sys})$ , which corresponds to 2.4 times the flux of the Crab-Nebula (the integral flux above 1 TeV observed from the Crab-Nebula is $(1.76\pm0.06_{\rm stat}\pm0.51_{\rm sys})$ [@2000ApJ...539..317A]).
Time averaged energy spectra {#spectrum}
----------------------------
The observational periods 1999/2000 and 2000/2001 are treated separately for two reasons: The average flux level and the amplitude of variability is largely different for the two seasons (see Fig. \[plot:lcurve\]). Secondly, the two measurements are separated by one year and allow to investigate secular changes by comparing the energy spectra for the two observational seasons
A fit of a power law function $dN/dE=N_0 (E/\mbox{TeV})^{-\alpha}$ to the energy spectrum derived from the observations in the years 1999/2000 results in $N_0=(3.8\pm0.2_{\rm stat}\pm0.4_{\rm sys})$ , $\alpha=3.19\pm0.04_{\rm stat}\pm0.04_{\rm sys}$ with $\chi^2_\mathrm{red}(d.o.f.)=2.9(8)$. This makes a power law fit seem unlikely given a probability of $5\cdot10^{-3}$ for a larger $\chi^2$-value. The soft spectral shape is consistent with previous measurements . A fit of a power law with an exponential cut-off \[eqn:fit\] &=& N\_0 ()\^[-]{}(-) results in a fit to the data with $N_0=4.3\pm0.3_\mathrm{stat}\pm0.5_\mathrm{sys}$ , $\alpha=2.5\pm0.1_\mathrm{stat}\pm0.04_\mathrm{sys}$, $E_0=3.8\,{+5\choose -1}_\mathrm{stat}\,{+0.9 \choose -0.8}_\mathrm{sys}$ TeV and $\chi^2_\mathrm{red}(d.o.f.)=0.9(7)$. The large statistical error on the position of the cut-off is a result of the strong correlation of the two parameters ($\alpha,E_0$) used to characterise the spectral shape.
The energy spectrum derived from observations in the period from November 2000 until May 2001 with a substantially larger number of photons and therefore reduced statistical fluctuation is not consistent with a pure power law function ($\chi^2=424$ with 8 degrees of freedom). A fit of a power law function with an exponential cut-off results in a smaller $\chi^2$-value of $8.8$ with 7 degrees of freedom and $N_0=11.4\pm0.3_\mathrm{stat}\pm0.4_\mathrm{sys}$, $\alpha=2.19\pm0.02_\mathrm{stat}\pm0.04_\mathrm{sys}$, $E_0=3.6\,{+0.4\choose-0.3}_\mathrm{stat}\,{+0.9\choose -0.8}_\mathrm{sys}$ TeV. Table \[table:flux\] shows the individual results on the power law fit and fitting a power law with an exponential cut-off to different time periods and flux levels.
The energy spectra derived individually from the two observational periods are presented in Fig. \[plot:time\_averaged\_spectrum\]. The differential flux is multiplied by $E^2$ to emphasise the subtle differences between the two spectra[^4]. Provided that the exponential cut-off is at the same energy for both years ($E_0=3.6$ TeV), the ratio of the two spectra should follow a power law. The lower panel of Fig. \[plot:time\_averaged\_spectrum\] shows the ratio of the two energy spectra. Systematic effects that apply to both data sets in a similar manner cancel out for the ratio of the two spectra. As indicated with a solid line in the lower panel of Fig. \[plot:time\_averaged\_spectrum\], a power law fit is a good description of the ratio. The photon indices differ by $\Delta\alpha=0.17\pm0.07$ indicating a trend for the spectrum to become harder at a higher flux level (see also Sect. \[sec:corr\]) . The dashed curve is the result of a fit with an exponential as expected for different cut-off energies: f\_i &:=&\
&=& N\_i()\^[-\_i]{} (-E/E\_i)\
\[eqn:ratio\] f\_1/f\_2 && E\^[-]{}(-cE)\
&=& \_1-\_2\
c &=& E\_1\^[-2]{}-E\_2\^[-2]{} The result of a $\chi^2$-fit to $f_1/f_2$ (see Fig. \[fig:hilo\]) results in a difference $c=(0.04\pm0.07)$ TeV$^{-1}$ ($\Delta E=(0.5\pm0.8)$ TeV) which is consistent with the two cut-off energies being identical.
Correlation of flux and photon index {#sec:corr}
------------------------------------
To investigate possible correlations of the flux level and the spectral shape, the data have been split in five flux intervals as indicated in Table \[table:flux\]. The energy spectra for three of the flux bins (labelled **L**ow, **M**edium, and **H**igh) are shown in Fig. \[fig:hilo\]. The different flux levels show different spectral slopes. The ratio of the energy spectra is again well-described by a power law indicating that the cut-off energy for the different flux levels remains constant within the statistical error, whereas the photon index changes by $\Delta\alpha=0.31\pm0.04$ for the ratio of the **H** and **M**. For an average flux of $F_{-11}<1$ (**L**ow), the spectrum softens such that the spectral index changes by $\Delta \alpha=0.9\pm0.2$ with respect to **H**.
The general trend of hardening of the energy spectrum with increased flux is shown in Fig. \[fig:corri\] where the photon index $\alpha$ is shown as a function of integral flux above 1 TeV. To parameterise the dependence of photon index and flux, a function of the form ()& = & \_[i=0]{}\^2a\_i\^[i]{}\_[10]{}()\
a\_0 &=& -2.70.1\
a\_1 &=& 1.10.3\
a\_2 &=& -0.40.2\
with $\chi^2_\mathrm{red}(d.o.f.)=1.1(2)$ represents the data points quite well. A simpler functional dependence of the form () &=& b\_0+b\_1\_[10]{}()\
b\_0 &=& 2.70.1\
b\_1 &=& 0.470.02 is compatible with the data with a $\chi^2_\mathrm{red}(d.o.f.)=2.3(3)$ (indicated by the dashed curve in Fig. \[fig:corri\]). A $\chi^2$-fit of a constant value is excluded by a very large $\chi^2_\mathrm{red}(d.o.f.)=14(4)$.
It is interesting to note that the energy spectrum of Mkn 421 observed during 1997 and 1998 fits nicely in this picture when applying the same fitting method to the archival data keeping the cut-off energy fixed at $3.6$ TeV (see Fig. \[fig:corri\]). This data-point has been excluded from the fitting of the functional dependence of the photon index.
Diurnal flux variations {#diurnal:flux}
-----------------------
Intra-night variations of the flux are seen during a large fraction of the observational nights. In the past, doubling times as short as 15 minutes have been observed [@1996Natur.383..319G]. To quantify the rise and decay times of flares, a simple profile consisting of two exponentials with a constant background is fit to the light curves: \[eqn:lcurve\] F(t) &=& f\_0 + f\_1{
[ll]{} (\_(t-t\_0)) & tt\_0\
(-\_(t-t\_0))& t>t\_0\
. An example of a night with a fast decay time (March 21/22 2001, MJD51989.93-51990.14) is shown in Fig. \[fig:decayandrise\]a. The exponential decay time as given by the result of a $\chi^2$-fit of a function $F(t)$ (Eqn. \[eqn:lcurve\]) to the light-curve results in $\tau_\mathrm{decay}=15(+9-3)$ min, whereas the rise time turns out to be $\tau_\mathrm{rise}=46(+60-15)$ min. This result is indicating an asymmetry of the light curve during a flare. However, the presence of a possible sub-structure in the light curve (at $\approx$1:00 UTC) which is not resolved could be responsible for an overestimated rise time. The night with the fastest rise time (March 22/23 2001, MJD51990.925-51991.12) of the data set shows a smooth increase of the flux by a factor of 5 within 3 hours (see Fig. \[fig:decayandrise\]b). Fitting again a function $F(t)$ to the light-curve results in $\tau_{rise}=0.52\pm0.03$ hr. This translates into a doubling time of $21\pm2$ minutes. The decaying part of the light curve shows indications for a decay time of $4.5$ hr. However, a fit of a Gaussian function with a constant background (dashed line in Fig \[fig:decayandrise\]b) describes the data equally well and would increase the estimated doubling time to $\approx 30$ min.
Another example of a diurnal light curve is shown in Fig. \[fig:diurnal1\] (February 24/25 2001). The light curve of this night is well described by a simple exponential decay with $\tau_{decay}=4.46(+0.54-0.39)$ hrs similar to the decay time seen in the night March 23/24 2001 (see Fig. \[fig:decayandrise\]b). It is conceivable that the observed light curve in this particular night covers the end of a flare for which the first part was not visible for HEGRA. Another peculiarity of this night is the small and constant hardness ratio indicating a soft spectrum at a comparably high flux level (see also next Sect.).
Diurnal spectral variations {#diurnal:spectrum}
---------------------------
Even with the large number of photons detected, the sensitivity for intra-night spectral changes is limited. In a few cases, the observational time and the number of detected $\gamma$-rays is sufficient to perform spectroscopy on time scales of hours. A particular set of flares already mentioned in the previous Sect. occurred during the night MJD 51989.9-51990.2 (March 21/22 2001) (see Fig. \[fig:decayandrise\]a) and the subsequent night MJD 51990.9-51991.2 (March 22/23 2001) (see Fig. \[fig:decayandrise\]b). The light curves show in both cases a rising flux with a doubling time of less than 1hr. At the end of the flare, the flux smoothly decreased. During the rising time of the flux, the hardness ratios exhibit in both cases a spectral hardening, well correlated with the increase of flux and a spectral softening at the end of the flare. For the night March 21/22 the hardness ratio as well as the flux level return after the flare to the initial state it had prior to the flare. For the night March 22/23, the spectrum softens to the level it had before the flare, whereas the flux remains at a high level at a factor of 3 higher than before the flare.
To gain further information on the spectral shape for the individual nights with indications for spectral variations, energy spectra were extracted for two time intervals (indicated by the vertical lines in Fig \[fig:decayandrise\]a-b) from each night. The time intervals were chosen to cover the preflare and the flare state. For both flares, the preflare energy spectrum is well described by a power law with a photon index of $\approx3.0$, and a hardening by $\Delta \alpha=0.75\pm0.25$ during the flare (see Fig. \[fig:1989\_sp\]a-b).
However, light curves with flux variations have been observed that are not accompanied by spectral variations. An example of a night with a varying flux and a constant spectral shape is given in Fig. \[fig:1964\_sp\]a, where the energy spectra for two time intervals from the night MJD 51964/51965 have been extracted (marked by vertical lines in Fig. \[fig:diurnal1\]). The spectrum is despite the large flux ($F(E>1~\mathrm{TeV})=(10.6\pm0.4)$ ) comparably soft with a photon index of $2.85\pm0.09$ for the first half of the night and $2.95\pm0.09$ for the second half of the night.
It is also interesting to note after comparing individual nights, that the spectral shape does not always relate directly to the absolute flux-level. As an example, Fig. \[fig:1964\_sp\]b displays the spectrum of the night with the highest diurnal flux (MJD 51991/51992) with a level only 20% higher than the night February 24/25 discussed earlier. During this night, no strong indication for flux variability or changes in the hardness ratio are evident. The energy spectrum is hard with a photon index of $\alpha=2.04\pm0.05$ (Fig. \[fig:1964\_sp\]b). The ratio of the two spectra indicate a difference of $\Delta \alpha=0.36\pm0.09$.
Mkn 501 and Mkn 421 cut-off energies
====================================
\[subsection:differences\]
In the presence of extragalactic background light (EBL) at wavelengths of $\lambda=0.5\ldots30\,\mu$m, TeV-energy photons are subject to absorption by pair-production processes with the low energy background photons [@1966PRL...16...252; @1992ApJ...390..L49]. As a result, the observed spectrum of extragalactic TeV sources is modified. A very likely feature in the observed spectrum is an exponential cut-off. However, the presence of a cut-off in the observed spectra could also be caused by processes related to the production of TeV photons (e.g. decrease of the Compton-scattering cross-section for large centre-of-momentum energies, absorption inherent to the source). In the simplest picture of a source spectrum that follows a power law, the position of the cut-off should be identical for all sources at the same red-shift provided that the EBL is isotropical.
The two well-established extragalactic sources of TeV-radiation (Mkn 421 & Mkn 501) happen to almost coincide in red shift, which makes an investigation of differences in the position of the cut-off energies meaningful.
In Fig. \[fig:mkn501\_comp\] the energy spectrum of Mkn 501 as measured during 1997 with a cut-off energy of and the energy spectrum derived from observations of Mkn 421 in 2000/2001 are compared. The upper panel contains the differential energy spectra multiplied by $E^2$ together with the respective fit functions indicated as solid lines. The dashed curve is the result of a fit of a power law with a fixed exponential cut-off at $E_0=6.2$ TeV to the Mkn 421 data points. Ignoring possible systematic errors, the resulting $\chi^2_\mathrm{red}(d.o.f.)=5.7(12)$ excludes $E_0=6.2$ TeV as a cut-off energy for the Mkn 421 spectrum. The lower panel of Fig. \[fig:mkn501\_comp\] shows the ratio of the two energy spectra. A pure power law fit to the data is very unlikely because of a large $\chi^2_\mathrm{red}(d.o.f.)=3.6(12)$ whereas a function as defined in Eqn. \[eqn:ratio\] results in a acceptable $\chi^2_\mathrm{red}(d.o.f.)=1.5(12)$ with $\Delta\alpha=0.35\pm0.08$ and $c=(0.12\pm0.03)$ TeV$^{-1}$. We note that constraining the fit region to energies above $2$ TeV where absorption starts to become important, a pure power law fit is not excluded any more for the ratio.
There is an overlap of the cut-off energies determined for Mkn 421 ($E_0=$) and Mkn 501 ($E_0=$) taking the combined systematic and statistical errors into account. However, the origin of the systematic errors are very similar for both results and therefore strongly correlated. As mentioned in Sect. \[sect:systematics\], a 6% systematic uncertainty on the relative energy scale remains when comparing the two energy spectra. Combining the independent statistical uncertainty and the estimated systematic uncertainty on the relative energy scale results in $\Delta E=2.6\pm0.6_\mathrm{stat}\pm0.6_\mathrm{sys}$ TeV for the difference on the energy of the cut-off position. A detailed investigation of possible differences of energy spectra will be the issue of a forthcoming paper.
Multi-wavelength observations {#MWL}
=============================
The flux increase observed from Mkn 421 in the TeV energy band has been accompanied by a strongly increased X-ray flux. Detailed studies of the correlation of the variability in both energy bands are under-way. The simultaneously taken data from January/February 2001 with RXTE and HEGRA [@ICRC2001...horns] reveal a largely different spectral evolution for Mkn 421 in the X-ray and TeV being almost opposite to what has been observed from Mkn 501: Whereas the shape of the TeV-spectrum of Mkn 501 remained constant during the 1997 activity period , the Mkn 421 spectrum showed a spectral hardening with increasing flux. A different spectral evolution is also seen in the X-ray spectrum for the two sources: Whereas the X-ray spectrum of Mkn 501 is very hard during the flare which is commonly interpreted as a shift of the synchrotron peak position $E^{Sy}_{peak}$ from below 1 keV to 100 keV and back [@2001ApJ...554..725T], the X-ray spectrum of Mkn 421 remains soft with a moderate shift of the synchrotron peak from below 1 keV at a flux level of $\approx10^{-10}$ erg/(cm$^2$s) [@2000MNRAS.312..123M] to $\approx 2-3$ keV on January 31,2001 when the flux measured between 2-10 keV reached $F_{2-10}=1.6\cdot10^{-9}$ erg/(cm$^2$s) [@ICRC2001...horns]. These observations confirm the energy dependence of the peak-position claimed to follow a power law behaviour with $E^{sy}_{peak}\propto F_{Sy}^{0.5}$ for Mkn 421. For Mkn 501 the relation of the peak position and flux follows $E^{sy}_{peak}\propto F_{Sy}^{2}$ underlining the very different behaviour of the two extreme BL Lac objects [@2000ApJ...541..166F].
The TeV data of Mkn 421 taken simultaneously with the X-ray observations in January/February 2001 indicate a hard spectrum with a photon index $\alpha\approx2$ that deviate substantially from the quiescent energy spectrum being much softer ($\alpha\approx3$). The constraints on a one-zone SSC model of the source derived from these multi-wavelength observations are severe. The X-ray spectrum with a peak-energy $<3$ keV constrains the electron-spectrum which is responsible for the synchrotron peak emission to have a low energy ($\gamma_{peak}=1.5\cdot10^{5}\delta_{50}^{-0.5}\,B_{0.2}^{-0.5}$). As a result, without invoking large $\delta$ and a small magnetic field, the SSC model predicts a peak energy for the Compton-scattered spectrum below 1 TeV and the spectral shape is expected to be soft in the energy region from 1-20 TeV because of the Klein-Nishina suppression of the Compton-scattering cross-section. However, the *observed* TeV-spectra are very hard and a small magnetic field $B<0.22$ G and a large Doppler-factor $\delta>50$ are required to achieve an adequate fit of the TeV data with a SSC-model (see e.g. @2001ApJ...559..187K). The absorption of TeV photons due to pair production with photons of the extragalactic radiation field adds to the difficulties, because the source spectra corrected for the absorption are intrinsically harder than the observed spectra which are not easily described with a simple SSC-one-zone model (see e.g. @felix_icrc).
Summary and conclusions
=======================
The observations of Mkn 421 at TeV-energies indicate significant spectral variations for different flux levels. The photon index $\alpha$ changes from $\approx3$ to $\approx2$ with a flux increasing from $F(E>1~\mathrm{TeV})=0.5\cdot F_{-11}\ldots10\cdot F_{-11}$. Diurnal flux variations have been observed with rise and decay times as short as $20$ min. During particular nights where a complete flare was observed, the photon index determined during the preflare and the flare period differ by as much as $\Delta \alpha=0.75\pm0.25$ getting harder while the flux increases. All energy spectra are well-fit by a power law with an exponential cut-off at $E_0=$ without indications for variations of the cut-off energy at different flux-levels. However, the sensitivity on changes of the cut-off energy is limited by strong correlations between the two parameters ($\alpha$,$E_0$) used to describe the energy spectra. A comparison of $E_0$ for Mkn 501 and Mkn 421 indicates that the position of the cut-off energy for Mkn 501 is larger by taking the combined systematic and statistical error into account. A difference of the cut-off energies for two sources at similar red shift precludes the possibility of the cut-off being only an absorption feature.
The support of the German ministry for research and technology (BMBF) and of the Spanish Research Council (CICYT) is gratefully acknowledged. GR acknowledges receipt of a Humboldt fellowship. We thank the Instituto de Astrofísica de Canarias for the use of the site and for supplying excellent working conditions at La Palma.
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[^1]: $\chi^2_{\mathrm{red}}:= \chi^2/d.o.f.$
[^2]: $F_{-11}=F(E>1\mathrm{TeV})/10^{-11}\mathrm{ph}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$
[^3]: The Crab-Nebula is frequently used as a standard candle for TeV astronomy
[^4]: This corresponds to $\nu\,F(\nu)$: 1 TeV/(cm$^2$ s)$=$1.6 erg/(cm$^2$ s).
|
---
abstract: 'We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning conic sections.'
author:
- Nikolai Ivanov Beluhov
title: 'The Mixed Poncelet–Steiner Closure Theorem'
---
A Sangaku Problem
=================
\[t1\]
Given are two circles $k$ and $\o$ such that $\o$ lies inside $k$. Let $t$ be any line tangent to $\o$. Let $\o_1$ and $\o_2$ be the two circles which are tangent externally to $\o$, internally to $k$, are tangent to $t$, and lie on the same side of $t$ as $\o$. If the circles $\o$, $\o_1$, and $\o_2$ have a second common external tangent $u$ for some position of the line $t$, then they have a second common external tangent $u$ for any position of the line $t$. (Fig. \[f1\])
![[]{data-label="f1"}](f1){width="55.00000%"}
The author discovered this fact while contemplating an 1839 Sangaku problem from a tablet found in Nagano Prefecture [@fp89]. The original problem asks for a proof that the product of the radii of the largest circles inscribed in the circular segments which $t$ and $u$ cut from $k$ in Fig. \[f1\] equals the square of the radius of $\omega$.
Theorem \[t1\] admits a straightforward algebraic proof.
Let $I$ and $r$ be the center and radius of $\o$ and $O$ and $R$ be the center and radius of $k$.
Let $\o_1$ and $\o_2$ be two circles inscribed in the annulus between $\o$ and $k$. Let $r_1$ and $r_2$ be their radii, $A$ and $B$ be their contact points with $\o$, and $C$ and $D$ be their contact points with $k$. Then the circles $\o$, $\o_1$, and $\o_2$ have two common external tangents exactly when $AB$ is a diameter of $\o$ and $r_1r_2 = r^2$.
Let $S$ be the internal similitude center of $\o$ and $k$; by the three similitude centers theorem, $S = AC \cap BD$. Let the lines $AS$ and $BS$ intersect $\o$ for the second time in $P$ and $Q$.
In this configuration, let $A$ and $B$ vary so that $AB$ is a diameter of $\o$. We wish to establish when $r_1r_2 = r^2$.
Let $s^2$ be the power of $S$ with respect to $\o$, $s^2 = SA \cdot SP = SB \cdot SQ$, and $m^2 = \frac{1}{2}(SA^2 + SB^2)$. Then both $s^2$ and $m^2$ remain constant as $AB$ varies: this is obvious for $s^2$, and for $m^2$ it follows from the fact that the side $AB$ and the median $SI$ to this side in $\t SAB$ remain constant.
Let $SA^2 = m^2 + x^2$ and $SB^2 = m^2 - x^2$. Notice that, when $AB$ varies, $x^2$ varies also.
We have $r : r_1 = AP : AC$, $r : r_2 = BQ : BD$, and $r : R = SP : SC = SQ : SD$. Therefore, $$r_1 = \frac{s^2R - m^2r - x^2r}{s^2 + m^2 + x^2} \textrm{ and } r_2 = \frac{s^2R - m^2r + x^2r}{s^2 + m^2 - x^2}.$$ It follows that $$\begin{split} r_1r_2 = r^2 &\Leftrightarrow (s^2R - m^2r)^2 - x^4r^2 = r^2[(s^2 + m^2)^2 - x^4]\\ &\Leftrightarrow (s^2R - m^2r)^2 = r^2(s^2 + m^2)^2, \end{split}$$ which does not depend on $x$. Therefore, if $\o$, $\o_1$, and $\o_2$ have two common external tangents for some position of $AB$, then they do so for any position of $AB$, as needed.
As a corollary we obtain a simple characterization of the pairs of circles satisfying the condition of Theorem \[t1\] which is strongly reminiscent of Euler’s formula [@euf]. Namely, two circles $k$ and $\o$ of radii $R$ and $r$ and intercenter distance $d$ satisfy the condition of Theorem \[t1\] exactly when $$d^2 = (R - r)^2 - 4r^2.$$
An additional curious property of the figure is as follows.
\[t2\]
In the configuration of Theorem \[t1\], the intersection $t \cap u$ of the common external tangents of $\o$, $\o_1$, and $\o_2$ describes a straight line when $t$ varies.
When $I \equiv O$, the line in question is the line at infinity. Let $I \not\equiv O$ and let the line $IO$ intersect $k$ in $A$ and $D$ and $\o$ in $B$ and $C$ so that $B$ and $C$ lie on the segment $AD$ in this order and $AB < CD$. Consider a coordinate system in which the coordinates of $A$, $B$, $C$, and $D$ are $(1, 0)$, $(a, 0)$, $(a^2, 0)$, and $(a^3, 0)$, respectively, for some real $a > 1$.
Consider an arbitrary circle $\o'$ of center $J$ of coordinates $(x, y)$ and radius $r'$ inscribed in the annulus between $\o$ and $k$. We have $$IJ = r + r' \Rightarrow \left (\frac{a^2 + a}{2} - x \right )^2 + y^2 = \left (\frac{a^2 - a}{2} + r' \right )^2$$ and $$OJ = R - r' \Rightarrow \left (\frac{a^3 + 1}{2} - x \right )^2 + y^2 = \left (\frac{a^3 - 1}{2} - r' \right )^2.$$
From these, we obtain $$\left (\frac{a^2 + a}{2} - x \right )^2 - \left (\frac{a^3 + 1}{2} - x \right )^2 = \left (\frac{a^2 - a}{2} + r' \right )^2 - \left (\frac{a^3 - 1}{2} - r' \right )^2 \Rightarrow r' = \frac{a - 1}{a + 1} \cdot x.$$
Therefore, the ratio $r : r'$ equals the ratio of the distances from $I$ and $J$ to the $y$-axis $l$, implying that the common external tangents of $\o$ and $\o'$ intersect on $l$ as $\o'$ varies, as needed.
The Mixed Poncelet–Steiner Closure Theorem
==========================================
Theorem \[t1\] admits a powerful generalization.
\[t3\]
Given are two circles $k$ and $\o$ such that $\o$ lies inside $k$. Let $\o_1$ be any circle inscribed in the annulus between $k$ and $\o$. Let $t_1$ be a common external tangent of $\o_1$ and $\o$. Let $\o_2$ be the second circle which is inscribed in the annulus between $k$ and $\o$, is tangent to $t_1$, and lies on the same side of $t_1$ as $\o$. Let $t_2$ be the second common external tangent of $\o_2$ and $\o$. Let $\o_3$ be the second circle which is inscribed in the annulus between $k$ and $\o$, is tangent to $t_2$, and lies on the same side of $t_2$ as $\o$, etc. If $\o_{n + 1} \equiv \o_1$ for some choice of the initial circle $\o_1$ and the initial tangent $t_1$, then $\o_{n + 1} \equiv \o_1$ for any choice of the initial circle $\o_1$ and the initial tangent $t_1$. (Fig. \[f2\])
![[]{data-label="f2"}](f2){width="55.00000%"}
This theorem is strongly reminiscent of the classical closure theorems of Poncelet and Steiner ([@pon], [@stein]). Poncelet’s theorem deals solely with chains of tangents, and Steiner’s theorem deals solely with chains of circles. Hence the name suggested by the author.
We proceed to show how Theorem \[t3\] can be reduced to Poncelet’s theorem.
\[t4\]
Given are an ellipse $\e$ and a parabola $p$ of a common focus $F$. Then, when $p$ rotates about $F$, the common chord of $\e$ and $p$ (for all positions of $p$ for which it exists) is tangent to some fixed conic $\gamma$.
Let $\alpha$ be the plane in which $\e$ and $p$ lie. Let $s$ be a sphere which is tangent to $\alpha$ at $F$. For any point $X$ in space outside $s$, let $t_X$ be the length of the tangent from $X$ to $s$, $c_X$ be the cone formed by the lines through $X$ which are tangent to $s$, $\pi_X$ be the plane which contains the contact circle of $c_X$ with $s$, and $l_X$ be the conic section in which $c_X$ intersects $\alpha$.
Let $E$ and $P$ be such that $l_E = \e$ and $l_P = p$. For all $s$ whose radius is small enough, $E$ and $P$ will both be finite points and the common chord of $\e$ and $p$ will lie in the opposite half-space with respect to $\pi_E$ and $\pi_P$ as $E$ and $P$, respectively (since the circle $c_X \cap s$ will separate $X$ and $l_X$ on the surface of $c_X$ for $X = E$ and $X = P$).
Let $d(X, \chi)$ denote the signed distance from the point $X$ to the plane $\chi$. Then, for all points $X$ in $\e$, $d(X, \pi_E) : t_X$ is some real constant $c_1$ and for all points $X$ in $p$, $d(X, \pi_P) : t_X$ is some, possibly different, real constant $c_2$. It follows, then, that the common chord of $\e$ and $p$ lies in the plane $\pi$ which is the locus of all points $X$ such that $d(X, \pi_E) : d(X, \pi_P) = c_1 : c_2$. Notice that $\pi$ also contains the straight line $m = \pi_E \cap \pi_P$, and that, when $p$ varies, $P$ and $\pi$ vary also, but $c_1$, $c_2$, and $c_1 : c_2$ remain fixed.
Let $l$ be the line through $F$ which is perpendicular to $\alpha$, and let $A = \pi_E \cap l$, $B = \pi_P \cap l$, and $C = \pi \cap l$. Then $A$ and $B$ remain fixed when $p$ and $P$ vary (since $p$ varies by rotating about $F$), $\pi_E$ and $\pi_P$ intersect $l$ at fixed angles, and the ratio $d(C, \pi_E) : d(C, \pi_P) = c_1 : c_2$ remains constant. It follows that the ratio of the signed distances $|CA| : |CB|$ remains constant also and that $C$ remains constant, too.
When $p$, $P$, and $\pi_P$ vary, the planes $\pi_P$ envelop some straight circular cone $d$. The intersection $d \cap \pi_E$ is then some fixed conic section $\beta$ tangent to the line $m$. Let $\gamma$ be the projection of $\beta$ through the fixed point $C$ onto $\alpha$. As we saw above, the projection of $m$ through $C$ onto $\alpha$ is a straight line which contains the common chord of $\e$ and $p$. Since projection preserves tangency, this common chord will always be tangent to the fixed conic $\gamma$, as needed.
Let $\o'$ of center $O'$ and radius $r'$ be any circle inscribed in the annulus between $k$ and $\o$ of centers $O$ and $I$ and radii $R$ and $r$. Then $OO' + O'I = R - r' + r' + r = R + r$ remains constant when $\o'$ varies, implying that $O'$ traces some fixed ellipse $\e$ of foci $O$ and $I$.
Let $t$ be any tangent to $\o$ and let $\o'$ and $\o''$ of centers $O'$ and $O''$ and radii $r'$ and $r''$ be the two circles inscribed in the annulus between $k$ and $\o$, tangent to $t$, and lying on the same side of $t$ as $\o$. Let $t'$ be the image of $t$ under homothety of center $I$ and coefficient $2$, and let $d(X, l)$ be the signed distance from the point $X$ to the line $l$. Then $O'I = r' + r = d(O', t')$ and $O''I = r'' + r = d(O'', t')$, implying that $O'$ and $O''$ both lie on the parabola $p$ of focus $I$ and directrix $t'$.
When $t$ varies, $t'$, and, consequently, $p$, rotates about $I$. Since $O'O''$ is the common chord of $\e$ and $p$, by Theorem \[t4\] $O'O''$ remains tangent to some fixed conic $\gamma$.
We are only left to apply Poncelet’s theorem to the two conics $\e$ and $\gamma$ and the chain $O_1$, $O_2$, ... $O_{n + 1}$ of the centers of the circles $\o_1$, $\o_2$, ... $\o_{n + 1}$.
More on Conic Sections
======================
Theorem \[t4\] remains true, its proof unaltered, when $p$ is taken to be an ellipse rather than a parabola. It also remains true when $p$ is an arbitrary conic section, provided that the common chord of $\e$ and $p$ is chosen “just right” and varies continuously when $p$ varies.
\[t5\]
In the configuration of the generalized Theorem \[t4\], $F$ is a focus of $\gamma$.
Let $c$ be a straight circular cone of axis $l$ and base the circle $k$ of center $O$ which lies in the plane $\alpha$. Let the plane $\beta$ intersect $c$ in the conic section $a$, $P$ be an arbitrary point on $l$, and $b$ be the projection of $a$ through $P$ onto $\alpha$. Then $O$ is a focus of $b$.
If $\alpha \parallel \beta$, then $b$ is a circle of center $O$ and we are done. Let $\alpha \not\,\parallel \beta$ and $s = \alpha \cap \beta$.
Let $A$ be the vertex of $c$, $X$ be any point on $b$, $Y = PX \cap a$, $Z = AY \cap k$, and $U$ be the orthogonal projection of $Y$ onto $\alpha$. Notice that $A$, $O$, $P$, $X$, $Y$, $Z$, and $U$ all lie in some plane perpendicular to $\alpha$ and that $O$, $X$, $Z$, and $U$ are collinear.
It suffices to show that $d(X, s)$ is a linear function of the length of the segment $OX$ when $X$ traces $b$. For, suppose that we manage to prove that $d(X, s) = m \cdot OX + n$ where $m$ and $n$ are some real constants. Without loss of generality, $m \ge 0$ (otherwise, we can change the direction of $s$). Let $s'$ be the line which is parallel to $s$ and at a signed distance of $n$ from $s$. Then $b$, the locus of $X$, is the conic section of focus $O$, directrix $s'$, and eccentricity $m$.
For simplicity, we assume that $a$ and $k$ do not intersect and that $P$ lies between the intersection points of $\alpha$ and $\beta$ with $l$. In this case, $P$ lies between $A$ and $O$ and between $X$ and $Y$, $Y$ lies between $A$ and $Z$, $U$ lies between $O$ and $Z$, and $O$ lies between $X$ and $Z$.
We have $$d(X, s) = \frac{ZX}{ZO} \cdot d(O, s) - \frac{OX}{ZO} \cdot d(Z, s) = \frac{R + OX}{R} \cdot d(O, s) - \frac{OX}{R} \cdot d(Z, s),$$ where $R$ is the radius of $k$.
We can forget about the coefficient $\frac{1}{R}$. Since $d(O, s)$ is constant, we can also forget about the term $(R + OX) \cdot d(O, s)$. This leaves us to prove that $$OX \cdot d(Z, s)$$ is linear in $OX$.
We have $$d(Z, s) = \frac{OZ}{OU} \cdot d(U, s) - \frac{ZU}{UO} \cdot d(O, s) = \frac{R}{OU} \cdot d(U, s) - \frac{ZU}{UO} \cdot d(O, s).$$
Since the shape of $\t YZU \sim \t AZO$ remains fixed when $X$ varies, the ratio $YU : ZU$ remains fixed as well. Since $Y$ varies in the fixed plane $\beta$, the ratio $d(U, s) : YU$ remains fixed also. It follows, then, that $d(U, s) : ZU$ remains fixed and that $$d(Z, s) = c_1 \cdot \frac{ZU}{UO}$$ for some real constant $c_1$.
Notice that $ZU : UO = ZY : YA$. By Menelaus’ theorem, then, we have $$\frac{ZY}{YA} \cdot \frac{AP}{PO} \cdot \frac{OX}{XZ} = 1 \Leftrightarrow \frac{ZY}{YA} = c_2 \cdot \frac{XZ}{OX},$$ where $c_2$ is some real constant.
From this, we obtain $$OX \cdot d(Z, s) = OX \cdot c_1 \cdot c_2 \cdot \frac{XZ}{OX} = c_1 \cdot c_2 \cdot (R + OX),$$ which, of course, is linear in $OX$, as needed.
In the notation of the proof of Theorem \[t4\], apply the lemma to the cone $d$, the planes $\alpha$ and $\pi_E$, and the projection point $C$.
![[]{data-label="f3"}](f3){width="75.00000%"}
Theorem \[t5\] has the following pleasing corollary.
\[t6\]
Three ellipses $\e_1$, $\e_2$, and $\e_3$ of a common focus $F$ intersect in two distinct points. The ellipses $\e_1$ and $\e_2$ start to revolve continuously about $F$ so that they still intersect in two distinct points. Then the ellipse $\e_3$ can also revolve continuously about $F$ so that, at all times, all three ellipses continue to intersect in two distinct points. (Fig. \[f3\])
Open Questions
==============
Let $w = a_1a_2 \ldots a_n$ be any $n$-letter word over the two-letter alphabet $\{c, s\}$. For any annulus $a$ delimited by an inner circle $\o$ and an outer circle $k$, we construct a chain $u_1$, $u_2$, ... $u_{n + 1}$ of circles and segments as follows:
\(a) $u_i$ is a circle inscribed in $a$ if $a_i = c$ and a segment tangent to $\o$ whose endpoints lie on $k$ if $a_i = s$ (for $i = n + 1$, we set $a_{n + 1} = a_1$);
\(b) If at least one of $u_i$ and $u_{i + 1}$ is a circle, then they are tangent;
\(c) If $u_i$ and $u_{i + 1}$ are both segments, then they have a common endpoint;
\(d) For $1 < i \le n$, the contact points of $u_i$ with $u_{i - 1}$ and $u_{i + 1}$ are separated on $u_i$ by the contact points of $u_i$ with $\o$ and $k$ if $u_i$ is a circle, and by the contact point of $u_i$ with $\o$ if $u_i$ is a segment.
If, for any annulus $a$, it is true that if $u_1 \equiv u_{n + 1}$ for some initial $u_1$ then $u_1 \equiv u_{n + 1}$ for all initial $u_1$, and there exists at least one annulus $a$ for which the premise of this implication is not void, then we say that the word $w$ is a *closure sequence*.
Poncelet’s theorem tells us that $s^n$ is a closure sequence for all $n \ge 3$. Steiner’s theorem tells us that $c^n$ is a closure sequence for all $n \ge 3$. Finally, the Mixed Poncelet-Steiner theorem tells us that $(cs)^n$ and $(sc)^n$ are closure sequences for all $n \ge 2$.
Are there any other closure sequences? Given a particular closure sequence $w$, what relation must the radii of $\o$ and $k$ and their intercenter distance satisfy in order for the chain described by $w$ to close? Is this relation always of the form $f(R, r, d) = 0$ for some rational function $f$? Finally, is it true that, for any closure sequence $w$ and any positive integer $n$, $w^n$ is also a closure sequence?
[9]{}
Fukagawa H., Pedoe D., *Japanese Temple Geometry Problems: San Gaku*, The Charles Babbage Research Center, 1989.
, or
[http://en.wikipedia.org/wiki/Euler’s\_theorem\_in\_geometry]{}
, or
[http://en.wikipedia.org/wiki/Poncelet’s\_closure\_theorem]{}
, or
[http://en.wikipedia.org/wiki/Steiner\_chain]{}
|
---
abstract: 'This paper compares two different ways of estimating statistical language models. Many statistical NLP tagging and parsing models are estimated by maximizing the (joint) likelihood of the fully-observed training data. However, since these applications only require the conditional probability distributions, these distributions can in principle be learnt by maximizing the conditional likelihood of the training data. Perhaps somewhat surprisingly, models estimated by maximizing the joint were superior to models estimated by maximizing the conditional, even though some of the latter models intuitively had access to “more information”.'
author:
- |
Mark Johnson\
Brown University\
[Mark\_Johnson@Brown.edu]{}
bibliography:
- 'mj.bib'
title: 'Joint and conditional estimation of tagging and parsing models[^1]'
---
Introduction
============
Many statistical NLP applications, such as tagging and parsing, involve finding the value of some hidden variable $Y$ (e.g., a tag or a parse tree) which maximizes a conditional probability distribution ${{\rm P}}_\theta(Y|X)$, where $X$ is a given word string. The model parameters $\theta$ are typically estimated by maximum likelihood: i.e., maximizing the likelihood of the training data. Given a (fully observed) training corpus $D = ((y_1,x_1), \ldots, (y_n,x_n))$, the [*maximum (joint) likelihood estimate*]{} (MLE) of $\theta$ is: $$\begin{aligned}
{\hat{\theta}}& = &
{\mathop{\rm argmax}}_\theta \, \prod_{i=1}^n {{\rm P}}_\theta(y_i,x_i). \label{e:mle}\end{aligned}$$
However, it turns out there is another maximum likelihood estimation method which maximizes the conditional likelihood or “pseudo-likelihood” of the training data [@Besag75]. Maximum conditional likelihood is consistent [*for the conditional distribution*]{}. Given a training corpus $D$, the [*maximum conditional likelihood estimate*]{} (MCLE) of the model parameters $\theta$ is: $$\begin{aligned}
{\hat{\theta}}& = &
{\mathop{\rm argmax}}_\theta \, \prod_{i=1}^n {{\rm P}}_\theta(y_i|x_i). \label{e:mcle}\end{aligned}$$
Figure \[f:mle\] graphically depicts the difference between the MLE and MCLE. Let $\Omega$ be the universe of all possible pairs $(y,x)$ of hidden and visible values. Informally, the MLE selects the model parameter $\theta$ which make the training data pairs $(y_i,x_i)$ as likely as possible relative to all other pairs $(y',x')$ in $\Omega$. The MCLE, on the other hand, selects the model parameter $\theta$ in order to make the training data pair $(y_i,x_i)$ more likely than other pairs $(y',x_i)$ in $\Omega$, i.e., pairs with the same visible value $x_i$ as the training datum.
In statistical computational linguistics, maximum conditional likelihood estimators have mostly been used with general exponential or “maximum entropy” models because standard maximum likelihood estimation is usually computationally intractable [@Berger96; @DellaPietra97; @Jelinek97]. Well-known computational linguistic models such as Maximum-Entropy Markov Models [@McCallum00] and Stochastic Unification-based Grammars [@Johnson99c] are standardly estimated with conditional estimators, and it would be interesting to know whether conditional estimation affects the quality of the estimated model. It should be noted that in practice, the MCLE of a model with a large number of features with complex dependencies may yield far better performance than the MLE of the much smaller model that could be estimated with the same computational effort. Nevertheless, as this paper shows, conditional estimators can be used with other kinds of models besides MaxEnt models, and in any event it is interesting to ask whether the MLE differs from the MCLE in actual applications, and if so, how.
Because the MLE is consistent for the joint distribution ${{\rm P}}(Y,X)$ (e.g., in a tagging application, the distribution of word-tag sequences), it is also consistent for the conditional distribution ${{\rm P}}(Y|X)$ (e.g., the distribution of tag sequences given word sequences) and the marginal distribution ${{\rm P}}(X)$ (e.g., the distribution of word strings). On the other hand, the MCLE is consistent for the conditional distribution ${{\rm P}}(Y|X)$ alone, and provides no information about either the joint or the marginal distributions. Applications such as language modelling for speech recognition and EM procedures for estimating from hidden data either explicitly or implicitly require marginal distributions over the visible data (i.e., word strings), so it is not statistically sound to use MCLEs for such applications. On the other hand, applications which involve predicting the value of the hidden variable from the visible variable (such as tagging or parsing) usually only involve the conditional distribution, which the MCLE estimates directly.
Since both the MLE and MCLE are consistent for the conditional distribution, both converge [*in the limit*]{} to the “true” distribution [*if the true distribution is in the model class*]{}. However, given that we often have insufficient data in computational linguistics, and there are good reasons to believe that the true distribution of sentences or parses cannot be described by our models, there is no reason to expect these asymptotic results to hold in practice, and in the experiments reported below the MLE and MCLE behave differently experimentally.
A priori, one can advance plausible arguments in favour of both the MLE and the MCLE. Informally, the MLE and the MCLE differ in the following way. Since the MLE is obtained by maximizing $\prod_{i}{{\rm P}}_\theta(y_i|x_i){{\rm P}}_\theta(x_i)$, the MLE exploits information about the distribution of word strings $x_i$ in the training data that the MCLE does not. Thus one might expect the MLE to converge faster than the MCLE in situations where training data is not over-abundant, which is often the case in computational linguistics.
On the other hand, since the intended application requires a conditional distribution, it seems reasonable to directly estimate this conditional distribution from the training data as the MCLE does. Furthermore, suppose that the model class is wrong (as is surely true of all our current language models), i.e., the “true” model ${{\rm P}}(Y,X)\neq {{\rm P}}_\theta(Y,X)$ for all $\theta$, and that our best models are particularly poor approximations to the true distribution of word strings ${{\rm P}}(X)$. Then ignoring the distribution of word strings in the training data as the MCLE does might indeed be a reasonable thing to do.
The rest of this paper is structured as follows. The next section formulates the MCLEs for HMMs and PCFGs as constrained optimization problems and describes an iterative dynamic-programming method for solving them. Because of the computational complexity of these problems, the method is only applied to a simple PCFG based on the ATIS corpus. For this example, the MCLE PCFG does perhaps produce slightly better parsing results than the standard MLE (relative-frequency) PCFG, although the result does not reach statistical significance.
It seems to be difficult to find model classes for which the MLE and MCLE are both easy to compute. However, often it is possible to find two closely related model classes, one of which has an easily computed MLE and the other which has an easily computed MCLE. Typically, the model classes which have an easily computed MLE define [*joint*]{} probability distributions over both the hidden and the visible data (e.g., over word-tag pair sequences for tagging), while the model classes which have an easily computed MCLE define [*conditional*]{} probability distributions over the hidden data given the visible data (e.g., over tag sequences given word sequences).
Section \[s:tagging\] investigates closely related joint and conditional tagging models (the latter can be regarded as a simplification of the Maximum Entropy Markov Models of ), and shows that MLEs outperform the MCLEs in this application. The final empirical section investigates two different kinds of stochastic shift-reduce parsers, and shows that the model estimated by the MLE outperforms the model estimated by the MCLE.
PCFG parsing
============
In this application, the pairs $(y,x)$ consist of a parse tree $y$ and its terminal string or yield $x$ (it may be simpler to think of $y$ containing all of the parse tree except for the string $x$). Recall that in a PCFG with production set $R$, each production $(A {\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha) \in R$ is associated with a parameter $\theta_{A {\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}$. These parameters satisfy a normalization constraint for each nonterminal $A$: $$\begin{aligned}
\sum_{\alpha {:}(A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha)\in R} \theta_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha} & = & 1 \label{e:pcfgc}\end{aligned}$$ For each production $r\in R$, let $f_r(y)$ be the number of times $r$ is used in the derivation of the tree $y$. Then the PCFG defines a probability distribution over trees: $$\begin{aligned}
{{\rm P}}_\theta(Y) & = & \prod_{(A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha)\in R} {\theta_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}}^{f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}(Y)}\end{aligned}$$
The MLE for $\theta$ is the well-known “relative-frequency” estimator: $$\begin{aligned}
{\hat{\theta}}_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha} & = & { \sum_{i=1}^n f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}(y_i) \over
\sum_{i=1}^n \sum_{\alpha'{:}(A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha')\in R} f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha'}(y_i)}.\end{aligned}$$
Unfortunately the MCLE for a PCFG is more complicated. If $x$ is a word string, then let $\tau(x)$ be the set of parse trees with terminal string or yield $x$ generated by the PCFG. Then given a training corpus $D = ((y_1,x_1), \ldots, (y_n,x_n))$, where $y_i$ is a parse tree for the string $x_i$, the log conditional likelihood of the training data $\log {{{\rm P}}(\vec{y}|\vec{x})}$ and its derivative are given by: $$\begin{aligned}
\hspace*{-1em}
\log {{{\rm P}}(\vec{y}|\vec{x})}& = & \sum_{i=1}^n \left(\log{{\rm P}}_\theta(y_i) -
\log\sum_{y\in\tau(x_i)}{{\rm P}}_\theta(y)\right) \label{e:pcfgcl} \\
\hspace*{-0.25em}
{\partial \log {{{\rm P}}(\vec{y}|\vec{x})}\over \partial \theta_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}}
&=&
{1 \over \theta_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}} \sum_{i=1}^n \left( f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}(y_i) -
{{\rm E}}_\theta(f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}|x_i) \right)\end{aligned}$$ Here ${{\rm E}}_\theta(f|x)$ denotes the expectation of $f$ with respect to ${{\rm P}}_\theta$ conditioned on $Y\in\tau(x)$. There does not seem to be a closed-form solution for the $\theta$ that maximizes ${{{\rm P}}(\vec{y}|\vec{x})}$ subject to the constraints (\[e:pcfgc\]), so we used an iterative numerical gradient ascent method, with the constraints (\[e:pcfgc\]) imposed at each iteration using Lagrange multipliers. Note that $\sum_{i=1}^n {{\rm E}}_\theta(f_{A{\raise 0.75pt \hbox{$\scriptscriptstyle\rightarrow$}}\alpha}|x_i)$ is a quantity calculated in the Inside-Outside algorithm [@Lari90] and ${{{\rm P}}(\vec{y}|\vec{x})}$ is easily computed as a by-product of the same dynamic programming calculation.
Since the expected production counts ${{\rm E}}_\theta(f|x)$ depend on the production weights $\theta$, the entire training corpus must be reparsed on each iteration (as is true of the Inside-Outside algorithm). This is computationally expensive with a large grammar and training corpus; for this reason the MCLE PCFG experiments described here were performed with the relatively small ATIS treebank corpus of air travel reservations distributed by LDC.
In this experiment, the PCFGs were always trained on the 1088 sentences of the ATIS1 corpus and evaluated on the 294 sentences of the ATIS2 corpus. Lexical items were ignored; the PCFGs generate preterminal strings. The iterative algorithm for the MCLE was initialized with the MLE parameters, i.e., the “standard” PCFG estimated from a treebank. Table \[t:pcfg\] compares the MLE and MCLE PCFGs.
-------------------------------------- ------- -------
MLE MCLE
$-\log {{{\rm P}}(\vec{y})}$ 13857 13896
$-\log {{{\rm P}}(\vec{y}|\vec{x})}$ 1833 1769
$-\log {{{\rm P}}(\vec{x})}$ 12025 12127
Labelled precision 0.815 0.817
Labelled recall 0.789 0.794
-------------------------------------- ------- -------
: \[t:pcfg\] The likelihood ${{{\rm P}}(\vec{y})}$ and conditional likelihood ${{{\rm P}}(\vec{y}|\vec{x})}$ of the ATIS1 training trees, and the marginal likelihood ${{{\rm P}}(\vec{x})}$ of the ATIS1 training strings, as well as the labelled precision and recall of the ATIS2 test trees, using the MLE and MCLE PCFGs.
The data in table \[t:pcfg\] shows that compared to the MLE PCFG, the MCLE PCFG assigns a higher conditional probability of the parses in the training data given their yields, at the expense of assigning a lower marginal probability to the yields themselves. The labelled precision and recall parsing results for the MCLE PCFG were slightly higher than those of the MLE PCFG. Because both the test data set and the differences are so small, the significance of these results was estimated using a bootstrap method with the difference in F-score in precision and recall as the test statistic [@Cohen95]. This test showed that the difference was not significant ($p \approx 0.1$). Thus the MCLE PCFG did not perform significantly better than the MLE PCFG in terms of precision and recall.
HMM tagging {#s:tagging}
===========
As noted in the previous section, maximizing the conditional likelihood of a PCFG or a HMM can be computationally intensive. This section and the next pursues an alternative strategy for comparing MLEs and MCLEs: we compare similiar (but not identical) model classes, one of which has an easily computed MLE, and the other of which has an easily computed MCLE. The application considered in this section is bitag POS tagging, but the techniques extend straight-forwardly to $n$-tag tagging. In this application, the data pairs $(y,x)$ consist of a tag sequence $y = t_1 \ldots t_m$ and a word sequence $x = w_1
\ldots w_m$, where $t_j$ is the tag for word $w_j$ (to simplify the formulae, $w_0$, $t_0$, $w_{m+1}$ and $t_{m+1}$ are always taken to be end-markers). Standard HMM tagging models define a [*joint*]{} distribution over word-tag sequence pairs; these are most straight-forwardly estimated by maximizing the likelihood of the joint training distribution. However, it is straight-forward to devise closely related HMM tagging models which define a [*conditional*]{} distribution over tag sequences given word sequences, and which are most straight-forwardly estimated by maximizing the conditional likelihood of the distribution of tag sequences given word sequences in the training data.
All of the HMM models investigated in this section are instances of a certain kind of graphical model that calls “Bayes nets”; Figure \[f:bayes\] sketches the networks that correspond to all of the models discussed here. (In such a graph, the set of incoming arcs to a node depicting a variable indicate the set of variables on which this variable is conditioned).
=[++\[o\]\[F-\]]{}
Recall the standard bitag HMM model, which defines a joint distribution over word and tag sequences: $$\begin{aligned}
{{\rm P}}(Y,X) &=& \prod_{j=1}^{m+1} {\hat{{{\rm P}}}}(T_j|T_{j-1}) {\hat{{{\rm P}}}}(W_j|T_j)
\label{e:hmm}\end{aligned}$$ As is well-known, the MLE for (\[e:hmm\]) sets ${\hat{{{\rm P}}}}$ to the empirical distributions on the training data.
Now consider the following [*conditional model*]{} of the conditional distribution of tags given words (this is a simplified form of the model described in ): $$\begin{aligned}
{{\rm P}}(Y|X) &=& \prod_{j=1}^{m+1} {{\rm P}}_0(T_j|W_j,T_{j-1}) \label{e:chmm}\end{aligned}$$ The MCLE of (\[e:chmm\]) is easily calculated: ${{\rm P}}_0$ should be set the empirical distribution of the training data. However, to minimize sparse data problems we estimated ${{\rm P}}_0(T_j|W_j,T_{j-1})$ as a mixture of ${\hat{{{\rm P}}}}(T_j|W_j)$, ${\hat{{{\rm P}}}}(T_j|T_{j-1})$ and ${\hat{{{\rm P}}}}(T_j|W_j,T_{j-1})$, where the ${\hat{{{\rm P}}}}$ are empirical probabilities and the (bucketted) mixing parameters are determined using deleted interpolation from heldout data [@Jelinek97].
These models were trained on sections 2-21 of the Penn tree-bank corpus. Section 22 was used as heldout data to evaluate the interpolation parameters $\lambda$. The tagging accuracy of the models was evaluated on section 23 of the tree-bank corpus (in both cases, the tag $t_j$ assigned to word $w_j$ is the one which maximizes the marginal ${{\rm P}}(t_j|w_1 \ldots w_m)$, since this minimizes the expected loss on a tag-by-tag basis).
The conditional model (\[e:chmm\]) has the worst performance of any of the tagging models investigated in this section: its tagging accuracy is 94.4%. The joint model (\[e:hmm\]) has a considerably lower error rate: its tagging accuracy is 95.5%.
One possible explanation for this result is that the way in which the interpolated estimate of ${{\rm P}}_0$ is calculated, rather than conditional likelihood estimation per se, is lowering tagger accuracy somehow. To investigate this possibility, two additional joint models were estimated and tested, based on the formulae below. $$\begin{aligned}
\hspace*{-2.9em}
{{\rm P}}(Y,X) &\!\!=\!\!& \prod_{j=1}^{m+1} {\hat{{{\rm P}}}}(W_j|T_j) {{\rm P}}_1(T_j|W_{j-1},T_{j-1}) \label{e:hmm1} \\
\hspace*{-2.9em}
{{\rm P}}(Y,X) &\!\!=\!\!& \prod_{j=1}^{m+1} {{\rm P}}_0(T_j|W_j,T_{j-1}) {\hat{{{\rm P}}}}(W_j|T_{j-1}) \label{e:hmm2}\end{aligned}$$ The MLEs for both (\[e:hmm1\]) and (\[e:hmm2\]) are easy to calculate. (\[e:hmm1\]) contains a conditional distribution ${{\rm P}}_1$ which would seem to be of roughly equal complexity to ${{\rm P}}_0$, and it was estimated using deleted interpolation in exactly the same way as ${{\rm P}}_0$, so if the poor performance of the conditional model was due to some artifact of the interpolation procedure, we would expect the model based on (\[e:hmm1\]) to perform poorly. Yet the tagger based on (\[e:hmm1\]) performs the best of all the taggers investigated in this section: its tagging accuracy is 96.2%.
(\[e:hmm2\]) is admitted a rather strange model, since the right hand term in effect predicts the [*following*]{} word from the current word’s tag. However, note that (\[e:hmm2\]) differs from (\[e:chmm\]) only via the presence of this rather unusual term, which effectively converts (\[e:chmm\]) from a conditional model to a joint model. Yet adding this term improves tagging accuracy considerably, to 95.3%. Thus for bitag tagging at least, the conditional model has a considerably higher error rate than any of the joint models examined here. (While a test of significance was not conducted here, previous experience with this test set shows that performance differences of this magnitude are extremely significant statistically).
Shift-reduce parsing
====================
The previous section compared similiar joint and conditional tagging models. This section compares a pair of joint and conditional parsing models. The models are both stochastic shift-reduce parsers; they differ only in how the distribution over possible next moves are calculated. These parsers are direct simplifications of the Structured Language Model [@Jelinek00]. Because the parsers’ moves are determined solely by the top two category labels on the stack and possibly the look-ahead symbol, they are much simpler than stochastic LR parsers [@Briscoe93; @Inui97]. The distribution over trees generated by the joint model is a probabilistic context-free language [@Abney99a]. As with the PCFG models discussed earlier, these parsers are not lexicalized; lexical items are ignored, and the POS tags are used as the terminals.
These two parsers only produce trees with unary or binary nodes, so we binarized the training data before training the parser, and debinarize the trees the parsers produce before evaluating them with respect to the test data [@Johnson98c]. We binarized by inserting $n-2$ additional nodes into each local tree with $n > 2$ children. We binarized by first joining the head to all of the constituents to its right, and then joining the resulting structure with constituents to the left. The label of a new node is the label of the head followed by the suffix “-1” if the head is (contained in) the right child or “-2” if the head is (contained in) the left child. Figure \[f:bin\] depicts an example of this transformation.
The Structured Language Model is described in detail in , so it is only reviewed here. Each parser’s stack is a sequence of node labels (possibly including labels introduced by binarization). In what follows, $s_1$ refers to the top element of the stack, or ‘${\star}$’ if the stack is empty; similarly $s_2$ refers to the next-to-top element of the stack or ‘${\star}$’ if the stack contains less than two elements. We also append a ‘${\star}$’ to end of the actual terminal string being parsed (just as with the HMMs above), as this simplifies the formulation of the parsers, i.e., if the string to be parsed is $w_1 \ldots w_m$, then we take $w_{m+1}
= {\star}$.
A shift-reduce parse is defined in terms of moves. A move is either ${{\rm shift}}(w)$, ${{\rm reduce}}_1(c)$ or ${{\rm reduce}}_2(c)$, where $c$ is a nonterminal label and $w$ is either a terminal label or ‘${\star}$’. Moves are partial functions from stacks to stacks: a ${{\rm shift}}(w)$ move pushes a $w$ onto the top of stack, while a ${{\rm reduce}}_i(c)$ move pops the top $i$ terminal or nonterminal labels off the stack and pushes a $c$ onto the stack. A shift-reduce parse is a sequence of moves which (when composed) map the empty stack to the two-element stack whose top element is ‘${\star}$’ and whose next-to-top element is the start symbol. (Note that the last move in a shift-reduce parse must always be a ${{\rm shift}}({\star})$ move; this corresponds to the final “accept” move in an LR parser). The isomorphism between shift-reduce parses and standard parse trees is well-known [@Hopcroft79], and so is not described here.
A (joint) shift-reduce parser is defined by a distribution ${{\rm P}}(m|s_1,s_2)$ over next moves $m$ given the top and next-to-top stack labels $s_1$ and $s_2$. To ensure that the next move is in fact a possible move given the current stack, we require that ${{\rm P}}({{\rm reduce}}_1(c)|{\star},{\star}) = 0$ and ${{\rm P}}({{\rm reduce}}_2(c)|c',{\star}) = 0$ for all $c, c'$, and that ${{\rm P}}(shift({\star})|s_1,s_2) = 0$ unless $s_1$ is the start symbol and $s_2 = {\star}$. Note that this extends to a probability distribution over shift-reduce parses (and hence parse trees) in a particularly simple way: the probability of a parse is the product of the probabilities of the moves it consists of. Assuming that ${{\rm P}}$ meets certain tightness conditions, this distribution over parses is properly normalized because there are no “dead” stack configurations: we require that the distribution over moves be defined for all possible stacks.
A conditional shift-reduce parser differs only minimally from the shift-reduce parser just described: it is defined by a distribution ${{\rm P}}(m|s_1,s_2,t)$ over next moves $m$ given the top and next-to-top stack labels $s_1$, $s_2$ and the next input symbol $w$ ($w$ is called the [*look-ahead symbol*]{}). In addition to the requirements on ${{\rm P}}$ above, we also require that if $w' \neq w$ then ${{\rm P}}({{\rm shift}}(w')
| s_1,s_2,w) = 0$ for all $s_1, s_2$; i.e., shift moves can only shift the current look-ahead symbol. This restriction implies that all non-zero probability derivations are derivations of the parse string, since the parse string forces a single sequence of symbols to be shifted in all derivations. As before, since there are no “dead” stack configurations, so long as ${{\rm P}}$ obeys certain tightness conditions, this defines a properly normalized distribution over parses. Since all the parses are required to be parses of of the input string, this defines a conditional distribution over parses given the input string.
It is easy to show that the MLE for the joint model, and the MCLE for the conditional model, are just the empirical distributions from the training data. We ran into sparse data problems using the empirical training distribution as an estimate for ${{\rm P}}(m|s_1,s_2,w)$ in the conditional model, so in fact we used deleted interpolation to interpolate ${\hat{{{\rm P}}}}(m|s_1,s_2,w)$, and ${\hat{{{\rm P}}}}(m|s_1,s_2)$ to estimate ${{\rm P}}(m|s_1,s_2,w)$. The models were estimated from sections 2–21 of the Penn treebank, and tested on the 2245 sentences of length 40 or less in section 23. The deleted interpolation parameters were estimated using heldout training data from section 22.
We calculated the most probable parses using a dynamic programming algorithm based on the one described in . Jelinek notes that this algorithm’s running time is $n^6$ (where $n$ is the length of sentence being parsed), and we found exhaustive parsing to be computationally impractical. We used a beam search procedure which thresholded the best analyses of each prefix of the string being parsed, and only considered analyses whose top two stack symbols had been observed in the training data. In order to help guard against the possibility that this stochastic pruning influenced the results, we ran the parsers twice, once with a beam threshold of $10^{-6}$ (i.e., edges whose probability was less than $10^{-6}$ of the best edge spanning the same prefix were pruned) and again with a beam threshold of $10^{-9}$. The results of the latter runs are reported in table \[t:sr\]; the labelled precision and recall results from the run with the more restrictive beam threshold differ by less than $0.001$, i.e., at the level of precision reported here, are identical with the results presented in table \[t:sr\] except for the Precision of the Joint SR parser, which was $0.665$. For comparision, table \[t:sr\] also reports results from the non-lexicalized treebank PCFG estimated from the transformed trees in sections 2-21 of the treebank; here exhaustive CKY parsing was used to find the most probable parses.
----------- ---------- ---------------- -------
Joint SR Conditional SR PCFG
Precision 0.666 0.633 0.700
Recall 0.650 0.639 0.657
----------- ---------- ---------------- -------
: Labelled precision and recall results for joint and conditional shift-reduce parsers, and for a PCFG. \[t:sr\]
All of the precision and recall results, including those for the PCFG, presented in table \[t:sr\] are much lower than those from a standard treebank PCFG; presumably this is because the binarization transformation depicted in Figure \[f:bin\] loses information about pairs of non-head constituents in the same local tree ( reports similiar performance degradation for other binarization transformations). Both the joint and the conditional shift-reduce parsers performed much worse than the PCFG. This may be due to the pruning effect of the beam search, although this seems unlikely given that varying the beam threshold did not affect the results. The performance difference between the joint and conditional shift-reduce parsers bears directly on the issue addressed by this paper: the joint shift-reduce parser performed much better than the conditional shift-reduce parser. The differences are around a percentage point, which is quite large in parsing research (and certainly highly significant).
The fact that the joint shift-reduce parser outperforms the conditional shift-reduce parser is somewhat surprising. Because the conditional parser predicts its next move on the basis of the lookahead symbol as well as the two top stack categories, one might expect it to predict this next move more accurately than the joint shift-reduce parser. The results presented here show that this is not the case, at least for non-lexicalized parsing. The [*label bias*]{} of conditional models may be responsible for this [@Bottou91; @Lafferty01].
Conclusion
==========
This paper has investigated the difference between maximum likelihood estimation and maximum conditional likelihood estimation for three different kinds of models: PCFG parsers, HMM taggers and shift-reduce parsers. The results for the PCFG parsers suggested that conditional estimation might provide a slight performance improvement, although the results were not statistically significant since computational difficulty of conditional estimation of a PCFG made it necessary to perform the experiment on a tiny training and test corpus. In order to avoid the computational difficulty of conditional estimation, we compared closely related (but not identical) HMM tagging and shift-reduce parsing models, for some of which the maximum likelihood estimates were easy to compute and for others of which the maximum conditional likelihood estimates could be easily computed. In both cases, the joint models outperformed the conditional models by quite large amounts. This suggests that it may be worthwhile investigating methods for maximum (joint) likelihood estimation for model classes for which only maximum conditional likelihood estimators are currently used, such as Maximum Entropy models and MEMMs, since if the results of the experiments presented in this paper extend to these models, one might expect a modest performance improvement.
As explained in the introduction, because maximum likelihood estimation exploits not just the conditional distribution of hidden variable (e.g., the tags or the parse) conditioned on the visible variable (the terminal string) but also the marginal distribution of the visible variable, it is reasonable to expect that it should outperform maximum conditional likelihood estimation. Yet it is counter-intuitive that joint tagging and shift-reduce parsing models, which predict the next tag or parsing move on the basis of what seems to be less information than the corresponding conditional model, should nevertheless outperform that conditional model, as the experimental results presented here show. The recent theoretical and simulation results of suggest that conditional models may suffer from [*label bias*]{} (the discovery of which Lafferty et. al. attribute to ), which may provide an insightful explanation of these results.
None of the models investigated here are state-of-the-art; the goal here is to compare two different estimation procedures, and for that reason this paper concentrated on simple, easily implemented models. However, it would also be interesting to compare the performance of joint and conditional estimators on more sophisticated models.
[^1]: I would like to thank Eugene Charniak and the other members of BLLIP for their comments and suggestions. Fernando Pereira was especially generous with comments and suggestions, as were the ACL reviewers; I apologize for not being able to follow up all of your good suggestions. This research was supported by NSF awards 9720368 and 9721276 and NIH award R01 MH60922-01A2.
|
---
abstract: |
In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping $$\left\{ \enspace
\begin{aligned}
&{\partial}_t\rho+\opdiv(\rho u)=0,\\
&{\partial}_t(\rho u)+\opdiv\left(\rho u\otimes u+p\,{\textup{\uppercase\expandafter{\romannumeral1}}}_d\right)=-{\alpha}(t)\rho u,\\
&\rho(0,x)=\bar \rho+{\varepsilon}\rho_0(x),\quad u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is ${\alpha}(t)=\frac{\mu}{(1+t)^{\lambda}}$ with ${\lambda}\ge0$ and $\mu>0$, $\bar\rho>0$ is a constant, $\rho_0,u_0 \in C_0^\infty({\mathbb R}^d)$, $(\rho_0,u_0)\not\equiv 0$, $\rho(0,x)>0$, and ${\varepsilon}>0$ is sufficiently small. One can totally divide the range of ${\lambda}\ge0$ and $\mu>0$ into the following four cases:
Case 1: $0\le{\lambda}<1$, $\mu>0$ for $d=2,3$;
Case 2: ${\lambda}=1$, $\mu>3-d$ for $d=2,3$;
Case 3: ${\lambda}=1$, $\mu\le 3-d$ for $d=2$;
Case 4: ${\lambda}>1$, $\mu>0$ for $d=2,3$.
We show that there exists a global $C^{\infty}-$smooth solution $(\rho, u)$ in Case 1, and Case 2 with $\opcurl u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(\rho, u)$ blows up in finite time. Therefore, ${\lambda}=1$ and $\mu=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(\rho, u)$ in $d-$dimensional compressible Euler equations with time-depending damping.
**Keywords.** Compressible Euler equations, damping, time-weighted energy inequality, Klainerman-Sobolev inequality, blowup, hypergeometric function.
**2010 Mathematical Subject Classification.** 35L70, 35L65, 35L67, 76N15.
author:
- |
Fei Hou$^{1, *}$ Huicheng Yin$^{2, }$[^1]\
\[12pt\] [1. Department of Mathematics and IMS, Nanjing University, Nanjing 210093, China]{}\
[2. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China]{}
title: 'On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping'
---
Introduction
============
In this paper, we are concerned with the global existence and blowup of $C^{\infty}-$smooth solution $(\rho, u)$ to the multi-dimensional compressible Euler equations with time-depending damping $$\label{euler-eqn}
\left\{ \enspace
\begin{aligned}
&{\partial}_t\rho+\opdiv(\rho u)=0,\\
&{\partial}_t(\rho u)+\opdiv(\rho u\otimes u+p\,{\textup{\uppercase\expandafter{\romannumeral1}}}_d)=-{\alpha}(t)\rho u,\\
&\rho(0,x)=\bar \rho+{\varepsilon}\rho_0(x),\quad u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $x=(x_1,\cdots,x_d)\in{\mathbb R}^d$, $d=2,3$, $\rho$, $u=(u_1,\cdots,u_d)$, and $p$ stand for the density, velocity and pressure, respectively, ${\textup{\uppercase\expandafter{\romannumeral1}}}_d$ is the $d\times d$ identity matrix, the frictional coefficient is ${\alpha}(t)=\frac{\mu}{(1+t)^{\lambda}}$ with ${\lambda}\ge0$ and $\mu>0$, and $u_0=(u_{1,0},\cdots,u_{d,0})$. The state equation of the gases is described by $p(\rho)=A\rho^{{\gamma}}$, where $A>0$ and ${\gamma}>1$ are constants. In addition, $\bar\rho>0$ is a constant, $\rho_0,u_0\in C_0^\infty({\mathbb R}^d)$, $\supp\rho_0,\supp u_0 \subseteq \{x\colon|x|\le M\}$, $(\rho_0, u_0)\not\equiv 0$, $\rho(0,x)>0$, and ${\varepsilon}>0$ is sufficiently small. For the physical background of , it can be found in [@Da] and the references therein.
For $\mu=0$ in ${\alpha}(t)$, is the standard compressible Euler equation. It is well known that smooth solution $(\rho, u)$ of will generally blow up in finite time. For examples, for a special class of initial data $(\rho(0,x), u(0,x))$, Sideris [@Sideris85] has proved that the smooth solution $(\rho, u)$ of in three space dimensions can develop singularities in finite time, and Rammaha in [@Rammaha89] has proved a blowup result in two space dimensions. For more extensive literature on the blowup results and the blowup mechanism for $(\rho, u)$, see [@Alinhac93; @Alinhac99a; @Alinhac99b; @Chr07; @CM14; @CL15; @DWY16; @Sideris97; @Speck14; @Yin04] and the references therein.
For ${\lambda}=0$ in ${\alpha}(t)$, it has been shown that admits a global smooth solution $(\rho, u)$, moreover, the long-term behavior of the solution $(\rho,u)$ has been established, see [@HL92; @HS96; @KY04; @Nish; @PZ09; @STW03; @TW12; @WY01; @WY07]. In particular, in [@STW03], the authors showed that the vorticity of velocity $u$ decays to zero exponentially in time $t$.
For $\mu>0$ and ${\lambda}>0$ in ${\alpha}(t)$, one naturally asks: does the smooth solution of blow up in finite time or does it exist globally? For the case of $\opcurl u_0\equiv0$, in [@HWY15], we have studied this problem in three space dimensions and proved that for $0\le{\lambda}\le1$ and $\mu>0$ there exists a global smooth solution $(\rho, u)$ of and while for ${\lambda}>1$, in general, the solution will blow up in finite time. In this paper, we will remove the assumption $\opcurl u_0\equiv0$ in [@HWY15] and systematically study this problem both in two and three space dimensions.
Obviously, one can divide ${\lambda}\ge0$, $\mu>0$ into four cases:
[**Case 1**]{}: $0\le{\lambda}<1$, $\mu>0$ for $d=2,3$;
[**Case 2**]{}: ${\lambda}=1$, $\mu>3-d$, for $d=2,3$;
[**Case 3**]{}: ${\lambda}=1$, $\mu\le 3-d$ for $d=2$;
[**Case 4**]{}: ${\lambda}>1$, $\mu>0$ for $d=2,3$.
0.1 true cm At first, we state the global existence results in this paper.
0.1 true cm
\[thm1\] If $0\le{\lambda}<1$ and $\mu>0$, then for small ${\varepsilon}>0$, admits a global $C^\infty-$ smooth solution $(\rho, u)$ which fulfills $\rho>0$ and which is uniformly bounded for $t\ge0$ together with all its derivatives. In addition, the vorticity $\opcurl u$ and its derivatives decay to zero in the rate $e^{-\frac{\mu}{3(1-{\lambda})}[(1+t)^{1-{\lambda}}-1]}$, where $\opcurl u={\partial}_1u_2-{\partial}_2u_1$ for $d=2$, and $\opcurl u=({\partial}_2u_3-{\partial}_3u_2, {\partial}_3u_1-{\partial}_1u_3, {\partial}_1u_2-{\partial}_2u_1)^T$ for $d=3$.
0.1 true cm
\[thm2\] If ${\lambda}=1$, $\mu>3-d$ and $\opcurl u_0\equiv 0$, then for small ${\varepsilon}>0$, admits a global $C^\infty-$ smooth solution $(\rho, u)$ which fulfills $\rho>0$ and which is uniformly bounded for $t\ge0$ together with all its derivatives.
Next we concentrate on Case 3 and Case 4. As in [@Rammaha89], we introduce the two functions $$\begin{aligned}
q_0(l)&\defeq \int_{x_1>l}(x_1-l)^2\left(\rho(0,x)-\bar\rho\right)dx, \\
q_1(l)&\defeq 2\int_{x_1>l}(x_1-l)(\rho u_1)(0,x)\,dx.\end{aligned}$$ Before stating our blowup result for problem , we require to introduce a special hypergeometric function $\Psi(a,b,c;z)$, where the constants $a$ and $b$ satisfy $a+b=1$ and $$ab=\left\{
\begin{aligned}
&\frac{\mu{\lambda}}{2}, && {\lambda}>1, \\
&\frac\mu2(1-\frac\mu2), && {\lambda}=1,
\end{aligned}
\right.$$ $c\in\Bbb R^+$, the variable $z\in\Bbb R$, and $$\Psi(a,b,c;z)={\displaystyle}\sum_{n=0}^{+\infty}{\frac}{(a)_n(b)_n}{n!(c)_n}z^n$$ with $(a)_n=a(a+1)\cdot\cdot\cdot(a+n-1)$ and $(a)_0=1$. It is known from [@EMOT] that $\Psi(a,b,c;z)$ is an analytic function of $z$ for $z\in(-1, 1)$ and $\Psi(a,b,c;0)=\Psi(a+1,b+1,c;0)=1$. In addition, there exists a small constant ${\delta}_0\in(0,1)$ depending on $\mu$ and ${\lambda}$ such that for $-\frac{{\delta}_0}{2}\le z\le 0$, $$\label{Psi-bound}
\frac12 \le \Psi(a,b,1;z), \Psi(a+1,b+1,2;z) \le \frac32.$$
\[thm3\] Suppose $\supp\rho_0,\supp u_0\subseteq \{x\colon|x|\le M\}$ and let $$\begin{aligned}
q_0(l)&>0, \label{q0-positive}\\
q_1(l)&\ge0 \label{q1-positive}\end{aligned}$$ hold for all $l\in ({\tilde}M, M)$, where ${\tilde}M$ is some fixed constant satisfying $0\le {\tilde}M<M$. Moreover, we assume that there exist two constants $M_0$ and $\Lambda$ with $\max\{{\tilde}M, M-{\delta}_0\}\le M_0<M$ and $\Lambda\ge 3ab$ such that $$\label{+condition}
q_1(l) \ge \Lambda q_0(l)$$ holds for all $l\in (M_0, M)$. If ${\lambda}=1$, $\mu\le1$ for $d=2$ or ${\lambda}>1$, $\mu>0$ for $d=2,3$, then there exists an ${\varepsilon}_0>0$ such that, for $0<{\varepsilon}\le{\varepsilon}_0$, the lifespan $T_{\varepsilon}$ of the smooth solution $(\rho, u)$ of is finite.
Our results in Theorem \[thm1\]-\[thm3\] are strongly motivated by considering the 1-D Burgers equation with time-depending damping term $$\label{burgers-eqn}
\left\{ \enspace
\begin{aligned}
&{\partial}_t v+v{\partial}_x v=-\,{\displaystyle}\frac{\mu}{(1+t)^{\lambda}}\,v,\qquad (t,x)\in{\mathbb R}_+\times{\mathbb R},\\
&v(0,x)={\varepsilon}v_0(x),
\end{aligned}
\right.$$ where ${\lambda}\ge0$ and $\mu>0$ are constants, $v_0\in C_0^{\infty}({\mathbb R})$, $v_0\not\equiv 0$, and ${\varepsilon}>0$ is sufficiently small. One may directly obtain that by the method of characteristics $$\left\{ \enspace
\begin{aligned}
&T_{{\varepsilon}}=\infty, & \text{if $0\le{\lambda}<1$ or ${\lambda}=1$, $\mu>1$,}\\
&T_{{\varepsilon}}<\infty, & \text{if ${\lambda}>1$ or ${\lambda}=1$, $0<\mu\le 1$,}
\end{aligned}
\right.$$ where $T_{{\varepsilon}}$ is the lifespan of the smooth solution $v$ of . Especially in the case of $0\le{\lambda}<1$, $v$ exponentially decays to zero with respect to the time $t$. This means that ${\lambda}=1$ and $\mu=1$ appear to be the critical power and critical value respectively, for the global existence of smooth solution $v$ of .
For the three dimensional problem and the case ${\lambda}=0$ in ${\alpha}(t)$, the authors in [@STW03] proved that the fluid vorticity decays to zero exponentially in time, while the solution $(\rho, u)$ does not decay exponentially. In [**Case 1**]{} of $0\le{\lambda}<1$ and $\mu>0$, we have precisely proved that the vorticity $\opcurl u$ decays to zero in the rate $e^{-\frac{\mu}{3(1-{\lambda})}[(1+t)^{1-{\lambda}}-1]}$ in Theorem \[thm1\].
In Theorem \[thm2\], we pose the assumption of $\opcurl u_0\equiv 0$ for [**Case 2**]{}. If not, it seems difficult for us to obtain the uniform control on the vorticity $\opcurl u$ by our method. Namely, so far we do not know whether the assumption of $\opcurl u_0\equiv 0$ can be removed in order to obtain the global existence of $(\rho, u)$ in [**Case 2**]{}.
It is not hard to find a large number of initial data $(\rho,u)(0,x)$ such that - are satisfied. For instance, choosing $\rho_0(x)>0$ and $u_{1,0}(x)=x_1\rho_0(x)\Lambda/\bar\rho$, then we get -.
\[rem1.6\] In [@Sideris85] and , the authors have shown the formation of singularities in multi-dimensional compressible Euler equations (corresponding $\mu=0$ in ) under the assumptions of -. However, in order to prove the blowup result of smooth solution $(\rho, u)$ to problem and overcome the difficulty arisen by the time-depending frictional coefficient ${\frac}{\mu}{(1+t)^{\lambda}}$ with $\mu>0$ and ${\lambda}\ge1$, we pose an extra assumption except -, which leads to the non-negativity lower bound of $P(t,l)$ in so that two ordinary differential blowup inequalities - can be established. One can see more details in $\S 5$.
If the damping term ${\alpha}(t)\rho u$ in is replaced by $({\alpha}_1(t)\rho u_1,\cdots,{\alpha}_d(t)\rho u_d)^T$ with ${\alpha}_i(t)=\frac{\mu_i}{(1+t)^{{\lambda}_i}}$ ($i=1,\cdots,d$), and there exists some $i_0$ $(1\le i_0\le d)$ such that ${\lambda}_{i_0}$ and $\mu_{i_0}$ satisfy [**Case 3**]{} or [**Case 4**]{}. In this case, one can define the new quantities $$\begin{aligned}
q_0(l)&= \int_{x_{i_0}>l}(x_{i_0}-l)^2\left(\rho(0,x)-\bar\rho\right)dx, \\
q_1(l)&= 2\int_{x_{i_0}>l}(x_{i_0}-l)(\rho u_2)(0,x)\,dx\end{aligned}$$ and $$P(t,l)=\int_{x_{i_0}>l}(x_{i_0}-l)^2\left(\rho(t,x)-\bar\rho\right)dx$$ instead of the ones in - and , respectively, we then obtain an analogous result in Theorem \[thm3\] by applying the same procedure in $\S 5$.
Let us indicate the proofs of Theorems \[thm1\]-\[thm3\]. Without loss of generality, from now on we assume that $\bar c=c(\bar\rho)=1$, where $c(\rho)=\sqrt{P'(\rho)}$ is the sound speed. At first, we reformulate problem . Set $$\label{theta-def}
\theta \defeq \frac1{{\gamma}-1}(A{\gamma}\rho^{{\gamma}-1}-1)=\frac1{{\gamma}-1}(c^2(\rho)-1).$$ Then problem can be rewritten as $$\label{euler-reform}
\left\{ \enspace
\begin{aligned}
&{\partial}_t\theta+u\cdot\nabla\theta+(1+({\gamma}-1)\theta)\opdiv u=0, \\
&{\partial}_tu+\frac\mu{(1+t)^{{\lambda}}}u+u\cdot\nabla u+\nabla\theta=0, \\
&\theta(0,x)=\frac{1}{{\gamma}-1}[(1+\frac{{\varepsilon}\rho_0(x)}{\bar\rho})^{{\gamma}-1}-1]
\defeq {\varepsilon}\theta_0(x)+{\varepsilon}^2g(x,{\varepsilon}), \\
& u(0,x)={\varepsilon}u_0(x),
\end{aligned}
\right.$$ where $\nabla=({\partial}_1,\cdots,{\partial}_d)=({\partial}_{x_1},\cdots,{\partial}_{x_d})$, $\theta_0(x)=\frac{\rho_0(x)}{\bar\rho}$ and $g(x,{\varepsilon})=({\gamma}-2)\frac{\rho_0^2(x)}{\bar\rho^2}\int_0^1 (1+\frac{\sigma{\varepsilon}\rho_0(x)}{\bar\rho})^{{\gamma}-3}(1-\sigma)\,d\sigma$. Note that $g(x,{\varepsilon})$ is smooth in $(x,{\varepsilon})$ and has compact support in $x$.
To prove Theorem \[thm1\], we introduce such a time-weighted energy $$\label{energy1}
\mathcal{E}_k[\Phi](t) \defeq (1+t)^{\lambda}\sum_{1\le|{\alpha}|+j\le k}
\|{\partial}_t^j\nabla^{\alpha}\Phi(t,\cdot)\|+\|\Phi(t,\cdot)\|,$$ where $k$ is a fixed positive number, and $\|\cdot\|$ stands for the $L_x^2$ norm on ${\mathbb R}^d$, i.e., $$\|\Phi(t,\cdot)\| \defeq \|\Phi(t,x)\|_{L_x^2({\mathbb R}^d)}=
\left(\int_{{\mathbb R}^d}|\Phi(t,x)|^2dx\right)^\frac12.$$ Denote by $$\label{energy+}
\mathcal{E}_k[\Phi_1,\Phi_2](t) \defeq \mathcal{E}_k[\Phi_1](t)+\mathcal{E}_k[\Phi_2](t).$$ For $0\le{\lambda}<1$ and $\mu>0$, one can choose a constant $t_0$ such that $$\label{critical-time}
(1+t_0)^{1-{\lambda}}=\max\,\{\frac2\mu, 1\},$$ so that problem has a local solution $(\theta, u)\in C^{\infty}([0, t_0]\times{\mathbb R}^3)$ by the smallness of ${\varepsilon}>0$ (see the local existence result for the multidimensional hyperbolic systems in [@Ma]). Making use of the vorticity $\opcurl u$ and the conditions of $0\le{\lambda}<1$ and $\mu>0$ in [**Case 1**]{}, and simultaneously taking the delicate analysis on the system , the uniform time-weighted energy estimates for $\mathcal{E}_4[\theta, u](t)$ are obtained. This, together with the continuity argument, yields the proof of Theorem \[thm1\].
Since we have proved Theorem \[thm2\] in [@HWY15] for the [**Case 2**]{} with $\opcurl u_0\equiv 0$ in three space dimensions, we only require to focus on the proof of Theorem \[thm2\] in two space dimensions. For this purpose, we define another energy $$\label{energy2}
E_k[\Phi](t) \defeq (1+t)^\frac12\sum_{0\le|{\alpha}|\le k-1}
\|{\partial}Z^{\alpha}\Phi(t,\cdot)\|+(1+t)^{-\frac12}\|\Phi(t,\cdot)\|,$$ where ${\partial}=({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2})$, $Z=(Z_0, Z_1, \dots, Z_6)=({\partial}, S, R, H)$ with the scaling field $S=t{\partial}_t+x_1{\partial}_1+x_2{\partial}_2$, the rotation field $R=x_1{\partial}_2-x_2{\partial}_1$, the Lorentz fields $H=(H_1, H_2)=(x_1{\partial}_t+t{\partial}_1, x_2{\partial}_t+t{\partial}_2)$ and $Z^{\alpha}=Z_0^{{\alpha}_0}Z_1^{{\alpha}_1}\cdots Z_6^{{\alpha}_6}$. From we may derive a damped wave equation of $\theta$ as follows $$\label{damped-wave}
{\partial}_t^2\theta+\frac\mu{1+t}{\partial}_t\theta-\Delta\theta=Q(\theta,u),$$ where the expression of $Q(\theta,u)$ will be given in below. Thanks to $\opcurl u\equiv0$, we can get the estimates of velocity $u$ from the equations in (see Lemma \[lem-Zvelocity\]). By $\mu>1$ and a rather technical analysis on the damped wave equation , we eventually show in $\S 4$ that $E_5[\theta,u](t) \le \frac12 \,K_3{\varepsilon}$ (see for the definition of $E_5[\theta,u](t)$) holds when $E_5[\theta,u](t) \le K_3{\varepsilon}$ is assumed for some suitably large constant $K_3>0$ and small ${\varepsilon}>0$. Based on this and the continuity argument, the global existence of $(\theta, u)$ and then Theorem \[thm2\] in two space dimensions are established for ${\lambda}=1$, $\mu>1$ and $\opcurl u_0\equiv0$.
To prove the blowup result in Theorem \[thm3\], as in [@Rammaha89; @Sideris85], we shall derive some blowup-type second-order ordinary differential inequalities in $\S 5$. From this and assumptions -, an upper bound of the lifespan $T_{{\varepsilon}}$ is derived by making use of ${\lambda}=1$, $\mu\le 3-d$ or ${\lambda}>1$, and then the proof of Theorem \[thm3\] is completed.
Here we point out that in [@HWY15], for the 3-d [**irrotational**]{} compressible Euler equations, it has been shown that for $0\le{\lambda}\le1$, there exists a global $C^{\infty}-$smooth small amplitude solution $(\rho, u)$, while for ${\lambda}>1$, the smooth solution $(\rho, u)$ generally blows up in finite time. This means that we have extended the global existence and blowup results in [@HWY15] for the 3-D irrotational flows to the 2-D and 3-D full Euler systems.
In the whole paper, we shall use the following convention:
- $C$ will denote a generic positive constant which is independent of $t$ and ${\varepsilon}$.
- $A\ls B$ or $B\gt A$ means $A\le CB$.
- $r=|x|=\sqrt{x_1^2+\cdots+x_d^2}$, $\sigma_-(t,x)\defeq
\sqrt{1+(r-t)^2}$.
- $\|\Phi(t,\cdot)\| \defeq \|\Phi(t,x)\|_{L_x^2({\mathbb R}^d)}$, $|\Phi(t,\cdot)|_\infty \defeq |\Phi(t,x)|_{L_x^\infty}=
{\displaystyle}\sup_{x\in{\mathbb R}^d}|\Phi(t,x)|$.
- $Z$ denotes one of the Klainerman vector fields $\{{\partial}, S, R, H\}$ on ${\mathbb R}_+\times{\mathbb R}^2$, where ${\partial}=({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2})$, $S=t{\partial}_t+x_1{\partial}_1+x_2{\partial}_2$, $R=x_1{\partial}_2-x_2{\partial}_1$ and $H=(H_1, H_2)=
(x_1{\partial}_t+t{\partial}_1, x_2{\partial}_t+t{\partial}_2)$.
- For two vector fields $X$ and $Y$, $[X,Y] \defeq XY-YX$ denotes the Lie bracket.
- Greek letters ${\alpha},\beta,\cdots$ denote multiple indices, i.e., ${\alpha}=({\alpha}_0,\cdots,{\alpha}_m)$, and $|{\alpha}|={\alpha}_0+\cdots+{\alpha}_m$ denotes its length, where ${\alpha}_i$ is some non-negative integer for all $i=0,\cdots,m$.
- For two multiple indices ${\alpha}$ and $\beta$, $\beta\le{\alpha}$ means $\beta_i\le{\alpha}_i$ for all $i=0,\cdots,m$ while $\beta<{\alpha}$ means $\beta\le{\alpha}$ and $\beta_i<{\alpha}_i$ for some $i$.
- For the differential operator $O=(O_0,\cdots,O_m)$, for example, $O=({\partial}_t,
{\partial}_{x_1},\cdots,{\partial}_{x_d})$ in $\S 3$ and $O=
({\partial}_t, {\partial}_{x_1}, {\partial}_{x_2}, S, R, H)$ in $\S 4$, denote $O^{\alpha}\defeq O_0^{{\alpha}_0}\cdots O_m^{{\alpha}_m}$, $O^{\le{\alpha}}
\defeq {\displaystyle}\sum_{0\le\beta\le{\alpha}}O^\beta$, $O^{<{\alpha}} \defeq {\displaystyle}\sum_
{0\le\beta<{\alpha}}O^\beta$ and $O^{\le k} \defeq {\displaystyle}\sum_{0\le|{\alpha}|\le k}
O^{\alpha}$ with $k$ is an integer.
- Leibniz’s rule: $O^{\alpha}(\Phi\Psi)={\displaystyle}\sum_{0\le\beta\le{\alpha}}C_{{\alpha},\beta}
O^\beta\Phi O^{{\alpha}-\beta}\Psi$ will be abbreviated as\
$O^{\alpha}(\Phi\Psi)={\displaystyle}\sum_{0\le\beta\le{\alpha}}O^\beta\Phi O^{{\alpha}-\beta}\Psi$.
- $\Xi$ is the solution of ${\displaystyle}\Xi'(t) = \frac{\mu}{(1+t)^{\lambda}}\,
\Xi(t)$ with $\Xi(0)=1$, i.e., $$\label{Xi-def}
\Xi(t)\defeq \begin{cases} e^{\frac{\mu}{1-{\lambda}}[(1+t)^{1-{\lambda}}-1]}, &
{\lambda}\ge 0,\,{\lambda}\neq1,\\ (1+t)^\mu, & {\lambda}=1. \end{cases}$$
- $c(\bar\rho)=1$ will be assumed throughout (otherwise, introduce $X=x/c(\bar\rho)$ as new space coordinate if necessary).
Some Preliminaries {#section2}
==================
At first, we derive the scalar equation of $\theta$ in . It follows from the first equation in that $$\label{dtdivu1}
{\partial}_t\opdiv u=-\frac1{(1+({\gamma}-1)\theta)} ({\partial}_t^2\theta+u\cdot\nabla{\partial}_t\theta
+{\partial}_tu\cdot\nabla\theta+({\gamma}-1){\partial}_t\theta\opdiv u).$$ Taking divergence on the second equation in yields $$\label{dtdivu2}
\opdiv{\partial}_t u+\frac\mu{(1+t)^{\lambda}}\opdiv u+\Delta\theta+
u\cdot\nabla\opdiv u+\sum_{i,j=1}^d {\partial}_iu_j{\partial}_ju_i=0,$$ where $\Delta={\partial}_1^2+\cdots+{\partial}_d^2$. Substituting into yields the damped wave equation of $\theta$ $$\label{theta-eqn}
{\partial}_t^2\theta+\frac\mu{(1+t)^{\lambda}}{\partial}_t\theta-\Delta\theta=Q(\theta,u),$$ where $$\begin{aligned}
Q(\theta,u) &\defeq Q_1(\theta,u)+Q_2(\theta,u), \label{Q-def}\\
Q_1(\theta,u) &\defeq ({\gamma}-1)\theta\Delta\theta-\frac\mu{(1+t)^{\lambda}}
u\cdot\nabla\theta-2u\cdot\nabla{\partial}_t\theta-\sum_{i,j=1}^d u_iu_j{\partial}_{ij}^2\theta \label{Q1-def},\\
Q_2(\theta,u) &\defeq -\sum_{i,j=1}^d u_i{\partial}_iu_j{\partial}_j\theta-{\partial}_tu\cdot\nabla\theta
+(1+({\gamma}-1)\theta) (\sum_{i,j=1}^d {\partial}_iu_j{\partial}_ju_i+({\gamma}-1)|\opdiv u|^2). \label{Q2-def}\end{aligned}$$ Let $$\label{vorticity-def}
w\defeq \opcurl u=
\begin{cases}
{\partial}_1u_2-{\partial}_2u_1, & d=2, \\
({\partial}_2u_3-{\partial}_3u_2, {\partial}_3u_1-{\partial}_1u_3, {\partial}_1u_2-{\partial}_2u_1)^T, & d=3.
\end{cases}$$ Then the second equation in implies that for $d=2$ $$\label{2dvorticity-eqn}
{\partial}_tw+\frac\mu{(1+t)^{\lambda}}w+u\cdot\nabla w+w\opdiv u=0$$ and for $d=3$ $$\label{3dvorticity-eqn}
{\partial}_tw+\frac\mu{(1+t)^{\lambda}}w+u\cdot\nabla w+w\opdiv u=w\cdot\nabla u.$$ To prove Theorem \[thm1\]-\[thm2\], we require to introduce the following lemma, which is easily shown.
\[lem-divcurl\] Let $U(x)=(U_1(x),\cdots,U_d(x))$ be a vector-valued function with compact support on ${\mathbb R}^d$ ($d=2,3$), then there holds that $$\label{divcurl-estimate}
\|\nabla U\|\le \|\opcurl U\|+\|\opdiv U\|.$$
The following Sobolev type inequality can be found in [@Kl87].
\[lem-Klainerman-ineq\] Let $\Phi(t,x)$ be a function on ${\mathbb R}^{1+2}$, then there exists a constant $C$ such that $$\label{Klainerman-ineq}
(1+t+r)\,\sigma_-(t,x)\,|\Phi(t,x)| \le C\sum_{|{\alpha}|\le2} \|Z^{\alpha}\Phi(t,\cdot)\|^2.$$
In addition, we have
\[lem-weight\] Let $\Phi(t,x)$ be a function on ${\mathbb R}^{1+2}$ and assume $\supp\Phi\subseteq\{(t,x)\colon |x|\le t+M\}$, then there exists a constant $C>0$ such that for $\nu\in (-\infty,1)$ $$\label{weight-pointwise}
|\sigma_-^{\nu-1}(t,\cdot)\Phi(t,\cdot)|_\infty \le C|\sigma_-^{\,\nu}
(t,\cdot)\nabla\Phi(t,\cdot)|_\infty$$ and for $\ell\neq 1$ $$\label{weight-L2}
\|\sigma_-^{-\ell}(t,\cdot)\Phi(t,\cdot)\| \le
C(t+M)^{(1-\ell)_+}\,\|\nabla\Phi(t,\cdot)\|,$$ where $(1-\ell)_+=\max\,\{1-\ell, 0\}$ and $\ell\in{\mathbb R}$.
For the purpose of completeness, we prove - here. In fact, for the proof of , one can also see [@Alinhac93 Lemma 2.2]. By introducing the polar coordinate $(r,\phi)$ such that $x_1=r\cos\phi$ and $x_2=r\sin\phi$, we then have $$\begin{aligned}
\Phi(t,x)&=\Phi(t,r\cos\phi,r\sin\phi)=-\int_r^{t+M} \frac{d}{d\xi}\,
\Phi(t,\xi\cos\phi,\xi\sin\phi) \,d\xi {\nonumber}\\
&=-\int_r^{t+M} (\cos\phi\,{\partial}_1\Phi(t,\xi\cos\phi,\xi\sin\phi)+
\sin\phi\,{\partial}_2\Phi(t,\xi\cos\phi,\xi\sin\phi) ) \,d\xi. \label{2.14}\end{aligned}$$ Together with the mean value theorem, this yields $$\Phi(t,x) \ls |\sigma_-^{\,\nu}(t,\cdot)\nabla\Phi(t,\cdot)|_\infty
\int_r^{t+M} (1+|t-\xi|)^{-\nu} \,d\xi,$$ which immediately derives . On the other hand, applying Cauchy-Schwartz inequality to derives $$|\Phi(t,r\cos\phi,r\sin\phi)|^2 \le \left( \int_r^{t+M}
|\nabla\Phi(t,\xi\cos\phi,\xi\sin\phi)|^2 \,\xi\,d\xi\right)
\left( \int_r^{t+M} \frac1\xi \,d\xi\right),$$ which yields $$\begin{aligned}
\left\|\frac{\Phi(t,\cdot)}{(t+2M-r)^\ell}\right\|^2
&= \int_0^{2\pi}\int_0^{t+M} \frac{|\Phi(t,r\cos\phi,r\sin\phi)|^2}
{(t+2M-r)^{2\ell}} \,r\,drd\phi {\nonumber}\\
&\le \int_0^{2\pi}\int_0^{t+M} \frac{r\int_r^{t+M} \frac1\xi \,d\xi}
{(t+2M-r)^{2\ell}} \,dr \int_r^{t+M} |\nabla\Phi(t,\xi\cos\phi,
\xi\sin\phi)|^2 \,\xi\,d\xi d\phi {\nonumber}\\
&\ls \int_0^{t+M} \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}} \,dr
\,\|\nabla\Phi(t,\cdot)\|^2. \label{2.15}\end{aligned}$$ On the other hand, it follows from direct computation that $$\label{2.16}
\int_0^\frac{t+M}{2}\, \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}} \,dr
\ls \frac1{(t+M)^{2\ell}} \int_0^\frac{t+M}{2}\, r\log\frac{t+M}r \,dr
\ls (t+M)^{2(1-\ell)}$$ and $$\label{2.17}
\int_\frac{t+M}{2}^{t+M}\, \frac{r\log\frac{t+M}r}{(t+2M-r)^{2\ell}}\,dr
=\int_0^\frac{t+M}{2}\, \frac{(t+M-\xi)\log\frac{t+M}{t+M-\xi} }{(M+\xi)^{2\ell}} \,d\xi
\le \int_0^\frac{t+M}{2} \,\frac\xi{(M+\xi)^{2\ell}} \,d\xi,$$ where we have used the fact of $\frac{t+M}{t+M-\xi}=1+\frac\xi{t+M-\xi} \le e^\frac\xi{t+M-\xi}$ in the last inequality.
Substituting - into and taking direct computation yield . Thus, the proof of Lemma \[lem-weight\] is completed.
Proof of Theorem 1.1. {#section3}
=====================
Throughout this section, we will always assume that $\mathcal{E}_4[\theta,u](t) \le K_1{\varepsilon}$ holds, where the definition of $\mathcal{E}_4[\theta,u](t)$ has been given in and . Together with the standard Sobolev embedding theorem, this yields $$\label{3.1}
|(\theta,u)(t,\cdot)|_\infty+(1+t)^{\lambda}|{\partial}{\partial}^{\le1}(\theta,u)(t,\cdot)|_\infty \ls K_1{\varepsilon}.$$ To prove Theorem \[thm1\], we now carry out the following parts.
Estimates of velocity $u$ and vorticity $w$.
--------------------------------------------
The following lemma is an application of Lemma \[lem-divcurl\] and .
\[lem-velocity1\] Under assumption , for all $t>0$, one has $$\label{3.2}
\mathcal{E}_4[u](t)\ls \|u(t,\cdot)\|+(1+t)^{\lambda}\left(\|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}{\partial}^{\le3}\theta(t,\cdot)\|\right),$$ where the definition of $w$ has been given in .
By the equations in , we see that $$\begin{aligned}
\opdiv u &= -\frac{{\partial}_t\theta+u\cdot\nabla\theta}{1+({\gamma}-1)\theta}, \label{3.3} \\
{\partial}_tu &= -\left(u\cdot\nabla u+\frac\mu{(1+t)^{\lambda}}u+\nabla\theta\right). \label{3.4}\end{aligned}$$ Taking $U={\partial}^{\alpha}u$ with $|{\alpha}|\le3$ in , we then arrive at $$\begin{aligned}
\|\nabla{\partial}^{\le3}& u(t,\cdot)\|
\ls \|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}^{\le3}\opdiv u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}^{\le3} w(t,\cdot)\|+\|{\partial}_t{\partial}^{\le3}\theta(t,\cdot)\|+ K_1{\varepsilon}\left( \|\nabla{\partial}^{\le3}\theta(t,\cdot)\|+
(1+t)^{-{\lambda}}\|{\partial}^{\le3} u(t,\cdot)\|\right), \label{3.5}\end{aligned}$$ where we have used and in the last inequality. Taking the $L^2$ norm of ${\partial}^{\alpha}$ yields $$\begin{aligned}
\|{\partial}_t{\partial}^{\alpha}u(t,\cdot)\|
&\ls \|\nabla{\partial}^{\alpha}\theta(t,\cdot)\|+(1+K_1{\varepsilon})(1+t)^{-{\lambda}}
\|{\partial}^{\le{\alpha}} u(t,\cdot)\| +K_1{\varepsilon}\|\nabla{\partial}^{\le{\alpha}} u(t,\cdot)\|. \label{3.6}\end{aligned}$$ Rewrite ${\partial}^{\alpha}={\partial}_t^k{\partial}_x^\beta$ with $0\le k+|\beta|\le3$. Summing up from $k=0$ to $k=3$ yields $$\begin{aligned}
\|{\partial}_t{\partial}^{\le3} u(t,\cdot)\|
&\ls \|\nabla{\partial}^{\le3}\theta(t,\cdot)\|+(1+t)^{-{\lambda}}\|u(t,\cdot)\|
+K_1{\varepsilon}\|\nabla{\partial}^{\le3} u(t,\cdot)\|. \label{3.7}\end{aligned}$$ By the smallness of ${\varepsilon}>0$, we immediately derive from and . This completes the proof of Lemma \[lem-velocity1\].
The following lemma shows the estimate of velocity $u$ itself.
\[lem-velocity2\] Let $\mu>0$. Under assumption , for all $t>0$, it holds that $$\label{3.8}
\frac{d}{dt}\|(\theta,u)(t,\cdot)\|^2+\frac\mu{(1+t)^{\lambda}}\|u(t,\cdot)\|^2
\ls (1+t)^{\lambda}|\theta(t,\cdot)|_\infty\|\nabla\theta(t,\cdot)\|^2.$$
Multiplying the second equation in by $u$ derives $$\label{3.9}
\frac12\,{\partial}_t|u|^2+\frac\mu{(1+t)^{\lambda}}|u|^2+u\cdot\nabla\theta=
-\frac12\,u\cdot\nabla|u|^2.$$ From the first equation in , we see that $$\begin{aligned}
u\cdot\nabla\theta &= \opdiv(\theta u)-\theta\opdiv u {\nonumber}\\
&= \opdiv(\theta u)+\theta({\partial}_t\theta+u\cdot\nabla\theta+
({\gamma}-1)\theta\opdiv u) {\nonumber}\\
&= \opdiv(\theta u+({\gamma}-1)\theta^2 u)+\frac12\,{\partial}_t|\theta|^2+
(3-2{\gamma})\theta\,u\cdot\nabla\theta. \label{3.10}\end{aligned}$$ Substituting into and integrating it over ${\mathbb R}^d$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\|(\theta,u)(t,\cdot)\|^2+\frac{2\mu}{(1+t)^{\lambda}}
\|u(t,\cdot)\|^2 {\nonumber}\\
&\ls |\theta(t,\cdot)|_\infty\|u(t,\cdot)\|\,\|\nabla\theta(t,\cdot)\|
+|\nabla u(t,\cdot)|_\infty\|u(t,\cdot)\|^2. \label{3.11}\end{aligned}$$ Substituting into and applying $\mu>0$ and the smallness of ${\varepsilon}$, we derive . This completes the proof of Lemma \[lem-velocity2\].
Next lemma shows the estimates of vorticity $w$ and its derivatives.
\[lem-vorticity\] Let $\mu>0$. Under assumption , for all $t>0$, it holds that $$\label{3.12}
\frac{d}{dt}\|{\partial}^{\le3} w(t,\cdot)\|^2+\frac\mu{(1+t)^{\lambda}}
\|{\partial}^{\le3} w(t,\cdot)\|^2
\ls |{\partial}^{\le1} w(t,\cdot)|_\infty\, \|{\partial}^{\le3} w(t,\cdot)\|\,
\|{\partial}{\partial}^{\le3} u(t,\cdot)\|.$$
It follows from vorticity equation - that for $|{\alpha}|\le3$, $$\begin{aligned}
& {\partial}_t{\partial}^{\alpha}w+\frac{\mu}{(1+t)^{\lambda}}{\partial}^{\alpha}w+u\cdot\nabla{\partial}^{\alpha}w {\nonumber}\\
&= -\sum_{0<\beta\le{\alpha}}\left[{\partial}^\beta\left(\frac\mu{(1+t)^{\lambda}}\right)
{\partial}^{{\alpha}-\beta} w+{\partial}^\beta u\cdot\nabla{\partial}^{{\alpha}-\beta} w\right]
-{\partial}^{\alpha}(w\opdiv u-w\cdot\nabla u),\label{3.13}\end{aligned}$$ here we point out that the last term $w\cdot\nabla u$ in does not appear when $d=2$ . Multiplying by ${\partial}^{\alpha}w$ and integrating it over ${\mathbb R}^d$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\|{\partial}^{\alpha}w(t,\cdot)\|^2+\frac{2\mu}{(1+t)^{\lambda}}
\|{\partial}^{\alpha}w(t,\cdot)\|^2 {\nonumber}\\
&\ls \frac1{(1+t)^{1+{\lambda}}}\|{\partial}^{\alpha}w(t,\cdot)\|\,\|{\partial}^{<{\alpha}}w(t,\cdot)\|+
|{\partial}{\partial}^{\le1} u(t,\cdot)|_\infty \|{\partial}^{\le{\alpha}}w(t,\cdot)\|^2 {\nonumber}\\
&\quad +|{\partial}^{\le1}w(t,\cdot)|_\infty \|{\partial}{\partial}^{\le3} u(t,\cdot)\|\,
\|{\partial}^{\le{\alpha}}w(t,\cdot)\|. \label{3.14}\end{aligned}$$ Note that when ${\alpha}=0$, the firs term $\|{\partial}^{\alpha}w(t,\cdot)\|\,\|{\partial}^{<{\alpha}}w(t,\cdot)\|$ in the right hand side of does not appear. Summing up from $|{\alpha}|=0$ to $|{\alpha}|=3$ and applying , $\mu>0$ and the smallness of ${\varepsilon}$, we then obtain . This completes the proof of Lemma \[lem-vorticity\].
\[rmk3.1\] The proof of Lemma \[lem-velocity2\] and Lemma \[lem-vorticity\] only depends on , $\mu>0$ and the smallness of ${\varepsilon}$.
Estimates of $\theta$ and its derivatives.
------------------------------------------
The next lemma shows the global estimates of $\theta$ and its derivatives for $t>t_0$, where $t_0$ is defined in .
\[lem-thetaglobal\] Let $0\le{\lambda}<1$, $\mu>0$. Under assumption , for all $t>t_0$, it holds that $$\begin{aligned}
&\quad \mathcal{E}_4^2[\theta](t)
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+K_1{\varepsilon}\int_{t_0}^t\left((1+s)^{-{\lambda}}
\|u(s,\cdot)\|^2+(1+s)^{\lambda}\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds. \label{3.15}\end{aligned}$$
Acting ${\partial}^{\alpha}$ with $|{\alpha}|\le3$ on both sides of equation shows $${\partial}_t^2{\partial}^{\alpha}\theta+\frac\mu{(1+t)^{\lambda}}{\partial}_t{\partial}^{\alpha}\theta-\Delta{\partial}^{\alpha}\theta
={\partial}^{\alpha}Q(\theta,u)+\sum_{\beta<{\alpha}} {\partial}^{{\alpha}-\beta}\left(\frac\mu{(1+t)^{\lambda}}
\right) {\partial}_t{\partial}^\beta\theta.$$ Multiplying this by $2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta$ derives $$\begin{aligned}
&{\partial}_t\left[(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu^2}2|{\partial}^{\alpha}\theta|^2-\frac{\mu{\lambda}}{2}(1+t)^{{\lambda}-1}|{\partial}^{\alpha}\theta|^2\right] {\nonumber}\\
& +\left[\mu(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1}\right]|{\partial}{\partial}^{\alpha}\theta|^2-
\opdiv\left[\nabla{\partial}^{\alpha}\theta\left(2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta
+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta\right) \right]{\nonumber}\\
&= \left(2(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta\right)
\left({\partial}^{\alpha}Q(\theta,u)+\sum_{\beta<{\alpha}} {\partial}^{{\alpha}-\beta}\left(
\frac\mu{(1+t)^{\lambda}}\right) {\partial}_t{\partial}^\beta\theta\right) {\nonumber}\\
&\qquad +\frac{\mu{\lambda}(1-{\lambda})}{2}(1+t)^{{\lambda}-2}|{\partial}^{\alpha}\theta|^2. \label{3.16}\end{aligned}$$ Thanks to $0\le{\lambda}<1$, $\mu>0$ and the choice of $t_0$ (see ), for all $t>t_0$ one easily knows that for the term in the first square bracket of the second line in , $$\label{3.17}
\mu(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1} \ge \mu(1-{\lambda})(1+t)^{\lambda}.$$ Furthermore, one gets that for the term in the square bracket of the first line in , $$\begin{aligned}
(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t)^{\lambda}{\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu^2}{2}|{\partial}^{\alpha}\theta|^2-\frac{\mu{\lambda}}{2}(1+t)^{{\lambda}-1}|{\partial}^{\alpha}\theta|^2 \\
=(1+t)^{2{\lambda}}\left(\frac{1-{\lambda}}{3-{\lambda}}|{\partial}_t{\partial}^{\alpha}\theta|^2+|\nabla{\partial}^{\alpha}\theta|^2\right)
+\frac{\mu^2(1-{\lambda})}{8}|{\partial}^{\alpha}\theta|^2 \\
+\left((1+t)^{\lambda}\sqrt\frac{2}{3-{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\frac\mu2\sqrt\frac{3-{\lambda}}{2}{\partial}^{\alpha}\theta\right)^2
+\frac{\mu{\lambda}}{4}(\mu-2(1+t)^{{\lambda}-1})|{\partial}^{\alpha}\theta|^2,\end{aligned}$$ which is equivalent to $(1+t)^{2{\lambda}}|{\partial}{\partial}^{\alpha}\theta|^2+|{\partial}^{\alpha}\theta|^2$ for $0\le{\lambda}<1$ and $t>t_0$. Consequently, integrating over $[t_0,t]\times{\mathbb R}^d$ gives $$\begin{aligned}
&\quad (1+t)^{2{\lambda}}\|{\partial}{\partial}^{\alpha}\theta(t,\cdot)\|^2+\|{\partial}^{\alpha}\theta(t,\cdot)\|^2
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\alpha}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+\int_{t_0}^t(1-{\lambda})(1+s)^{{\lambda}-2}\|\theta(s,\cdot)\|^2\,ds
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& +\left|\int_{t_0}^t\int_{{\mathbb R}^d} \left({\partial}^{\alpha}Q_1(\theta,u)+{\partial}^{\alpha}Q_2(\theta,u)\right) \left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+
\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right|. \label{3.18}\end{aligned}$$ Next we deal with the last term in the right hand side of . It follows from - and $|{\alpha}|\le3$ that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} {\partial}^{\alpha}Q_2(\theta,u) \left(2(1+s)^{2{\lambda}}
{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}(\theta,u)(s,\cdot)\|^2\,ds {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds
+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.19}\end{aligned}$$ Now we turn our attention to the term ${\partial}^{\alpha}Q_1(\theta,u)$. It is easy to get $$\begin{aligned}
{\partial}^{\alpha}Q_1(\theta,u) \defeq -{\partial}^{\alpha}\left(\frac\mu{(1+t)^{\lambda}}u\cdot\nabla\theta\right)+({\gamma}-1)\theta\Delta{\partial}^{\alpha}\theta {\nonumber}\\
-2u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta-\sum_{i,j=1}^d u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta+Q^{\alpha}_1(\theta,u). \label{3.20}\end{aligned}$$ One easily checks that still holds if ${\partial}^{\alpha}Q_2(\theta,u)$ is replaced by $Q^{\alpha}_1(\theta,u)$. In addition, for ${\alpha}=0$, we see that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \frac\mu{(1+s)^{\lambda}}u\cdot\nabla\theta
\left(2(1+s)^{2{\lambda}}{\partial}_t\theta+\mu(1+s)^{\lambda}\theta\right) \,dxds\right| {\nonumber}\\
&\ls \int_{t_0}^t \left(|\theta(s,\cdot)|_\infty \|u(s,\cdot)\|+(1+s)^{\lambda}|u(s,\cdot)|_\infty\|{\partial}_t\theta(s,\cdot)\|\right) \|\nabla\theta(s,\cdot)\|\,ds {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t \left( (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2+(1+s)^{\lambda}\|{\partial}\theta(s,\cdot)\|^2\right) \,ds, \label{3.21}\end{aligned}$$ where we have used again. If ${\alpha}>0$, by we find that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} {\partial}^{\alpha}\left(\frac\mu{(1+s)^{\lambda}}u\cdot\nabla\theta\right)
\left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.22}\end{aligned}$$ On the other hand, direct computation derives the following identities $$\begin{aligned}
(1+t)^{2{\lambda}}\theta\Delta{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
\opdiv\left[(1+t)^{2{\lambda}}\theta\nabla{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right]
-(1+t)^{2{\lambda}}\nabla\theta\cdot\nabla{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta \\
-\frac12\,{\partial}_t\left[(1+t)^{2{\lambda}}\theta\,|\nabla{\partial}^{\alpha}\theta|^2\right]
+{\lambda}(1+t)^{2{\lambda}-1}\theta\,|\nabla{\partial}^{\alpha}\theta|^2+\frac12(1+t)^{2{\lambda}}{\partial}_t\theta\,|\nabla{\partial}^{\alpha}\theta|^2\end{aligned}$$ and $$(1+t)^{\lambda}\theta\Delta{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=\opdiv\left[(1+t)^{\lambda}\theta\nabla{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]-(1+t)^{\lambda}(\theta\,
|\nabla{\partial}^{\alpha}\theta|^2+\nabla\theta\cdot\nabla{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta).$$ Integrating these two identities over $[t_0,t]\times{\mathbb R}^d$ yields $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \theta\Delta{\partial}^{\alpha}\theta \left(2
(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right)\,dxds\right| {\nonumber}\\
& \ls K_1{\varepsilon}\left(\mathcal{E}_4^2[\theta](t)+\mathcal{E}_4^2[\theta](t_0)\right)
+K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2\,ds, \label{3.23}\end{aligned}$$ where we have used and ${\lambda}\le1$.
Analogously for the remaining items $u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta$ and ${\displaystyle}\sum_{i,j=1}^du_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta$ in ${\partial}^{\alpha}Q_1(\theta,u)$ (see ), direct computations show $$\begin{aligned}
2(1+t)^{2{\lambda}}u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
& \opdiv\left[(1+t)^{2{\lambda}}u\,|{\partial}_t{\partial}^{\alpha}\theta|^2\right]
-(1+t)^{2{\lambda}}\opdiv u\,|{\partial}_t{\partial}^{\alpha}\theta|^2, \\
(1+t)^{\lambda}u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=
& \opdiv\left[(1+t)^{\lambda}u\,{\partial}_t{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]
-(1+t)^{\lambda}{\partial}_t{\partial}^{\alpha}\theta(u\cdot\nabla{\partial}^{\alpha}\theta+\opdiv u\,
{\partial}^{\alpha}\theta)\end{aligned}$$ and $$\begin{aligned}
2(1+t)^{2{\lambda}}u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta=
&{\partial}_i\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_j{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right]
+{\partial}_j\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_t{\partial}^{\alpha}\theta\right] \\
& -(1+t)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta\Big[{\partial}_i(u_iu_j){\partial}_j{\partial}^{\alpha}\theta+
{\partial}_j(u_iu_j){\partial}_i{\partial}^{\alpha}\theta\Big] \\
& -{\partial}_t\left[(1+t)^{2{\lambda}}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta\right]
+(1+t)^{2{\lambda}}{\partial}_t(u_iu_j){\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta \\
& +2{\lambda}(1+t)^{2{\lambda}-1}u_iu_j{\partial}_i{\partial}^{\alpha}\theta{\partial}_j{\partial}^{\alpha}\theta, \\
(1+t)^{\lambda}u_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta=
& {\partial}_i\left[(1+t)^{\lambda}u_iu_j{\partial}_j{\partial}^{\alpha}\theta{\partial}^{\alpha}\theta\right]
-(1+t)^{\lambda}{\partial}_j{\partial}^{\alpha}\theta{\partial}_i(u_iu_j{\partial}^{\alpha}\theta).\end{aligned}$$ Then we have that $$\begin{aligned}
&\quad \left|\int_{t_0}^t\int_{{\mathbb R}^d} \left(2u\cdot\nabla{\partial}_t{\partial}^{\alpha}\theta
+\sum_{i,j=1}^du_iu_j{\partial}_{ij}^2{\partial}^{\alpha}\theta\right) \left(2(1+s)^{2{\lambda}}{\partial}_t{\partial}^{\alpha}\theta
+\mu(1+s)^{\lambda}{\partial}^{\alpha}\theta\right) \,dxds\right| {\nonumber}\\
&\ls K_1{\varepsilon}\left(\mathcal{E}_4^2[\theta](t)+\mathcal{E}_4^2[\theta](t_0)\right)
+K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds. \label{3.24}\end{aligned}$$ Substituting - into yields $$\begin{aligned}
&\quad (1+t)^{2{\lambda}}\|{\partial}{\partial}^{\alpha}\theta(t,\cdot)\|^2+\|{\partial}^{\alpha}\theta(t,\cdot)\|^2
+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{\alpha}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
& \ls \mathcal{E}_4^2[\theta](t_0)+K_1{\varepsilon}\,\mathcal{E}_4^2[\theta](t)
+\int_{t_0}^t(1-{\lambda})(1+s)^{{\lambda}-2}\|\theta(s,\cdot)\|^2\,ds+\int_{t_0}^t(1+s)^{\lambda}\|{\partial}{\partial}^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_1{\varepsilon}\int_{t_0}^t(1+s)^{\lambda}\left(\|{\partial}{\partial}^{\le3}\theta(s,\cdot)\|^2
+\|{\partial}^{\le3} w(s,\cdot)\|^2\right)\,ds +K_1{\varepsilon}\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds. \label{3.25}\end{aligned}$$ Summing up from $|{\alpha}|=0$ to $|{\alpha}|=3$ and applying Gronwall’s inequality for ${\lambda}<1$ yield provided that ${\varepsilon}>0$ is small enough. This completes the proof of Lemma \[lem-thetaglobal\].
Proof of Theorem \[thm1\].
--------------------------
First, we assume that $\mathcal{E}_4[\theta,u](t) \le K_1{\varepsilon}$ holds. Multiplying by $(1+t)^{2{\lambda}}$ yields $$\begin{aligned}
&\quad \frac{d}{dt}\|(1+t)^{\lambda}{\partial}^{\le3}w(t,\cdot)\|^2+\left(\mu
(1+t)^{\lambda}-2{\lambda}(1+t)^{2{\lambda}-1}\right)\|{\partial}^{\le3}w(t,\cdot)\|^2 {\nonumber}\\
&\ls K_1{\varepsilon}(1+t)^{\lambda}\|{\partial}^{\le3}w(t,\cdot)\|\,\|{\partial}{\partial}^{\le3} u(t,\cdot)\|, \label{3.26}\end{aligned}$$ where we have used . In view of , the second term on the first line in is bounded below by $(1+t)^{\lambda}\|{\partial}^{\le3}w(t,\cdot)\|^2$. Integrating and over $[t_0,t]\times{\mathbb R}^d$ derives $$\label{3.27}
\|(\theta,u)(t,\cdot)\|^2+\int_{t_0}^t (1+s)^{-{\lambda}}\|u(s,\cdot)\|^2\,ds
\ls \|(\theta,u)(t_0,\cdot)\|^2+K_1{\varepsilon}\int_{t_0}^t (1+s)^{\lambda}\|\nabla\theta(s,\cdot)\|^2\,ds$$ and $$\begin{aligned}
&\quad \|(1+t)^{\lambda}{\partial}^{\le3}w(t,\cdot)\|^2+\int_{t_0}^t (1+s)^{\lambda}\|{\partial}^{\le3}w(s,\cdot)\|^2\,ds {\nonumber}\\
&\ls \|{\partial}^{\le3}w(t_0,\cdot)\|^2+K_1{\varepsilon}\int_{t_0}^t (1+s)^{\lambda}\|{\partial}{\partial}^{\le3} u(s,\cdot)\|^2\,ds. \label{3.28}\end{aligned}$$ Collecting , - with , we infer $\mathcal{E}_4[\theta,u](t) \le C_1\mathcal{E}_4[\theta,u](t_0)$. It follows from the local existence of the hyperbolic system that $\mathcal{E}_4[\theta,u](t_0) \le C_2{\varepsilon}$. Let $K_1=2C_1C_2$ and choose ${\varepsilon}>0$ sufficiently small. Then, we conclude that $\mathcal{E}_4[\theta,u](t) \le \frac12 K_1{\varepsilon}$, which implies that admits a global solution $(\theta,u)$ for Case 1. It follows from the definition of $\theta$ (i.e. ) that there exists a global solution $(\rho,u)$ to for Case 1.
Next we show $$\label{vorticity-decay}
\|{\partial}^{\le3}w(t,\cdot)\| \ls \Xi(t)^{-\frac13} \,{\varepsilon},$$ where $\Xi(t)=e^{\frac{\mu}{1-{\lambda}}[(1+t)^{1-{\lambda}}-1]}$ has been defined in .
For this purpose, we assume that $\| \Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|\le K_2{\varepsilon}$ holds for sufficiently large constant $K_2>0$ and small ${\varepsilon}>0$. This immediately implies $$\label{3.30}
\Xi(t)^\frac13 |{\partial}^{\le1}w(t,\cdot)|_\infty \ls K_2{\varepsilon}.$$ Multiplying by $ \Xi(t)^\frac23$ yields $$\begin{aligned}
&\frac{d}{dt} \|\Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|^2
+\frac{\mu}{3(1+t)^{\lambda}} \,\|\Xi(t)^\frac13 {\partial}^{\le3}w(t,\cdot)\|^2 {\nonumber}\\
&\ls \Xi(t)^\frac23 |{\partial}^{\le1} w(t,\cdot)|_\infty\,
\|{\partial}^{\le3}w(t\cdot)\|\,\|{\partial}{\partial}^{\le3} u(t,\cdot)\|. \label{3.31}\end{aligned}$$ Substituting and into and applying Young’s inequality, we then have $$\begin{aligned}
&\quad \frac{d}{dt} \|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2+
\frac{\mu}{3(1+t)^{\lambda}}\|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2 {\nonumber}\\
&\ls \Xi(t)^\frac23 |{\partial}^{\le1}w(t,\cdot)|_\infty \,\|{\partial}^{\le3}w(t\cdot)\|
\left((1+t)^{-{\lambda}}\|u(t,\cdot)\|+\|{\partial}^{\le3}w(t\cdot)\|
+\|{\partial}{\partial}^{\le3}\theta(t\cdot)\| \right) {\nonumber}\\
&\ls K_2{\varepsilon}\left((1+t)^{-{\lambda}}\|u(t,\cdot)\|^2+\big((1+t)^{-{\lambda}}
+\Xi(t)^{-\frac13}\big)\|\Xi(t)^\frac13 {\partial}^{\le3} w(t,\cdot)\|^2
+(1+t)^{\lambda}\|{\partial}{\partial}^{\le3}\theta(t,\cdot)\|^2 \right). \label{3.32}\end{aligned}$$ Collecting , , and applying the same argument as in the proof for the global existence of $(\rho, u)$, we infer . This completes the proof of . Thus the proof of Theorem \[thm1\] is completed.
\[rmk3.2\] The proof of Theorem \[thm1\] can be applied to the case of ${\lambda}=1$, $\mu>2$. In this case, Lemma \[lem-velocity1\]-\[lem-vorticity\] still hold and the coefficient of the first term in the second line of is $(\mu-2)(1+t)$, which plays the same role as . Instead of the identity below , we have $$\begin{aligned}
(1+t)^2|{\partial}{\partial}^{\alpha}\theta|^2+\mu(1+t){\partial}^{\alpha}\theta
{\partial}_t{\partial}^{\alpha}\theta+\frac{\mu(\mu-1)}{2}|{\partial}^{\alpha}\theta|^2 \\
=(1+t)^2\left(\frac{\mu-2}{3\mu-2}|{\partial}_t{\partial}^{\alpha}\theta|^2+|\nabla{\partial}^{\alpha}\theta|^2\right)
+\frac{\mu(\mu-2)}{8}|{\partial}^{\alpha}\theta|^2 \\
+\left((1+t)\sqrt\frac{2\mu}{3\mu-2}{\partial}_t{\partial}^{\alpha}\theta+\sqrt\frac{\mu(3\mu-2)}{8}{\partial}^{\alpha}\theta\right)^2,\end{aligned}$$ which is equivalent to $(1+t)|{\partial}{\partial}^{\alpha}\theta|^2+|{\partial}^{\alpha}\theta|^2$ for $\mu>2$. However, we cannot obtain the exponential decay of the vorticty $w$.
Proof of Theorem 1.2. {#section4}
=====================
Theorem \[thm2\] in three space dimensions has been proved in [@HWY15]. In this section, we fix $d=2$ and assume that $$\label{4.1}
E_5[\theta,u](t) \le K_3{\varepsilon}$$ holds. By the finite propagation speed property of hyperbolic systems, one easily knows that $(\theta, u)$ and their derivatives are supported in $\{(t,x)\colon |x|\le t+M\}$, which implies that for ${\alpha}>0$, $$\label{4.2}
|Z^{\alpha}(\theta,u)(t,x)| \ls (1+t) |{\partial}Z^{<{\alpha}}(\theta,u)(t,x)|.$$ On the other hand, collecting - with assumption derives the following pointwise estimate $$\label{4.3}
|\sigma_-^{-\frac12}(t,\cdot)Z^{\le2}(\theta,u)(t,\cdot)|_\infty+
|\sigma_-^\frac12(t,\cdot){\partial}Z^{\le2}(\theta,u)(t,\cdot)|_\infty
\ls \frac{K_3{\varepsilon}}{1+t}.$$ To prove Theorem \[thm2\] for $d=2$, we shall focus on the following parts.
Estimates of velocity $u$ and its derivatives.
----------------------------------------------
The following lemma is an application of Lemma \[lem-divcurl\] and -.
\[lem-Zvelocity\] Under assumption , for all $t\ge0$, it holds that $$\label{4.4}
E_5[u](t)\ls (1+t)^{-\frac12}\|u(t,\cdot)\|+(1+t)^\frac12\|{\partial}Z^{\le4}\theta(t,\cdot)\|.$$
In view of $\opcurl u_0\equiv0$ and , it is easy to know that $\opcurl u(t,x)\equiv0$ always holds for any $t\ge0$ as long as the smooth solution $(\theta, u)$ of exists. Then, it follows from $\opcurl u\equiv0$ that $$\begin{aligned}
\opcurl Z^{\alpha}u= Z^{\alpha}\opcurl u+\sum_{\beta<{\alpha}}C_{{\alpha},\beta}{\partial}Z^\beta u= \sum_{\beta<{\alpha}}C_{{\alpha},\beta}{\partial}Z^\beta u,\end{aligned}$$ which can be abbreviated as $$\label{4.5}
\opcurl Z^{\alpha}u={\partial}Z^{<{\alpha}}u.$$ Analogously, we have $$\label{4.6}
\opdiv Z^{\alpha}u=Z^{\alpha}\opdiv u+{\partial}Z^{<{\alpha}}u.$$ Taking $U=Z^{\alpha}u$ with $|{\alpha}|\le4$ in and applying - yield $$\begin{aligned}
\|\nabla Z^{\alpha}u(t,\cdot)\|
&\ls \|Z^{\alpha}\opdiv u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+K_3{\varepsilon}(1+t)^{-1}\sum_{0<\beta\le{\alpha}}
\|Z^\beta(\theta,u)(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\|, \label{4.7}\end{aligned}$$ where we have used the first equation in and -. On the other hand, one easily gets $$\begin{aligned}
\|{\partial}_t Z^{\alpha}u(t,\cdot)\|
&\ls \|Z^{\alpha}{\partial}_t u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\| {\nonumber}\\
&\ls \|{\partial}Z^{\le{\alpha}}\theta(t,\cdot)\|+K_3{\varepsilon}\|{\partial}Z^{\le{\alpha}}u(t,\cdot)\|
+(1+t)^{-1}\|u(t,\cdot)\|+\|{\partial}Z^{<{\alpha}}u(t,\cdot)\|. \label{4.8}\end{aligned}$$ Summing up - from $|{\alpha}|=0$ to $|{\alpha}|=4$, then is obtained by the smallness of ${\varepsilon}$. This completes the proof of Lemma \[lem-Zvelocity\].
\[lem-velocity3\] Let $\mu>0$. Under assumption , for all $t\ge0$, it holds that $$\label{4.9}
\frac{d}{dt}\left[(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2\right]
+\frac1{2(1+t)^2}\|(\theta,u)(t,\cdot)\|^2 \le 0.$$
Multiplying the second equation in by $(1+t)^{-1}u$ derives $$\label{4.10}
\frac12\,{\partial}_t\left[(1+t)^{-1}|u|^2\right]+\frac{\mu+\frac12}{(1+t)^2}|u|^2
+(1+t)^{-1}u\cdot\nabla\theta=-\frac12\,(1+t)^{-1}u\cdot\nabla|u|^2.$$ From the first equation in , we see that $$\begin{aligned}
(1+t)^{-1}u\cdot\nabla\theta
&= \opdiv\left[(1+t)^{-1}(\theta u+({\gamma}-1)\theta^2 u)\right]+
\frac12\,{\partial}_t\left[(1+t)^{-1}|\theta|^2\right] {\nonumber}\\
&\quad +\frac1{2(1+t)^2}|\theta|^2+(3-2{\gamma})(1+t)^{-1}\theta\,
u\cdot\nabla\theta, \label{4.11}\end{aligned}$$ which is similar to the expression in . Substituting into and integrating it over ${\mathbb R}^2$ yield $$\begin{aligned}
&\quad \frac{d}{dt}\left[(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2\right]+
\frac{1}{2(1+t)^2}\|(\theta,u)(t,\cdot)\|^2 {\nonumber}\\
&\ls (1+t)^{-1}|\nabla\theta(t,\cdot)|_\infty\|u(t,\cdot)\|\,\|\theta(t,\cdot)\|
+(1+t)^{-1}|\nabla u(t,\cdot)|_\infty\|u(t,\cdot)\|^2. \label{4.12}\end{aligned}$$ Substituting into , then can be obtained from the smallness of ${\varepsilon}$. This completes the proof of Lemma \[lem-velocity3\].
Estimates of $\theta$ and its derivatives.
------------------------------------------
The following lemma shows the estimates of $\theta$.
\[lem-Ztheta\] Let $\mu>1$. Under assumption , for all $t\ge0$, it holds that $$\begin{aligned}
&\quad E_5^2[\theta](t)+\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2
+(1+s)^{-2}\|Z^{\le4}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
& \ls E_5^2[\theta](0)+K_3{\varepsilon}\int_0^t (1+s)^{-2}\|u(s,\cdot)\|^2 \,ds. \label{4.13}\end{aligned}$$
Acting $Z^{\alpha}$ with $|{\alpha}|\le4$ on both sides of equation implies $$\label{4.14}
{\partial}_t^2Z^{\alpha}\theta+\frac\mu{1+t}{\partial}_tZ^{\alpha}\theta-\Delta Z^{\alpha}\theta=Q^{\alpha}_2(\theta,u),$$ where $$\begin{aligned}
Q^{\alpha}_2(\theta,u) &\defeq Z^{\alpha}Q(\theta,u)+Q^{\alpha}_{21}+Q^{\alpha}_{22}+
Q^{\alpha}_{23} {\nonumber}\\
&\defeq Z^{\alpha}Q(\theta,u)+[{\partial}_t^2-\Delta,Z^{\alpha}]\,\theta+
\frac\mu{1+t}[{\partial}_t,Z^{\alpha}]\,\theta
+\sum_{0<\beta\le{\alpha}}Z^\beta\left(\frac\mu{1+t}\right)
Z^{{\alpha}-\beta}{\partial}_t\theta. \label{4.15}\end{aligned}$$ Multiplying by $2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta$ yields $$\begin{aligned}
&{\partial}_t\left[\mu(1+t)|{\partial}Z^{\alpha}\theta|^2+(2\mu-1)Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta+
\frac{\mu(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2\right] {\nonumber}\\
&\quad +(\mu-1)(2\mu-1)|{\partial}_tZ^{\alpha}\theta|^2+(\mu-1)|\nabla Z^{\alpha}\theta|^2+
\frac{\mu(2\mu-1)}{2(1+t)^2}|Z^{\alpha}\theta|^2 {\nonumber}\\
&=\opdiv\left[\nabla Z^{\alpha}\theta \left(2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)
Z^{\alpha}\theta\right) \right]+Q^{\alpha}_2(\theta,u)\left(2\mu(1+t){\partial}_t Z^{\alpha}\theta+(2\mu-1)
Z^{\alpha}\theta\right). \label{4.16}\end{aligned}$$ Thanks to $\mu>1$, we see that for the term in the square bracket of the first line in $$\begin{aligned}
&\quad \mu(1+t)|{\partial}Z^{\alpha}\theta|^2+(2\mu-1)Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta+
\frac{\mu(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2 \\
&=(1+t)\Big(\frac12|{\partial}_tZ^{\alpha}\theta|^2+\mu|\nabla Z^{\alpha}\theta|^2\Big)+
\frac{(\mu-1)(2\mu-1)}{2(1+t)}|Z^{\alpha}\theta|^2+\frac{2\mu-1}{2(1+t)}
\Big( (1+t){\partial}_tZ^{\alpha}\theta+Z^{\alpha}\theta\Big)^2,\end{aligned}$$ which is equivalent to $(1+t)|{\partial}Z^{\alpha}\theta|^2+\frac1{1+t}|Z^{\alpha}\theta|^2$. Integrating over $[0,t]\times{\mathbb R}^2$ yields $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+\left|\int_0^t\int_{{\mathbb R}^2} Q^{\alpha}_2(\theta,u)\Big(2\mu
(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\right)\,dxds\Big|. \label{4.17}\end{aligned}$$ It follows from a direct computation that $$\label{4.18}
Q^{\alpha}_{21}=Z^{<{\alpha}}({\partial}_t^2-\Delta)\theta=Z^{<{\alpha}}Q(\theta,u)-Z^{<{\alpha}}\left(
\frac{\mu}{1+t}{\partial}_t\theta\right) \defeq Z^{<{\alpha}}Q(\theta,u)+Q^{\alpha}_{24}.$$ For ${\alpha}=0$, we find that $Q^{\alpha}_{22}=Q^{\alpha}_{23}=Q^{\alpha}_{24}=0$. For ${\alpha}>0$ and by , we see that $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2}\left(Q^{\alpha}_{22}+Q^{\alpha}_{23}+Q^{\alpha}_{24}\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls \int_0^t\|{\partial}Z^{\le{\alpha}}\theta(s,\cdot)\|\,\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|\,ds. \label{4.19}\end{aligned}$$ Recall the definition of $Q(\theta,u)$ in - as follows $$\begin{aligned}
Q(\theta,u) &=Q_1(\theta,u)+Q_2(\theta,u), \\
Q_1(\theta,u) &=-\frac\mu{1+t}u\cdot\nabla\theta+({\gamma}-1)\theta\Delta\theta
-2u\cdot\nabla{\partial}_t\theta-\sum_{i,j=1,2} u_iu_j{\partial}_{ij}^2\theta, \\
Q_2(\theta,u) &= -\sum_{i,j=1,2}u_i{\partial}_iu_j{\partial}_j\theta-{\partial}_tu\cdot\nabla\theta+(1+({\gamma}-1)\theta)
(\sum_{i,j=1,2}{\partial}_iu_j{\partial}_ju_i+({\gamma}-1)|\opdiv u|^2).\end{aligned}$$ Then it follows from - and $|{\alpha}|\le4$ that $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Z^{\le{\alpha}}Q_2(\theta,u)\Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls \int_0^t \Big(|\theta(s,\cdot)|_\infty+(1+s)|{\partial}Z^{\le2}(\theta,u)
(s,\cdot)|_\infty\Big)\,\|{\partial}Z^{\le4}(\theta,u)(s,\cdot)\|^2 \,ds {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big)\,ds. \label{4.20}\end{aligned}$$ Substituting - into derives $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+\int_0^t\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|^2\,ds
+K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big)\,ds {\nonumber}\\
&\quad +\left|\int_0^t \int_{{\mathbb R}^2} Z^{\le{\alpha}}Q_1(\theta,u)\Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right|. \label{4.21}\end{aligned}$$ Next we focus on the treatment of $Z^{\le{\alpha}}Q_1(\theta,u)$. In view of $|{\alpha}|\le4$, we find that $$Z^{\le{\alpha}}(\theta\Delta\theta)=\theta\Delta Z^{\alpha}\theta+\sum_{1\le|\beta|\le2}
Z^\beta\theta\,{\partial}^2Z^{{\alpha}-\beta}\theta+\sum_{|\beta|\ge3}
Z^\beta\theta\,{\partial}^2Z^{{\alpha}-\beta}\theta,$$ which can be abbreviated as $$Z^{\le{\alpha}}(\theta\Delta\theta)=\theta\Delta Z^{\alpha}\theta+Z^{\le2}\theta
\,{\partial}^2Z^{\le3}\theta+Z^{\le4}\theta\,{\partial}^2Z^{\le1}\theta.$$ From this and the definition of $Q_1(\theta,u)$, we see that $$\begin{aligned}
&\quad Z^{\le{\alpha}}Q_1(\theta,u) {\nonumber}\\
&=-Z^{\le{\alpha}}\left(\frac\mu{1+t}u\cdot\nabla\theta\right)+({\gamma}-1)
\theta\Delta Z^{\alpha}\theta-2u\cdot\nabla{\partial}_tZ^{\alpha}\theta
-\sum_{i,j=1,2}u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta+Q_3(\theta,u),\end{aligned}$$ where $$\begin{aligned}
Q_3(\theta,u) \defeq (Z^{\le2}\theta+Z^{\le2}u){\partial}^2Z^{\le3}\theta+
(Z^{\le4}\theta+Z^{\le4}u){\partial}^2Z^{\le1}\theta {\nonumber}\\
+Z^{\le2}u\,Z^{\le2}u\,{\partial}^2Z^{\le3}\theta+Z^{\le2}u\,Z^{\le4}u\,
{\partial}^2Z^{\le1}\theta.\end{aligned}$$ If ${\alpha}=0$, applying yields $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \frac\mu{1+s}u\cdot\nabla\theta
\Big(2\mu(1+s){\partial}_t\theta+(2\mu-1)\theta\Big) \,dxds\right| {\nonumber}\\
&\ls \int_0^t \Big(|u(s,\cdot)_\infty\|{\partial}\theta(s,\cdot)\|^2+(1+s)^{-1}
|\theta(s,\cdot)|_\infty\|u(s,\cdot)\|\,\|\nabla\theta(s,\cdot)\|\Big) \,ds {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big) \,ds. \label{4.22}\end{aligned}$$ If ${\alpha}>0$, from - we see that $$\begin{aligned}
&\quad \left\|Z^{\le{\alpha}}\left(\frac\mu{1+t}u\cdot\nabla\theta\right) \right\| \\
&\ls (1+t)^{-1} |Z^{\le2}u(t,\cdot)|_\infty\|{\partial}Z^{\le4}\theta(t,\cdot)
\|+(1+t)^{-1} |{\partial}Z^{\le2}\theta(t,\cdot)|_\infty\|Z^{\le4}u(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+K_3{\varepsilon}(1+t)^{-2}\|u(t,\cdot)\|,\end{aligned}$$ which derives $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Z^{\le{\alpha}}\left(\frac\mu{1+s}u\cdot\nabla\theta\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.23}\end{aligned}$$ As in Lemma \[lem-thetaglobal\], direct computation derives the following identities $$\begin{aligned}
(1+t)\theta\Delta Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
& \opdiv\left[(1+t)\theta\nabla Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\right]-(1+t)
\nabla\theta\cdot\nabla Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta \\
& -\frac12\,{\partial}_t\left[(1+t)\theta\,|\nabla Z^{\alpha}\theta|^2\right]+\frac12\,
\theta\,|\nabla Z^{\alpha}\theta|^2+\frac12(1+t){\partial}_t\theta\,|\nabla Z^{\alpha}\theta|^2, \\
\theta\Delta Z^{\alpha}\theta Z^{\alpha}\theta=
&\opdiv\left[\theta\nabla Z^{\alpha}\theta Z^{\alpha}\theta\right]-\theta\,
|\nabla Z^{\alpha}\theta|^2-\nabla\theta\cdot\nabla Z^{\alpha}\theta Z^{\alpha}\theta,\end{aligned}$$ together with -, this yields $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \theta\Delta Z^{\alpha}\theta \Big(2\mu(1+s)
{\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big)\,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\Big(E_5^2[\theta](0)+E_5^2[\theta](t)\Big)+K_3{\varepsilon}\int_0^t
\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2 \,ds. \label{4.24}\end{aligned}$$ Analogously, we have $$\begin{aligned}
2(1+t)u\cdot\nabla{\partial}_tZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
& \opdiv\left[(1+t)u\,|{\partial}_tZ^{\alpha}\theta|^2\right]-(1+t)\opdiv u\,|{\partial}_tZ^{\alpha}\theta|^2, \\
u\cdot\nabla{\partial}_tZ^{\alpha}\theta Z^{\alpha}\theta=
& \opdiv\left[u\,{\partial}_tZ^{\alpha}\theta Z^{\alpha}\theta\right]
-{\partial}_tZ^{\alpha}\theta(u\cdot\nabla Z^{\alpha}\theta+\opdiv u\,Z^{\alpha}\theta)\end{aligned}$$ and $$\begin{aligned}
2(1+t)u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta{\partial}_tZ^{\alpha}\theta=
&{\partial}_i\Big[(1+t)u_iu_j{\partial}_jZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\Big]+
{\partial}_j\Big[(1+t)u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_tZ^{\alpha}\theta\Big] \\
& -(1+t){\partial}_tZ^{\alpha}\theta\Big[{\partial}_i(u_iu_j){\partial}_jZ^{\alpha}\theta+
{\partial}_j(u_iu_j){\partial}_iZ^{\alpha}\theta\Big] \\
& -{\partial}_t\Big[(1+t)u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta\Big]+
u_iu_j{\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta \\
& +(1+t){\partial}_t(u_iu_j){\partial}_iZ^{\alpha}\theta{\partial}_jZ^{\alpha}\theta, \\
u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta Z^{\alpha}\theta=
& {\partial}_i\left[u_iu_j{\partial}_jZ^{\alpha}\theta Z^{\alpha}\theta\right]-
{\partial}_jZ^{\alpha}\theta{\partial}_i(u_iu_jZ^{\alpha}\theta),\end{aligned}$$ which derives $$\begin{aligned}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} \left(2u\cdot\nabla{\partial}_tZ^{\alpha}\theta+
\sum_{i,j=1,2} u_iu_j{\partial}_{ij}^2Z^{\alpha}\theta\right)
\Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\Big(E_5^2[\theta](0)+E_5^2[\theta](t)\Big)+K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|u(s,\cdot)\|^2\Big) \,ds. \label{4.25}\end{aligned}$$ Next we turn our attention to $Q_3(\theta,u)$. Note that $Q_3(\theta,u)=0$ when ${\alpha}=0$. It follows from direct calculation that for any function $\Phi(t,x)$ $$|\sigma_-(t,x){\partial}\Phi(t,x)| \ls |Z\Phi(t,x)|.$$ From this, and -, we see that $$\begin{aligned}
&\quad \|Q_3(\theta,u)\| {\nonumber}\\
&\ls \|Z^{\le2}(\theta,u){\partial}^2Z^{\le3}\theta\|+
\|Z^{\le4}(\theta,u){\partial}^2Z^{\le1}\theta\| \\
&\ls |\sigma_-^{-1}(t,\cdot)Z^{\le2}(\theta,u)(t,\cdot)|_\infty
\|\sigma_-(t,\cdot){\partial}^2Z^{\le3}\theta(t,\cdot)\| \\
&\quad +|\sigma_-^{\frac32}(t,\cdot){\partial}^2Z^{\le1}\theta(t,\cdot)|_\infty
\|\sigma_-^{-\frac32}(t,\cdot)Z^{\le4}(\theta,u)(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+|\sigma_-^{\frac12}(t,\cdot)
{\partial}Z^{\le2}\theta(t,\cdot)|_\infty \|\nabla Z^{\le4}(\theta,u)(t,\cdot)\| \\
&\ls K_3{\varepsilon}\,(1+t)^{-1}\|{\partial}Z^{\le4}\theta(t,\cdot)\|+K_3{\varepsilon}\,
(1+t)^{-2}\|u(t,\cdot)\|,\end{aligned}$$ which implies $$\begin{aligned}
\label{}
&\quad \left|\int_0^t\int_{{\mathbb R}^2} Q_3(\theta,u) \Big(2\mu(1+s){\partial}_t Z^{\alpha}\theta
+(2\mu-1)Z^{\alpha}\theta\Big) \,dxds\right| {\nonumber}\\
&\ls K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.26}\end{aligned}$$ Substituting - into derives $$\begin{aligned}
&\quad (1+t)\|{\partial}Z^{\alpha}\theta(t,\cdot)\|^2+(1+t)^{-1}\|Z^{\alpha}\theta(t,\cdot)\|^2
+\int_0^t \Big(\|{\partial}Z^{\alpha}\theta(s,\cdot)\|^2+(1+s)^{-2}
\|Z^{\alpha}\theta(s,\cdot)\|\Big)\,ds {\nonumber}\\
&\ls E_5^2[\theta](0)+K_3{\varepsilon}\,E_5^2[\theta](t)+\int_0^t\|{\partial}Z^{<{\alpha}}\theta(s,\cdot)\|^2\,ds {\nonumber}\\
&\quad +K_3{\varepsilon}\int_0^t \Big(\|{\partial}Z^{\le4}\theta(s,\cdot)\|^2+(1+s)^{-2}\|u(s,\cdot)\|^2\Big) \,ds. \label{4.27}\end{aligned}$$ Summing up from $|{\alpha}|=0$ to $|{\alpha}|=4$ yields . This completes the proof of Lemma \[lem-Ztheta\].
Proof of Theorem \[thm2\].
--------------------------
Integrating over $[0,t]$ yields that $$(1+t)^{-1}\|(\theta,u)(t,\cdot)\|^2+\int_0^t (1+s)^{-2}\|(\theta,u)(s,\cdot)\|^2\,ds \ls \|(\theta,u)(0,\cdot)\|^2.$$ Collecting this with and , we conclude that $E_5[\theta,u](t) \le C_3{\varepsilon}$. Let $K_3=2C_3$, and choose ${\varepsilon}>0$ sufficiently small. Then, we infer $E_5[\theta,u](t) \le \frac12 K_3{\varepsilon}$, which implies that admits a global solution for Case 2 with $\opcurl u_0(x)\equiv 0$. Thus we complete the proof of Theorem \[thm2\].
Proof of Theorem \[thm3\]. {#section5}
==========================
In this section, we shall only prove Theorem \[thm3\] for $d=2$ since the corresponding blowup result for Case 4 with $\opcurl u_0(x)\equiv 0$ in three space dimensions has been proved in [@HWY15].
We divide the proof into two parts.
### Part : ${\gamma}=2$. {#part-gamma2. .unnumbered}
Let $(\rho, u)$ be a $C^{\infty}-$smooth solution of . For $l>0$, we define $$\label{5.1}
P(t,l)=\int_{x_1>l}\eta(x,l)\left(\rho(t,x)-\bar\rho\right)dx,$$ where $$\eta(x,l)=(x_1-l)^2.$$ Employing the first equation in and an integration by parts, we see that $$\begin{aligned}
\label{5.2}
{\partial}_tP(t,l)&=\int_{x_1>l}\eta(x,l){\partial}_t\left(\rho(t,x)-\bar\rho\right)dx=
-\,\int_{x_1>l}\eta(x,l)\opdiv(\rho u)(t,x)\,dx {\nonumber}\\
&=\int_{x_1>l}({\partial}_{x_1}\eta)(x,l)(\rho u_1)(t,x)\,dx,\end{aligned}$$ where we have used the facts of $\eta(x,l)=0$ on $x_1=l$ and $u(t,x)=0$ for $|x|\ge t+M$. By differentiating ${\partial}_tP(t,l)$ in again and using the equation of $u_1$ in , we find that $$\begin{gathered}
{\partial}_t^2P(t,l) =\int_{x_1>l}({\partial}_{x_1}\eta)(x,l){\partial}_t(\rho u_1)(t,x)\,dx
=-\sum_{j=1,2}\int_{x_1>l}({\partial}_{x_1}\eta)\,{\partial}_{x_j}(\rho u_1u_j)(t,x)\,dx \\
-\int_{x_1>l}({\partial}_{x_1}\eta)(x,l){\partial}_{x_1}(p(t,x)-\bar p)\,dx-\frac\mu{(1+t)^{\lambda}}
\int_{x_1>l}({\partial}_{x_1}\eta)(x,l)(\rho u_1)(t,x)\,dx,\end{gathered}$$ where $\bar p=p(\bar\rho)$. It follows from the integration by parts that $$\begin{aligned}
\label{5.3}
{\partial}_t^2P(t,l)+ \frac\mu{(1+t)^{\lambda}}\,{\partial}_tP(t,l)&=\sum_{j=1,2}
\int_{x_1>l}({\partial}_{x_1x_j}^2\eta)\rho u_1u_j\,dx+\int_{x_1>l}2(p-\bar p)\,dx {\nonumber}\\
&= \int_{x_1>l}2\rho u_1^2\,dx+\int_{x_1>l}2(p-\bar p)\,dx,\end{aligned}$$ here we have used that ${\partial}_{x_1}\eta(x,l)=0$ on $x_1=l$ and $p(t,x)-\bar p$ vanishes for $|x|\ge t+M$. Note that $${\partial}_l^2\eta(x,l)=\Delta_x\eta(x,l)=2.$$ Then we have $$\label{5.4}
\int_{x_1>l}2(p-\bar p)\,dx=\int_{x_1>l}{\partial}_l^2\eta(x,l)(p(t,x)-\bar p)\,dx=
{\partial}_l^2\int_{x_1>l}\eta(x,l)(p(t,x)-\bar p)\,dx,$$ where we have used the fact that $\eta$ and ${\partial}_l\eta$ vanish on $x_1=l$. Collecting -, we arrive at $$\label{5.5}
{\partial}_t^2P(t,l)-{\partial}_l^2P(t,l)+\frac\mu{(1+t)^{\lambda}}\,{\partial}_tP(t,l)\defeq f(t,l)=\int_{x_1>l}2\rho u_1^2\,dx+G(t,l)\ge G(t,l),$$ where $$\label{5.6}
G(t,l)=\int_{x_1>l}2\left(p-\bar p-(\rho-\bar\rho)\right)dx
={\partial}_l^2\int_{x_1>l}\eta(x,l)\left(p-\bar p-(\rho-\bar\rho)\right)dx
\defeq {\partial}_l^2{\tilde}G(t,l).$$ Due to ${\gamma}=2$ and the sound speed $\bar c=\sqrt{2A\bar\rho}=1$, we have $$\label{5.7}
p-\bar p-(\rho-\bar\rho)=A\left(\rho^2-\bar\rho^2
-2\bar\rho\left(\rho-\bar\rho\right)\right)=A(\rho-\bar\rho)^2.$$ Substituting into gives $$G(t,l),\,{\tilde}G(t,l) \ge 0.$$ For $M_0$ satisfying the condition , let $\Sigma\defeq \{(t,l)\colon t\ge0, t+M_0\le l\le t+M\}$ be the strip domain. By applying Riemann’s representation (see [@CH §5.5 of Chapter 5]) with the assumptions -, we have the following lower bound of the solution $P(t,l)$ to for $(t,l)\in\Sigma$ $$\label{5.8}
P(t,l)\ge \frac14\Xi(t)^{-\frac12}q_0(l-t)+\frac14\int_0^t\int_{l-t+\tau}^{l+t-\tau} \left(\frac{\Xi(\tau)}{\Xi(t)}\right)^\frac12 f(\tau,y)\,dyd\tau.$$ We put the proof of in Appendix. Define the function $$\label{5.9}
F(t)\defeq\int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}P(\tau,l)\,\frac{dl}{\sqrt l} d\tau.$$ From the definition of $\Xi(t)$, i.e., for ${\lambda}=1$, $\mu\le1$ or ${\lambda}>1$, we have $\Xi(t)^{-\frac12}\gt(t+M)^{-\frac12}$ and $\frac{\Xi(\tau)}{\Xi(t)}\gt\frac{\tau+M}{t+M}$. Then, by , we arrive at $$\begin{aligned}
&F''(t)=\int_{t+M_0}^{t+M}P(t,l)\,\frac{dl}{\sqrt l} \gt
(t+M)^{-\frac12}\int_{t+M_0}^{t+M} q_0(l-t)\,\frac{dl}{\sqrt l} {\nonumber}\\
&\quad +\int_{t+M_0}^{t+M}\int_0^t\int_{l-t+\tau}^{l+t-\tau} \left(\frac{\tau+M}{t+M}\right)^\frac12 G(\tau,y)\,dyd\tau
\frac{dl}{\sqrt l} \defeq J_1+J_2. \label{5.10}\end{aligned}$$ From assumption , we see that $$\label{5.11}
J_1\gt \frac{1}{t+M}\int_{t+M_0}^{t+M}q_0(l-t)\,dl=\frac{1}{t+M}\int_{M_0}^M q_0(l)\,dl \gt \frac{{\varepsilon}}{t+M}.$$ To bound $J_2$ from below, we write $$\begin{aligned}
J_2&=\int_0^{t-M_1}\int_{\tau+M_0}^{\tau+M} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{t+M_0}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\quad+\int_{t-M_1}^t\int_{\tau+M_0}^{2t-\tau+M_0} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{t+M_0}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\quad+\int_{t-M_1}^t\int_{2t-\tau+M_0}^{\tau+M} \left(\frac{\tau+M}{t+M}\right)^\frac12
G(\tau,y) \int_{y-t+\tau}^{y+t-\tau} \,\frac{dl}{\sqrt l}dyd\tau {\nonumber}\\
&\defeq J_{2,1}+J_{2,2}+J_{2,3}, \label{5.12}\end{aligned}$$ where $M_1=\left(M-M_0\right)/2$. For $t<M_1$, $t-M_1$ in the limits of integration will be replaced by $0$. For the integrand in $J_{2,1}$ we have that $$\label{5.13}
\int_{t+M_0}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{y-\tau-M_0}{(t+M)^\frac12}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}.$$ Analogously, for the integrands in $J_{2,2}$ and $J_{2,3}$ we have that $$\label{5.14}
\int_{t+M_0}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}$$ and $$\label{5.15}
\int_{y-t+\tau}^{y+t-\tau} \frac{dl}{\sqrt l}
\gt \frac{t-\tau}{(t+M)^\frac12}
\gt \frac{(t-\tau)(y-\tau-M_0)^2}{(t+M)^\frac32}.$$ Substituting - into yields $$\begin{aligned}
J_2 \gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12
\int_{\tau+M_0}^{\tau+M} (y-\tau-M_0)^2{\partial}_y^2{\tilde}G(\tau,y)\,dyd\tau,\end{aligned}$$ where ${\tilde}G(\tau,y)=\int_{x_1>y} (x_1-y)^2 \left(p(\tau,x)-\bar p-(\rho(\tau,x)-\bar\rho)\right)dx$. Note that ${\tilde}G(\tau,y)={\partial}_y{\tilde}G(\tau,y)=0$ for $y=\tau+M$. Then it follows from the integration by parts together with - that $$\begin{aligned}
J_2&\gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M}{\tilde}G(\tau,y)\,dyd\tau {\nonumber}\\
&\gt \frac{1}{(t+M)^2}\int_0^t (t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M} \int_{x_1>y} (x_1-y)^2
\left(\rho(\tau,x)-\bar\rho\right)^2dxdyd\tau {\nonumber}\\
&\defeq \frac{c}{(t+M)^2}\,J_3. \label{5.16}\end{aligned}$$ By applying the Cauchy-Schwartz inequality to $F(t)$ defined by , we arrive at $$\label{5.17}
F^2(t) \le J_3\int_0^t (t-\tau)(\tau+M)^{-\frac12}\int_{\tau+M_0}^{\tau+M}
\int_{{\tilde}\Omega} (x_1-y)^2 \,dx\frac{dy}{y}d\tau\defeq J_3J_4,$$ where ${\tilde}\Omega \defeq \{x\colon x_1>y,~|x|<\tau+M\}$. Note that $$\begin{aligned}
J_4 &\ls \int_0^t (t-\tau)(\tau+M)^{-\frac12}\int_{\tau+M_0}^{\tau+M}\int_y^{\tau+M}
(x_1-y)^2 [(\tau+M)^2-x_1^2]^\frac12 \,dx_1\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}(\tau+M-y)^\frac72\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M}\frac{dy}{y}d\tau {\nonumber}\\
&\ls \int_0^t\frac{t-\tau}{\tau+M}\,d\tau \ls (t+M)\log(t/M+1). \label{5.18}\end{aligned}$$ Combining - and - gives the following ordinary differential inequalities $$\begin{aligned}
F''(t) &\gt \frac{{\varepsilon}}{t+M}, && t\ge0, \label{5.19} \\
F''(t) &\gt \left[(t+M)^3\log(t/M+1)\right]^{-1} \,F^2(t), && t\ge0. \label{5.20}\end{aligned}$$ Next, we apply - to prove that the lifespan $T_{\varepsilon}$ of smooth solution $F(t)$ is finite for all $0<{\varepsilon}\le{\varepsilon}_0$. The fact that $F(0)=F'(0)=0$, together with , yields $$\begin{aligned}
F'(t) &\gt {\varepsilon}\log(t/M+1), && t\ge0, \label{5.21} \\
F(t) &\gt {\varepsilon}(t+M)\log(t/M+1), && t\ge t_1\defeq Me^2. \label{5.22}\end{aligned}$$ Substituting into derives $$F''(t) \gt {\varepsilon}^2(t+M)^{-1}\log(t/M+1), \qquad t\ge t_1,$$ which leads to the improvement $$\label{5.23}
F(t) \gt {\varepsilon}^2(t+M)\log^2(t/M+1), \qquad t\ge t_2 \defeq Me^3>t_1.$$ Substituting this into yields $$\label{5.24}
F''(t) \gt {\varepsilon}^2(t+M)^{-2}\log(t/M+1)\,F(t), \qquad t\ge t_2.$$ It follows from that $F'(t)\ge0$ for $t\ge0$. Then multiplying by $F'(t)$ and integrating from $t_3$ (which will be chosen later) to $t$ derive $$F'(t)^2 \ge F'(t_3)^2+C_4{\varepsilon}^2\int_{t_3}^t (s+M)^{-2}\log(s/M+1)\,[F(s)^2]'ds.$$ It follows from the integration by parts that $$\begin{gathered}
\label{5.25}
F'(t)^2 \ge
F'(t_3)^2+C_4{\varepsilon}^2 \left((t+M)^{-2}\log(t/M+1)F(t)^2-(t_3+M)^{-2}\log(t_3/M+1)F(t_3)^2\right)\\
-C_4{\varepsilon}^2\int_{t_3}^t \left(\frac{\log(s/M+1)}{(s+M)^2}\right)' F(s)^2\,ds, \quad t\ge t_3,\end{gathered}$$ where $\left(\frac{\log(s/M+1)}{(s+M)^2}\right)'\le0$ for $s\ge t_3\ge t_2$. Since $F''(t)\ge 0$ and $F(0)=0$, the mean value theorem yields $$\label{5.26}
F(t_3)=\int_0^{t_3}F'(s)ds \le t_3F'(t_3).$$ Choose $$\label{5.27}
t_3=Me^\frac1{2C_4{\varepsilon}^2}-M,$$ which satisfies $C_4{\varepsilon}^2\log(t_3/M+1)=\frac12$. Together with -, this yields $$\label{5.28}
F'(t) \ge \sqrt{C_4}{\varepsilon}(t+M)^{-1}\log^\frac12(t/M+1)\,F(t), \quad t\ge t_3.$$ By integrating from $t_3$ to $t$, we arrive at $$\log\frac{F(t)}{F(t_3)} \ge \sqrt{C_4}{\varepsilon}\log^\frac32\left(\frac{t+M}
{t_3+M}\right), \quad t\ge t_3.$$ If $t\ge t_4\defeq Ct_3^2$, then we have $$\log\frac{F(t)}{F(t_3)} \ge 8\log(t+M).$$ Together with for $F(t_3)$, this yields $$\label{5.29}
F(t) \gt {\varepsilon}^2(t+M)^8, \quad t\ge t_4.$$ Substituting this into derives $$F''(t) \gt {\varepsilon}F(t)^\frac32, \quad t\ge t_4.$$ Multiplying this differential inequality by $F'(t)$ and integrating from $t_4$ to $t$ yield $$F'(t)^2 \gt {\varepsilon}\left(F(t)^\frac52-F(t_4)^\frac52\right).$$ On the other hand, $F(t)\ge 0$, $F''(t)\ge 0$, and the mean value theorem imply that, for $t\ge t_4$, $$F(t)=F'(\xi)(t-t_4)+F(t_4) \ge F'(t_4)(t-t_4) \ge F(t_4)\frac{t-t_4}{t_4},$$ where $t_4\le\xi\le t$. For $t\ge t_5\defeq Ct_4$, we have $$F(t)^\frac52-F(t_4)^\frac52 \ge \frac{1}{2}F(t)^\frac52.$$ Thus $$\label{5.30}
F'(t) \gt \sqrt{\varepsilon}F(t)^\frac54, \quad t\ge t_5.$$ If $T_{\varepsilon}>2t_5$, then integrating from $t_5$ to $T_{\varepsilon}$ derives $$F(t_5)^{-\frac14}-F(T_{\varepsilon})^{-\frac14} \gt \sqrt{\varepsilon}T_{\varepsilon}.$$ We see from and $t_5=Ct_3^2$ that $$F(t_5)\gt {\varepsilon}^2e^\frac{C}{{\varepsilon}^2},$$ which together with $F(T_{\varepsilon})>0$ is a contradiction. Thus, $T_{\varepsilon}\le 2t_5=Ct_3^2$. From the choice of $t_3$ in , we see that $T_{\varepsilon}\le e^{C/{\varepsilon}^2}$.
### Part : ${\gamma}>1$ and ${\gamma}\not=2$. {#part-gamma1-and-gammanot2. .unnumbered}
In view of $\bar c=\sqrt{{\gamma}A\bar\rho^{{\gamma}-1}}=1$, instead of we have $$p-\bar p-(\rho-\bar\rho)=
A\left(\rho^{\gamma}-\bar\rho^{\gamma}-{\gamma}\bar\rho^{{\gamma}-1}(\rho-\bar\rho)\right)
\defeq A\psi(\rho,\bar\rho).$$ The convexity of $\rho^{\gamma}$ for ${\gamma}>1$ implies that $\psi(\rho,\bar\rho)$ is positive for $\rho\neq\bar\rho$. Applying Taylor’s theorem, we have $$\psi(\rho,\bar\rho) \ge C_{{\gamma},\bar\rho} \,\Phi_{\gamma}(\rho,\bar\rho),$$ where $C_{{\gamma},\bar\rho}$ is a positive constant and $\Phi_{\gamma}$ is given by $$\Phi_{\gamma}(\rho,\bar\rho)=
\begin{cases}
(\bar\rho-\rho)^{\gamma}, & \rho< \frac12\bar\rho,\\
(\rho-\bar\rho)^2, & \frac12\bar\rho\le\rho\le2\bar\rho,\\
(\rho-\bar\rho)^{\gamma}, & \rho>2\bar\rho.\\
\end{cases}$$ For ${\gamma}>2$, we have that $(\bar\rho-\rho)^{\gamma}=(\bar\rho-\rho)^2(\bar\rho-\rho)^{{\gamma}-2}\ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$ for $2\rho<\bar\rho$ and $(\rho-\bar\rho)^{\gamma}=(\rho-\bar\rho)^2(\rho-\bar\rho)^{{\gamma}-2}\ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$ for $\rho> 2\bar\rho$. Thus, $\Phi_{\gamma}(\rho,\bar\rho) \ge C_{{\gamma},\bar\rho} (\rho-\bar\rho)^2$. In this case, Theorem \[thm3\] can be shown completely analogously to Part .
Next we treat the case $1<{\gamma}<2$. We define $F(t)$ as in $$F(t)\defeq \int_0^t(t-\tau)\int_{\tau+M_0}^{\tau+M} \int_{x_1>l}
(x_1-l)^2\left(\rho(\tau,x)-\bar\rho\right)\,dx\frac{dl}{\sqrt l}d\tau.$$ Similarly to Part , we have $$\label{5.31}
F''(t)\ge J_1+J_2,$$ where $$\begin{aligned}
J_1 &\gt \frac{{\varepsilon}}{t+M},\\
J_2 &\gt (t+M)^{-2}{\tilde}J_3\end{aligned}$$ and $${\tilde}J_3 =\int_0^t(t-\tau)(\tau+M)^\frac12 \int_{\tau+M_0}^{\tau+M}
\int_{x_1>y}(x_1-y)^2\,\Phi_{\gamma}(\rho(\tau,x), \bar\rho)\,dxdyd\tau.$$ Denote $\Omega_1=\{(\tau,x)\colon \bar\rho\le\rho(\tau,x)\le2\bar\rho\}$, $\Omega_2=\{(\tau,x)\colon \rho(\tau,x)>2\bar\rho\}$, and $\Omega_3=\{(\tau,x)\colon \rho(\tau,x)<\bar\rho\}$. Divide $F(t)$ into the following three integrals over the domains $\Omega_i$ $(1\le i\le 3)$ $$F(t)=F_1(t)+F_2(t)+F_3(t)\defeq \int_{\Omega_1}\cdots+\int_{\Omega_2}\cdots+\int_{\Omega_3}\cdots.$$ Corresponding to the three parts of $F(t)$, we define ${\tilde}J_3\defeq{\tilde}J_{3,1}+{\tilde}J_{3,2}+{\tilde}J_{3,3}$. In view of $F(t)\ge0$ and $F_3(t)\le0$, we have $$F(t)\le F_1(t)+F_2(t).$$ Applying Hölder’s inequality for the domains $\Omega_1$ and $\Omega_2$, we obtain that $$\begin{aligned}
F(t)&\le {\tilde}J_{3,1}^\frac12\left(\int_0^t(t-\tau)(\tau+M)^{-\frac12}
\int_{\tau+M_0}^{\tau+M}\frac1y\int_{{\tilde}\Omega}(x_1-y)^2\,dxdyd\tau\right)^\frac12 \\
&\quad +{\tilde}J_{3,2}^\frac1{\gamma}\left(\int_0^t(t-\tau)(\tau+M)^{-\frac1{2({\gamma}-1)}}
\int_{\tau+M_0}^{\tau+M}\frac{1}{y^\frac{{\gamma}}{2({\gamma}-1)}}\int_{{\tilde}\Omega}
(x_1-y)^2\,dxdyd\tau\right)^\frac{{\gamma}-1}{\gamma}\\
&\ls {\tilde}J_3^\frac12(t+M)^\frac12\log^\frac12 (t/M+1)
+{\tilde}J_3^\frac1{\gamma}(t+M)^\frac{{\gamma}-1}{\gamma}\\
&=\left({\tilde}J_3(t+M)^{-1}\right)^\frac12(t+M)\log^\frac12 (t/M+1)
+\left({\tilde}J_3(t+M)^{-1}\right)^\frac1{\gamma}(t+M).\end{aligned}$$ In view of $1<{\gamma}<2$, we have ${\displaystyle}\frac1{2{\gamma}}<\frac12<\frac1{\gamma}$. Applying Young’s inequality yields $$F(t) \ls \Big(\big({\tilde}J_3(t+M)^{-1}\big)^\frac1{2{\gamma}}+
\big({\tilde}J_3(t+M)^{-1}\big)^\frac{1}{{\gamma}}\Big)(t+M)\log^\frac12 (t/M+1),
\quad t\ge {\tilde}t_1\defeq Me.$$ Together with the fact that $F(t)\gt {\varepsilon}(t+M)\log(t/M+1)$, this yields $${\tilde}J_3 \gt F(t)^{\gamma}(t+M)^{1-{\gamma}}\log^{-\frac{{\gamma}}{2}}(t/M+1), \quad t\ge {\tilde}t_1.$$ Substituting this into yields $$\begin{aligned}
F''(t) &\gt \frac{\varepsilon}{t+M}, && t\ge0, \label{5.32}\\
F''(t) &\gt F(t)^{\gamma}(t+M)^{-1-{\gamma}}\log^{-\frac{{\gamma}}{2}}(t/M+1), && t\ge {\tilde}t_1. \label{5.33}\end{aligned}$$ Substituting $F(t)\gt {\varepsilon}(t+M)\log(t/M+1)$ into derives $$F''(t) \gt {\varepsilon}^{\gamma}(t+M)^{-1}\log^\frac{{\gamma}}{2}(t/M+1).$$ Integrating this yields $$F(t) \gt {\varepsilon}^{\gamma}(t+M)\log^\frac{{\gamma}+2}{2}(t/M+1).$$ Substituting this into again gives $$F''(t) \gt {\varepsilon}^{{\gamma}^2}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}+1)}{2}(t/M+1)
={\varepsilon}^{{\gamma}^2}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}^2-1)}{2({\gamma}-1)}(t/M+1).$$ Repeating this process $k$ times, we see that $$\label{5.34}
F''(t) \gt {\varepsilon}^{{\gamma}^k}(t+M)^{-1}\log^\frac{{\gamma}({\gamma}^k-1)}{2({\gamma}-1)}(t/M+1),$$ where $k=\left[\log_{\gamma}2\right]$. Solving yields $$F(t) \gt {\varepsilon}^{{\gamma}^k}(t+M)\log^{\frac{{\gamma}({\gamma}^k-1)}{2({\gamma}-1)}+1}(t/M+1), \quad t\ge {\tilde}t_2,$$ where ${\tilde}t_2>0$ is a constant only depending on ${\gamma}$ and $M$. Substituting this into derives $$\label{5.35}
F''(t) \gt F(t){\varepsilon}^{{\gamma}^k({\gamma}-1)}(t+M)^{-2}\log^\frac{{\gamma}^{k+1}-2}{2}(t/M+1), \quad t\ge {\tilde}t_2,$$ where $\frac{{\gamma}^{k+1}-2}{2}>0$ by the choice of $k=\left[\log_{\gamma}2\right]$. Since is analogous to , as in Part , we can choose ${\tilde}t_3\defeq O\Big(e^{C{\varepsilon}^{-\frac{2{\gamma}^k({\gamma}-1)}{{\gamma}^{k+1}-2}}}\Big)$ such that $$F'(t) \gt {\varepsilon}^\frac{{\gamma}^k({\gamma}-1)}2(t+M)^{-1}\log^\frac{{\gamma}^{k+1}-2}{4}(t/M+1)\,F(t), \quad t\ge {\tilde}t_3,$$ which is similar to and yields $$\label{5.36}
F(t) \gt {\varepsilon}^{C_{\gamma}}(t+M)^\frac{2({\gamma}+2)}{{\gamma}-1}, \quad t\ge {\tilde}t_4\defeq C{\tilde}t_3^2,$$ where $C_{{\gamma}}>0$ is a constant depending on ${\gamma}$. Substituting into yields $$\label{5.37}
F''(t) \gt {\varepsilon}^{C_{\gamma}} F(t)^\frac{{\gamma}+1}2, \qquad t\ge {\tilde}t_4.$$ Multiplying by $F'(t)$ and integrating over the variable $t$ as in Part , we have $$F'(t) \gt {\varepsilon}^{C_{\gamma}}F(t)^\frac{{\gamma}+3}4, \quad t\ge {\tilde}t_5\defeq C{\tilde}t_4.$$ Together with ${\gamma}>1$ and the choice of ${\tilde}t_3$, this yields $T_{\varepsilon}<\infty$.
Collecting Part and Part completes the proof of Theorem \[thm3\].
Proof on the lower bound of $P(t,l)$ in $\Sigma\equiv \{(t,l)\colon
t\ge0, t+M_0\le l\le t+M\}$. {#appendix}
===================================================================
We fixed a point $A=(t_A,l_A)\in\Sigma$. In the characteristic coordinates $\xi=1+t-l$ and $\zeta=1+t+l$, can be written as $$\label{A.1}
\mathscr{L}\bar P\defeq {\partial}_{\xi\zeta}^2\bar P+\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}
({\partial}_\xi\bar P+{\partial}_\zeta\bar P)=\frac{\bar f}4,$$ where $\bar P(\xi,\zeta)\defeq P(\frac{\zeta+\xi}2-1,\frac{\zeta-\xi}2)$. The adjoint operator $\mathscr{L}^*$ of $\mathscr{L}$ has the form $$\label{A.2}
\mathscr{L}^*{\mathcal{R}}\defeq {\partial}_{\xi\zeta}^2{\mathcal{R}}-\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}
({\partial}_\xi{\mathcal{R}}+{\partial}_\zeta{\mathcal{R}})+\frac{2^{{\lambda}-1}\mu{\lambda}}{(\xi+\zeta)^{{\lambda}+1}}{\mathcal{R}}.$$ For the point $A=(\xi_A,\zeta_A)$ with $\xi_A+\zeta_A=2(1+t_A)\ge2$, denote $B=(2-\zeta_A,\zeta_A)$, $C=(\xi_A,2-\xi_A)$ and $\mathscr{D}$, the domain surrounded by the triangle $ABC$ (see Figure 1 below).
Let the numbers $a$ and $b$ satisfy $a+b=1$ and $$ab=\left\{
\begin{aligned}
&\frac{\mu{\lambda}}{2}, && {\lambda}>1, \\
&\frac\mu2(1-\frac\mu2), && {\lambda}=1.
\end{aligned}
\right.$$ We define $$\label{A.3}
z\defeq -\frac{(\xi_A-\xi)(\zeta_A-\zeta)}{(\xi_A+\zeta_A)(\xi+\zeta)}$$ and $$\label{A.4}
{\mathcal{R}}(\xi,\zeta;\xi_A,\zeta_A)\defeq \Big[\frac{\Xi(\xi+\zeta-1)}{\Xi
(\xi_A+\zeta_A-1)}\Big]^{2^{{\lambda}-2}} \Psi(a,b,1;z),$$ here the definition of function $\Xi$ is given in and $\Psi$ is the hypergeometric function.
![**$(\xi, \zeta)-$plane**[]{data-label="fig:1"}](graph1.eps){width="9cm" height="6.5cm"}
From this and direct calculation, we infer $$\label{A.5}
\mathscr{L}^*{\mathcal{R}}=[\frac{2^{{\lambda}-2}\mu{\lambda}}{(\xi+\zeta)^{{\lambda}+1}}-
\frac{ab}{(\xi+\zeta)^2}-\frac{4^{{\lambda}-2}\mu^2}{(\xi+\zeta)^{2{\lambda}}}]{\mathcal{R}}.$$ On the other hand, from - we arrive at $${\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}}={\partial}_\zeta({\mathcal{R}}{\partial}_\xi\bar P
+\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}{\mathcal{R}}\bar P)-{\partial}_\xi(\bar P{\partial}_\zeta{\mathcal{R}}-\frac{2^{{\lambda}-2}\mu}{(\xi+\zeta)^{\lambda}}{\mathcal{R}}\bar P).$$ Integrating this over $\mathscr{D}$ yields $$\begin{aligned}
\label{A.6}
\bar P(A)&=\frac12{\mathcal{R}}(C;A)\bar P(C)+\frac12{\mathcal{R}}(B;A)\bar P(B)+{\int\!\!\!\!\!\int}_\mathscr{D}
({\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}})\,d\xi d\zeta {\nonumber}\\
&+\int_{BC}(\frac12{\mathcal{R}}{\partial}_\xi\bar P-\frac12\bar P{\partial}_\xi{\mathcal{R}}+\frac\mu4{\mathcal{R}}\bar P)\,d\xi
+(\frac12\bar P{\partial}_\zeta{\mathcal{R}}-\frac12{\mathcal{R}}{\partial}_\zeta\bar P-\frac\mu4{\mathcal{R}}\bar P)\,d\zeta.\end{aligned}$$ Returning to the variable $(t,l)$ (see Figure 2 below), we find in the second line of that $$\begin{aligned}
\label{A.7}
\int_{BC}\cdots=\int_B^C[\frac14{\mathcal{R}}({\partial}_t-{\partial}_l)P-\frac14P({\partial}_t-{\partial}_l)
{\mathcal{R}}+\frac\mu4{\mathcal{R}}P]\,(-dl) {\nonumber}\\
+[\frac14P({\partial}_t+{\partial}_l){\mathcal{R}}-\frac14{\mathcal{R}}({\partial}_t+{\partial}_l)P-\frac\mu4{\mathcal{R}}P]\,dl {\nonumber}\\
=\int_{l_A-t_A}^{l_A+t_A}\left.[\frac\mu2{\mathcal{R}}P+\frac12{\mathcal{R}}{\partial}_tP
-\frac12P{\partial}_t{\mathcal{R}}]\right|_{t=0}dl {\nonumber}\\
=\int_{l_A-t_A}^{l_A+t_A} \Xi(t_A)^{-\frac12} \Big[\Psi(a,b,1;z|_{t=0})
\Big(\frac\mu4q_0(l)+\frac12q_1(l)\Big) {\nonumber}\\
-\frac{ab}{2}\Psi(a+1,b+1,2;z|_{t=0})q_0(l)z_t|_{t=0}\Big]dl,\end{aligned}$$
![**$(t, l)-$plane**[]{data-label="fig:2"}](graph2.eps){width="8.5cm" height="6.5cm"}
where we have used the formula $\Psi'(a,b,c;z)=\frac{ab}{c}\Psi(a+1,b+1,c+1;z)$ (see page 58 of [@EMOT]). From the definition , we arrive at $$z=-\frac{(t_A-l_A-t+l)(t_A+l_A-t-l)}{4(1+t_A)(1+t)}$$ and $$\label{A.8}
z_t|_{t=0}=\frac{t_A}{2(1+t_A)}-z|_{t=0}.$$ If $(t, l)\in\Sigma\cap\overline{\mathscr{D}}$, we infer $$\label{A.9}
0\ge z \ge -\frac12(M-M_0)\ge -\frac12\delta_0,$$ which implies that holds. This, together with , - and the assumption of $\Lambda \ge 3ab$, yields that the integral in the second line of is non-negative. Next we prove that $P(t,l)\ge0$ for all $(t, l)\in\Sigma$. Define $$\bar t\equiv\inf \{t\colon \exists~l\in(t+M_0,t+M)~s.t.~P(t,l)<0\}.$$ From assumption , we get $\bar t>0$. If $\bar t<+\infty$, we see that there exists $\bar l\in(\bar t+M_0,\bar t+M)$ such that $P(\bar t,\bar l)=0$. Moreover, we have $P(t,l)\ge0$ for $t<\bar t$. Choose $A=(t_A,l_A)=(\bar t,\bar l)$ in . From - and we infer $\mathscr{L}^*{\mathcal{R}}\le0$ for ${\lambda}\ge1$ and $(t,l)\in\Sigma\cap\mathscr{D}$ ($\mathscr{L}^*{\mathcal{R}}\equiv0$ if ${\lambda}=1$). It follows from $f(t,l)\ge0$ in , - and that $$\begin{aligned}
P(\bar t,\bar l)\ge \frac12{\mathcal{R}}(C;A)P(0,\bar l-\bar t)+{\int\!\!\!\!\!\int}_{\Sigma\cap\mathscr{D}}
({\mathcal{R}}\mathscr{L}\bar P-\bar P\mathscr{L}^*{\mathcal{R}})\,d\xi d\zeta \ge \frac14\Xi(\bar t)^{-\frac12}q_0(\bar l-\bar t)>0,\end{aligned}$$ which is a contradiction with $P(\bar t,\bar l)=0$. Consequently, we conclude that $\bar t=+\infty$ and $P(t,l)\ge0$ for all $(t, l)\in\Sigma$. It follows from -, , , $P(t,l)\ge0$ and $\mathscr{L}^*{\mathcal{R}}\le0$ that $$P(t_A,l_A)\ge \frac14\Xi(t_A)^{-\frac12}q_0(l_A-t_A)+\frac14\int_0^{t_A}\int_{l_A-t_A+\tau}^{l_A+t_A-\tau} \left(\frac{\Xi(\tau)}{\Xi(t_A)}\right)^\frac12 f(\tau,y)\,dyd\tau,$$ which is .
Yin Huicheng wishes to express his gratitude to Professor Ingo Witt, University of Göttingen, and Professor Michael Reissig, Technical University Bergakademie Freiberg, for their interests in this problem and some very fruitful discussions in the past.
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[^1]: Fei Hou (`houfeimath@gmail.com`) and Huicheng Yin (`huicheng@nju.edu.cn`) were supported by the NSFC (No. 11571177) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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---
abstract: 'In this paper, we investigate various types of Fubini instantons in the Dilatonic Einstein–Gauss–Bonnet theory of gravitation which describes the decay of the vacuum state at a hilltop potential through tunneling without barrier. It is shown that the vacuum states are modified by the non-minimally coupled higher-curvature term. Accordingly, we present the new solutions which describe the tunneling from new vacuum states in anti-de Sitter and de Sitter backgrounds. The decay probabilities of the vacuum states are also influenced. We thus show that the semiclassical exponents can be decreased for specific parameter ranges, thereby increasing tunneling probability.'
address:
- 'Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea'
- 'Center for Quantum Spacetime, Sogang University, Seoul 04107, Korea'
- 'Department of Physics, Sogang University, Seoul 04107, Korea'
author:
- 'Bum-Hoon Lee'
- Wonwoo Lee
- Daeho Ro
bibliography:
- 'References.bib'
title: 'Fubini instantons in Dilatonic Einstein–Gauss–Bonnet theory of gravitation'
---
Fubini instanton ,DEGB theory ,Tunneling without barrier 04.62.+v ,98.80.Cq\
Report Number: APCTP Pre2016-017
\[sec:1\]Introduction
=====================
If an initial state of a potential is a false vacuum or metastable vacuum, the vacuum state can decay through tunneling [@Coleman:1977py; @Kobzarev:1974cp; @Goto:2016ofi] or rolling as a consequence of quantum and classical mechanics, respectively. The bounce solution describes the quantum tunneling process in Euclidean space, which has a negative mode [@Callan:1977pt]. The effect is more subtle in the presence of gravitation [@Koehn:2015hga; @Lee:2014uza] The bounce solution determines the semi-classical exponent of the decay rate of an initial state. The decay rate per unit time per unit volume is given by $\Gamma/V = A e^{-B}$, where the pre-exponential factor $A$ is a functional determinant around the classical solution and the semi-classical exponent $B$ is the Euclidean action difference between the bounce solution and the background. The mechanism of the false vacuum decay is later developed with gravitation [@Coleman:1980aw; @Parke:1982pm] and expanded upon in more detail [@Lee:2008hz].
In the quantum tunneling process, a particle can penetrate a potential barrier to a region which is classically considered forbidden. The phenomenon of tunneling without barrier can also occur and it is very peculiar, because the tunneling occurs in a classically allowed region. If the time-scale of tunneling on the potential is shorter than that of rolling, the tunneling phenomenon can occure despite the lack of a potential barrier. This phenomenon was studied first as a Fubini instanton [@Fubini:1976jm; @Lipatov:1976ny], a bounce solution on the hilltop potential consisting of a negative quartic scalar field. The solution is used in the case of tunneling without a potential barrier between the top of a potential and an arbitrary state. The explicit form of the Fubini instanton was obtained through the conformal invariance of scalar field theory which allows the existence of a one-parameter family of a Euclidean solution of an arbitrary size but with the same probability. The conformal invariance can be broken by introducing a mass term. This makes the solution disappear [@Affleck:1980mp]. There have been intensive studies on tunneling without barrier in the absence of gravitation [@Lee:1985uv] and in the presence of gravitation [@Lee:1986saa] which are expanded upon later [@Kanno:2012zf; @Lee:2012ug; @Battarra:2013rba; @Lee:2014ula].
The inflationary universe scenario [@Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Starobinsky:1980te] was proposed to solve a number of questions in the standard model of hot big-bang cosmology and it is now well established as a leading theory describing the early universe. Inflationary models with many different types of a potential have been studied and some of them have continued to stay relevant even after precision observation. Recently, it has been found that Planck data [@Ade:2015lrj] prefers hilltop and plateau inflationary models [@Kohri:2007gq; @Martin:2013tda; @Barenboim:2013wra; @Coone:2015fha; @Vennin:2015egh; @Barenboim:2016mmw]. In addition, the instanton solutions in those potentials have renewed interest in the AdS/CFT correspondence [@deHaro:2006ymc; @Papadimitriou:2007sj]. In line with these scenarios, it is worthwhile to observe quantum tunneling in a model with a hilltop potential as a case of Fubini instanton.
If one considers the very early universe of the Plank scale, the effects of the spacetime curvature become significant such that the early universe was in the quantum gravity regime. For this reason, we consider the higher-curvature Gauss–Bonnet (GB) term as a correction, in which the GB coefficient $\alpha$ is a dimensionless parameter. In four-dimensional spacetime, the theory using the GB term does not have a ghost particle or negative energy states [@Zwiebach:1985uq]. It also does not affect the equations of motion and the solutions. In order to introduce the contribution from the GB term, the term is coupled with a dilaton field with the coupling constant $\gamma$ which has a length dimension [@Kawai:1999pw; @Guo:2010jr; @Koh:2014bka]. The nonminimally coupled higher-curvature term appears in the first order $\alpha'$-correction (16$\alpha \kappa$ in the present paper) of the string effective action [@Boulware:1986dr]. This is called the Dilatonic Einstein–Gauss–Bonnet theory of gravitation (DEGB theory). It may provide a chance to avoid the initial singularity of the universe [@Antoniadis:1993jc; @Rizos:1993rt; @Easther:1996yd]. When taking into account the model of the Fubini instanton in the DEGB theory, the shape of the effective potential may change significantly depending on the signs of $\alpha$ and $\gamma$.
The paper is organized as follows: in the next section, we set up the basic framework with the action and the equations of motion for the DEGB theory. In section \[sec:3\], we analyze how the vacuum states are modified by the dilaton coupling with higher-order curvature terms. In section \[sec:4\], we present various types of solutions in DEGB theory. Finally in section \[sec:5\], we summarize our results and discuss their implication for cosmology.
\[sec:2\]Set up
===============
Let us consider the action that the scalar field is interacting with the Gauss–Bonnet (GB) term as follows: $$\begin{gathered}
\label{eq:action}
S = \int_{\cal M} d^4x \sqrt{-g} \bigg[ \dfrac{R}{2\kappa} - \dfrac{1}{2} \partial_\mu \phi \partial^\mu \phi - U(\phi) + f(\phi) R_{\text{GB}}^2 \bigg]
\\ + S_{\text{YGH}},\end{gathered}$$ where $g = \det g_{\mu\nu}$ with the signs $(-,+,+,+)$, $\kappa = 8\pi G$, $R$ is the scalar curvature of a space-time manifold ${\cal M}$, $U(\phi)$ is a scalar field potential, and GB term is given by $R_{\text{GB}}^2 = R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} - 4R^{\mu\nu}R_{\mu\nu} + R^2$. The function $f(\phi)$ is a coupling function between the scalar field and GB term. $S_{\text{YGH}}$ is the generalized York-Gibbons-Hawking boundary term [@York:1972sj; @Gibbons:1976ue; @Myers:1987yn; @Brihaye:2008xu].
By adopting the analytic continuation from Lorentzian to Euclidean, the action is changed with omitting boundary term as follows: $$\label{eq:action.eu}
S_E = \int_{\cal M} d^4x \sqrt{g} \left[ \dfrac{-R}{2\kappa} + \dfrac{1}{2} \partial_\mu \phi \partial^\mu \phi + U(\phi) - f(\phi) R_{\text{GB}}^2 \right],$$ where the signs of metric are now $(+,+,+,+)$ and the boundary term will be cancelled in the calculation of the difference between the action of the solution and background. The equation of motion for scalar field and Einstein’s equation are $$\begin{aligned}
\label{eq:eom.phi}
0 &=& \nabla^2 \phi - U'(\phi) + f'(\phi) R_{\text{GB}}^2,
\\ \nonumber
0 &=& \dfrac{1}{2\kappa} \left( R_{\mu\nu} - \dfrac{1}{2} g_{\mu\nu} R \right) - \dfrac{1}{2} \partial_\mu \phi \partial_\nu \phi
\\ \label{eq:eom.gr}
&& + \dfrac{1}{4} g_{\mu\nu} \partial_\rho \phi \partial^\rho \phi + \dfrac{1}{2} g_{\mu\nu} U(\phi) + (\text{GB})_{\mu\nu},\end{aligned}$$ where we use the primed notation for the derivative with respect to the scalar field $\phi$. The last term of second equation is obtained by GB term variation that in four-dimensional space is $$\begin{gathered}
\label{eq:gb.mn}
(\text{GB})_{\mu\nu} = - 2 (\nabla_\mu \nabla_\nu f(\phi)) R + 2 g_{\mu\nu} (\nabla^2 f(\phi)) R
\\
+ 4 (\nabla_\rho \nabla_\mu f(\phi)) R_\nu{}^\rho + 4 (\nabla_\rho \nabla_\nu f(\phi)) R_\mu{}^\rho
\\
- 4 (\nabla^2 f(\phi)) R_{\mu\nu} - 4 g_{\mu\nu} (\nabla_\rho \nabla_\sigma f(\phi)) R^{\rho\sigma}
\\
+ 4 (\nabla^\rho \nabla^\sigma f(\phi)) R_{\mu\rho\nu\sigma}.\end{gathered}$$ There were the terms linear in $f(\phi)$ but those terms are cancelled each other especially in four-dimensional space [@deWitt:1964]. Thus, $(\text{GB})_{\mu\nu}$ can be zero when the coupling function $f(\phi)$ is given by a constant which means that the scalar field and GB term are minimally coupled each other.
We consider Euclidean $O(4)$ symmetriy for the dominant contribution to the decay probability [@Coleman:1977th]. Then, the field $\phi$ as well as $\rho$ depends only on $\eta$ which is the radial coordinate of Euclidean space. The geometry is written as $$\label{eq:o4.metric}
ds^2 = d\eta^2 + \rho(\eta)^2 \big(d\chi^2 + \sin^2\chi (d\theta^2 + \sin^2\theta d\psi^2)\big).$$ The scalar curvature and GB term turn out to be $$\label{eq:r.rgb}
R = - 6\dfrac{(\dot{\rho}^2 - 1 + \rho \ddot{\rho})}{\rho^2}, \quad \text{and} \quad R_{\text{GB}}^2 = 24\dfrac{\ddot{\rho}(\dot{\rho}^2 - 1)}{\rho^3},$$ where the dotted notation denotes the derivative with respect to $\eta$. The equations of motion for $\phi$ and $(\eta,\eta)$, $(\chi, \chi)$ components of Einstein’s equation are obtained from plugging Eq. into Eqs. and as follows: $$\begin{aligned}
\label{eq:eom2.phi}
0 =&\ \ddot{\phi} + 3\dfrac{\dot{\rho}}{\rho} \dot{\phi} - U'(\phi) + 24 f'(\phi)\dfrac{\ddot{\rho}(\dot{\rho}^2 - 1)}{\rho^3},
\\ \label{eq:eom2.gt}
0 =&\ \dfrac{3}{2\kappa}\dfrac{(\dot{\rho}^2 - 1)}{\rho^2} - \dfrac{1}{4} \dot{\phi}^2 + \dfrac{1}{2}U(\phi) - 12 \dot{f}(\phi) \dfrac{\dot{\rho} (\dot{\rho}^2 - 1)} {\rho^3},
\\ \nonumber
0 =&\ \dfrac{\dot{\rho}^2 - 1 + 2 \rho \ddot{\rho}}{2\kappa} + \dfrac{\rho^2}{4} \dot{\phi^2} + \dfrac{\rho^2}{2} U(\phi) - 8 \dot{f}(\phi) \dot{\rho}\ddot{\rho}
\\ \label{eq:eom2.gtheta}
&\ - 4 \ddot{f}(\phi) (\dot{\rho}^2 - 1).\end{aligned}$$ The last equation is different from the result obtained in Ref. [@Cai:2008ht].
In order to solve the equations of motion, we should impose appropriate boundary conditions. In the presence of gravity, there need boundary conditions for not only $\phi$ but also $\rho$. These boundary conditions are divided into two types depending on the sign of $U(\phi_v)$. The maximum value of $\eta$ is $\eta_{\text{max}}=\infty$ for AdS and flat backgrounds, while it is finite for dS background which satisfies $\rho(\eta_{\text{max}})=0$.
For the flat and AdS space, we can impose the boundary conditions as follows: $$\rho(0) = 0, \quad \dot{\rho}(0) = 1, \quad \dot{\phi}(0) = 0, \quad \text{and} \quad \phi(\eta_{\text{max}}) = \phi_v,$$ where the first condition is for a geodesically complete space and second condition comes from the Eq. and $\phi_v$ is the vacuum value of scalar field which was zero in Einstein’s theory of gravitation (Einstein theory). For dS space, we can impose the boundary conditions as follows: $$\rho(0) = 0, \ \ \rho(\eta_{\text{max}}) = 0, \ \ \dot{\phi}(0) = 0, \ \ \text{and} \ \ \dot{\phi}(\eta_{\text{max}}) = 0.$$
We consider the potential $$\label{eq:po}
U(\phi) = - \dfrac{\lambda}{4} \phi^4 + U_0,$$ where $\lambda$ is a positive constant [@Fubini:1976jm] and $U_0$ is a vacuum value of potential in Einstein theory. It plays a roll of cosmological constant as $\Lambda = \kappa U_0$ [@Lee:2012ug] while it is modified as $\Lambda=\kappa U(\phi_v)$ in DEGB theory. The coupling function is $$\label{eq:cc}
f(\phi) = \alpha e^{-\gamma \phi},$$ where $\alpha$ is the GB coefficient and $\gamma$ is the dilaton coupling constant. Here, we adopt the dimensionless parameters [@Lee:2012ug]. The change of variables still remain the equations of motion, Eqs. , , and , same as before but the parameter $\lambda$ is disappeared. For numerical calculations we also set the initial value of $\eta$ by $0+\epsilon$ where $\epsilon \ll 1$ to avoid the initial divergence of equations of motion at $\eta=0$. Then, the initial values of $\phi$ and $\rho$ up to second order of $\epsilon$ are changed into $$\begin{gathered}
\phi(\epsilon) \approx \phi_0 - \dfrac{\epsilon^2}{8} \phi_0^3 + \cdots, \qquad \phi'(\epsilon) \approx - \dfrac{\epsilon}{4} \phi_0^3 + \cdots, \\
\rho(\epsilon) \approx \epsilon + \cdots, \hspace{24pt} \rho'(\epsilon) \approx 1 - \dfrac{\epsilon^2}{6} \kappa U(\phi_0) + \cdots.\end{gathered}$$
\[sec:3\]Vacuum states
======================
We first determine the shape of the effective potential and vacuum states by solving the equations of motion with the given potential and GB coupling function in DEGB theory. This state is simply at $\phi=0$ in Einstein theory [@Lee:2014ula]. However, it becomes quite complicated depending on $\alpha$ and $\gamma$ in DEGB theory. For this reason, we analyze how the vacuum state can be modified by the dilaton coupling with higher-order curvature terms.
In order to find the value of $\phi_v$, the scalar field is supposed to initially at the vacuum state. Then, the scalar field stays at the vacuum state. In this regime, derivatives of the scalar field are vanished as follows: $$\label{eq:vac.con.1}
\dot{\phi}|_{\phi=\phi_v} = 0, \qquad \text{and} \qquad \ddot{\phi}|_{\phi=\phi_v} = 0.$$ By substituting this condition into Eqs. and , the analytic form of $\rho(\eta)$ is obtained for AdS and dS background as follows: $$\label{eq:vac.con.2}
\rho(\eta) = \sqrt{\dfrac{3}{|\Lambda|}} \sinh \sqrt{\dfrac{|\Lambda|}{3}} \eta, \quad \text{and} \quad \rho(\eta) = \sqrt{\dfrac{3}{\Lambda}} \sin \sqrt{\dfrac{\Lambda}{3}} \eta,$$ respectively. Recall that the cosmological constant is defined by $\Lambda = \kappa U(\phi_v)$. Then, the scalar field equation, Eq. , can be simplified by using Eqs. and such as $$\label{eq:new}
0 = \phi_v^3 - \dfrac{8}{3} \alpha \gamma e^{-\gamma \phi_v} \Lambda^2.$$ From this equation, the cosmological constant also satisfies $\Lambda = \pm \sqrt{3\phi_v^3/8\alpha\gamma e^{-\gamma \phi_v}}$, where the negative and positive sign correspond to the cases in the AdS and dS backgrounds, respectively. Finally, we can obtain the information about the vacuum state of scalar field by solving this equation with given parameters. For example, $\phi_v$ should be zero when $\alpha = 0$ which is the case of Einstein theory. If $\alpha \neq 0$, the equation cannot be solved analytically because it is higher-order and non-linear equation for $\phi_v$. Therefore, the values of $\phi_v$ are obtained numerically. In the numerical computation, we always choose the value of $\gamma$ to be positive for simplification of numerical results. Also, we fixed the value of $\kappa$ to be $0.1$ for all calculations.
Fig. \[fig:vac\] represents the modified vacuum states in DEGB theory. For the negative values of $\alpha$, there exist two vacuum states and the effective potential is changed significantly, which makes the potential bounded below for the negative direction of the scalar field. However, for the positive values of $\alpha$, there exists only one vacuum state and the effective potential has the same form as the original hilltop potential. Note that the effective potential is not exactly obtained, but expected to be given form. It shows that the potential $U(\phi)$ is more affected by the GB term when $\alpha$ has different sign with $\gamma$.
\
Fig. \[fig:alpha.vac\] represents the value of $\phi_v$ with respect to $\alpha$ for AdS and dS backgrounds. First two plots look very similar, but the exact values of $\phi_v$ are different. For example, the first and second vacuum states are at $\phi_v = -0.0634741$ and $\phi_v = -3.0678$ for AdS background, but those for dS background at $\phi_v = -0.0634729$ and $\phi_v = -3.08818$ when $\alpha = -0.1$. In this case, the black line indicates the change of vacuum states and the magnitude depend on the value of $\alpha$. The red line only appears in the region of negative $\alpha$. Both lines are becoming closer when $\alpha$ decreases and expected to meet at some point. It can be seen by increasing $\gamma$ as the other two plots. Two lines eventually meet at the specific negative value of $\alpha$ and there is no vacuum state when $\alpha$ further decreases in the AdS background. However in the dS background, the number of vacuum states increases from one up to four with the specific range of $\alpha$ which makes the effective potential be more complicated. Both lines in the dS background are also expected to meet at some point by decreasing $\alpha$.
Fig. \[fig:new.vac\] shows the limitation of the initial scalar field values with $\alpha=-0.1$. This figure represents the evolution of scalar fields with specific initial values. When the scalar fields have more higher values than the second vacuum state, they converge to the first vacuum state in the AdS background and oscillate near the first vacuum state in the dS background. However, the scalar fields grow into the negative infinity when they initially have the lower value than the second vacuum state in both AdS and dS background. Then, they cannot satisfy the boundary condition.
\[sec:4\]Numerical solutions and decay rates
============================================
In this section, we present the solutions by solving the equations of motion in the Ads and dS backgrounds. We first briefly review the results of the solutions in Einstein theory [@Lee:2014ula]. There are three parameters to determine a solution, $\kappa$, $\phi_0$ and $U_0$. The crossing number of the top (or the vacuum state) in the inverted potential is the number of oscillations [@Hackworth:2004xb; @Lee:2011ms; @Battarra:2012vu]. In the AdS background, any set of parameters allows a oscillating solution with different number of oscillations. The solutions with the specific number of oscillations have the maximum or minimum initial value of scalar field, we call it the marginal solution. In general, all the solutions have infinite exponent $B$ while the marginal solutions have finite values. In the dS background, the specific set of parameters allows a oscillating solution with the different number of oscillations and the equations of motion diverge in other cases. There exist two types of solutions which are $Z_2$-symmetric and $Z_2$-asymmetric. When the solution starts at $\phi_0$ and ends at $\pm\phi_0$, then it is called $Z_2$-symmetric. For $Z_2$-asymmetric, the solution starts at $\phi_1$ or $\phi_2$ and ends at $\phi_2$ or $\phi_1$ where $\phi_1 \neq \phi_2$.
If we plot all the data with respect to $\alpha$, it is numerous. Thus, we select two solutions in the AdS background which are the solutions with $\phi_0 = \pm 1.5$ and one solution in the dS background which corresponds to the $Z_2$-symmetric solution in Einstein theory. In this paper, we concentrate on how those solutions are modified by the dilaton coupling with higher-order curvature terms.
\
\
Figs. \[fig:ads.alpha\] and \[fig:ads.alpha2\] represent two selected solutions in the AdS background with respect to $\alpha$. The gray line of a solution becomes lighten or darken when $\alpha$ increases or decreases, respectively. The solutions converge into the vacuum states which are determined by the values of $\alpha$. The sign of the first vacuum state, $\phi_v$ follows that of $\alpha$ and the magnitude of $\phi_v$ increases when the absolute value of $\alpha$ increases. In order to see the details near the vacuum state, small inset figures are added in each figure. Fig. \[fig:ads.alpha\] shows the solutions of $\phi(\eta)$ with respect to $\eta$ and Fig. \[fig:ads.alpha2\] shows the solutions of $\phi'(\eta)$ with respect to $\phi(\eta)$. Most of the solutions do not have oscillatory behavior. However, the solutions only with $\phi_0=\pm 1.5$ and $\alpha=\mp 0.10$ have one oscillation. This fact can be easily recognized by the inset plots in Fig. \[fig:ads.alpha2\].
All the solutions of scalar fields are behaving very similar until $\eta \sim 10$ and separating after that into there own vacuums. Therefore, the magnitude of vacuum value determines how many times the solution oscillate for fixed $\phi_0$, in other words, it is determined by $\alpha$. Increase of $|\alpha|$ which has the same sign with $\phi_0$ thus enhance those oscillatory behavior. However, it can be suppressed if $\alpha$ and $\phi_0$ have different sign. In addition, the oscillatory behavior is more enhanced when $\alpha$ has the negative value than positive, which can be seen by comparing the darkest line of Fig. \[fig:ads.alpha2\](a) and lightest line of Fig. \[fig:ads.alpha2\](b). The difference comes from the existence of the second vacuum state which is only appeared in the region of the negative $\alpha$. The second vacuum state restricts the selection of $\phi_0$ and leads to significant change of the effective potential. It is expected that the second vacuum state highly affects the oscillatory behavior.
$\alpha$ $-0.10$ $-0.06$ $-0.03$ $-0.01$ $0$ $0.01$ $0.03$ $0.06$ $0.10$
---------- ------------ ------------ ------------ ------------ ---------- ----------- ----------- ----------- -----------
$\phi_v$ $-0.06347$ $-0.05336$ $-0.04219$ $-0.02913$ $0$ $0.02857$ $0.04104$ $0.05152$ $0.06090$
$\phi_0$ $5.2629$ $5.0842$ $4.8807$ $4.6274$ $3.9355$ $2.8734$ $2.1955$ $1.5536$ $1.0391$
$B$ $32.7$ $32.9$ $32.7$ $32.0$ $29.8$ $27.2$ $26.1$ $26.1$ $29.1$
The solutions with arbitrary parameters in the AdS background have infinite exponent $B$ as we discussed earlier. Thus, we find the marginal solutions which does not oscillate with respect to $\alpha$. Table \[tab:1\] represents the exponent $B$ of decay rate for those marginal solutions. The exponent is defined by $B = S_{\text{bs}} - S_{\text{bg}}$ where $S_{\text{bs}}$ and $S_{\text{bg}}$ are Euclidean action for the bounce solution and background, respectively. The exponent $B$ turns out to be $$\begin{gathered}
B = 2\pi^2 \int_0^{\rho_{\text{m}}} d\rho \dfrac{\rho^3}{\dot{\rho}}
\\
\bigg( S_E|_{\eta\rightarrow \rho^{-1}} - S_E|_{\eta\rightarrow \sqrt{\frac{3}{|\Lambda|}}\sinh^{-1}\sqrt{\frac{|\Lambda|}{3}}\rho, \phi\rightarrow\phi_v}\bigg)
\label{eq:bounce}\end{gathered}$$ where $\rho_{\text{m}}$ is the value of the maximum $\rho$ which is infinite, in general. However, for the marginal solution, it can be finite because the integration values after $\rho_{\text{m}}$ are not significant.
The exponent $B$ and $\phi_0$ increase when $\alpha$ decreases in the vicinity of $\alpha = 0$. The increase of the difference between $\phi_0$ and $\phi_v$ is expected to give more higher value during the integration of exponent $B$. However, the exponent $B$ decreases when $\alpha$ decreases further. Similarly, the exponent $B$ and $\phi_0$ decrease when $\alpha$ increases, but those values increase when $\alpha$ increases further. Since the decay rate is proportional to $e^{-B}$, the small exponent value means the case is more probable. Thus, the case with the small positive or large negative values of $\alpha$ is dominant.
Fig. \[fig:ds.alpha.phi\] represent a selected solution in the dS background with respect to $\alpha$. It is $Z_2$-symmetric one as the case for $\alpha=0$ in Einstein theory, however, GB term brake $Z_2$-symmetry. The gray line of solution becomes lighten or darken when $\alpha$ increases or decreases, the same as the AdS solution. In Fig. \[fig:ds.alpha.phi\](a), the initial (or final) value of the scalar field increase (or decreases) when $\alpha$ decreases. In Fig. \[fig:ds.alpha.phi\](b), the absolute value of $\phi'(\eta)$ increases when $|\alpha|$ increases. The solutions with $\alpha = \pm0.10$ look symmetric under the inversion of y-axis, but there exists a little difference. The difference is enhanced by increasing $\gamma$.
In the numerical computation to find the solutions, we noticed that there exists a region, in which the solution is forbidden under the specific value of $\phi_0$ and $\alpha$. For those parameters, the equations of motion seem to diverge in the computation. Fig. \[fig:forbidden\] represent the forbidden region for AdS and dS background with respect to $\phi_0$ and $\alpha$. These regions only appear when $\alpha$ has the negative value. The forbidden region becomes bigger when $\phi_v$ decreases. In addition, the region is going to be far from zero when $\alpha$ increases. We cannot obtain the exact forbidden region in the vicinity of $\alpha=0$. Thus, there remain the upper bound with the dashed line. The forbidden region seems to appear due to the solution is imaginary. Similar things happen for the black holes in DEGB theory [@Ohta:2009tb].
Until now, we have only considered the solutions with $\gamma=1.0$ by changing $\alpha$, which solutions are already shown in Einstein theory. In order to find a new type of solution which is expected from the shape of the effective potential as in Fig. \[fig:alpha.vac\](b), we increase the gamma value into $\gamma=8.0$ and try to find solution. As a result, Figs. \[fig:new.sol\] and \[fig:new.sol2\] represent the new solution in the AdS and dS backgrounds. The black and red dashed lines indicate the first and second vacuum in both figures. In this new solution, the scalar field directly go to the second vacuum state and stop there, while the other solutions are converge or oscillate only for the first vacuum state.
\[sec:5\]Summary and Discussion
===============================
In this paper, we have investigated various types of Fubini instanton in DEGB theory. It is worthwhile to consider the hilltop potential in line with the recently preferred inflationary scenarios. In particular, we have taken into consideration the hilltop potential which is made only by the quartic scalar field term. The vacuum state is at $\phi=0$ in Einstein theory, $\phi_v$ in DEGB theory. Furthermore, the second vacuum state is formed when there is a negative coefficient $\alpha$. The existence of the second vacuum state affects the shape of the potential. Thus, the effective potential is changed to be bounded below as a result of the negative direction of potential which significantly changes the solution. This change becomes substantial when we set a higher value fir the dilaton coupling constant $\gamma$. For the consistent comparison of the solutions, we have fixed $\gamma = 1.0$.
We had investigated various types of solutions in Einstein theory in our previous paper [@Lee:2014ula]. In this paper, we presente solutions with respect to the GB coefficient $\alpha$ as an extension of our previous work. Since the vacuum state is shifted, the oscillation numbers of the solutions in the AdS background have changed. If the solution has a positive initial value, it displays a more oscillatory behavior when $\alpha$ increases and vice versa. In order to check the change in probability, we have calculated the exponent of the decay rate $B$ for the marginal solution, which only has a finite exponent $B$ in the AdS background [@Lee:2014ula], with respect to $\alpha$. By doing so, the exponent $B$ decreases and increases again when $\alpha$ increases. In the other way around, the exponent $B$ decreases and increases again when $\alpha$ decreases. As a consequence of decreasing this exponent, it is possible to choose the more probable case by selecting a specific set of parameters. In the dS background, we only consider the $Z_2$-symmetric solution, and the symmetry is broken when $\alpha$ is introduced. We have also found that the initial values of the scalar field are restricted to be smaller than the second vacuum states, due to the face that the scalar field grows into negative infinity. Furthermore, the equations of motion cannot allow for regular behavior during numerical calculation for certain values of the initial scalar field. Thus, the solutions are forbidden for the specific parameter region for $\phi_0$ and $\alpha$. We have found a new solution using a second vacuum state which can be predicted by the form of the effective potential. The scalar field is neither oscillating nor converging to the specific value of scalar field, i.e. the first vacuum state, but instead exactly stops at the second vacuum state which is never shown in Einstein theory.
Recent inflation scenarios are focused on the hilltop or plateau inflationary models which are preferred for precision observations. In line with this, we have considered quantum tunneling on hilltop potential. We may compare the probability between rolling as an inflation scenario and tunneling as we have done in this paper. The timescale (or the decay probability) of tunneling on the potential may be larger (or smaller) than that of rolling in Einstein theory. However, we have found that tunneling probability can be improved by choosing the proper set of GB parameters in DEGB theory. By employing this method, the tunneling event is improved sufficiently and the effects can be seen in the observational data. In order to examine this point, we should consider the decay modes, i.e. rolling and tunneling simultaneously in a specific model. We hope to adopt our result into cosmology and consider this decay mode as well as the rolling mode in an inflationary scenario for future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Seoktae Koh for his hospitality during our visit to Jeju National University. BHL was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No.2014R1A2A1A01002306). WL was supported by Basic Science Research Program through National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B01010234). DH was supported by the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City.
References {#references .unnumbered}
==========
|
---
abstract: |
Here we define a procedure for evaluating KL-projections (I- and rI-projections) of channels. These can be useful in the decomposition of mutual information between input and outputs, e.g. to quantify synergies and interactions of different orders, as well as information integration and other related measures of complexity.
The algorithm is a generalization of the standard iterative scaling algorithm, which we here extend from probability distributions to channels (also known as transition kernels).
[**Keywords:**]{} Markov Kernels, Hierarchy, I-Projections, Divergences, Interactions, Iterative Scaling, Information Geometry.
author:
- 'Paolo Perrone[^1]'
- Nihat Ay
title: Iterative Scaling Algorithm for Channels
---
Introduction
============
Here we present an algorithm to compute projections of channels onto exponential families of fixed interactions.
The decomposition is geometrical, and it is based on the idea that, rather than joint distributions, the quantities we work with are channels, or conditionals (or Markov kernels, stochastic kernels, transition kernels, stochastic maps). Our algorithm can be considered a channel version of the iterative scaling of (joint) probability distributions, presented in [@csiszar].
Exponential and mixture families (of joints and of channels) have a duality property, shown in Section \[fam\]. By fixing some marginals, one determines a mixture family. By fixing (Boltzmann-type) interactions, one determines an exponential family. These two families intersect in a single point, which means that (Theorem \[dualmk\]) *there exists a unique element which has the desired marginals and the desired interactions*.
As a consequence, Theorem \[dualmk\] translates projections onto exponential families (which are generally hard to compute) to projections onto fixed-marginals mixture families (which can be approximated by an iterative procedure). Section \[algo\] explains how this is done.
Projections onto exponential families are becoming more and more important in the definition of measures of statistical interaction, complexity, synergy, and related quantities. In particular, the algorithm can be used to compute decompositions of mutual information, as for example the ones defined in [@olbrich] and [@us], and it was indeed used to compute all the numerical examples in [@us]. Another application of the algorithm is explicit computations of complexity measure as treated in [@preprint2], [@amaripreprint], and [@amaribook]. Examples of both applications can be found in Section \[applic\].
For all the technical details about the iterative scaling algorithm in its traditional version, we refer the interested reader to Chapters 3 and 5 of [@csiszar].
All proofs can be found in the Appendix.
Technical Definitions
---------------------
We take the same definitions and notations as in [@us], except that we let the output be multiple. More precisely, we consider a set of $N$ input nodes $V$, taking values in the sets $X_1,\dots,X_N$, and a set of $M$ output nodes $W$, taking values in the sets $Y_1,\dots,Y_M$. We write the input globally as $X:=X_1\times\dots\times X_N$, and the output globally as $Y:=Y_1\times\dots\times Y_M$. We denote by $F(Y)$ the set of real functions on $Y$, and by $P(X)$ the set of probability measures on $X$.
Let $I\subseteq V$ and $J\subseteq W$. We call $F_{IJ}$ the space of functions which only depend on $X_I$ and $Y_J$: $$\begin{aligned}
F_{IJ} := \big\{ f &\in F(X,Y)\;\big| \notag \\
&f(x_I,x_{I^c},y_J,y_{J^c})=f(x_I,x'_{I^c},y_J,y'_{J^c})\;\forall x_{I^c}, x'_{I^c},y_{J^c},y'_{J^c} \big\}\;.
\end{aligned}$$
We can model the channel from $X$ to $Y$ as a Markov kernel $k$, that assigns to each $x\in X$ a probability measure on $Y$ (for a detailed treatment, see [@kakihara]). Here we will consider only finite systems, so we can think of a channel simply as a transition matrix (or stochastic matrix), whose rows sum to one. $$\label{stoc}
k(x;y)\ge 0\quad \forall x,y; \qquad \sum_{y} k(x;y) =1 \quad \forall x\;.$$ The space of channels from $X$ to $Y$ will be denoted by $K(X;Y)$. We will denote by $X$ and $Y$ also the corresponding random variables, whenever this does not lead to confusion.
Conditional probabilities define channels: if $p(X,Y)\in P(X,Y)$ and the marginal $p(X)$ is strictly positive, then $p(Y|X)\in K(X;Y)$ is a well-defined channel. Viceversa, if $k\in K(X;Y)$, given $p\in P(X)$ we can form a well-defined joint probability: $$pk(x,y):= p(x)\,k(x;y)\quad \forall x,y\;.$$
To extend the notion of divergence from probability distributions to channels, we need an “input distribution”:
Let $p\in P(X)$, let $k,m\in K(X;Y)$. Then: $$\label{mkdiv}
D_p(k||m):= \sum_{x,y} p(x)\,k(x;y)\,\log\dfrac{k(x;y)}{m(x;y)}\;.$$
Let $p,q$ be joint probability distributions on $X\times Y$, and let $D$ be the KL-divergence. Then the following “chain rule” holds: $$\label{pkdiv}
D(p(X,Y)||q(X,Y)) = D(p(X)||q(X)) + D_{p(X)}(p(Y|X)||q(Y|X))\;.$$
Families of Channels {#fam}
====================
Suppose we have a family $\operatorname{\mathcal{E}}$ of channels, and a channel $k$ that may not be in $\operatorname{\mathcal{E}}$. Then we can define the “distance” between $k$ and $\operatorname{\mathcal{E}}$ in terms of $D_p$.
Let $p$ be an input distribution. The divergence between a channel $k$ and a family of channels $\operatorname{\mathcal{E}}$ is given by: $$D_p(k||\operatorname{\mathcal{E}}):=\inf_{m\in\operatorname{\mathcal{E}}} D_p(k||m)\;.$$ If the minimum is uniquely realized, we call the channel $$\pi_{\operatorname{\mathcal{E}}}k:=\operatorname*{arg\,min}_{m\in\operatorname{\mathcal{E}}} D_p(k||m)\;$$ the *rI-projection* of $k$ on $\operatorname{\mathcal{E}}$ (and simply “an” rI-projection if it is not unique).
The families considered here are of two types, dual to each other: linear and exponential. For both cases, we take the closures, so that the minima defined above always exist.
A *mixture family* of $K(X;Y)$ is a subset of $K(X;Y)$ defined by one or several affine equations, i.e., the locus of the $k$ which satisfy a (finite) system of equations in the form: $$\label{mixture}
\sum_{x,y} k(x;y) f_i(x,y) = c_i\;,$$ for some functions $f_i\in F(X,Y)$, and some constants $c_i$.
#### Example.
Consider a channel $m\in K(X;Y_1,Y_2)$. We can form the marginal: $$m(x;y_1):= \sum_{y_2}m(x;y_1,y_2)\;.$$ The channels $k\in K(X;Y_1,Y_2)$ such that $k(x;y_1)=m(x;y_1)$ form a mixture family, defined by the system of equations (for all $x'\in X$, $y'_1\in Y_1$): $$\sum_{x,y_1,y_2}k(x;y_1,y_2)\,\delta(x,x')\delta(y_1,y'_1) = m(x';y'_1)\;,$$ where the function $\delta(z,z')$ is equal to 1 for $z=z'$, and zero for any other case.
More in general, let $\operatorname{\mathcal{L}}$ be a (finite-dimensional) linear subspace of $F(X,Y)$, and let $k\in K(X;Y)$. Then: $$\label{m}
\operatorname{\mathcal{M}}(k,\operatorname{\mathcal{L}}) := \bigg\{ m\in K(X;Y)\,\bigg| \sum_{x,y}m(x;y)l(x,y)=\sum_{x,y}k(x;y)l(x,y) \; \forall l\in \operatorname{\mathcal{L}}\bigg\}\,$$ is a mixture family, which we call *generated by $k$ and $\operatorname{\mathcal{L}}$*.
A (closed) *exponential family* of $K(X;Y)$ is (the closure of) a subset of $K(X;Y)$ of channels in the form: $$\dfrac{e^{f(x,y)}}{Z(x)}\, k(x;y) \;,$$ where $f$ satisfies affine constraints, $k$ is fixed, and: $$Z(x):=\sum_y e^{f(x,y)}\,k(x;y)\;$$ so that the channel is correctly normalized.
This is a sort of multiplicative equivalent of mixture families, as the exponent satisfies constraints similar to .
#### Example.
Let $\operatorname{\mathcal{L}}$ be a (finite-dimensional) linear subspace of $F(X,Y)$, and let $k\in K(X;Y)$. Then the closure: $$\label{e}
\operatorname{\mathcal{E}}(k,\operatorname{\mathcal{L}}) := \bigg\{ \dfrac{e^{l(x,y)}}{Z(x)}\, k(x;y) \,\bigg|\,Z(x)=\sum_y e^{l(x,y)}\,k(x;y),\,l\in \operatorname{\mathcal{L}}\bigg\}\;$$ is an exponential family, which again we call *generated by $k$ and $\operatorname{\mathcal{L}}$*.
This family is in some sense “dual” to the family in . The duality is expressed more precisely by the following result.
\[dualmk\] Let $\operatorname{\mathcal{L}}$ be a subspace of $F(X,Y)$. Let $p\in P(X)$ be strictly positive. Let $k_0\in K(X;Y)$ be a strictly positive “reference” channel. Let $\operatorname{\mathcal{E}}:= \operatorname{\mathcal{E}}(k_0,\operatorname{\mathcal{L}})$ and $\operatorname{\mathcal{M}}:= \operatorname{\mathcal{M}}(k,\operatorname{\mathcal{L}})$. For $k'\in K(X;Y)$, the following conditions are equivalent:
1. $k'\in \operatorname{\mathcal{M}}\cap\operatorname{\mathcal{E}}$.
2. $k'\in \operatorname{\mathcal{E}}$, and $D_p(k||k')=\inf_{m\in\operatorname{\mathcal{E}}} D_p(k||m)$.
3. $k'\in \operatorname{\mathcal{M}}$, and $D_p(k'||k_0)=\inf_{m\in\operatorname{\mathcal{M}}} D_p(m||k_0)$.
In particular, $k'$ is unique, and it is exactly $\pi_{\operatorname{\mathcal{E}}}k$.
Geometrically, we are saying that $k'=\pi_{\operatorname{\mathcal{E}}} k$, the rI-projection of $k$ on $\operatorname{\mathcal{E}}$. We call the mapping $k\to k'$ the *rI-projection operator*, and the mapping $k_0\to k'$ the *I-projection operator* These are the channel equivalent of the I-projections introduced in [@iproj] and generalized in [@iriv]. The result is illustrated in Figure \[fig:dualmk\].
![Illustration of Theorem \[dualmk\]. The point $k'$ at the intersection minimizes on $\operatorname{\mathcal{E}}$ the distance from $k$, and minimizes on $\operatorname{\mathcal{M}}$ the distance from $k_0$.[]{data-label="fig:dualmk"}](./em.pdf)
As suggested by Figure \[fig:dualmk\], I- and rI-projections on exponential families satisfy a Pythagoras-type equality. For any $m\in \operatorname{\mathcal{E}}$, with $\operatorname{\mathcal{E}}$ exponential family: $$\label{py}
D_p(k||m)= D_p(k||\pi_{\operatorname{\mathcal{E}}}k) + D_p( \pi_{\operatorname{\mathcal{E}}}k||m)\;.$$ This statement follows directly from the analogous statement for probability distribution found in [@hierarchy], after applying the chain rule .
Algorithm {#algo}
=========
The algorithm can be considered as a channel equivalent of the iterative scaling procedure for joint distributions, which can be found in Chapter 5 of [@csiszar]. Translated into our language, that theorem says the following:
\[jointit\] Let $\operatorname{\mathcal{L}}_1,\dots,\operatorname{\mathcal{L}}_n$ be mixture families of joint distributions with nonempty intersection $\operatorname{\mathcal{L}}$. Denote by $\Sigma_iq$ the $I$-projection of a joint $q$ onto the family $\operatorname{\mathcal{L}}_i$. Consider the sequence that starts at $q^0$ and is defined iteratively by: $$q^j := \Sigma_{(j \operatorname{mod}n)} q^{j-1}\;.$$
Then $\{q^j\}$ converges, and the limit point is the $I$-projection of $q^0$ onto $\operatorname{\mathcal{L}}$, i.e. if we call: $$\lim_{i\to\infty} q^i := q\;,$$ then $q\in \operatorname{\mathcal{L}}$, and for any $\bar q\in \operatorname{\mathcal{L}}$: $$D(q^0||\bar q) = D(q^0||q) + D(q||\bar q)\;.$$
Our result depends on the theorem above, in the following way. We define a marginal procedure for channels, which in general depends on the choice of an input distribution. We define mixture families of channels with fixed marginals in a way compatible with the equivalent for joints. We then define scalings of channels, and prove that they give the desired result at the joint level. This makes it possible to translate the statement of Theorem \[jointit\] to an analogous statement for channels, Theorem \[convergence\].
Unless otherwise stated, all the input distributions here will be assumed strictly positive. All our proofs can be found in the appendix.
Consider an input distribution $p\in P(X)$. Let $I\subseteq [N], J\subseteq [M], J\ne\emptyset$. We define the *marginal operator* for channels as: $$\label{margop}
k(x;y)\mapsto k(x_I;y_J):= \sum_{x_{I^c},y_{J^c}}p(x_{I^c}|x_I)\,k(x_I,x_{I^c};y_J,y_{J^c})\;,$$ given the input $p$.
\[jeq\] Defined as above, $k(X_I;Y_J)$ is exactly the conditional probability for the marginal $pk(X_I,Y_J)$. In other words, $k(X,Y)$ has marginal $k(X_I;Y_J)$ if and only if $pk(X,Y)$ has marginal $pk(X_I,Y_J)$.
Consider an input distribution $p\in P(X)$. Let $I\subseteq [N], J\subseteq [M]$, $J\ne\emptyset$. We define the mixture families $\operatorname{\mathcal{M}}_{IJ}(\bar{k})$ as: $$\operatorname{\mathcal{M}}_{IJ}(\bar{k}) := \big\{k(x_{1\dots n};y_{1\dots m}) \,\big|\, p(x_I)\,k(x_I;y_J) = p(x_I)\, \bar k(x_I;y_J) \big\}\;,$$ where the $\bar k(x_I;y_J)$ are prescribed channel marginals.
Analogously, let $\bar q$ be a probability distribution in $P(X_I,Y_J)$. We define the mixture families: $$\mathcal{J}_{IJ}(\bar q) := \big\{q(x_{1\dots n},y_{1\dots m}) \,\big|\, q(x_I,y_J) = \bar q(x_I,y_J) \big\}\;.$$
Proposition \[jeq\] says that, for any $k\in K(X;Y)$, for any (strictly positive) $p\in P(X)$, and for any $I\subseteq [N], J\subseteq [M]$ : $$\label{corresp}
k\in \operatorname{\mathcal{M}}_{IJ}(\bar{k}) \quad \Leftrightarrow \quad pk \in \mathcal{J}_{IJ}(p \bar k)\;.$$
\[presc\] $\operatorname{\mathcal{M}}_{IJ}(\bar{k})$ is exactly the set $\operatorname{\mathcal{M}}(\bar{k},\operatorname{\mathcal{L}})$ of equation , where as $\operatorname{\mathcal{L}}$ we take the space $F_{IJ}$ of functions which only depend on the nodes in $I,J$.
Just as in [@csiszar], the I-projections for single marginals can be obtained by scaling. For joint distributions the scaling is done in this way: if $\bar p(X_I,Y_J)$ is a “prescribed” marginal, then: $$\operatorname{\sigma}^{\bar q}_{IJ} p\,(X,Y) := p(X,Y)\,\dfrac{\bar q(X_I,Y_J)}{p(X_I,Y_J)}$$ will have the prescribed marginals, and even be the I-projection of $p$ on $\mathcal{J}_{IJ}(\bar q)$. i.e., $\operatorname{\sigma}^{\bar q}_{IJ} p \in\mathcal{J}_{IJ}(\bar q)$, and: $$D(p||\bar q) = D(p||\operatorname{\sigma}^{\bar q}_{IJ} p ) + D(\operatorname{\sigma}^{\bar q}_{IJ} p ||\bar q)\;.$$ For the proof, see Chapter 3 and Section 5.1 of [@csiszar].
In the case of channels, the scaling is instead done in two steps.
We define the (unnormalized) *$IJ$-scaling* as the operator $\operatorname{\sigma}_{IJ}^{\bar k}:K(X;Y)\to F(X,Y)$, mapping $k$ to: $$\label{usk}
\operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y) := k(x_I , x_{I^c} ; y_J , y_{J^c} ) \frac{ \bar{k}(x_I ; y_J) }{ k(x_I ; y_J) } \;.$$
We have that $\operatorname{\sigma}_{IJ}^{\bar k} k$ is *not* an element of $ \operatorname{\mathcal{M}}_{IJ}$, as in general it is not even in $K(X;Y)$ (i.e. a correctly normalized channel). However, at the joint level this corresponds exactly to the joint scaling:
\[jlevel\] Let $p\in P(X)$, $k\in K(X;Y)$, and $\bar k \in K(X_I;Y_J)$. Then: $$p (\operatorname{\sigma}_{IJ}^{\bar k} k) = \operatorname{\sigma}_{IJ}^{p \bar k} pk\;.$$
This implies that $p \operatorname{\sigma}_{IJ}^{\bar k} k$ is the $I$-projection of $pk$ to the family $\mathcal{J}_{IJ} (p \bar k)$.
We define the *normalized $IJ$-scaling* as the operator $N \operatorname{\sigma}_{IJ}^{\bar k}:K(X,Y)\to K(X;Y)$, mapping $k$ to: $$\label{sk}
N \operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y) := \dfrac{1}{Z(x)}\, \operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y) \;,$$ where: $$Z(x) := \sum_{y'}\operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y')\;.$$
At the joint level, this corresponds to scaling of the input in the following way:
\[inscale\] Let $p\in P(X)$, $k\in K(X;Y)$, and $\bar k \in K(X_I;Y_J)$. Then: $$p( N \operatorname{\sigma}_{IJ}^{\bar k} k) = \operatorname{\sigma}_{[N]}^{p} \operatorname{\sigma}_{IJ}^{p \bar k} pk\;.$$
This implies that $p N \operatorname{\sigma}_{IJ}^{\bar k} k$ is the $I$-projection of $p\operatorname{\sigma}_{IJ}^{\bar k} k$ to the family with prescribed input $p(X)$. For brevity, let’s call this family $\mathcal{J}_{[N]}(p)$.
Now $N \operatorname{\sigma}_{IJ}^{\bar k} k$ is an element of $K(X;Y)$, but still *not* of $ \operatorname{\mathcal{M}}_{IJ}$. However, if we iterate the operator $N \operatorname{\sigma}_{IJ}^{\bar k}$, the resulting sequence will converge to the projector on $ \operatorname{\mathcal{M}}_{IJ}$. More in general, the following result holds:
\[convergence\] For $1\le i \le n$, let $I_i\subseteq [N]$ be subsets of $[N]$ and $J_i\subseteq [M]$ be nonempty subsets of $[M]$. Take an input distribution $p\in P(X)$ and a channel $\bar k\in K(X,Y)$. Define the mixture families of prescribed marginals: $$\operatorname{\mathcal{M}}_i = \operatorname{\mathcal{M}}_{I_iJ_i}(\bar k(X_{I_i},Y_{J_i}))\;,$$ and their intersection, which is also a mixture family (nonempty, as it contains at least $\bar k$): $$\operatorname{\mathcal{M}}:= \bigcap_i \operatorname{\mathcal{M}}_i\;.$$ Choose a (different) channel $k^0\in K(X,Y)$ and consider the sequence of normalized scalings starting at $k^0$ and defined iteratively by: $$k^j := N\sigma_{I_{(j \operatorname{mod}n)}J_{(j\operatorname{mod}n)}} k^{j-1}\;.$$ Then:
- $k^i$ converges to a limit channel: $$\lim_{i\to\infty} k^i := l\;;$$
- The limit $l$ is the $I$-projection of $k^0$ on $\operatorname{\mathcal{M}}$, i.e. $l\in\operatorname{\mathcal{M}}$ and: $$D_p(k^0||\bar k) = D_p(k^0||l) + D_p(l||\bar k)\;.$$
The proof can be found in the appendix.
To apply the Theorem \[convergence\] in our algorithm, we choose as initial channel $k^0$ exactly the reference channel $k_0$ of Theorem \[dualmk\], usually the uniform channel. As $\bar k$ we take exactly the “prescription channel” $k$ of Theorem \[dualmk\], i.e. the channel which has the desired marginals. The result of the iterative scaling will be the rI-projection of $k$ on the desired exponential family.
Applications {#applic}
============
Synergy Measures
----------------
The algorithm presented here permits to compute the decompositions of mutual information between inputs and outputs in [@olbrich] and [@us]. We give here examples of computations of *pairwise synergy* as an rI-projection for channels, as described in [@us]. It is not within the scope of this article to motivate this measure, we rather want to show how it can be computed.
Let $k$ be a channel from $X=(X_1,X_2)$ to $Y$. Let $p\in P(X)$ be a strictly positive input distribution. We define in [@us] the synergy of $k$ as: $$\label{syn}
d_2(k):= D_p(k||\operatorname{\mathcal{E}}_1)\;,$$ where $\operatorname{\mathcal{E}}_1$ is the (closure of the) family of channels in the form: $$m(x_1,x_2;y)=\dfrac{1}{Z(X)} \exp\big( \phi_0(x_1,x_2)+\phi_1(x_1,y)+\phi_2(x_2,y) \big)\;,$$ where: $$Z(x):=\sum_y\exp\big( \phi_0(x_1,x_2)+\phi_1(x_1,y)+\phi_2(x_2,y) \big)\;,$$ and: $$\phi_0 \in F_{\{1,2\}\emptyset}\,,\quad \phi_1 \in F_{\{1\}\{1\}}\,,\quad \phi_2 \in F_{\{2\}\{1\}}\;.$$ According to Theorem \[dualmk\], the rI-projection of $k$ on $\operatorname{\mathcal{E}}_1$ is the unique point $k'$ of $\operatorname{\mathcal{E}}_1$ which has all the prescribed marginals: $$k'(x_1;y) = k(x_1;y)\,,\quad k'(x_2;y) = k(x_2;y)\;,$$ and can therefore be computed by iterative scaling, either of the joint distribution (as it is traditionally done, see [@csiszar]), or of the channels (our algorithm).
Here we present a comparison of the two algorithms, implemented similarly and in the same language (Mathematica). The red dots represent our (channel) algorithm, and the blue dots represent the joint rescaling algorithm.
For the easiest channels (see Figure \[fig:xor\]), both algorithm converge instantly.
![Comparison of convergence times for the synergy of the XOR gate. Both algorithms get immediatly the desired result. (The dots here are overlapping, the red ones are not visible.)[]{data-label="fig:xor"}](./xor10.pdf)
A more interesting example is a randomly generated channel (Figure \[fig:rand\]), in which both method need 5-10 iterations to get to the desired value. However, the channel method is slightly faster.
![Comparison of convergence times for the synergy of a randomly generated channel. The channel method (red) is slightly faster.[]{data-label="fig:rand"}](./rand10.pdf)
The most interesting example is the synergy of the AND gate, which should be zero according to the procedure [@us]. In that article, we mistakenly wrote a different value, that here we would like to correct (it is zero). The convergence to zero is very slow, of the order of $1/n$ (Figure \[fig:and\]). It is clearly again slightly faster for the channel method in terms of iterations.
![Comparison of convergence times for the synergy of the AND gate. The channel method (red) tends to zero proportionally to $n^{-1.05}$, the joint method (blue) proportionally to $n^{-0.95}$.[]{data-label="fig:and"}](./and1000.pdf)
It has to be noted, however, that rescaling a channel requires more elementary operations than rescaling a joint distribution. Because of this, one single iteration with our method takes longer than with the joint method. (As explained in Section \[algo\], a scaling for the channel corresponds to two scalings for the joint.) In the end, despite the need of fewer iterations, the total computation time of a projection with our algorithm can be longer (depending on the particular problem). For example, again for the synergy of the AND gate, we can plot the computation time as a function of the accuracy (distance to actual value), down to $10^{-3}$. The results are shown in Figure \[fig:comp\].
![Comparison of total computation times for the synergy of the AND gate. The channel method (red) is slightly slower than the joint method (blue).[]{data-label="fig:comp"}](./comp.pdf)
To get to the same accuracy, though, the channel approach used less iterations. In summary, our algorithm is better in terms of iteration complexity, but generally worse in terms of computing time.
Complexity Measures
-------------------
Iterative scaling can also be used to compute measures of complexity, as defined in [@preprint2], [@amaripreprint], and in Section 6.9 of [@amaribook]. For simplicity, consider two inputs $X_1,X_2$, two outputs $Y_1,Y_2$ and a generic channel between them. In general, any sort of interaction is possible, which in terms of graphical models (see [@lauritzen]) can be represented by diagrams such as those in Figure \[fig:graphs1\].
[.5]{} ![a) The graphical model corresponding to conditionally independent outputs $Y_1$ and $Y_2$ are indeed correlated, but only indirectly, via the inputs. b) The graphical model corresponding to a non-complex system. []{data-label="fig:graphs1"}](./graph2.pdf "fig:") \[fig:graph2\]
[.45]{} ![a) The graphical model corresponding to conditionally independent outputs $Y_1$ and $Y_2$ are indeed correlated, but only indirectly, via the inputs. b) The graphical model corresponding to a non-complex system. []{data-label="fig:graphs1"}](./gp1.pdf "fig:") \[fig:gp1\]
Any line in the graph indicates an interaction between the nodes. In [@preprint2] the outputs are assumed to be conditionally independent, i.e. they do not directly interact (or, their interaction can be *explained away* by conditioning on the inputs). In this case the graph looks like Figure \[fig:graphs1\]a, and the maginals to preserve are those of the family of pairs $(X_{I_i},Y_{J_i})$, $i=1,2$ with: $X_{I_1}=X_{\{1,2\}}$, $Y_{J_1}=Y_{\{1\}}$, $X_{I_2}=X_{\{1,2\}}$, $Y_{J_2}=Y_{\{2\}}$.
Suppose now that $Y_1,Y_2$ correspond to $X_1,X_2$ at a later time. In this case it is natural to assume that the system is not complex if $Y_1$ does not depend (directly) on $X_2$, and $Y_2$ does not depend (directly) on $X_1$. Intuitively, in this case “the whole is exactly the sum of its parts”. In terms of graphical models, this means that our system is represented by Figure \[fig:graphs1\]b, meaning that the subsets of nodes in question are now only the ones given by $X_{I_1}=X_{\{1\}}$, $Y_{J_1}=Y_{\{1\}}$, $X_{I_2}=X_{\{2\}}$, $Y_{J_2}=Y_{\{2\}}$. These channels (or joints) form an exponential family (see [@preprint2]) which we call $\operatorname{\mathcal{F}}_1$.
[.5]{} ![a) The graphical model of Figure \[fig:graphs1\]a, with correlation between the outputs. b) The non-complex model of Figure \[fig:graphs1\]b, with correlation between the outputs. []{data-label="fig:graphs3"}](./graph32.pdf "fig:") \[fig:graph32\]
[.45]{} ![a) The graphical model of Figure \[fig:graphs1\]a, with correlation between the outputs. b) The non-complex model of Figure \[fig:graphs1\]b, with correlation between the outputs. []{data-label="fig:graphs3"}](./gp2.pdf "fig:") \[fig:gp2\]
Suppose now, though, that the outputs are not conditionally independent anymore, because of some “noise” (see [@amaripreprint] and [@amaribook]). This way the interaction structure would look like Figure \[fig:graphs3\], i.e. the “complete” subset given by $(X_I,Y_J)$ with $X_I=X_{\{1,2\}}$ and $Y_J=Y_{\{1,2\}}$. In particular, a non-complex but “noisy” system would be represented by Figure \[fig:graphs3\]b, and have subsets of nodes given by the pairs $(X_{I_i},Y_{J_i})$, $i=1,2,3$ with: $X_{I_1}=X_{\{1\}}$, $Y_{J_1}=Y_{\{1\}}$, $X_{I_2}=X_{\{2\}}$, $Y_{J_2}=Y_{\{2\}}$, $X_{I_3}=X_\emptyset$, $Y_{J_3}=Y_{\{1,2\}}$. Such channels form again an exponential family, which we call $\operatorname{\mathcal{F}}_2$.
We would like now to have a measure of complexity for a channel (or joint). In [@preprint2], the measure of complexity is defined as the divergence from the family $\operatorname{\mathcal{F}}_1$ represented in Figure \[fig:graphs1\]b. We will call such a measure $c_1$. In case of noise, however, it is argued in [@amaripreprint] and [@amaribook] that the divergence should be computed from the family $\operatorname{\mathcal{F}}_2$ represented in \[fig:graphs3\]b (for example, as written in the cited papers, because such a complexity measure should be required to be upper bounded by the mutual information between $X$ and $Y$). We will call such a measure $c_2$.
Both divergences can be computed with our algorithm. As an example, we have considered the following channel: $$k(x_1,x_2,x_3;y_1,y_2) = \dfrac{1}{Z(x)}\,\exp\bigg(\big( \alpha\,x_1\,x_2 + \beta x_3\big) (y_1-y_2) \bigg)\;,$$ with: $$Z(x) = \sum_{y'_1,y'_2} \exp\bigg(\big( \alpha\,x_1\,x_2 + \beta x_3\big) (y'_1-y'_2) \bigg)\;.$$ Here $X_3$ represents a node of “unknown input noise” that adds correlation between the outputs (of unknown form) when if it is not observed. We have chosen $\alpha=1$ and $\beta=2$, and a uniform input probability $p$. After marginalizing out $X_3$ (obtaining then an element of the type of Figure \[fig:graphs3\]a), we can compute the two divergences:
- $c_1(k)=D_p(k||\operatorname{\mathcal{F}}_1)=0.519$.
- $c_2(k)=D_p(k||\operatorname{\mathcal{F}}_2)=0.110$.
This could indicate that $c_1$ is incorporating part the correlation of the output nodes due to the “noise”, and therefore probably overestimating the complexity, at least in this case.
One could nevertheless also argue that $c_2$ can underestimate complexity, as we can see in the following “control” example. Consider the channel: $$h(x_1,x_2;y_1,y_2) = \dfrac{1}{Z(x)}\,\exp\bigg(\big( \alpha\,x_1\,x_2 \big) (y_1-y_2) \bigg)\;,$$ with: $$Z(x) = \sum_{y'_1,y'_2} \exp\bigg(\big( \alpha\,x_1\,x_2 \big) (y'_1-y'_2) \bigg)\;,$$ which is represented by the graph in Figure \[fig:graphs1\]a. If the difference between $c_1$ and $c_2$ were just due to the noise, then for our new channel $c_1(h)$ and $c_2(h)$ should be equal. This is not the case:
- $c_1(h)=D_p(h||\operatorname{\mathcal{F}}_1)=0.946$.
- $c_1(h)=D_p(h||\operatorname{\mathcal{F}}_2)=0.687$.
The divergences are still different. This means that there is an element $h_2$ in $\operatorname{\mathcal{F}}_2$, which does *not* lie in $\operatorname{\mathcal{F}}_1$, for which: $$D_p(h||h_2) < D_p(h||h_1)\quad \forall h_1\in f_1\;.$$
The difference is this time smaller, which could mean that noise still does play a role, but in rigor it is hard to say, since none of these quantities is linear, and divergences do not satisfy a triangular inequality.
We do not want to argue here in favor or against any of these measures. We would rather like to point out that such considerations can be done mostly after explicit computations, which can be done with iterative scaling.
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Proofs
======
$1\Leftrightarrow2$: Choose a basis $f_1,\dots,f_d$ of $\operatorname{\mathcal{L}}$. Define the map $\theta\mapsto k_\theta$, with: $$k_\theta(x;y)=k(\theta_1,\dots,\theta_d)(x;y) := \dfrac{1}{Z_\theta(x)}\, k_0(x;y)\,\exp\left( \sum_{j=1}^d \theta_j f_j(x,y)\right)\;,$$ and: $$Z_\theta(x) := \sum_{y} k_0(x;y)\,\exp\left( \sum_{i=1}^d \theta_i f_i(x,y)\right)\;.$$ Then: $$D_p(k||k_\theta) = D_p(k||k_0) -\sum_{j=1}^d \theta_j\,\operatorname{\mathbb{E}}_{pk}[f_j] + \operatorname{\mathbb{E}}_p[\log Z_\theta]\;.$$ Deriving (where $\operatorname{\partial}_j$ is w.r.t. $\theta_j$): $$\label{pder}
\operatorname{\partial}_jD_p(k||k_\theta) = - \operatorname{\mathbb{E}}_{pk}[f_j] + \operatorname{\mathbb{E}}_p\left[ \dfrac{\operatorname{\partial}_j Z_\theta}{Z_\theta} \right]\;.$$ The term in the last brackets is equal to: $$\begin{aligned}
\dfrac{\operatorname{\partial}_j Z_\theta}{Z_\theta} &= \dfrac{1}{Z_\theta}\,\sum_{y} k_0(x;y)\,\exp\left( \sum_{i=1}^d \theta_i f_i(x,y)\right) f_j(x,y)\\
&= \sum_y k_\theta(x;y) f_j(x,y) \;,
\end{aligned}$$ so that now reads: $$\label{extremum}
\operatorname{\partial}_jD_p(k||k_\theta) = - \operatorname{\mathbb{E}}_{pk}[f_j] + \operatorname{\mathbb{E}}_{pk_\theta}[f_j]\;.$$ This quantity is equal to zero for every $j$ if and only if $k_\theta\in\operatorname{\mathcal{M}}$. Now if $k_\theta$ is a minimizer, it satisfies , and so $k_\theta\in\operatorname{\mathcal{M}}$. Viceversa, suppose $k_\theta\in\operatorname{\mathcal{M}}$, so that it satisfies for every $j$. To prove that it is a global minimizer, we look at the Hessian: $$\operatorname{\partial}_i\operatorname{\partial}_jD_p(k||k_\theta) = \operatorname{\partial}_i\operatorname{\partial}_jD(pk||pk_\theta)\;.$$ This is precisely the covariance matrix of the joint probability measure $pk_\theta$, which is positive definite.
$1\Leftrightarrow3$: For every $m\in\operatorname{\mathcal{M}}$, we have: $$\begin{aligned}
\label{mk0}
D_p(m||k_0) &= \sum_{x,y}p(x)\,m(x;y)\log \dfrac{m(x;y)}{k_0(x;y)} = \operatorname{\mathbb{E}}_{pm}\left[ \log \dfrac{m}{k_0} \right]\;.
\end{aligned}$$ If $k'\in\operatorname{\mathcal{E}}$, then: $$\label{mk1}
D_p(m||k_0) = \operatorname{\mathbb{E}}_{pm}\left[ \log \dfrac{m}{k'}+\log\dfrac{k'}{k_0} \right] = D_p(m||k') + \operatorname{\mathbb{E}}_{pm}\left[ \log\dfrac{k'}{k_0} \right]\;.$$ By definition of $\operatorname{\mathcal{E}}$, the logarithm in the last brackets belongs to $\operatorname{\mathcal{L}}$, and since $m\in\operatorname{\mathcal{M}}$: $$\operatorname{\mathbb{E}}_{pm}\left[ \log\dfrac{k'}{k_0} \right] = \operatorname{\mathbb{E}}_{pk}\left[ \log\dfrac{k'}{k_0} \right] = \operatorname{\mathbb{E}}_{pk'}\left[ \log\dfrac{k'}{k_0} \right]\;.$$ Inserting in : $$\label{mk2}
D_p(m||k_0) = D_p(m||k') + \operatorname{\mathbb{E}}_{pk'}\left[ \log\dfrac{k'}{k_0} \right] = D_p(m||k') + D_p(k'||k_0)\;.$$ Since $D_p(m||k')\geq0$, shows that $k'$ is a minimizer. Since $D_p(m||k_0)=D(pm||pk_0)$ is strictly convex in the first argument, its minimizer is unique.
$$\begin{aligned}
p(x_I)k(x_I;y_J) &= p(x_I)\sum_{x_{I^c},y_{J^c}}p(x_{I^c}|x_I)\,k(x_I,x_{I^c};y_J,y_{J^c}) \\
&= \sum_{x_{I^c},y_{J^c}}p(x_I)\,p(x_{I^c}|x_I)\,k(x_I,x_{I^c};y_J,y_{J^c}) \\
&= \sum_{x_{I^c},y_{J^c}}p(x_I,x_{I^c})\,k(x_I,x_{I^c};y_J,y_{J^c}) \\
&= \sum_{x_{I^c},y_{J^c}}pk(x_I,x_{I^c},y_J,y_{J^c}) \\
&= pk(x_I,y_J)\;.
\end{aligned}$$
For $f$ in $F_{IJ}$: $$\label{mtwo1}
\operatorname{\mathbb{E}}_{pk}[f] = \sum_{x,y} p(x)\,k(x;y)\,f(x,y) = \sum_{x_I,y_J} p(x_I)\,k(x_I;y_J)\,f(x_I;y_J)\;,$$ and just as well: $$\label{mtwo2}
\operatorname{\mathbb{E}}_{p\bar{k}}[f] = \sum_{x,y} p(x)\,\bar{k}(x;y)\,f(x,y) = \sum_{x_I,y_J} p(x_I)\,\bar{k}(x_I;y_J)\,f(x_I;y_J)\;.$$ The definition in (with strict positivity of $p$) requires exactly that: $$\operatorname{\mathbb{E}}_{pk}[f]=\operatorname{\mathbb{E}}_{p\bar{k}}[f]$$ for every $f\in F_{IJ}$. Using and , the equality becomes: $$\sum_{x_I,y_J} p(x_I)\,k(x_I;y_J)\,f(x_I;y_J) = \sum_{x_I,y_J} p(x_I)\,\bar{k}(x_I;y_J)\,f(x_I;y_J)$$ for every $f$ in $F_{IJ}$, which means that $k(x_I;y_J)=\bar{k}(x_I;y_J)$.
We have: $$\begin{aligned}
p \operatorname{\sigma}_{IJ}^{\bar k} k\,(x_I,y_J) &= p \operatorname{\sigma}_{IJ}^{\bar k} k\,(x_I,x_{I^c},y_J,y_{J^c})\\
&= p(x_I,x_{I^c}) k(x_I , x_{I^c} ; y_J , y_{J^c} ) \, \frac{ \bar{k}(x_I ; y_J) }{ k(x_I ; y_J) } \\
&= pk(x_I , x_{I^c} , y_J , y_{J^c} ) \, \frac{p(x_I) \, \bar{k}(x_I ; y_J) }{p(x_I) \, k(x_I ; y_J) } \\
&= pk(x_I , x_{I^c} , y_J , y_{J^c} ) \, \frac{p\bar{k}(x_I , y_J) }{pk(x_I , y_J) } \\
&= \sigma_{IJ}^{p\bar k} pk\,(x,y)\;.
\end{aligned}$$
The first member is equal to: $$\begin{aligned}
p(x) N \operatorname{\sigma}_{IJ}^{\bar k} k(x,y) &= p(x)\, \dfrac{\operatorname{\sigma}_{IJ} k \, (x,y)}{\sum_{y'}\operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y')} \\
&= p(x)\, \dfrac{\operatorname{\sigma}_{IJ}^{p \bar k} pk (x)\, \operatorname{\sigma}_{IJ} k \, (x,y)}{\sum_{y'}\operatorname{\sigma}_{IJ}^{p \bar k} pk (x)\,\operatorname{\sigma}_{IJ}^{\bar k} k \, (x,y')} \\
&= p(x)\, \dfrac{\operatorname{\sigma}_{IJ}^{p \bar k} pk\,(x,y)}{\sum_{y'}\operatorname{\sigma}_{IJ}^{p \bar k} pk \, (x,y')} \\
&= \operatorname{\sigma}_{IJ}^{p \bar k} pk\,(x,y)\,\dfrac{p(x)}{\operatorname{\sigma}_{IJ}^{p \bar k} pk \, (x)} \\
&= \operatorname{\sigma}_{[N]}^{p} \operatorname{\sigma}_{IJ}^{p \bar k} pk\;.
\end{aligned}$$
In the hypothesis and notation of Theorem \[jointit\], take the collection $\operatorname{\mathcal{L}}_1,\dots,\operatorname{\mathcal{L}}_{2n}$ in the following way, for $i=1,\dots,n$:
- $\operatorname{\mathcal{L}}_{2i-1}:=\mathcal{J}_{[N]}(p)$;
- $\operatorname{\mathcal{L}}_{2i}:=\mathcal{J}_{I_iJ_i} (p \bar k)$.
Their intersection $\operatorname{\mathcal{L}}$ is nonempty, as it contains at least $p\bar k$.
Take as initial distribution $q^0:=pk^0$ and form as in Theorem \[jointit\] the sequence $\{q^j\}$ of $I$-projections. According to Theorem \[jointit\], this sequence converges to the $I$-projection of $pk^0$ on $\operatorname{\mathcal{L}}$. Since $\operatorname{\mathcal{L}}\subseteq \mathcal{J}_{[N]}(p)$, this projection will have input marginal equal to $p(X)$, and so we can write it as $p(X)\,l(X;Y)$ for some uniquely defined channel $l$. We have, for $j\to\infty$: $$q^j \to pl\;,$$ so in particular, for the subsequence of even-numbered terms also: $$q^{2j} \to pl\;.$$ This subsequence is defined iteratively by: $$q^{2j} = \operatorname{\sigma}_{[N]}^{p} \operatorname{\sigma}_{I_jJ_J}^{p \bar k} q^{2(j-1)}\;.$$ Propositions \[jlevel\] and \[inscale\] imply then that: $$q^{2j} = p\,k^j$$ for every $j$, where $\{k^j\}$ is the sequence defined in the statement of Theorem \[convergence\]. Therefore this sequence converges: $$k^j \to l\;.$$
Since $pl\in\operatorname{\mathcal{L}}\subseteq \operatorname{\mathcal{L}}_i$ for all $i$, $l\in \operatorname{\mathcal{M}}_i$ for all $i$ because of , which by definition means that $l\in \operatorname{\mathcal{M}}$. Moreover, $pl$ is the $I$-projection $q$ of $pk^0$ on $\operatorname{\mathcal{L}}$, which means that: $$D(pk^0||p\bar k) = D(pk^0||pl) + D(pl||p\bar k)\;.$$ Using the chain rule of the KL-divergence , we get: $$D_p(k^0||\bar k) = D_p(k^0||l) + D_p(l||\bar k)\;,$$ which means that $l$ is the $I$-projection of $k^0$ on $\operatorname{\mathcal{M}}$.
[^1]: Correspondence: perrone@mis.mpg.de
|
---
abstract: 'This paper reports the analysis of audio and visual features in predicting the continuous emotion dimensions under the seventh Audio/Visual Emotion Challenge (AVEC 2017), which was done as part of a B.Tech. 2nd year internship project. For visual features we used the HOG (Histogram of Gradients) features, Fisher encodings of SIFT (Scale-Invariant Feature Transform) features based on Gaussian mixture model (GMM) and some pretrained Convolutional Neural Network layers as features; all these extracted for each video clip. For audio features we used the Bag-of-audio-words (BoAW) representation of the LLDs (low-level descriptors) generated by openXBOW provided by the organisers of the event. Then we trained fully connected neural network regression model on the dataset for all these different modalities. We applied multimodal fusion on the output models to get the Concordance correlation coefficient on Development set as well as Test set.'
author:
-
bibliography:
- 'bib.bib'
title: Continuous Multimodal Emotion Recognition Approach for AVEC 2017
---
HOG, SIFT, GMM, Fisher, Neural Networks.
Introduction
============
Emotions are one of the most important and complex aspects of human consciousness. Researchers have been trying to classify or measure human emotions, as this research can lead to a significant improvement in human-computer interaction. Many studies have defined emotions as discrete categories. In discrete emotion theory, humans are thought to have some basic emotions. These basic emotions include ‘anger’, ‘disgust’, ‘fear’, ‘happiness’, ‘sadness’ and ‘surprise’ [@discrete]. While some other studies represent emotions using continuous dimensional models. Dimensional models of emotion attempt to represent emotions as a particular continuous region in 2-D or 3-D continuum. Harold Schlosberg in his study defined the three dimensions of emotion: ‘pleasantness-unpleasantness’, ‘attention-rejection’, ‘level of activation’ [@dim1]. Dimensional analysis of emotions suggest that each dimension/factor is related to different regions in our brain, and are processed independently. Most studies incorporate two dimensions that are ‘arousal’ and ‘valence’. ‘Valence’ as used in psychology of emotions represents ‘pleasantness-unpleasantness’ of an emotion, while ‘Arousal’ signifies the level of activation. Another dimension is ‘liking’ which points the wanting of a particular experience [@liking]. Recently the dimensional models are being widely used for research purposes, thanks to their practicality and ease of representing complex emotions. The basic motivation behind this challenge is the observation that a lot of human emotions can be judged by the physical gestures of human face while interacting with other people. [@psychology] Psychologists have shown that various facial activities like eye movements, lip movement, wrinkles around eye-corners etc. have special semantic meanings in emotional context, and conversely by detecting these facial signs one can judge the emotional state of a person. Various studies have shown that it is in fact how humans understand others emotions.\
This paper implements our work in the challenge and our main focus is on the visual and audio features. We have extracted the visual features in the form of HOG features, deep visual features and fisher vector representation of SIFT [@sift] features and the audio features are used as it is provided by the organizers of the challenge. Then we have trained models using Neural Networks.
Related Works
=============
Affective Computing Analysis has been attracting many researchers since it was first introduced into modern computing by Rosalind Picard in 1995 [@affective]. Since then researchers have been introducing new techniques to predict human emotions. Regarding the extraction of features using audio signals and from human speech, feature sets such that LPC [@dim] have been used. MFCC [@fishk] features have also been used by some getting better results. Teams which participated in AVEC 2016 competition [@Povolny:2016:MER:2988257.2988268] also used physiological features like the heart rate (HR), ECG signals, the skin conductance response (SCR), the skin conductance level (SCL), etc. Apart from the visual features like HOG [@hog] and SIFT [@sift] features, PHOG are also successful in human detection [@huk]. HOG [@hog] and LBP (Local binary patterns) features can also be used in combination like in [@lop], where they complemented each other. LBP tend to reduce the noise of hog noisy edges. LSTM (Long Short Term Memory) is a type of Recurrent neural Network proposed in 1997 which was used for affect recognition in [@arf]. This network was also used by teams in the AVEC 2015 competition [@He:2015:MAD:2808196.2811641]. Talking of neural networks, deep neural networks like Deep Convolutional Neural Network in [@nnn] used transductive learning approach for emotion recognition along with the hypergraphs.\
Multimodal techniques are nowadays and are being used widely in research areas [@multi]. Researchers dealing with the different types of modalities very often want to get better results by combining their results. Many such techniques exist, one of them being Markov fusion Network [@mfn]. MFN combines classifier outputs of different modalities with respect to its temporal relationship. Researchers also created apps and softwares for sentiment prediction that analyses facial expressions, gestures, speech, etc. One such tool is Kairos Emotions API[^1] which is available online.
Dataset
=======
This challenge is based on a novel database of human-human interactions recorded ‘in-the-wild’: Automatic Sentiment Analysis in the Wild (SEWA[^2]) [@schuller2016multimodal] data set. The dataset contains the video data, audio data and manual transcriptions of the speech of the corresponding subjects for training, development and testing with emotion dimensions values : arousal, valence and liking. Among the audio data are the audio files of the subjects, 23 LLDs from the eGeMAPSv0.1a feature set, 88 acoustic features from the eGeMAPSv0.1a feature set and Bag-of-audio-words representation (BoAW) of the LLDs for all the subjects. Among the video data are the video files of the subjects, video features (pixel coordinates of facial landmark points and head orientation) : normalised and unnormalised, and Bag-of-video-words representation of the normalised video features.
Features
========
Audio Features
--------------
The BoAW representation (generated by openXBOW [@xbow]) of 23 LLDs from eGeMAPSv0.1a [@eg] feature set extracted using openSmile [@opensmile] are provided in the sub challenge dataset. We used BoAW with block size 6 seconds as it is for the training process using neural network.
Visual Features
---------------
Before extracting any visual feature, we detected the human faces from the frames of the videos and applied the affine transformation using the facial landmark coordinates provided in the dataset to get the vertical faces. Faces are detected using the Viola-Jones face detection [@Viola01robustreal-time] algorithm and then cropped. All the visual features further discussed are extracted from these affine-warped faces. Most of the visual features are trained after normalisation process except in the cases where it increased the error. Images are resized and converted to gray scale as per need depending on the model requirements.
### Histogram of Gradients (HOG) features
Faces are resized to the default window size of HOG [@hog] (64x128) so as to get a single feature from the frames. Then trained on the neural network after normalisation.
### SIFT-Fisher Vector representation with GMM
Bag of words model is a quite successful approach of simplifying the representation of text/documents, used in Natural Language Processing. In the context of computer vision it has proved to be a very effective representation of an image [@david]. We have used the BOW representation with GMM. GMM is a generative and rather complex clustering model as compared to simpler models such as k-means. We used GMM model for final cluster formation as it takes into account the distance of a data point from the centroid and also the weight of a particular cluster, which is otherwise ignored in simple k-means. We first calculated the SIFT [@sift] descriptors for each frame and only the descriptors having highest response value were chosen (top 50 descriptors). Then we clustered our entire data into 32 clusters[^3] using K-means algorithm. After dividing the data into 32 clusters, we calculated mean, covariance matrix and weights for each cluster. Weights are calculated as the fraction of data points allotted to that cluster. After that these values are used as the initial values of means, covariance matrices and weights in the EM (Expectation-Maximization) algorithm for the generation of the GMM.\
 
#### Fisher Vector Extraction
Following the fisher vector approach in [@7344580], we took the previously calculated SIFT features of each frame and got the Fisher vector encodings [@fish1] of these sift features based on the pre-calculated GMM model. The resultant fisher vector has dimensions 128x32x2. Dimension reduction technique Principal Component Analysis (PCA) is used to convert the fisher encoding into final 4508 dimensions (99 percent variance). Thus a final fisher vector is obtained for each frame of each video.
### Deep Visual Features
An output of a particular layer of pretrained models : VGG-Face [@vgg] and ResNet-50-dag [@He2015] are used as features. These models are deep convolutional neural networks obtained from MatConvNet toolbox [@vedaldi15matconvnet]. Output of 35^th^ layer of VGG-Face and output of 174^th^ layer of ResNet-50-dag [@He2015] are used as features of the frames of each video clip. These are activation layer (ReLU) and a fully connected layer in their respective models.
Training
========
Three different models for arousal,valence and liking are trained for different modalities. We used Neural Networks for this regression task instead of LibSVM because it was very time consuming for such a large dataset. More or less the number of layers used in the network is the same in all the models. ReLU (rectified linear unit) activation function is used between layers except the last layer where $tanh$ function is used to get the final output. We observed that 4-5 layers were sufficient and increasing the layers more than this was computationally expensive as well as it did not improve the results.
Results
=======
For each emotion dimension, the models trained on the Train Set, are used for the prediction on the Development (Dev) Set. Then CCC is calculated for each model. Here $ \rho$ is the correlation coefficient between two variables $x$ and $y$, and $\sigma_x^2 , \sigma_y^2$ is the variance of $x$ and $y$ respectively while $\mu_x , \mu_y$ are respective means. Here CCC is the Concordance Correlation Coefficient which is calculated as:$$CCC = \frac{2\rho\sigma_x\sigma_y}{\sigma_x^2 + \sigma_y^2 + (\mu_x-\mu_y)^2}$$
\[table:1\]
--------------------- --------- --------- --------
Model Arousal Valence Liking
\[0.5ex\] Dev-Audio 0.344 0.351 0.081
Dev-Video 0.466 0.400 0.155
Dev-Text 0.373 0.390 0.314
Dev-Multimodal 0.525 0.507 0.235
Test-Audio 0.225 0.244 -0.020
Test-Video 0.308 0.455 0.002
Test-Text 0.375 0.425 0.246
Test-Multimodal 0.306 0.466 0.048
--------------------- --------- --------- --------
: AVEC Baseline [@ringeval2017avec] : Concordance Correlation Coefficient Table for Development (Dev) Set and Test Set.
\[table:2\]
--------------- --------- --------- --------
Model Arousal Valence Liking
\[0.5ex\] HOG 0.289 0.310 0.056
VGG FACE 0.277 0.336 0.021
RESNET 0.165 0.274 0.010
SIFT-fisher 0.075 0.123 -0.021
AUDIO 0.212 0.218 -0.089
--------------- --------- --------- --------
: Obtained: Concordance Correlation Coefficient Table for Development (Dev) Set.
Multimodal Fusion
=================
Now we can perfom multimodal fusion on these features to get final fused results.
\[table:3\]
-------------------------- --------- --------- --------
Fused Features Arousal Valence Liking
\[0.5ex\] Dev-Multimodal 0.294 0.346 0.013
Test-Multimodal 0.276 0.365 0.00
-------------------------- --------- --------- --------
: Concordance Correlation Coefficient Table of the Fused results.
--------------------------- --------- --------- --------
Feature Arousal Valence Liking
\[0.5ex\] Test-Multimodal 0.361 0.437 -0.001
--------------------------- --------- --------- --------
: Linear Correlation Coefficient for Test Data.
Conclusion
==========
This report summarizes our second year internship project which proved very informative as beginners and this was our first time handling such large datasets. Although the field of emotion recognition is not in very advanced state, still the rate of growth ensures a quite bright future.\
From our project we can say that visual features such as HOG, SIFT, etc. are capable of capturing the important facial information. We observed that the HOG features and VGG-Face features performed quite well as compared to the others.We were expecting better results from the SIFT-GMM features but the results obtained are not very encouraging. However the overall results have shown that these visual features are quite dependable, for most of the cases and can be used as a way for emotion detection. Moreover, it was observed that the multi-modal fusion provides even better results by giving suitable weights to the respective features. Further we have observed that the weights varied from model to model, which suggests that the success of detecting a particular emotion is dependent on the type of feature we are extracting, and simpler models can have more impact as compared to complex models.
Acknowledgment {#acknowledgment .unnumbered}
==============
We are greatly thankful to our teacher and the entire AVEC 2017 organising team, who gave us the opportunity to participate in this informative event. This project has helped us in gaining more knowledge and insight regarding the field of computer vision as well as machine learning and some basic knowledge of deep learning.
[^1]: https://www.kairos.com/emotion-analysis-api
[^2]: https://sewaproject.eu/
[^3]: The cluster number value 32 was chosen empirically.
|
---
abstract: 'This paper studies the non-spherical perturbations of the continuously self-similar critical solution of the gravitational collapse of a massless scalar field (the Roberts solution). The exact analysis of the perturbation equations reveals that there are no growing non-spherical perturbation modes.'
address: |
Physics Department, University of Alberta\
Edmonton, Alberta, Canada, T6G 2J1
author:
- 'Andrei V. Frolov [^1]'
title: |
Critical Collapse Beyond Spherical Symmetry:\
General Perturbations of the Roberts Solution
---
Introduction {#sec:intro}
============
Choptuik’s discovery of critical phenomena in the gravitational collapse of a scalar field [@Choptuik:93] sparked a surge of interest in gravitational collapse just at the threshold of black hole formation. The discovery of critical behavior in several other matter models quickly followed . Despite the fact that the evolution equations are very complex and highly non-linear, the dynamics of the near-critical field evolution is relatively simple and, in some important aspects, universal. The critical solution, which depends on the matter model only, serves as an intermediate attractor in the phase space of solutions, and often has an additional peculiar symmetry called self-similarity. The mass of the black hole produced in supercritical evolution scales as a power law $$\label{eq:scaling}
M_{\text{BH}}(p) \propto |p-p^*|^\beta,$$ with parameter $p$ describing initial data, and mass-scaling exponent $\beta$ is dependent only on the matter model, but not on the initial data family. An interesting consequence of mass scaling which has direct bearing on the cosmic censorship conjecture is the fact that arbitrarily small black holes can be produced in near-critical collapse, with the critical solution exhibiting a curvature singularity and no event horizon.
The explanation of the universality of the critical behavior lies in perturbation analysis and renormalization group ideas . It turns out that critical solutions generally have only one unstable perturbation mode, making them the most important solutions for understanding the dynamics of field evolution, after the stable ones (flat space and Schwarzschild or Kerr-Newman black hole). As the near-critical field configuration evolves, all its perturbation modes decay, losing information about the initial data and bringing the solution closer to critical, except the one growing mode which will eventually drive the solution to black hole formation or dispersal, depending on its content in the initial data. Thus the critical solution acts as an intermediate attractor (of codimension one) in the phase space of field configurations. Finding the eigenvalue of the growing perturbation mode allows one to calculate important parameters of the critical evolution, the mass-scaling exponent in particular.
An important question is how generic the critical behavior is with respect to initial data, or, in phase space language, how big is the basin of attraction of the critical solution. So far most of the work on critical gravitational collapse, numerical or analytic, has been restricted to the case of spherical symmetry, simply because of the enormous difficulties in treating fully general non-symmetric solutions of Einstein equations. A natural concern is whether the critical phenomena observed so far are limited to spherical symmetry, and whether the evolution of non-spherical data will lead to the same results. The numerical study of Abrahams and Evans on axisymmetric gravitational wave collapse and recent numerical perturbation calculations by Gundlach [@Gundlach:97; @Gundlach:98] give numerical evidence for the claim that critical phenomena are not restricted to spherical symmetry, and that the critical solutions are indeed attractors in the full phase space. In this paper we search for analytical evidence to support that claim.
One of the few known closed form solutions related to critical phenomena is the Roberts solution, originally constructed as a counterexample to the cosmic censorship conjecture [@Roberts:89], and later rediscovered in the context of critical gravitational collapse . It is a continuously self-similar solution of a spherically symmetric gravitational collapse of a minimally coupled massless scalar field. While it is not a proper critical solution, as it has more than one growing mode [@Frolov:97], it is still a good (and simple) toy model of the critical collapse of the scalar field.
This paper considers fully general perturbations of the Roberts solution in a gauge-invariant formalism. Due to the symmetries of the background, the linear perturbation equations decouple and the variables separate, so an exact analytical treatment is possible. We find that there are no growing perturbation modes apart from spherically symmetric ones described earlier [@Frolov:97]. So all the non-sphericity of the initial data decays in the collapse of the scalar field, and only the spherically symmetric part will play a role in the critical behavior.
To our knowledge, this is the first paper to obtain analytical results on non-spherical critical collapse.
The Roberts Solution {#sec:roberts}
====================
The spacetime we will use as a background in our calculations is a continuously self-similar spherically symmetric solution of the gravitational collapse of a massless scalar field (the Roberts solution). The Einstein-scalar field equations $$\begin{aligned}
R_{\mu\nu} &=& 2 \phi_{,\mu} \phi_{,\nu}, \label{eq:r}\\
\Box\phi &=& 0 \label{eq:box}\end{aligned}$$ can be solved analytically in spherical symmetry by imposing continuous self-similarity on the solution, i.e. by assuming that there exists a vector field $\xi$ such that $$\label{eq:ss}
{{\pounds}}_\xi g_{\mu\nu} = 2g_{\mu\nu}, \hspace{1em}
{{\pounds}}_\xi \phi = 0,$$ where ${{\pounds}}$ denotes Lie derivative. Self-similar solutions form a one-parameter family, which is most easily derived in null coordinates . The critical solution is given by the metric $$\label{eq:metric}
ds^2 = - 2\, du\, dv + r^2\, d\Omega^2,$$ where $$\label{eq:crit}
r = \sqrt{u^2 - uv}, \hspace{1em}
\phi = \frac{1}{2} \ln \left[1 - \frac{v}{u}\right].$$ The global structure of the critical spacetime is shown in Fig. \[fig:roberts\]. The influx of the scalar field is turned on at the advanced time $v=0$, so that the spacetime is Minkowskian to the past of this surface. The initial conditions for the field equations (\[eq:r\]) and (\[eq:box\]) are specified there by the continuity of the solution.
It is instructive to rewrite Roberts solution in new coordinates so that the self-similarity becomes apparent. For this purpose we introduce scaling coordinates $$\label{eq:coord:xs}
x = \frac{1}{2} \ln \left[1 - \frac{v}{u}\right], \hspace{1em}
s = - \ln(-u),$$ with the inverse transformation $$\label{eq:coord:uv}
u = - e^{-s}, \hspace{1em}
v = e^{-s} (e^{2x} - 1).$$ The signs are chosen to make the arguments of the logarithm positive in the region of interest ($v>0$, $u<0$), where the field evolution occurs. In these coordinates the metric (\[eq:metric\]) becomes $$\label{eq:metric:xs}
ds^2 = 2 e^{2(x - s)}
\left[(1 - e^{-2x}) ds^{2} - 2 ds dx\right] +
r^2\, d\Omega^2,$$ and the critical solution (\[eq:crit\]) is simply $$\label{eq:crit:xs}
r = e^{x-s}, \hspace{1em}
\phi = x.$$ Observe that the scalar field $\phi$ does not depend on the scale variable $s$ at all, and the only dependence of the metric coefficients on the scale is through the conformal factor $e^{-2s}$. This is a direct expression of the geometric requirement (\[eq:ss\]) in scaling coordinates; the homothetic Killing vector $\xi$ is simply $-\frac{\partial\ }{\partial s}$.
Gauge-Invariant Perturbations {#sec:pert}
=============================
To avoid complicated gauge issues of fully general perturbations, we will use the gauge-invariant formalism developed by Gerlach and Sengupta . This formalism describes perturbations around a general spherically symmetric background $$\label{eq:pert:background}
g_{\mu\nu} dx^\mu dx^\nu = g_{AB} dx^A dx^B
+ r^2 \gamma_{ab} dx^a dx^b,$$ which in our case we take to be the Roberts solution (\[eq:metric\]). Here and later capital Latin indices take values $\{0,1\}$, and lower-case Latin indices run over angular coordinates. $g_{AB}$ and $r$ are defined on a spacetime two-manifold, while $\gamma_{ab}$ is the metric of the unit two-sphere.
Because the background spacetime is spherically symmetric, perturbations around it can be decomposed in spherical harmonics. Scalar spherical harmonics $Y_{lm}(\theta, \varphi)$ have even parity under spatial inversion, while vector spherical harmonics $S_{lm\,a}(\theta, \varphi) \equiv \epsilon_a^{~b} Y_{lm,b}$ have odd parity. We will only concern ourselves with even-parity perturbations here, since odd-parity perturbations can not couple to scalar field perturbations. We will focus on non-spherical perturbation modes ($l \ge 1$), as the spherically symmetric case ($l=0$) was studied earlier [@Frolov:97]. For clarity, angular indices $l, m$ and the summation over all harmonics will be suppressed from now on. The most general even-parity metric perturbation is $$\begin{aligned}
\label{eq:pert:metric}
\delta g_{\mu\nu} dx^\mu dx^\nu &=& h_{AB} Y dx^A dx^B \nonumber\\
&& + h_A Y_{,b} (dx^A dx^b + dx^b dx^A) \nonumber\\
&& +r^2 [K Y \gamma_{ab} + G Y_{:ab}] dx^a dx^b,\end{aligned}$$ and the scalar field perturbation is $$\label{eq:pert:field}
\delta \phi = F Y.$$ As you can see, metric perturbations are described by a two-tensor $h_{AB}$, a two-vector $h_A$, and two two-scalars $K$ and $G$; the scalar field perturbation is described by a two-scalar $F$. However, these perturbation amplitudes do not have direct physical meaning, as they change under the (even-parity) gauge transformation induced by the infinitesimal vector field $$\label{eq:pert:gauge}
\xi_\mu dx^\mu = \xi_A Y dx^A + \xi Y_{,a} dx^a.$$ One can construct two gauge-invariant quantities from the metric perturbations $$\begin{aligned}
\label{eq:pert:gi:metric}
k_{AB} &=& h_{AB} - 2 p_{(A|B)},\nonumber\\
k &=& K - 2 v^A p_A,\end{aligned}$$ and one from the scalar field perturbation $$\label{eq:pert:gi:field}
f = F + \phi^{,A} p_A,$$ where $$v_A = \frac{r_{,A}}{r}, \hspace{1em}
p_A = h_A - \frac{r^2}{2}\, G_{,A}.$$ Only gauge-invariant quantities have physical meaning in the perturbation problem. All physics of the problem, including the equations of motion and boundary conditions, should be written in terms of these gauge-invariant quantities. Once gauge-invariant quantities have been identified, one is free to convert between gauge-invariant perturbation amplitudes and their values in whatever gauge choice one desires.
We will work in longitudinal gauge ($h_A = G = 0$), which is particularly convenient since perturbation amplitudes in it are just equal to the corresponding gauge-invariant quantities. The above condition fixes the gauge uniquely for non-spherical modes. (There is some gauge freedom left over for the $l=0$ mode, but remember that we are only concerned with higher $l$ modes.) Expressions for the components of the linear perturbation equations $$\begin{aligned}
&\delta R_{\mu\nu} = 4 \phi_{(,\mu} \delta \phi_{,\nu)}&, \\
&\delta (\Box\phi) = 0&\end{aligned}$$ for a fully general perturbation in longitudinal gauge are collected in Appendix \[sec:long\]. By inspection of the $\theta\varphi$ component of the equations, it is clear that the equations of motion require that $h_{uv}=0$ for $l \ge 1$. With the change of notation $h_{uu}=U$ and $h_{vv}=V$, the remaining equations of motion for non-spherical modes are
\[eq:pert\] $$\label{eq:pert:box}
4 (u^2-uv) F_{,vu} - u U_{,v} - u K_{,u} + v K_{,v} + v V_{,u}
- 2 u F_{,u} + 2 (2u-v) F_{,v} + 2 l(l+1) F = 0,$$ $$\label{eq:pert:r:uu}
- 2 (u^2-uv) K_{,uu} + u U_{,u} + (2u-v) (U_{,v} - 2 K_{,u})
- 4 v F_{,u} + l(l+1) U = 0,$$ $$\label{eq:pert:r:uv}
- (u^2-uv) (U_{,vv} + 2 K_{,vu} + V_{,uu})
+ u U_{,v} + u K_{,u} - (2u-v) (K_{,v} + V_{,u})
+ 2 u F_{,u} - 2 v F_{,v} = 0,$$ $$\label{eq:pert:r:vv}
- 2 (u^2-uv) K_{,vv} + 2 u K_{,v}
- u V_{,u} - (2u-v) V_{,v} + 4 u F_{,v} + l(l+1) V = 0,$$ $$\label{eq:pert:r:tt}
2 (u^2-uv) K_{,vu} - u U_{,v} - 2 u K_{,u} + (2u-v) (2 K_{,v} + V_{,u})
- 2 K + l(l+1) K + 2 V = 0,$$ $$\label{eq:pert:r:ut}
(u^2-uv) (U_{,v} + K_{,u}) + 2 v F = 0,$$ $$\label{eq:pert:r:vt}
(u-v) (V_{,u} + K_{,v}) - 2 F = 0.$$
Equation (\[eq:pert:box\]) comes from the scalar wave equation, and equations (\[eq:pert:r:uu\]–\[eq:pert:r:vt\]) are the $uu$, $uv$, $vv$, $\theta\theta$, $u\theta$, and $v\theta$ components of the Einstein equations, correspondingly. As usual with a scalar field, the system (\[eq:pert\]) has one redundant equation, so equation (\[eq:pert:r:uv\]) is satisfied automatically by virtue of other equations. Equations (\[eq:pert:r:ut\]) and (\[eq:pert:r:vt\]) are constraints, and the remaining four equations are dynamic equations for four perturbation amplitudes $U$, $V$, $K$, and $F$.
Boundary conditions for the system (\[eq:pert\]) are specified at $v=0$ and the spatial infinity. Continuity of matching with flat spacetime at the hypersurface $v=0$ requires the vanishing of the perturbations there. We also require well-behavedness of the perturbations at ${\cal I}^-$ and ${\cal I}^+$, so that the perturbation expansion holds. Thus, the boundary conditions are $$\begin{aligned}
\label{eq:pert:bc}
&U=V=K=F=0 \text{ at } v=0,& \nonumber\\
&U, V, K, F \text{ are bounded at } u=-\infty \text{ and } v=+\infty.&\end{aligned}$$ Equations (\[eq:pert\]) together with boundary conditions (\[eq:pert:bc\]) constitute our eigenvalue problem.
Decoupling of Perturbation Equations {#sec:decouple}
====================================
It is possible to decouple the dynamic equations (\[eq:pert:box\]–\[eq:pert:r:tt\]) by combining them with the constraints (\[eq:pert:r:ut\]) and (\[eq:pert:r:vt\]), and their first derivatives. After somewhat cumbersome algebraic manipulations, which we will not show here, the system of linear perturbation equations (\[eq:pert\]) can be rewritten as
\[eq:uv\] $$\label{eq:uv:f}
2 (u^2-uv) F_{,vu} - u F_{,u} + (2u-v) F_{,v}
+ \frac{2vF}{u-v} + l(l+1) F = 0,$$ $$\label{eq:uv:u}
2 (u^2-uv) U_{,vu} + u U_{,u} + 3 (2u-v) U_{,v} + l(l+1) U = 0,$$ $$\label{eq:uv:v}
2 (u^2-uv) V_{,vu} - 3 u V_{,u} - (2u-v) V_{,v} + l(l+1) V = 0,$$ $$\label{eq:uv:k}
2 (u^2-uv) K_{,vu} - u K_{,u} + (2u-v) K_{,v} - 2 K + l(l+1) K =
- 2 V - \frac{4uF}{u-v},$$ $$\label{eq:uv:ut}
u U_{,v} + u K_{,u} + \frac{2vF}{u-v} = 0,$$ $$\label{eq:uv:vt}
V_{,u} + K_{,v} - \frac{2F}{u-v} = 0.$$
This decoupled system of partial differential equations can be further simplified by exploiting continuous self-similarity of the background to separate spatial and scale variables. With this intent, we rewrite equations (\[eq:uv\]) in terms of the scaling coordinates (\[eq:coord:xs\])
\[eq:xs\] $$\label{eq:xs:f}
\frac{1}{2}\, (1-e^{-2x}) F_{,xx} + F_{,xs} + F_{,s} - 2 (1-e^{-2x}) F + l(l+1) F = 0,$$ $$\label{eq:xs:u}
\frac{1}{2}\, (1-e^{-2x}) U_{,xx} + U_{,xs} - 2 U_{,x} - U_{,s} + l(l+1) U = 0,$$ $$\label{eq:xs:v}
\frac{1}{2}\, (1-e^{-2x}) V_{,xx} + V_{,xs}+ 2 V_{,x} + 3 V_{,s} + l(l+1) V = 0,$$ $$\label{eq:xs:k}
\frac{1}{2}\, (1-e^{-2x}) K_{,xx} + K_{,xs} + K_{,s} - 2 K + l(l+1) K = - 2 V - 4 e^{-2x} F,$$ $$\label{eq:xs:ut}
U_{,x} - (1-e^{2x}) K_{,x} + 2 e^{2x} K_{,s} - 4 (1-e^{2x}) F = 0,$$ $$\label{eq:xs:vt}
K_{,x} - (1-e^{2x}) V_{,x} + 2 e^{2x} V_{,s} + 4 F = 0.$$
We decompose the perturbation amplitudes into modes that grow exponentially with the scale $s$ (which amounts to doing Laplace transform on them) $$\begin{aligned}
\label{eq:laplace}
F(x,s) &=& \sum_\kappa F_\kappa(x)\, e^{\kappa s}, \nonumber\\
U(x,s) &=& \sum_\kappa U_\kappa(x)\, e^{\kappa s}, \nonumber\\
V(x,s) &=& \sum_\kappa V_\kappa(x)\, e^{\kappa s}, \nonumber\\
K(x,s) &=& \sum_\kappa K_\kappa(x)\, e^{\kappa s}.\end{aligned}$$ The summation runs over the perturbation mode eigenvalues $\kappa$, which could, in general, be complex. Modes with ${\text{Re} }\,\kappa>0$ grow and are relevant for critical behavior, while modes with ${\text{Re} }\,\kappa<0$ decay and are irrelevant. The growing perturbation mode amplitudes vanish at $s=-\infty$, so the boundary condition at ${\cal I}^-$ is satisfied automatically. For clarity, the perturbation mode subscript $\kappa$ and the explicit summation over all modes will be suppressed from now on, so henceforth $F$, $U$, $V$, and $K$ will mean $F_\kappa$, $U_\kappa$, $V_\kappa$, and $K_\kappa$ for the mode with eigenvalue $\kappa$.
The decomposition (\[eq:laplace\]) converts the system of partial differential equations (\[eq:xs\]) into a system of ordinary differential equations, which is much easier to analyze:
\[eq:x\] $$\label{eq:x:f}
\frac{1}{2}\, (1-e^{-2x}) F'' + \kappa F' + \kappa F - 2 (1-e^{-2x}) F + l(l+1) F = 0,$$ $$\label{eq:x:u}
\frac{1}{2}\, (1-e^{-2x}) U'' + (\kappa-2) U' - \kappa U + l(l+1) U = 0,$$ $$\label{eq:x:v}
\frac{1}{2}\, (1-e^{-2x}) V'' + (\kappa+2) V' + 3 \kappa V + l(l+1) V = 0,$$ $$\label{eq:x:k}
\frac{1}{2}\, (1-e^{-2x}) K'' + \kappa K' + (\kappa-2) K + l(l+1) K = - 2 V - 4 e^{-2x} F,$$ $$\label{eq:x:ut}
U' - (1-e^{2x}) K' + 2 \kappa e^{2x} K - 4 (1-e^{2x}) F = 0,$$ $$\label{eq:x:vt}
K' - (1-e^{2x}) V' + 2 \kappa e^{2x} V + 4 F = 0.$$
The prime denotes a derivative with respect to spatial variable $x$. These equations can be converted into standard algebraic form by the change of variable $$\label{eq:coord:y}
y = e^{2x}, \hspace{1em} x = \frac{1}{2} \ln y,$$ so that the system (\[eq:x\]) becomes
\[eq:y\] $$\label{eq:y:f}
y(1-y) \ddot{\Phi} + [3 - (\kappa+3)y] \dot{\Phi} - [3\kappa/2 + l(l+1)/2] \Phi = 0,$$ $$\label{eq:y:u}
y(1-y) \ddot{U} + [1 - (\kappa-1)y] \dot{U} - [-\kappa/2 + l(l+1)/2] U = 0,$$ $$\label{eq:y:v}
y(1-y) \ddot{V} + [1 - (\kappa+3)y] \dot{V} - [3\kappa/2 + l(l+1)/2] V = 0,$$ $$\label{eq:y:k}
y(1-y) \ddot{K} + [1 - (\kappa+1)y] \dot{K} - [\kappa/2 - 1 + l(l+1)/2] K = 2 \Phi + V,$$ $$\label{eq:y:ut}
\dot{U} + (y-1) \dot{K} + \kappa K + 2 y \Phi - 2 \Phi = 0,$$ $$\label{eq:y:vt}
\dot{K} + (y-1) \dot{V} + \kappa V + 2 \Phi = 0.$$
The dot denotes a derivative with respect to $y$, and we redefined the scalar field perturbation amplitude as $F = y\Phi$ to cast the equations into standard table form. The boundary conditions (\[eq:pert:bc\]) are $$\begin{aligned}
\label{eq:y:bc}
&U=V=K=\Phi=0 \text{ at } y=1,& \nonumber\\
&U, V, K, y\Phi \text{ are bounded at } y=+\infty.&\end{aligned}$$ Imposed on the decoupled system of ordinary differential equations (\[eq:y\]), these boundary conditions give an eigenvalue problem for the perturbation spectrum $\kappa$.
Perturbation Spectrum {#sec:spectrum}
=====================
In the previous section we formulated an eigenvalue problem for the spectrum of non-spherical perturbations of the critical Roberts solution. We now proceed to solve it. Observe that equations (\[eq:y:f\]–\[eq:y:k\]) governing the dynamics of the perturbations are hypergeometric equations of the form $$\label{eq:hypergeom}
y(1-y) \ddot{X} + [c - (a+b+1)y] \dot{X} - ab X = 0.$$ Equation (\[eq:y:k\]) is not homogeneous, but we will deal with that shortly. The hypergeometric equation coefficients are different for equations describing the perturbations $\Phi$, $U$, $V$, and $K$; they are summarized in the table below. $$\label{eq:coeffs}
\setlength{\arraycolsep}{1em}
\renewcommand{\arraystretch}{1.4}
\begin{array}{c|ccc}
& c & a+b & ab\\ \hline
\Phi & 3 & \kappa+2 & \frac{3}{2} \kappa + \frac{1}{2} l(l+1)\\
U & 1 & \kappa-2 & -\frac{1}{2} \kappa + \frac{1}{2} l(l+1)\\
V & 1 & \kappa+2 & \frac{3}{2} \kappa + \frac{1}{2} l(l+1)\\
K & 1 & \kappa & \frac{1}{2} \kappa + \frac{1}{2} l(l+1)\\
\end{array}$$ Hypergeometric equations have been extensively studied; for complete description of their properties see, for example, . Hypergeometric equation (\[eq:hypergeom\]) has three singular points at $y=0,1,\infty$, and its general solution is a linear combination of any two different solutions from the set $$\begin{aligned}
\label{eq:solns}
X_1 &=& {\cal F}(a, b; a+b+1-c; 1-y),\nonumber\\
X_2 &=& (1-y)^{c-a-b} {\cal F}(c-a, c-b; c+1-a-b; 1-y),\nonumber\\
X_3 &=& (-y)^{-a} {\cal F}(a, a+1-c; a+1-b; y^{-1}),\nonumber\\
X_4 &=& (-y)^{-b} {\cal F}(b+1-c, b; b+1-a; y^{-1}),\end{aligned}$$ where ${\cal F}(a, b; c; y)$ is the hypergeometric function, which is regular at $y=0$ and has ${\cal F}(a, b; c; 0) = 1$. Any three of the functions (\[eq:solns\]) are linearly dependent with constant coefficients. In particular, $$\begin{aligned}
\label{eq:connection}
X_2 &=&
\frac{\Gamma(c+1-a-b) \Gamma(b-a)}{\Gamma(1-a) \Gamma(c-a)}\, e^{-i\pi(c-b)} X_3 +\nonumber\\
&&\frac{\Gamma(c+1-a-b) \Gamma(a-b)}{\Gamma(1-b) \Gamma(c-b)}\, e^{-i\pi(c-a)} X_4.\end{aligned}$$ The functions $X_1$, $X_2$ are appropriate for discussing the behavior of solution near $y=1$, while $X_3$, $X_4$ give the behavior at infinity $$\begin{aligned}
\label{eq:asymptotics}
X_1 = 1,\ X_2 = (1-y)^{c-a-b} &\text{ near }& y=1, \nonumber\\
X_3 = (-y)^{-a},\ X_4 = (-y)^{-b} &\text{ near }& y=\infty.\end{aligned}$$
As we said before, imposing the boundary conditions (\[eq:y:bc\]) on solutions of equations (\[eq:y\]) leads to a perturbation spectrum. We will now investigate what restrictions the boundary conditions place on the hypergeometric equation coefficients. The vanishing of perturbation amplitudes at $y=1$ rules out $X_1$ as a component of the solution and requires that ${\text{Re} }(c-a-b) > 0$ to make $X_2$ go to zero. The solution $X_2$ has non-zero content of both $X_3$ and $X_4$ by virtue of (\[eq:connection\]), hence for it to be bounded at infinity, both ${\text{Re} }\,a$ and ${\text{Re} }\,b$ must be positive to guarantee convergence of $X_3$ and $X_4$. So, unless there is degeneracy, the boundary conditions translate to the following conditions on the hypergeometric equation coefficients:
\[eq:bc:coeff\] $$\begin{aligned}
&{\text{Re} }(c-a-b) > 0,& \label{eq:bc:coeff:1}\\
&{\text{Re} }\, a, {\text{Re} }\, b > 0.& \label{eq:bc:coeff:infty}\end{aligned}$$
We are now ready to take on system (\[eq:y\]). Take a look at equation (\[eq:y:v\]) for $V$. Condition (\[eq:bc:coeff:1\]) for it is ${\text{Re} }\,\kappa<-1$, i.e. there are no growing $V$ modes! With the amplitude of relevant $V$ perturbation modes being zero, the constraints (\[eq:y:ut\]) and (\[eq:y:vt\]) become $$\label{eq:constraints}
K = - \frac{\dot{U}}{\kappa}, \hspace{1em}
\Phi = \frac{\ddot{U}}{2\kappa},$$ and right hand side of equation (\[eq:y:k\]) can be absorbed by the left hand side, making the equation for $K$ homogeneous (with $c=2$). Indeed equations (\[eq:y:k\]) and (\[eq:y:f\]) for $K$ and $\Phi$ are just derivatives of equation (\[eq:y:u\]) for $U$ $$\label{eq:master}
y(1-y) \ddot{U} + [1 - (\kappa-1)y] \dot{U} - [-\kappa/2 + l(l+1)/2] U = 0,$$ which is the homogeneous hypergeometric equation with coefficients $$\label{eq:master:coeff}
c = 1, \hspace{1em}
a+b = \kappa-2, \hspace{1em}
ab = -\frac{1}{2} \kappa + \frac{1}{2} l(l+1).$$ Imposing the boundary condition at $y=1$ for the solution of the above equation and its derivatives, which behave like $$\left.\begin{array}{r}
U \propto (1-y)^{3-\kappa}\\
K \propto (1-y)^{2-\kappa}\\
\Phi \propto (1-y)^{1-\kappa}\\
\end{array}\right\} \text{ near } y=1,$$ produces restriction on the non-spherical mode eigenvalue $${\text{Re} }\, \kappa < 1,$$ which is the strongest of restrictions (\[eq:bc:coeff:1\]) for equations for $U$, $K$, and $\Phi$. But then $${\text{Re} }\, a + {\text{Re} }\, b = {\text{Re} }\, \kappa - 2 < -1,$$ and hence ${\text{Re} }\,a$ and ${\text{Re} }\,b$ can not be both positive, and so the boundary condition at infinity can not be satisfied. A more careful investigation of degenerate cases of relation (\[eq:connection\]) shows that the contradiction between boundary conditions at $y=1$ and infinity still persists if $V=0$. It can only be resolved by the trivial solution $U=K=\Phi=0$. Thus we have shown that there are no growing non-spherical perturbation modes around the critical Roberts solution.
In fact, an even stronger statement is true. The contradiction between boundary conditions at $y=1$ and infinity can not be resolved by a non-trivial solution so long as $V=0$, i.e. so long as ${\text{Re} }\,\kappa \ge -1$. Hence non-trivial non-spherical perturbation modes of critical Roberts solution must decay faster than $e^{-s}$.
Conclusion {#sec:conclusion}
==========
In this paper we used the gauge-invariant perturbation formalism to explore the critical behavior in the gravitational collapse of a massless scalar field. Perturbing around a continuously self-similar critical solution (the Roberts solution), we obtained an eigenvalue problem for the spectrum of perturbations. The remarkable feature of this model of critical scalar field collapse is that it allows an exact analytical treatment of the perturbations as well as of the critical solution, due to the highly symmetric background.
An exact analysis of the perturbation eigenvalue problem reveals that there are no growing non-spherical perturbation modes. However, there are growing spherical perturbation modes. Their spectrum is continuous and occupies a big chunk of the complex plane [@Frolov:97]. In view of these findings, the following picture of dynamics of scalar field evolution near self-similarity emerges: As we evolve generic initial data which is sufficiently close to the critical Roberts solution, non-spherical modes decay and the solution approaches the spherically symmetric one. Asymmetry of the initial data does not play a role in the collapse. The growing spherical modes, on the other hand, drive the solution farther away from the continuously self-similar one. In this sense, the critical Roberts solution is an intermediate attractor for non-spherical initial data.
An interesting question, which is not answered by perturbative calculations, is the further fate of the scalar field evolution as it gets away from the Roberts solution. In all likelihood, it evolves towards the discretely self-similar Choptuik solution, which is a local attractor of lower codimension (one), as the continuous self-similarity is broken by oscillatory growing modes. After staying near the Choptuik solution for a while, the scalar field will eventually disperse or settle into a black hole, with these final states being global attractors in the phase space of field configurations. This evolution from attractor to attractor in phase space is somewhat analogues to a ball rolling down the stairs, going from a step to a lower step, until it reaches the bottom.
The results of this paper shed some light at the complicated problem of critical collapse of generic initial data from the analytical viewpoint, confirming the hypothesis that critical phenomena are not restricted to spherical symmetry. Investigation of the fate of a scalar field solution as it breaks away from continuous self-similarity, as outlined above, will further our understanding of the dynamics of scalar field collapse, and presents an interesting (and challenging) analytical problem. Numerical simulations might also help to establish a clearer picture of near-critical scalar field evolution.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Killam Trust.
Perturbations in Longitudinal Gauge {#sec:long}
===================================
In this appendix we collect expressions for components of the perturbed Einstein-scalar equations calculated in longitudinal gauge ($h_A=G=0$). The perturbed metric in longitudinal gauge is $$ds^2 =
h_{uu} Y du^2 - 2(1 - h_{uv} Y) du\, dv + h_{vv} Y dv^2
+ (1 + K Y) r^2 d\Omega^2,$$ and the perturbed scalar field is $$\phi = \frac{1}{2} \left[1 - \frac{v}{u}\right] + F Y.$$ The Einstein equations for scalar field are equivalent to the vanishing of the tensor $E_{\mu\nu} = R_{\mu\nu} - 2 \phi_{,\mu} \phi_{,\nu}$. Its non-trivial components, calculated to the first order in the perturbation amplitude using the above metric and scalar field, are
$$\begin{aligned}
&& E_{uu} =
\frac{1}{2} \Big[
- 2 (u^2-uv) K_{,uu} + u h_{uu,u} \nonumber\\ && \mbox{\hspace{4.3em}}
+ (2u-v) (h_{uu,v} - 2 h_{uv,u} - 2 K_{,u}) \nonumber\\ && \mbox{\hspace{4.3em}}
- 4 v F_{,u} + l(l+1) h_{uu}
\Big]\, \frac{Y}{u^2-uv},\end{aligned}$$
$$\begin{aligned}
&& E_{uv} =
- \frac{1}{2} \Big[
(u^2-uv) (h_{uu,vv} - 2 h_{uv,vu} + h_{vv,uu} + 2 K_{,vu}) \nonumber\\ && \mbox{\hspace{4.3em}}
- u h_{uu,v} + (2u-v) (h_{vv,u} + K_{,v}) - u K_{,u} \nonumber\\ && \mbox{\hspace{4.3em}}
+ 2 v F_{,v} - 2 u F_{,u} - l(l+1) h_{uv}
\Big]\, \frac{Y}{u^2-uv},\end{aligned}$$
$$\begin{aligned}
&& E_{vv} =
\frac{1}{2} \Big[
- 2 (u^2-uv) K_{,vv} + 2 u h_{uv,v} \nonumber\\ && \mbox{\hspace{4.3em}}
- u h_{vv,u} - (2u-v) h_{vv,v} + 2 u K_{,v} \nonumber\\ && \mbox{\hspace{4.3em}}
+ 4 u F_{,v}
+ l(l+1) h_{vv}
\Big]\, \frac{Y}{u^2-uv},\end{aligned}$$
$$\begin{aligned}
&& E_{u\theta} =
- \frac{1}{2} \Big[
(u^2-uv) (h_{uu,v} - h_{uv,u} + K_{,u}) \nonumber\\ && \mbox{\hspace{4.3em}}
+ (2u-v) h_{uv} + 2 v F
\Big]\, \frac{Y_{,\theta}}{u^2-uv},\end{aligned}$$
$$\begin{aligned}
&& E_{u\varphi} =
- \frac{1}{2} \Big[
(u^2-uv) (h_{uu,v} - h_{uv,u} + K_{,u}) \nonumber\\ && \mbox{\hspace{4.3em}}
+ (2u-v) h_{uv} + 2 v F
\Big]\, \frac{Y_{,\varphi}}{u^2-uv},\end{aligned}$$
$$\begin{aligned}
&& E_{v\theta} =
- \frac{1}{2} \Big[
(u-v) (- h_{uv,v} + h_{vv,u} + K_{,v}) \nonumber\\ && \mbox{\hspace{4.3em}}
- h_{uv} - 2 F
\Big]\, \frac{Y_{,\theta}}{u-v},\end{aligned}$$
$$\begin{aligned}
&& E_{v\varphi} =
- \frac{1}{2} \Big[
(u-v) (- h_{uv,v} + h_{vv,u} + K_{,v}) \nonumber\\ && \mbox{\hspace{4.3em}}
- h_{uv} - 2 F
\Big]\, \frac{Y_{,\varphi}}{u-v},\end{aligned}$$
$$\begin{aligned}
&& E_{\theta\theta} =
\frac{1}{2} \Big[
2 (u^2-uv) K_{,vu} - u h_{uu,v} \nonumber\\ && \mbox{\hspace{4.3em}}
+ (2u-v) (h_{vv,u} + 2 K_{,v}) - 2 u K_{,u} \nonumber\\ && \mbox{\hspace{4.3em}}
- 2 h_{uv} + 2 h_{vv} - 2 K + l(l+1) K
\Big]\, Y \nonumber\\ && \mbox{\hspace{3em}}
+ h_{uv} Y_{,\theta\theta},\end{aligned}$$
$$E_{\varphi\varphi} = \sin^2\theta\, E_{\theta\theta},$$
$$E_{\theta\varphi} =
h_{uv} (Y_{,\theta\varphi} - \cot\theta\, Y_{,\varphi}).$$
The scalar field equation requires the vanishing of $\Box\phi$, which, calculated to first order using the above metric and scalar field, is $$\begin{aligned}
&& \Box\phi =
\frac{1}{2} \Big[
- 4 (u^2-uv) F_{,vu} + u h_{uu,v} - v h_{vv,u} \nonumber\\ && \mbox{\hspace{4.3em}}
+ u K_{,u} - v K_{,v} + 2 u F_{,u} - 2 (2u-v) F_{,v} \nonumber\\ && \mbox{\hspace{4.3em}}
- 2 l(l+1) F
\Big]\, \frac{Y}{u^2-uv}.\end{aligned}$$
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-------------------- -----------------------
=4 10 =4 10
(null coordinates) (scaling coordinates)
-------------------- -----------------------
[^1]: Email: `andrei@phys.ualberta.ca`
|
---
abstract: |
Given any finite index quadrilateral $(N, P, Q, M)$ of $II_1$-factors, the notions of interior and exterior angles between $P$ and $Q$ were introduced in [@BDLR2017]. We determine the possible values of these angles when the quadrilateral is irreducible and the subfactors $N \subset P$ and $N \subset Q$ are both regular in terms of the cardinalities of the Weyl groups of the intermediate subfactors. For a more general quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained, which enforce some restrictions on the index of $N
\subset Q$ in terms of that of $N \subset P$.
address:
- 'Chennai Mathematical Institute, Chennai, INDIA'
- 'School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA'
author:
- Keshab Chandra Bakshi
- Ved Prakash Gupta
title: 'A note on irreducible quadrilaterals of $II_1$ factors'
---
[^1]
Introduction
============
A quadrilateral is a quadruple $(N, P, Q, M)$ of $II_1$-factors such that $N \subset P, Q \subset M$, $N = P \wedge Q$, $M = P \vee Q$ and $[M:N] <\infty$; it is called irreducible if $N \subset M$ is irreducible. The Weyl group of a finite index $II_1$-subfactor $N \subset M$ is the quotient group $G:=\mathcal{N}_M(N)/\mathcal{U}(N)$. This article concentrates mainly on the analysis of such quadrilaterals from the perspectives of (a) calculating the interior and exterior angles between $P$ and $Q$ as was introduced in [@BDLR2017], (b) understanding the Weyl group of $N \subset M$ in terms of those of $N \subset P$ and $N \subset Q$, and (c) establishing a relationship between the above two aspects.
Unlike the notion of set of angles by Sano and Watatani ([@SW]), the interior and exterior angles are both single entities and are seemingly more calculable, as we show in by making some explicit calculations. As an important application of the notion of interior angle, the authors in [@BDLR2017] were able to improve a result of Longo [@Lon] by providing a better bound for the number of intermediate subfactors of a given subfactor.
A natural question that struck us, after the appearance of [@BDLR2017], was to determine the possible set of values that interior and exterior angles can attain. This article is devoted to this theme. In general, it looks like a tough nut to crack. However, in the irreducible set up, we see that these angles take definitive values.
In , we discuss various generalities and formulae related to the interior and exterior angles and employ them to compute angles between two intermediate subfators associated with a quadruple of crossed product algebras.
In , our main focus is on irreducible quadrilaterals $(N, P, Q, M)$ for which $N\subset P$ and $N\subset Q$ are both regular. Jones, in [@Jon], had asked whether an irreducible regular subfactor is always a group subfactor. Making use of a theorem of Sutherland [@Sut] on vanishing of cohomologies, Popa [@PiPo2] and Kosaki [@Kos] (for properly infinite case) answered Jones’ question in the affirmative, which was announced earlier for the hyperfinite case by Ocnenanu in 1986. Later, Hong gave an explicit realization of the same in [@Hong]. Using Hong’s technique, we deduce (in ) that an irreducible quadrilateral $(N, P, Q,
M)$ with regular $N \subset P$ and $N \subset Q$ can be realized as a quadrilateral of crossed product algebras through outer actions of Weyl groups. Using this realization and the calculations of , we provide a direct relationship between the interior and exterior angles between $P$ and $Q$ and the Weyl groups of $N \subset P$ and $N \subset Q$ in:
[****]{}
*Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $N\subset P$ and $N\subset Q$ are both regular. Then, $\alpha(P,Q)=\pi/2$, i.e., $(N, P, Q, M)$ is a commuting square, and $$\cos \beta(P,Q)=\displaystyle \frac{\frac{|G|}{\lvert H\rvert
\lvert K\rvert}-1}{\sqrt{[G:H]-1}\sqrt{[G:K]-1}},$$ where $H, K$ and $G$ denote the Weyl groups of $N \subset P$, $N
\subset Q$ and $N \subset M$, respectively.*
In particular, $(N, P, Q, M)$ is a cocommuting square if and only if $G = HK$.
dwells around the main theme of this article, viz., to determine the possible values of the interior and exterior angles. We first derive expressions for the angles in terms of the common norm $\lambda$ of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple (in ). Then, in the irreducible setup, we exploit these expressions to obtain some definitive values for angles by making use of above relationship between angles and Weyl groups, a theorem of Popa [@pop] wherein he determines the possible values taken by the set $\Lambda(M, N)$ of relative dimensions of projections, and the values attained by the polynomials $P_n(x), n \geq 0$, as introduced by Jones in [@Jon]. The results that we prove are:
[****]{} [*Let $(N,P,Q,M)$ be a quadruple with $N\subset M$ irreducible and let $0<t\leq 1/2$ be such that $t(1-t)=\tau$. If $r/\lambda \geq t$, then, $$\cos(\alpha(P,Q)) \leq
\frac{[P:N][Q:N](1-t)-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and $$\cos(\beta(P,Q))\leq \displaystyle
\frac{\frac{1}{t}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}.$$ And, if $r/\lambda < t$, then, $$\cos(\alpha(P,Q)) =
\displaystyle\frac{[P:N][Q:N]\frac{P_k(\tau)}{P_{k-1}(\tau)}-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and $$\cos(\beta(P,Q))=\displaystyle \frac{\frac{P_k(\tau)}{\tau
P_{k-1}(\tau)}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}$$ for some $k\geq
0,$ where $r:= \frac{[Q:N]}{[M:P]}$.*]{}
[****]{} [*Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $N\subset P$ and $N\subset Q$ are both regular and suppose $[P:N]=2$. Then, $\cos(\beta(P,Q))=
\displaystyle \frac{P_2(m/2)}{\sqrt{P_2(\delta^2/2)P_3(m/2)}}$, where $m = [M:Q] \in{\mathbb N}$ and, as usual $\delta:=\sqrt{[M:N]}.$*]{}
As a ‘geometric’ consequence, in , we see that if both $N \subset P$ and $N \subset Q$ have index $2$, then the exterior angle $\beta(P,Q) > \pi/3$.
Finally, while analyzing a quadrilateral intuitively as a picture in the plane (), loosely speaking, we realize in that the angles impose some sort of rigidity on the lengths of its sides. This could be inferred as a direct consequence of certain bounds on interior and exterior angles that we obtain in:
[****]{} [*Let $(N,P,Q,M)$ be a finite index irreducible quadruple such that $N\subset P$ is regular. Then, $$\cos\big(\alpha(P,Q)\big)\leq
\Bigg(\sqrt{\frac{[P:N]-1}{[Q:N]-1}}\Bigg)$$ and $$\cos(\beta(P,Q))\leq \Bigg(\sqrt{\frac{[P:N]-r}{[Q:N]-r}}\Bigg).$$* ]{}
The flow of the article revolves around the results mentioned above, more or less in the same order.
Interior and Exterior angles between intermediate subfactors {#angle-generalities}
============================================================
\[angles\] In this section, we first recall the notions of interior and exterior angles between intermediate subfactors of a given subfactor as introducted by Bakshi [*et al.*]{} in [@BDLR2017] and some useful formulae related to them. This will be followed by some further generalities and explicit calculations related to these angles.
In this article, we will be dealing only with subfactors and quadruples of type $II_1$ with finite Jones’ index. Given any such quadruple $$\begin{matrix}
Q &\subset & M \cr \cup &\ &\cup\cr N &\subset & P,
\end{matrix}$$ consider the basic constructions $N\subset M \subset M_1$, $P \subset M \subset P_1$ and $Q \subset M \subset Q_1$. As is standard, we denote by $e_1$ the Jones projection $e^M_N$. It is easily seen that, as $II_1$-factors acting on $L^2(M)$, both $ P_1$ and $Q_1$ are contained in $ M_1$. In particular, if $e_P: L^2(M) {\rightarrow}L^2(P)$ denotes the orthogonal projection, then $e_P \in M_1$. Likewise, $e_Q \in M_1$. Thus, we naturally obtain a dual quadruple $$\begin{matrix} P_1 &\subset & M_1
\cr \cup &\ &\cup\cr M &\subset & Q_1.
\end{matrix}$$
[@BDLR2017]\[alpha-angle\]\[beta-angle\] Let $P$ and $Q$ be two intermediate subfactors of a subfactor $N
\subset M$. Then, the interior angle $\alpha^N_M(P,Q)$ between $P$ and $Q$ is given by $$\alpha^N_M(P, Q) = \cos^{-1} {\langle v_P,v_Q\rangle}_2,$$ where $v_P := \frac{e_P-e_1}{{\lVert e_P-e_1\rVert}_2}$, ${\langle x, y\rangle}_2 := tr(y^*x)$ and ${\lVert x\rVert}_2 := (tr(x^*x))^{1/2}$. And, the exterior angle between $P$ and $Q$ is given by $\beta^N_M(P, Q) = \alpha^M_{M_1}(P_1, Q_1)$.
We will avoid being pedantic and often drop the superscript $N$ and the subscript $M$ when the subfactor $N \subset M$ is clear from the context.
The following useful relationship between $\alpha(P,Q)$ and $\beta(P,Q)$ was mentioned in [@BDLR2017], following Definition 3.6, without any proof. For the sake of completeness, we include a proof (though, only for the extremal case, which will be enough for our requirements).
\[angle-duality\] For an extremal subfactor $N \subset M$ with intermediate subfactors $P$ and $Q$, we have $$\alpha^N_M (P, Q) = \beta^M_{M_1} (P_1, Q_1).$$
We have a tower $$N \subset P \subset M \subset P_1 \subset M_1 \subset P_2 \subset M_2$$ where $P_i \subset M_i \subset P_{i+1} = \la M_i, e_{P_i} \ra$ is a basic construction with Jones projection $e_{P_i} = e_{0, i+1}:
L^2(M_i) {\rightarrow}L^2(P_i)$, and $P_0:=P$ - see [@BL $\S$ 3]. Likewise, we have another tower $$N \subset Q \subset M \subset Q_1 = \la M, e_{Q}\ra \subset M_1 \subset Q_2 = \la M_1, {e_{Q_1}}\ra \subset M_2 .$$ From [@BL Lemma 4.2], we have $ e_{P_2} =
\vcenter{\psfrag{e}{$e_P$}\includegraphics[scale=0.4]{ep2.eps}.}$ We have a similar figure for $e_{Q_2}$ with respect to $e_Q$. From this pictorial description, it is readily seen through pictures that $$tr(e_P
e_Q) = tr (e_{P_2} e_{Q_2}), tr(e_P) = tr(e_{P_2})\ \text{and}\ tr (e_Q) =
tr (e_{Q_2}).$$ From , we have $
\cos \big(\alpha^N_M (P, Q) \big) = \frac{tr(e_P e_Q) - \tau}{\sqrt{\tau_P - \tau} \sqrt{\tau_Q - \tau}}
$ and, similarly, $
\cos \big(\alpha^{M_1}_{M_2} (P_2, Q_2) \big) = \frac{tr(e_{P_2} e_{Q_2}) - \tau}{\sqrt{\tau_{P_2} - \tau} \sqrt{\tau_{Q_2} - \tau}}.
$ Finally, employing the above equalities obtained through pictures, we obtain $$\cos \big( \beta^M_{M_1}(P_1, Q_1)\big) = \cos \big(\alpha^{M_1}_{M_2} (P_2, Q_2) \big) = \cos \big(\alpha^N_M (P, Q) \big),$$ as was desired.
![Intuitive picture of a quadrilateral along with its dual[]{data-label="fig-2"}](quad.eps)
Some useful formulae related to interior and exterior angles.
-------------------------------------------------------------
We first list some plausible facts from [@BDLR2017] that make computations of $\alpha(P,Q)$ and $\beta(P,Q)$ more amenable.
[@BDLR2017]\[Thm: alpha-beta\] For a quadruple $(N,P,Q,M)$, let $\tau_P=tr(e_P)$ and $\tau_Q=tr(e_Q)$. Then, the interior angle $\alpha (P,Q)$ satisfies $$\begin{aligned}
\label{Equ:alpha}
\cos \alpha(P, Q)
&=\frac{tr(e_Pe_Q)-\tau}{\sqrt{\tau_P-\tau}\sqrt{\tau_Q-\tau}},\end{aligned}$$ which, then, yields that $$\label{alpha-eqn-basis}
\cos \big(\alpha (P, Q) \big) = \frac{\sum_{i,j} tr_M\big( E^M_N
(\lambda_i^* \mu_j) \mu_j^* \lambda_i\big) -1}{\sqrt{[P:N]
-1}\sqrt{[Q:N] -1}}$$ for any two Pimsner-Popa bases $\{\lambda_i\}$ and $\{\mu_j\}$ of $P/N$ and $Q/N$, respectively. And, if the quadruple is extremal, i.e., $N
\subset M$ is extremal, then the exterior angle $\beta(P,Q)$ satisfies $$\begin{aligned}
\label{Equ:beta}
\cos \beta(P, Q) &=
\frac{tr(e_Pe_Q)-\tau_P\tau_Q}{\sqrt{\tau_P-\tau_P^2}\sqrt{\tau_Q-\tau_Q^2}}.\end{aligned}$$
The following useful expression for $tr(e_P e_Q)$ is quite evident from and ; the details can be readily extracted from the proof of [@BDLR2017 Proposition 2.14].
\[tr-epeq\] Let $N \subset M$ be a subfactor and $P$ and $Q$ be two intermediate subfactors. Then, $$tr_{M_1}(e_P e_Q) = \tau \sum_{i,j}\|E_N(\lambda_i^* \mu_j)\|_2^2$$ for any two Pimsner-Popa bases $\{\lambda_i\}$ and $\{\mu_j\}$ of $P/N$ and $Q/N$, respectively, where $\tau := [M:N]^{-1}$.
Recall that two subfactors $N \subset M$ and $
\mathcal{N}\subset\mathcal{M}$ are said to be isomorphic (denoted as $(N\subset M)\cong (\mathcal{N}\subset \mathcal{M})$) if there exists a $*$-isomorphism $\varphi$ from $M$ onto $\mathcal{M}$ such that $\varphi(N)=\mathcal{N}.$ Likewise, two quadruples $(N, P, Q, M)$ and $(\mathcal{N}, \mathcal{P}, \mathcal{Q}, \mathcal{M}) $ are said to be isomorphic if there is an isomorphism $\varphi$ between the subfactors $N \subset M$ and $ \mathcal{N}\subset\mathcal{M}$ such that $\varphi
(P) = \mathcal{P}$ and $\varphi (P) = \mathcal{P}$.
\[isomorphism\] Since Pimsner-Popa bases are preserved by isomorphisms of subfactors, in view of and , we observe that an isomorphism between two quadruples preserves interior and exterior angles.
Recall that a quadruple $(N, P, Q, M)$ is said to be a commuting square if $e_P e_Q = e_1 = e_Q e_P$. It is said to be a cocommuting square if the dual quadruple $(M, P_1, Q_1, M)$ is a commuting square. It is said to be non-degenerate (resp., irreducible) if $\overline{\text{span}PQ} = M$ (resp., $N'\cap M= {\mathbb C}$). Further, it is said to be a parallelogram if $\tau_P \tau_Q = \tau$ or, equivalently, if $[M: P ] = [Q:N]$ or $[M:Q] = [P:N]$. And, a quadruple $(N,P,Q,M)$ is said to be a quadrilateral if $P\vee Q=M$ and $P\wedge Q=N.$
\[commuting-cocommuting\] Commuting and cocommuting conditions have very natural interpretations in terms of above angles, viz., a quadruple $(N, P, Q,
M)$ is a commuting (resp., co-commuting) square if and only if $\alpha(P,Q)$ (resp., $\beta(P,Q)$) equals $\pi/2$ - see [@BDLR2017 $\S
2$].
Computation of angles for quadruples of crossed product algebras {#examples}
----------------------------------------------------------------
\[ex-1\] Let $G$ be a finite group acting outerly on a $II_1$-factor $S$. Let $H, K $ and $L$ be subgroups of $G$ such that $H \subseteq K \cap L$ and $K$ and $L$ are non-trivial. Consider the quadruple $(N = S
\rtimes H, P = S \rtimes K, Q = S \rtimes L, M= S \rtimes G )$. Then, $$tr(e_P e_Q) = \frac{ |K \cap L|}{|G|},$$ $$\cos\Big( \alpha(P, Q)\Big) = \frac{|K \cap L| -|H|}{\sqrt{|K|-1} \sqrt{|L|-1}}$$ and $$\cos\Big( \beta(P, Q)\Big) = \frac{\frac{|G|}{|KL|} - 1}{\sqrt{ [G:K]- 1} \sqrt{[G:L] -1}}.$$ In particular, as is well known, $(N,P,Q,M)$ is a commuting (resp., cocommuting) square if and only if $K \cap L = H$ (resp., $G = KL$).
Note that if $\alpha : G {\rightarrow}\text{Aut}(S)$ denotes the action of $G$ on $S$, then there is a unitary representation $G \ni t \mapsto
u_t \in B(L^2(S))$, such that $u_t(x\Omega) = \alpha_t(x) \Omega$ for all $x \in S$ - see [@JS $\S$ A.4].
Fix left coset representatives $\{k_i : 1 \leq i \leq [K : H]\}$ and $
\{l_j : 1 \leq j \leq [L : H]\}$ of $H$ in $K$ and $L$, respectively. Since $E_N(\sum_g x_g u_g) = \sum_h x_h u_h$, it follows that $ \{k_i: 1 \leq i \leq [K:H]\}$ and $\{l_j : 1 \leq j \leq
[L:H]\}$ are (right) orthonormal bases for $P/N$ and $Q/N$, respectively. So, by , we obtain $$\begin{aligned}
{\mathrm{tr}}(e_P e_Q) & = & [G:H]^{-1} \sum_{i,j}tr_{M} \Big(
E_N\big( u_{k_i}^{-1} u_{l_j}\big)u_{l_j}^{-1} u_{k_i} \Big) \\
& = & [G:H]^{-1} \, \sum_{ \{ i,j : k_i^{-1}l_j \in H
\}}tr_{M} \Big( u_{k_i}^{-1} u_{l_j} u_{l_j}^{-1} u_{k_i} \Big)\quad \Big(\text{since }
E_N\big(\sum_{g\in G}x_g u_g\big) = \sum_{h\in H} x_h u_h \Big)\\
& = &[G:H]^{-1} |\{(i, j) : l_j \in k_iH\}|\\
& = &[G:H]^{-1}\,|\{ (i, j) : k_iH \cap l_jH \neq
\emptyset \}|;\end{aligned}$$ and note that the map $$\{(i, j) : k_i H \cap l_j H \neq \emptyset\} \ni (i, j) \mapsto k_i H = l_j H \in (K \cap L)/H$$ is a natural bijection; so that, $ tr(e_P e_Q) = \frac{ |K \cap
L|}{|G|}$. Then, from , we immediately obtain $$\cos\Big( \alpha(P, Q)\Big) = \frac{|K \cap L| -|H|}{\sqrt{|K|-1} \sqrt{|L|-1}}$$ and, from , through an elementary simplification, we deduce that $$\cos\Big( \beta(P, Q)\Big) = \frac{\frac{|G|}{|KL|} - 1}{\sqrt{ [G:K]- 1} \sqrt{[G:L] -1}}.$$ The commuting and cocommuting conditions follow from .
\[fixed-angle\] Let $K, L, G$ and $S$ be as in . Consider the quadruple $(N= S^G, P = S^K, Q = S^L, M = S)$. Then, $$\cos (\alpha (P, Q)) = \frac{ \frac{| G|}{|KL|}-1}{\sqrt{{[G:K]}-1}\sqrt{{[G:L]}-1}}$$ and $$\cos (\beta(P,Q)) = \frac{|K \cap L| - 1}{\sqrt{|K| -1}\sqrt{|L| -1}}.$$ In particular, $(N, P , Q , M )$ is a commuting square if and only if $K L = G$. And, it is a cocommuting square if and only if $K \cap L = \{ e\}$.
Since $S^G \subset S$ is extremal, by , we have $$\alpha^N_M (P, Q) = \beta^M_{M_1} (P_1, Q_1).$$ Outhere, we have $M = S, P_1 = S \rtimes K$, $Q_1= S \rtimes L$ and $M_1 = S \rtimes G$; so, by (taking $H$ to be the trivial subgroup), we obtain $$\cos \big(\alpha^N_M (S^K, S^L)\big) = \cos \big(\beta^M_{M_1} (S
\rtimes K, S \rtimes L)\big) = \frac{ \frac{|
G|}{|KL|}-1}{\sqrt{{[G:K]}-1}\sqrt{{[G:L]}-1}} .$$ On the other hand, by definition, we have $\beta^N_M(P,Q) =
\alpha^M_{M_1}(P_1, Q_1)$. Hence, by , we obtain $$\cos (\beta(P,Q)) = \frac{|K \cap L| - 1}{\sqrt{|K| -1}\sqrt{|L| -1}}.$$
It was shown in [@BDLR2017 $\S$ 5] that the notion of Sano-Watatani’s set of angles does not agree with the notion of interior angle. Using , we add to that list and show that the Sano-Watatani’s set of angles and the interior angle may not be equal even if the former is a singleton.
Consider the quadruple $(N= R^G, P = R^H, Q = R^K, M = R)$ with the assumption that $H \cap K =\{e\}$, $|H\backslash G/H| = 2$, $H$ and $K$ are both non-trivial subgroups. Then, the Sano-Watatani’s set of angles $
\mathrm{Ang}_M(P, Q)$ is a singleton and $ \{\alpha (P, Q)\} \neq
\mathrm{Ang}_M(P, Q).$
From [@SW Lemma 5.3 and Proposition 5.2], we have $\mathrm{Ang}_M(P, Q)$ is a singleton, namely, $$\mathrm{Ang}_M(P, Q) = \left\{\cos^{-1}
\left(\frac{|G| - |H||K|}{|K|(|G| -|H|)}\right)^{1/2}\right\}.$$ And, by , we have $$\cos\big(\alpha(P, Q)\big) = \frac{ \frac{|
G|}{|HK|}-1}{\sqrt{{[G:H]}-1}\sqrt{{[G:K]}-1}} = \frac{ \frac{|
G|}{|H||K|}-1}{\sqrt{|G|/|H|-1}\sqrt{|G| |K|-1}},$$ where the second equality follows because $H \cap K = \{e\}$ gives $|HK| =|H||K|$. Thus, $
\{\alpha (P, Q)\} =
\mathrm{Ang}_M(P, Q)
$ if and only if $$\label{equality}
\left(\frac{|G| - |H||K|}{|K|(|G| -|H|)}\right)^{1/2} = \frac{ \frac{|
G|}{|H||K|}-1}{\sqrt{|G|/|H|-1}\sqrt{|G| |K|-1}}.$$ Note that $RHS = \left(\frac{|G| - |H||K|}{(|G| -|H|)}\right)^{1/2}
\cdot \left(\frac{|G| - |H||K|}{(|G| -|K|)
|H||K|}\right)^{1/2}$. Hence (\[equality\]) is true if and only if $$\frac{1}{|K|} = \frac{|G| - |H||K|}{(|G| -|K|)
|H||K|},$$ which is then true if and only if $|H| = 1$, which is not true since $H$ is not the trivial subgroup.
We conclude this subsection by deducing the following well known fact.
Let $N \subset M$ be a subfactor and $G$ be a finite group acting outerly (through $\alpha$) on $M$ . Then, $(N, N\rtimes G,M, M\rtimes G)$ is a commuting square.
Note that $E^{M\rtimes G}_N\big(\sum_g a_gu_g\big)= E^M_N(a_e).$ Indeed, for any $b\in N$, we have $$tr\Big(\big(\sum a_gu_g\big)b\Big)= tr\Big(\sum a_g{\alpha}_g(b)u_g\Big)=tr(a_eb),$$ and, on the other hand, $tr\big(E^M_N(a_e)b\big)=tr\big(E^M_N(a_eb)\big)=tr(a_eb).$
Let $\{\lambda_i\}$ be a (right) basis for $M/N$. Then, from , we obtain $$\cos\Big(\alpha(N \rtimes G, M)\Big)= \frac{\sum_{i,g}
tr\big(E_N({\lambda}^*_iu_g) u_g^*
{\lambda}_i\big)-1}{\sqrt{[M:N]-1}\sqrt{|G|-1}}= \frac{tr\big(\sum_i
E_N({\lambda}^*_i){\lambda}_i\big)-1}{\sqrt{[M:N]-1}\sqrt{|G|-1}} = 0,$$ because $\sum_i \lambda_i E_N(\lambda_i^*) = 1$. Thus, $\alpha= \pi/2$, and hence, $(N, N \rtimes G, M, M \rtimes G)$ is a commuting square.
Weyl group, Quadrilaterals and regularity {#quadrilaterals-regularity}
=========================================
In this section we focus on the analysis of irreducible subfactors and quadrilaterals from the perspectives of Weyl group and interior and exterior angles between intermediate subfactors.
First, we make some useful observations related to Pimsner-Popa bases and regularity.
\[intermediatebasis1\] Let $P$ be an intermediate $II_1$-factor of a subfactor $N \subset M$. Let $e_P$ denote the canonical Jones projection for the basic construction $P \subset M \subset P_1$ and $\{{\lambda}_i\}$ be a finite set in $P$. Then, $\{{\lambda}_i\}$ is a Pimsner-Popa basis for $P/N$ if and only if $\sum_i \lambda_i e_1{\lambda}^*_i=e_P$
If $\{{\lambda}_i\}$ is a Pimsner-Popa basis for $P/N$, then we know that $\sum_i \lambda_i e_1 \lambda_i^* = e_P$ - see , for instance, the proof of [@BDLR2017 Proposition 2.14]. This proves necessity.
To prove sufficiency, consider the basic construction $N\subset
P\subset N_1$ with Jones projection $e^P_N$. Recall, from [@BL], that this tower is isomorphic to the tower $Ne_P\subset
e_PMe_P=Pe_P \subset e_PM_1e_P$ via a map $\phi: N_1 {\rightarrow}e_P
M_1 e_P$ satisfying $\phi(x)=xe_P$ for all $x\in P$. The Jones projection for the second tower is given by $e_Pe_1=e_1$. Note that $\sum_i {\lambda}_ie_Pe_1e_P{\lambda}^*_i= \sum_i
{\lambda}_ie_1{\lambda}^*_i=e_P.$ Thus, we obtain $$\phi\big(\sum_i
{\lambda}_ie^P_N{\lambda}^*_i\big)=\sum_i
({\lambda}_ie_P)e_1(e_P{\lambda}^*_i)=e_P=\phi(1).$$ This implies that $\sum \lambda_ie^P_N{\lambda}^*_i=1$ and, hence, $\{\lambda_i\}$ is a Pimsner-Popa basis for $P/N$.
\[p=ep\] Let $N\subset M$ be an irreducible subfactor and $P:=\big
\{\mathcal{N}_M(N)\big\}''$. If $\{u_g: g = [u_g] \in G\}$ denotes a set of coset representatives of $G$ in $\mathcal{N}_M(N)$, then $\{u_g: g \in G\} \subset \mathcal{N}_P(N)$ and it forms a two sided orthonormal basis for $P/N.$
Since $N \subset M$ is irreducible, it follows that $P$ is a $II_1$-factor. By definition, we have $\mathcal{N}_P(N) \subset
\mathcal{N}_M(N)$. On the other hand, $\mathcal{N}_M(N) \subset
P$. So, if $u \in \mathcal{N}_M(N)$, then $u \in
\mathcal{N}_P(N)$. Hence, $\mathcal{N}_P(N) =
\mathcal{N}_M(N)$. Therefore, we conclude that $N \subset P$ is a regular subfactor and also that the Weyl group of $N \subset P$ is the same as that of $N \subset M$.
Then, since $N \subset P$ is regular and irreducible, we conclude, from [@Hong Lemma 3.1], that $\{u_g:g\in G\}$ forms a two-sided orthonormal basis for $P/N$.
Above two propositions yield the following improvement of [@Hong Lemma 3.1]:
\[reg-onb\] Let $N\subset M$ be an irreducible subfactor and $\{u_g : g \in G\}$ be a set of coset representatives of $G$ in $\mathcal{N}_M(N)$. Then, the following are equivalent:
1. $[M:N] =|G|$.
2. $\{ u_g : g \in G\}$ is a two sided orthonormal basis for $M/N$.
3. $N \subset M$ is regular.
$(1) \Leftrightarrow (2):$ Let $p :=\sum_g u_g e_1 u_g^*$. Then, $tr(p) = \sum_{g \in G} tr(u_g
e_1 u_g^*) = |G| [M:N]^{-1} $. Thus, $\{ u_g : g \in G\}$ is an orthonormal basis for $M/N$ if and only if $[M:N] = |G|$.
$(2) \Leftrightarrow (3):$ This equivalence follows immediately from and .
Next, we move to Weyl groups. Let $N \subset M$ be a subfactor and let $\mathcal{U}(N)$ (resp., $\mathcal{U}(M)$) denote the group of unitaries of $N$ (resp., $M$) and $\mathcal{N}_M(N) := \{ u \in
\mathcal{U}(M): u N u^* = N\}$ denote the group of unitary normalizers of $N$ in $M$. Clearly, $\mathcal{U}(N)$ is a normal subgroup of $\mathcal{N}_M(N)$. For a finite index subfactor $N \subset M$, one associates the so-called Weyl group, which we shall denote by $G$, defined as the quotient group $\mathcal{N}_M(N)/\mathcal{U}(N)$ (see, for example [@Hong; @JoPo; @Kos; @Cho; @PiPo2]).
\[weyl-example\] Let $G$ be a finite group acting outerly on a $II_1$-factor $N$ and $H$ be a normal subgroup of $G$. Then, the Weyl group of the subfactor $N\rtimes H
\subset N \rtimes G$ is isomorphic to the quotient group $G/H$.
Fix a set of coset representatives $\{g_i: 1 \leq i \leq n=[G:H]\}$ of $H$ in $G$. Then, $\{u_{g_i}: 1 \leq i \leq n\}$ forms a two sided orthonormal basis for $(N \rtimes G )/N$ (where $u_g$’s are as in ). Clearly, the map $G/H \ni g_i H
\stackrel{\varphi}{\longmapsto} [u_{g_i}] \in G$ is a bijection. Then, note that $$\varphi(g_i H)\varphi( g_j H) = [u_{g_i}][u_{g_j}] = [u_{g_i}u_{g_j}]= [u_{g_i g_j}].$$ On the other hand, if $g_i g_j H = g_k H$, then $g_i g_k = g_k h$ for some $h \in H$, which implies that $ u_{g_i g_j} = u_{g_k}u_h $, i.e., $[u_{g_i g_j} ] = [u_k] $ in $G$. Thus, $\varphi(g_i H\cdot g_j H) =
\varphi(g_k H) = [u_{g_k}]= [u_{g_i g_j} ] = \varphi(g_i H)\varphi(
g_j H) $ for all $1 \leq i, j \leq n$. Hence, $K/H \cong G$.
Analogous to the well known Goldman’s Theorem for a subfactor with index $2$, it is known that an irreducible regular subfactor $N \subset M$ can be realized as the group subfactor $N \subset N \rtimes G$, where $G$ is the Weyl group of $N \subset M$ which acts outerly on $N$ - see [@Hong; @PiPo2; @Kos] and the references therein. As a consequence, we deduce the following version of Goldman’s type Theorem for irreducible quadrilaterals.
\[hk-generate-g\] Let $(N, P, Q, M)$ be an irreducible quadrilateral such that $N\subset
P$ and $N \subset Q$ are both regular. Then, $G$ acts outerly on $N$ and $(N, P, Q, M) = (N, N \rtimes H, N \rtimes K, N \rtimes G)$, where $H, K$ and $G$ are the Weyl groups of $N \subset P$, $N \subset Q$ and $N \subset M$, respectively.
First, note that $N \subset M$ is regular because $$M= P \vee Q =
\mathcal{N}_P(N)'' \vee \mathcal{N}_Q(N)'' \subseteq
\{\mathcal{N}_P(N) \cup \mathcal{N}_Q(N)\}'' \subseteq
\mathcal{N}_M(N)'' \subseteq M.$$ Then, since $N \subset M$ is regular and irreducible, Hong [@Hong] had shown that if $N_{-1} \subset N \subset M$ is an instance of downward basic construction with Jones projection $e_{-1}$, then there is a representation $G\ni g \mapsto v_g \in \mathcal{U}(N_{-1}'\cap
M)$ such that $v_g \in \mathcal{N}_M(N)$ for all $g\in G$, $M = \{N,
v_g: g \in G\}''$ and $ G \in g \mapsto Ad_{v_g}\in Aut(N)$ is an outer action of $G$ on $N$, i.e., $(N \subset M) = (N \subset N
\rtimes G)$ - see [@Hong Lemma 3.3 and Theorem 3.1]. Also, for each $g \in G$, the coset $ v_g\, \mathcal{U}(N) = g$ in $G$.
So, by Galois correspondence, $P = N \rtimes H'$ and $Q = N \rtimes K'$ for unique subgroups $H'$ and $K'$ of $G$. We assert that $H = H'$ and $K = K'$.
We have $P = N \rtimes H' = \{N, v_{h'}: h' \in H'\}''$. So, for each $h' \in H'$, $v_{h'} \in P$; thus, $\{v_{h'}: h' \in H'\} \subset
\mathcal{N}_P(N)$. Also, $h'= [v_{h'}]\in
\mathcal{N}_P(N)/\mathcal{U}(N) = H $, so that $H' \subset H$. As seen in , $H' \cong H$; so, we must have $|H'| = |H|$ and hence $H' = H$. Likewise, we obtain $K = K'$. Hence, $(N, P, Q, M) = (N, N
\rtimes H, N \rtimes K, N \rtimes G)$.
\[hk-generate-g2\] Let $(N, P, Q, M)$ be an irreducible quadrilateral such that $N\subset P$ and $N \subset Q$ are both regular. Then, the Weyl groups of $N \subset P$ and $N \subset Q$ together generate the Weyl group of $N \subset M$.
Let $G'$ be the subgroup of $G$ generated by $H$ and $K$, then $N
\rtimes G' \subseteq N \rtimes G$. Also, since $M = P\vee Q$, we have $$N \rtimes G = (N \rtimes H)\vee (N \rtimes K) \subseteq N \rtimes G'\subseteq N \rtimes G.$$ Hence, by Galois correspondence again, we must have $G = G'$, i.e., $G$ is generated by its subgroups $H$ and $K$.
We have the following partial converse of .
Let $(N,P,Q,M)$ be an irreducible quadruple such that $N\subset P$ and $N\subset Q$ are both regular. If $N\subset M$ is regular and the Weyl groups of $N\subset P$ and $N\subset Q$ together generate the Weyl group of $N\subset M$, then $M=P\vee Q$.
Fix any set of coset representatives $\{u_g: g \in G\}$ of $G$ in $\mathcal{N}_M(N)$. Since $N \subset M$ is regular, $\{u_g: g \in
G\}$ forms a two sided orthonormal basis for $M/N$, by . Note that each $g$ in $G$ is a word in $H \cup K$ and for any pair $g, g' \in G$, we have $[u_{gg'}] = gg' =
[u_g][u_{g'}]= [u_g u_{g'}]$, so that $u_{gg'} = v u_g u_g' $ for some $v\in
\mathcal{U}(N)$. Thus, $M = \sum_g N u_g \subseteq (\sum_h N u_h)
\vee (\sum_k u_k) = P \vee Q$.
Following corollary first appeared implicitly in the proof of [@SW Theorem 6.2]. We include it here, as an application of and .
Let $(N, P, Q, M)$ be an irreducible quadrilateral with $[P:N] = 2 =
[Q:N]$. Then, $[M:N]$ is an even integer and the Weyl group of $N \subset
M$ is isomorphic to the Dihedral group of order $2n$, where $n = [M:P]=[M:Q]$.
By Goldman’s Theorem, we know that $(N \subset P)\cong (N \subset N
\rtimes_{\sigma} {\mathbb Z}_2)$ and $ (N \subset Q)\cong (N \subset N
\rtimes_{\tau} {\mathbb Z}_2)$ for some outer actions $\sigma$ and $ \tau$ of ${\mathbb Z}_2$ on $N$ and hence both $N \subset P$ and $N \subset Q$ are regular. Then, by , we obtain $|H| = [P:N] =2$ and $|K| =[Q:N] =2$, where $H$ and $K$ are as in . So, $H$ and $K$ are both cyclic of order $2$. By , $N \subset M$ is regular and hence $[M:N] =
|G|$ by . Also, by , $G$ is a finite group generated by $H$ and $K$. Thus, $G$ is generated by two elements which are both of order $2$. Hence, by [@Suz Theorem 6.8], $G$ is isomorphic to the Dihedral group of order $2n$.
We conclude this section with the demonstration of a direct relationship between angles and Weyl groups of interemdiate subfactors of an irreducible quadruple. As above, for a quadruple $(N,P,Q,M)$, we denote by $G,H$ and $K$ the Weyl groups of $N\subset M, N\subset P$ and $N\subset Q$, respectively. First, we deduce the relationship for an irreducible quadrilateral.
\[betacomutationforregularquadrilateral\] Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $N\subseteq
P$ and $N\subseteq Q$ are both regular. Then, $\alpha(P,Q)=\pi/2$, i.e., $(N, P, Q, M)$ is a commuting square, and $$\cos \beta(P,Q)=\displaystyle \frac{\frac{\lvert G \rvert}{\lvert
H\rvert \lvert K\rvert}-1}{\sqrt{[G:H]-1}\sqrt{[G:K]-1}}.$$
In particular, $(N, P, Q, M)$ is a commuting square if and only if $G = HK$.
It follows from and .
More generally, we have the following relationship.
\[main1\] Let $(N,P,Q,M)$ be an irreducible quadruple such that $N\subseteq P$ and $N\subseteq Q$ are both regular. Then, $$\cos \alpha(P,Q)= \frac{|H \cap K| - 1}{\sqrt{|H| -1}\sqrt{|K| -1}
}$$ and $$\cos\beta(P,Q) = \frac{\frac{[M:N]}{|HK|} -1}{\sqrt{[M:P]
-1}\sqrt{[M:Q] -1}}.$$ In particular, $(N, P, Q, M)$ is a commuting square, if and only if, $H \cap K$ is trivial, if and only, if $P \cap Q = $. And, $(N, P, Q,
M)$ is a cocommuting square if and only if $|G| = |HK| =
[M:N]$.
Since $N \subset M$ is irreducible, $P \vee Q$ is a $II_1$-factor. Consider the irreducible quadruple $(N, P, Q, P\vee Q)$. Then, by (2), $(N, P, Q, P\vee Q) = (N, N \rtimes H, N
\rtimes K, N \rtimes G')$ where $G'$ is the Weyl group of $N \subset
P \vee Q$. Hence, by , we obtain $$\cos\big( \alpha^N_{P \vee Q}(P,Q)\big)= \frac{|H \cap K| -
1}{\sqrt{|H| -1}\sqrt{|K| -1} }.$$ And, it is known that $\alpha^N_{P \vee Q}(P,Q) = \alpha^N_{M}(P,Q)$ - see [@BDLR2017 Proposition 2.16].
On the other hand, being irreducible, $N \subset M$ is extremal. So, by , the exterior angle between $P$ and $Q$ is given by $$\begin{aligned}
\cos \beta(P, Q) & = & \frac{tr(e_P e_Q) -\tau_P \tau_Q}{\sqrt{\tau_P - \tau_p^2} \sqrt{\tau_Q - \tau_Q^2}}
\\
& = & \frac{[M:N]^{-1} |H \cap K| - [M:P]^{-1}[M:Q]^{-1}}{\sqrt{[M:P]^{-1} - [M:P]^{-2}}\sqrt{[M:Q]^{-1} - [M:Q]^{-2}}}
\\
& = & \frac{|H \cap K| [P:N]^{-1} [M:Q] -1 }{\sqrt{[M:P] - 1}\sqrt{[M:Q] - 1}}
\\
& = & \frac{\frac{[M:N]}{|HK|} -1}{\sqrt{[M:P] -1}\sqrt{[M:Q] -1}},
\end{aligned}$$ where we have used the equalities $tr(e_P e_Q) = \tau \sum_{h \in H, k
\in K} \| E_N(u_h^* u_k)\|_2^2 = [M:N]^{-1}|H \cap K|$ by and the well known formula $|HK| = |H| |K||H \cap
K|^{-1}$. Hence, by [@BDLR2017 Proposition 2.7], $(N, P, Q, M)$ is a cocommuting square if and only if $\beta (P,Q) = \pi/2$, i.e., if and only if $|HK| = [M:N]$. Note that $|G| \leq [M:N]$ (see [@Hong; @PiPo2]) and $HK \subseteq G$. So, $|HK| = [M:N]$ if and only if $|G| = |HK|= [M:N]$.
Sano and Watatani ([@SW Theorem 6.1]) had proved that an irreducible quadrilateral $(N,P,Q,M)$ with $[P:N] = 2 =[Q:N]$ is always a commuting square. Thus, Theorem \[betacomutationforregularquadrilateral\] can also be thought of as a generalization of that result.
Possible values of interior and exterior angles {#values}
===============================================
It is a very natural curiousity to know the possible values of interior and exterior angles between intermediate subfactor. As a first attempt in this direction, we make some calculations in the irreducible set up.
Prior to that we recall two auxiliary positive operators associated to a quadruple (from [@BDLR2017; @SW]) whose norms are equal, and show that this common entity has a direct relationship with the possible values of interior and exterior angles.
Two auxiliary operators associated to a qudruple {#auxiliary}
------------------------------------------------
Consider a quadruple $(N,P,Q,M)$. Let $\{\lambda_i:i\in I\}$ and $\{\mu_j:j\in J\}$ be (right) Pimsner-Popa bases for $P/N$ and $Q/N$, respectively. Consider two positive operators $p(P,Q)$ and $p(Q,P)$ given by $$p(P,Q)= \sum_{i,j}{\lambda_i}\mu_j e_1 {\mu}^*_j{\lambda}^*_i\quad \text{and}\quad
p(Q,P)= \sum_{i,j}\mu_j \lambda_i e_1 {\lambda}^*_i {\mu}^*_j.$$
By [@BDLR2017 Lemma 2.18], $p(P,Q)$ and $p(Q,P)$ are both independent of choices of bases. And, by [@BDLR2017 Proposition 2.22], $Jp(P,Q)J = p(Q,P)$, where $J$ is the usual modular conjugation operator on $L^2(M)$; so that, $\|p(P,Q)\| =
\|p(Q,P)\|$.
For a quadruple $(N, P, Q, M)$, let $r:=\frac{[P:N]}{[M:Q]}=\frac{[Q:N]}{[M:P]}$ and $\lambda:=\lVert p(P,Q)\rVert = \|p(Q,P)\|$.
Recall that for a self adjoint element $x$ in a von Neumann algebra $\mathcal{M}$, its support is given by $s(x):= \inf\{ p \in
\mathcal{P}(\mathcal{M}): px = x =xp\}$. We will need the following useful lemma which follows from [@BDLR2017 Proposition 2.25 $\&$ Lemma 3.2].
[@BDLR2017]\[sp-lemma\] If $(N,P,Q,M)$ is a quadruple such that $N \subset M$ is irreducible, then $ \lambda = [Q:N]\, tr(p(P,
Q) e_Q) $ and $s(p(P,Q)) = p(P,Q)/ \lambda$. In particular, $tr
(s(p(P,Q))) = r/\lambda$ and $p(P,Q)$ is a projection if and only if $\lambda = 1$.
It turns out that $s(p(P,Q))$ is a minimal projection in $P'\cap Q_1$ which is central as well.
Let $(N,P,Q,M)$ be quadruple such that $N\subset M$ irreducible. Then, $\frac {p(P,Q)}{\lambda}$ (resp., $\frac {p(Q,P)}{\lambda}$) is a minimal projection in $P^{\prime}\cap Q_1$ (resp., $Q^{\prime}\cap P_1$) which is also central.
By , $\frac{1}{\lambda}p(P,Q)$ is a projection. Further, by [@BDLR2017 Proposition 2.25], we have $p(P,Q)= [P:N]E^{N^{\prime}}_{P^{\prime}}(e_Q)\in P^{\prime}\cap
Q_1$. We first show that $\frac{1}{\lambda} p(P,Q)$ is minimal in $P'\cap Q_1$. Consider any projection $q \in P^{\prime}\cap Q_1$ satisfying $0\leq q \leq\frac{1}{\lambda} p(P,Q)$. Then, $q=\frac{q}{\lambda}p(P,Q).$ We also have $qe_Q= [M:Q] E_M^{Q_1}(q
e_Q)e_Q$ (by the Pushdown Lemma [@PiPo Lemma 1.2]). Clearly, $E_M(qe_Q)\in N^{\prime}\cap M$. Thus, irreduciblility of $N\subseteq M$ implies that $qe_Q=te_Q$ for the scalar $t = [M:Q]
E_M^{Q_1}(q e_Q) $. Therefore, $$\begin{aligned}
q
& = & \frac{q}{\lambda} p(P,Q)\\
& = & \frac{q}{\lambda} [P:N] E^{N^{\prime}}_{P^{\prime}}(e_Q)\\
& = & \frac{[P:N]}{\lambda} E^{N^{\prime}}_{P^{\prime}}(qe_Q)\\
& = & \frac{[P:N]}{\lambda}tE^{N^{\prime}}_{P^{\prime}}(e_Q)\\
& = & \frac{t}{\lambda} p(P,Q).\end{aligned}$$ Since $q$ and $\frac{1}{\lambda} p(P,Q)$ are projections we conclude that $t^2=t$. Therefore, $q=0$ or $p(P,Q)$. Since $q$ was arbitrary, this proves the minimality of $\frac{p(P,Q)}{\lambda}.$
We now prove that $\frac{1}{\lambda}p(P,Q)$ is a central projection in $P^{\prime}\cap Q_1.$ For this, we first show that $e_Q$ is a minimal central projection in $N^{\prime}\cap Q_1$. Let $u$ be an arbitrary unitary in $N^{\prime}\cap Q_1$. Then, by the Pushdown Lemma again, we have $ue_Q=[M:Q]E_M(ue_Q)e_Q$. But clearly $E_M(ue_Q)\in
N^{\prime}\cap M={\mathbb C}$. Thus, $ue_Qu^*=te_Q$ for some scalar $t$. Since $te_Q$ is a non-zero projection, we must have $t = 1$; so that, $ue_Q =
e_Q u$ for all $u \in \mathcal{U}(N'\cap Q_1)$, thereby implying that $e_Q$ is central in $N^{\prime}\cap Q_1.$ This shows that $$vp(P,Q)v^* = [P:N] E^{N^{\prime}}_{P^{\prime}}(ve_Qv^*) =
[P:N]E^{N^{\prime}}_{P^{\prime}}(e_Q) = p(P,Q)$$ for all $v \in \mathcal{U}(P'\cap Q_1)$ and we are done.
Assertion about $p(Q,P)$ then follows from the fact that $p(Q,P) = J
p(P,Q) J$ - see [@BDLR2017 Proposition 2.22].
As asserted above, we now present the direct relationship that exists between the values of the interior and exterior angles and the common norm of the above two auxiliary operators.
\[formulaalphabeta\] Let $(N,P,Q,M)$ be a finite index quadruple of $II_1$-factors with $N\subset M$ irreducible. Then, $$\label{alpha-lambda}
\cos(\alpha(P,Q))=
\displaystyle\frac{(\lambda-1)}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and $$\label{beta-lambda-r} \cos(\beta(P,Q)) =
\displaystyle\frac{(\lambda-r)}{\sqrt{[P:N]-r}\sqrt{[Q:N]-r}}.$$
From , we have $$\cos(\alpha(P,Q))= \frac{tr(e_P
e_Q)-\tau}{\sqrt{tr(e_P)-\tau}\sqrt{tr(e_Q)-\tau}}=\frac{[M:N]tr(e_Pe_Q)-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}.$$ It can be shown that $p(P,Q) =[Q:N]E^{M_1}_{Q_1}(e_P)$ - see the proof of [@BDLR2017 Proposition 2.25]; hence, $p(P,Q)
e_Q=[Q:N]E^{M_1}_{Q_1}(e_Pe_Q).$ Thus, $tr(p(P,Q)e_Q)=[Q:N]tr(e_Pe_Q)$. This, along with the fact that $p(P,Q) e_Q=\lambda e_Q$ (because $N \subset M$ is irreducible - see the proof of [@BDLR2017 Lemma 3.2]), yields $tr(e_Pe_Q)=\frac{\lambda tr(e_Q)}{[Q:N]} = \lambda \tau,$ that is $[M:N]tr(e_Pe_Q)=\lambda$. Thus, we obtain $$\cos(\alpha(P,Q))=
\frac{\lambda-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}.$$ On the other hand, being irreducible, $N
\subset M$ is extremal. So, from , we have $$\begin{aligned}
\cos(\beta(P,Q)) & = & \frac{tr(e_P e_Q) - \tau_P
\tau_Q}{\sqrt{\tau_P - \tau_P^2}\sqrt{\tau_Q - \tau_Q^2}}\\ & = &
\frac{\tau \lambda - \tau_P \tau_Q}{\sqrt{\tau_P -
\tau_P^2}\sqrt{\tau_Q - \tau_Q^2}}\\ & = &
\frac{\lambda-r}{\sqrt{[P:N]-\frac{[P:N]}{[M:P]}}\sqrt{[Q:N]-\frac{[Q:N]}{[M:Q]}}}.
\end{aligned}$$ Then, note that $$\begin{aligned}
&
\left([P:N]-\frac{[P:N]}{[M:P]}\right)\left([Q:N]-\frac{[Q:N]}{[M:Q]}\right)
\\ & \qquad=
[P:N][Q:N]-\frac{[P:N][Q:N]}{[M:Q]}-\frac{[P:N][Q:N]}{[M:P]}+\frac{[P:N][Q:N]}{[M:P][M:Q]}\\ &\qquad=
[P:N][Q:N]-r[Q:N]-r[P:N]+r^2\\ & \qquad=([Q:N]-r)([P:N]-r),\end{aligned}$$ and we are done.
Let $(N,P,Q,M)$ be a commuting square with $N\subset M$ irreducible. Then, $$\cos(\beta(P,Q))= \displaystyle
\frac{r^{-1}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}.$$ And, if $(N,P,Q,M)$ is a a cocommuting square with $N\subset M$ irreducible, then $$\cos(\alpha(P,Q))=\displaystyle \frac{r-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}.$$
The formula for $\beta(P,Q)$ is easy and is left to the reader. Cocommuting square implies $\beta(P,Q)=\pi/2.$ Thus, $tr(e_Pe_Q)=\tau_P\tau_Q$. Now simply use $\frac{\tau_P\tau_Q}{\tau}=r$ and the formula follows from the definition of $\alpha(P,Q)$.
Values of angles in the irreducible setup
-----------------------------------------
In order to determine the values of interior and exterior angles between intermeidate subfactors, as is evident from , it becomes important to know the possible values of $r$ and $\lambda$. Recall, from [@pop], Popa’s set of relative-dimensions of projections in $M$ relative to $N$ given by $$\Lambda(M,N)=\{\alpha\in \mathbb{R}: \exists~~
\text{a projection}~~ f_0 \in M ~~\text{such that}~~ E_N(f_0)=\alpha
1_N\}.$$
\[rbylambda\] For an irreducible quadruple $(N, P, Q, M)$, $tr\big(s\big(p(P,Q)\big)\big)=\frac{r}{\lambda}\in \Lambda(M_1,M).$ Also, $(1-\frac{r}{\lambda})\in \Lambda(M_1,M)$.
Let $\{\lambda_i\}$ be a basis for $P/N$. Then, $p(P,Q) : = \sum
\lambda_i e_Q{\lambda_i}^* \in M_1$ and clearly $
E^{M_1}_M\big(p(P,Q)\big)=\sum
\lambda_iE^{M_1}_M(e_Q){\lambda}^*_i=\displaystyle\frac{\sum
\lambda_i{\lambda}^*_i}{[M:Q]}=\frac{[P:N]}{[M:Q]}=r. $ Thus, $E^{M_1}_M(\frac{1}{\lambda}p(P,Q))=\frac{r}{\lambda}$. And, by , the operator $\frac{1}{\lambda}p(P,Q) =
s(p(P,Q)) $ is a projection. This completes the proof.
For a commuting square $(N,P,Q,M)$ with $N\subset M$ irreducible, it is known that $\lambda = 1$ - see [@BDLR2017 Proposition 2.20]. Thus, we deduce the following:
If $(N,P,Q,M)$ is a commuting square with $N\subset M$ irreducible, then $r\in \Lambda(M_1,M)$. And, if $(N,P,Q,M)$ is a parallelogram, then ${\lambda}^{-1}\in
\Lambda(M_1,M).$
Consider the polynomials $P_n(x)$ for $n \geq 0$ (introduced by Jones in [@Jon] and) defined recursively by $P_{0}=1, P_1=1,
P_{n+1}(x)=P_n(x)-xP_{n-1}(x), n>0.$ Thus, $P_2(x)=1-x, P_3(x)=1-2x$ and so on. From [@Jon], we know that $P_k\big(
\frac{1}{4{\cos}^2\pi/(n+2)}\big)>0$ for $0\leq k\leq n-1$, and $P_n\big(\frac{1}{4{\cos}^2\pi/(n+2)}\big)=0.$ Furthermore, $P_k(\epsilon)>0$ for all $\epsilon \leq 1/4$ and $k\geq 0$. Also, by definition, we have $\frac{\tau
P_{n-1}(\tau)}{P_n(\tau)}=1-\frac{P_{n+1}(\tau)}{P_n(\tau)}$ for all $n\geq 1$, where, as is standard, $\tau:=[M:N]^{-1}$.
While trying to determine the possible entries of the set $\Lambda(M,
N)$, Popa [@pop] proved the following theorem:
[@pop] \[popa\] Let $N\subset M$ be a subfactor of finite index.
1. If $[M:N]=4 {\cos}^2\big(\frac{\pi}{n+2}\big)$ for some $n\geq 1$, then $$\Lambda(M,N)=\big\{0\big\}\cup \bigg\{\frac{\tau P_{k-1}(\tau)}{P_k(\tau)}:0\leq k\leq n-1\bigg\}=\bigg\{\frac{P_k(\tau)}{P_{k-1}(\tau)}:0\leq k\leq n\bigg\}.$$
2. If $[M:N]\geq 4$ and $t\leq 1/2$ is so that $t(1-t)=\tau$, then $$\Lambda(M,N)\cap (0,t) = \bigg\{\frac{\tau P_{k-1}(\tau)}{P_k(\tau)}:k\geq 0\bigg\}.$$
Since a subfactor with index less than $4$ does not admit any intermediate subfactor, for any non-trivial quadrilateral $(N, P, Q,
M)$, we always have $[M:N] \geq 4$.
\[rlambdacomputation\] Let $(N,P,Q,M)$ be an irreducible quadruple and let $0<t\leq 1/2$ be such that $t(1-t)=\tau$. Then, either $\frac{r}{\lambda}\geq t$ or $\frac{r}{\lambda}=\frac{\tau P_{k-1}(\tau)}{P_{k}(\tau)}$ for some $k\geq 0$.
This follows from and .
We may thus compute the interior and exterior angles in this specific situation, as follows.
\[valuesofangle\] Let $(N,P,Q,M)$ be a quadruple with $N\subset M$ irreducible and let $0<t\leq 1/2$ be such that $t(1-t)=\tau$. If $r/\lambda \geq t$, then, $$\cos(\alpha(P,Q)) \leq
\frac{[P:N][Q:N](1-t)-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and $$\cos(\beta(P,Q))\leq \displaystyle
\frac{\frac{1}{t}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}.$$ And, if $r/\lambda < t$, then, $$\cos(\alpha(P,Q)) =
\displaystyle\frac{[P:N][Q:N]\frac{P_k(\tau)}{P_{k-1}(\tau)}-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and $$\cos(\beta(P,Q))=\displaystyle \frac{\frac{P_k(\tau)}{\tau
P_{k-1}(\tau)}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}$$ for some $k\geq
0.$
First, suppose that $r/\lambda\geq t$. Thus, $\lambda\leq
r/t$. Observe that $r/t=[P:N][Q:N](1-t)$; so, from , we obtain the first inequality. Also, $\lambda-r \leq r(1/t-1)$. Thus, from , we obtain $$\cos(\beta(P,Q))\leq \displaystyle
\frac{r(1/t-1)}{\sqrt{[P:N]-r}\sqrt{[Q:N]-r}}=\displaystyle
\frac{(1/t-1)}{\sqrt{[M:Q]-1}\sqrt{[M:P]-1}}.$$
Next, suppose that $N \subset M$ is irreducible and $r/\lambda <t$. By , we have $\lambda=\frac{r}{\tau}\frac{P_k(\tau)}{P_{k-1}(\tau)}$, for some $k\geq 0.$ Observe that $r/\tau=[P:N][Q:N].$ Thus, by , we obtain $$\cos(\alpha(P,Q)) =
\displaystyle\frac{[P:N][Q:N]\frac{P_k(\tau)}{P_{k-1}(\tau)}-1}{\sqrt{[P:N]-1}\sqrt{[Q:N]-1}}$$ and, by , we obtain $$\cos(\beta(P,Q))=
\displaystyle \frac{r\bigg(\frac{P_k(\tau)}{\tau
P_{k-1}(\tau)}-1\bigg)}{\sqrt{[P:N]-r}\sqrt{[Q:N]-r}}=\displaystyle
\frac{\frac{P_k(\tau)}{\tau
P_{k-1}(\tau)}-1}{\sqrt{[M:P]-1}\sqrt{[M:Q]-1}}.$$
\[valuesofbeta\] Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $N\subset
P$ and $N\subset Q$ are both regular and suppose $[P:N]=2$. Then, $\cos(\beta(P,Q))= \displaystyle
\frac{P_2(m/2)}{\sqrt{P_2(\delta^2/2)P_3(m/2)}}$, where $m = [M:Q]
\in {\mathbb N}$ and, as usual, $\delta:=\sqrt{[M:N]}.$
Let $H, K$ and $G$ denote the Weyl groups of $N \subset P$, $N \subset
Q$ and $N \subset M$, respectively. We have $m = [M:Q] = \frac{
[M:N]}{[Q:N]} = \frac{|G|}{|K|}$ (by and ) and $\lvert H\rvert=2$. Thus, by , we obtain
$$\begin{aligned}
\cos(\beta(P,Q)) & = & \frac{\frac{\lvert G\rvert}{\lvert
H\rvert \lvert K\rvert}-1}{\sqrt{\frac{\vert G\rvert}{\lvert
H\rvert}-1}\sqrt{\frac{\lvert G \rvert}{\lvert K\rvert}-1}}\\
& = & \displaystyle
\frac{\frac{m}{2}-1}{\sqrt{(\delta^2/2)-1}\sqrt{m-1}}\\ &
= &
\frac{P_2(m/2)}{\sqrt{P_2(\delta^2/2)}\sqrt{P_3(m/2)}}.\end{aligned}$$
\[valuesofbeta2\] Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $[P:N] = 2
= [Q:N]$. Then, $\alpha(P,Q)=\pi/2$ and $\beta(P,Q)={\cos}^{-1}\left(\frac{P_2(m/2)}{P_3(m/2)}\right)$, where $m = [M:P]=[M:Q]$ is an integer. In particular, $\beta(P,Q)>\pi/3.$
This follows from and the fact that $P_2(\delta^2/2)=P_2(m)=P_3(m/2)$, where $[M:P]=[M:Q]=m$.
\[valuesofbeta3\] Let $(N,P,Q,M)$ be an irreducible quadrilateral such that $N\subseteq P$ and $N\subseteq Q$ are both regular with $[P:N]=2$ and $[Q:N]=3.$ Then, $\alpha(P,Q)=\pi/2$ and $\beta(P,Q)>
{\cos}^{-1}\bigg(\frac{1}{\sqrt{6}}\bigg).$
By , we have $\alpha(P,Q)
= \pi/2$ and taking $m = [G:K]$, we obtain $${\cos}^2(\beta(P,Q))=\displaystyle
\frac{m^2/4-m+1}{3m^2/2-5m/2+1}< 1/6,$$ where the inequality follows from a routine comparison using the fact that $m \geq 2$.
Certain bounds on angles and their implications {#angle-bounds}
===============================================
In this section, we observe that when one leg of an irreducible quadruple is assumed to be regular, it enforces certain bounds on interior and exterior angles, which then imposes some bounds on the index of the other leg.
\[inequality\] Let $(N,P,Q,M)$ be an irreducible quadruple such that $N\subset P$ is regular. Then, $$\cos\big(\alpha(P,Q)\big)\leq \Bigg(\sqrt{\frac{[P:N]-1}{[Q:N]-1}}\Bigg)$$ and $$\cos(\beta(P,Q))\leq \Bigg(\sqrt{\frac{[P:N]-r}{[Q:N]-r}}\Bigg).$$
For any $u\in \mathcal{N}_M(P)$ we have $uE_Q(u^*)\in N^{\prime}\cap
M ={\mathbb C}$. To see this, let $n\in N$ be arbitrary. Then, $uE_Q(u^*).n=uE_Q(u^*n)=uE_Q(u^*nu.u^*)=n.uE_Q(u^*).$
Let $uE_Q(u^*)=t \in {\mathbb C}.$ If $t\neq 0$ we get $E_Q(u^*) = t u^*$ so that $u\in Q$, which implies that $t= uE_Q(u^*) = u u^* = 1$. Thus, $uE_Q(u^*) \in \{0, 1\}$ for all $u \in \mathcal{N}_M(P)$.
Using , fix an orthonormal basis $\{u_i\} \subset \mathcal{N}_P(N)$ for $P/N$. Consider the auxiliary operator $p(P,Q) =\sum_iu_ie_Qu^*_i$. Since $N \subset M$ is irreducible, we have $p(P,Q) e_Q = \lambda
e_Q$ (see [@BDLR2017 Lemma 3.2]), where $\lambda = \|p(P,Q)\|$. Also, $p(P,Q)e_Q=\sum_iu_ie_Qu^*_ie_Q=\sum_iu_iE_Q(u^*_i)e_Q\in {\mathbb C}e_Q$, which yields $ \sum_i u_i E_Q(u_i^*) = \lambda$. In particular, we obtain $ 0 \leq \lambda\leq \lvert
\mathcal{N}_P(N)/\mathcal{U}(N)\rvert = [P:N]$. Thus, $\lambda-1 \leq
[P:N]-1$ and the lower bound for $\alpha$ follows from (1).
We also have $\lambda-r\leq [P:N]-r.$ So the lower bound for $\beta$ follows from .
Note that in above proof, we also observed that $\|p(P,Q)\|$ is an integer less than or equal to $[P:N]$.
Above theorem imposes some immediate bounds on $[Q:N]$ in terms of $[P:N]$, as follows.
Let $(N, P, Q, M)$ be as in . Then, we have the following:
1. If $\alpha(P,Q) \leq \pi/3$, then $[Q:N] \leq 4[P:N] -3$.
2. If $\alpha(P,Q) \leq \pi/4$, then $[Q:N] \leq 2[P:N] -1$.
3. If $\alpha(P,Q) \leq \pi/6$, then $[Q:N] \leq 4/3 [P:N] -1/3$.
If $\alpha (P, Q) \leq \pi/3$, then $\cos \alpha (P,Q) \geq \cos
(\pi/3) = 1/2$. So, $\frac{[P:N] - 1}{[Q:N] - 1} \geq 1/4$ and hence $[Q:N] \leq 4 [P:N]
-3$. Others follow similarly.
As a consequence, when we intuitively try to visualize an irreducible quadruple $(N, P, Q, M)$ with $[P:N] = 2$ as a $4$-sided structure in plane (as in ), then it seems that the smaller is the interior angle between $P$ and $Q$ the shorter is the length (or index) of $N \subset Q$. This assertion is supported by the following observations:
If $[Q:N] > 7/3$, then it follows from above corollary that $\alpha(P,Q) > \pi/6$. Likewise, If $[Q:N] > 3$, then $\alpha(P,Q) >
\pi/4$. And, if $[Q:N] > 5$, then $\alpha(P,Q) > \pi/3$. In particular, since $P$ is a minimal subfactor, i.e., $N \subset
P$ admits no intermediate subfactor, in the last scenario, $Q$ cannot be a minimal subfactor because, by [@BDLR2017], we know that the interior angle between two minimal intermedite subfactors is always less than $\pi/3$. Also, observe that if $\alpha(P,Q) \leq \pi/4$, then $[Q:N] \leq 3$ and hence $N \subset Q$ must be a Jones’ subfactor.
The reader can get a better feeling of above assertion by making similar calculations for an irreducible quadruple $(N, P, Q, M)$ such that $P \subset N$ is regular with $[P:N] = n$, for arbitrary $n$.
[89]{}
K. C. Bakshi, S. Das, Z. Liu and Y. Ren, Angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc., to appear. B. Bhattacharya and Z. Landau, Intermediate standard ivariants and intermediate planar algebras, preprint. M. Choda, A characterization of crossed product of factors by discrete outer automorphism groups, J. Math. Soc. Japan, 31, 1979, 257 - 261. J. Hong, Characterization of crossed product without cohomology, J. Korean Math. Soc., 32 (2), 1995, 183 - 192. V. F. R. Jones, Index for subfactors, Invent. Math., 72 (1), 1983, 1 - 25. V. F. R. Jones, and S. Popa, Some properties of MASA’s in factors, Invariant Subspaces and Other Topics, Operator Theory: Advances and Applications, Birkhauser, Basel, vol 6., 1982, 89 - 102. V. F. R. Jones and V. S. Sunder, Introduction to Subfactors, London Mathematical Society Lecture Note Series, Cambridge University Press, vol. 234, 1997. H. Kosaki, Characterization of crossed product (properly infinite case), Pacific J. Math., 137 (1), 1989, 159-167.
R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys., 237 (1-2), 2003, 7-30. M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup., Series 4, 19 (1), 1986, 57 - 106. M. Pimsner, S. Popa, Finite dimensional approximation of algberas and obstructions for the index, J. Funct. Anal. 98, 1991, 270 - 291. S. Popa, Relative dimension, towers of projections and commuting squares of subfactors, Pacific J. Math., 137(1), 1989, 181-207. T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory, 32 (2), 1994, 209 - 241. C. E. Sutherland, Cohomology and extensions of von Neumann algebras. II, Publ. RIMS, Kyoto Univ., 16, 135 - 174, 1980. M. Suzuki, Group Theory I, Grundlehren der mathematischen Wissenschaften, Springer, 1982.
[^1]: The first named author was supported by a postdoctoral fellowship of the National Board for Higher Mathematics (NBHM), India.
|
---
abstract: 'Scene modeling is very crucial for robots that need to perceive, reason about and manipulate the objects in their environments. In this paper, we adapt and extend Boltzmann Machines (BMs) for contextualized scene modeling. Although there are many models on the subject, ours is the first to bring together objects, relations, and affordances in a highly-capable generative model. For this end, we introduce a hybrid version of BMs where relations and affordances are incorporated with shared, tri-way connections into the model. Moreover, we introduce a dataset for relation estimation and modeling studies. We evaluate our method in comparison with several baselines on object estimation, out-of-context object detection, relation estimation, and affordance estimation tasks. Moreover, to illustrate the generative capability of the model, we show several example scenes that the model is able to generate, and demonstrate the benefits of the model on a humanoid robot. The code and the dataset are publicly made available at: <https://github.com/bozcani/COSMO>'
address: 'KOVAN Research Lab, Dept. of Computer Engineering, Middle East Technical University, Ankara, Turkey'
author:
- İlker Bozcan
- Sinan Kalkan
bibliography:
- 'references.bib'
title: 'COSMO: Contextualized Scene Modeling with Boltzmann Machines'
---
Scene Modeling ,Context ,Boltzmann Machines.
Conclusion
==========
In this paper, we proposed a novel method (COSMO) for contextualized scene modeling. For this purpose, we extended Boltzmann Machines (BMs) to include spatial relations and affordances via tri-way edges in the model. For integrating spatial relations and affordances into the model, we introduced shared nodes into BMs, allowing the concept of relations and affordances to be shared among different objects pairs. We evaluated and compared our model on several tasks on a real dataset and a real robot platform.
On several challenging tasks, we demonstrated that our model is very suitable for scene modeling purposes with its generative and explicit nature. Being generative, we showed that a single COSMO model allows reasoning about many aspects of the scene given any partial information. On these tasks, COSMO performed consistently better in comparison to the baseline methods (general BMs and restricted BMs) and relational networks [@santoro2017simple].
Limitations and Future Work
---------------------------
they try to address these issues by formulating and evaluating a novel BM model for a medium-scale scene modeling task.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through project called “Context in Robots” (project no 215E133). We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research.
References {#references .unnumbered}
==========
|
harvmac.tex
David I. Olive$^1$, Neil Turok$^2$ and Jonathan W.R. Underwood$^3$
$^1$ Department of Mathematics, University College of Swansea, Swansea SA2 8PP, Wales, UK.
$^2$ Joseph Henry Laboratories, Princeton University, Princeton, NJ08544, USA.
$^3$ The Blackett Laboratory, Imperial College, London SW7, UK.
**Abstract**
Affine Toda theories with imaginary couplings associate with any simple Lie algebra ${\bf g}$ generalisations of Sine Gordon theory which are likewise integrable and possess soliton solutions. The solitons are created" by exponentials of quantities $\hat F^i(z)$ which lie in the untwisted affine Kac-Moody algebra ${\bf\hat g}$ and ad-diagonalise the principal Heisenberg subalgebra. When ${\bf g}$ is simply-laced and highest weight irreducible representations at level one are considered, $\hat
F^i(z)$ can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions can be directly related to the algebraic structure. For example, a classical version of Dorey’s fusing rule follows from the operator product expansion of two $\hat F$’s, at least when ${\bf
g}$ is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields.
*[E\_M]{} [**1. Introduction**]{} This work continues that of our previous paper \[\] in which we identified the manner whereby solutions describing any number of solitons fitted into the more general class of solutions of the affine Toda theories associated with an affine untwisted Kac-Moody algebra $\gh$. Substitution into the energy-momentum tensor confirmed that the energy was real and positive, being of the form appropriate to a set of moving, relativistic particles. This was despite the fact that the solutions were complex as a consequence of the coupling constant being imaginary in order to provide topological stability. Although the energy density was likewise complex, the reality of its integral was a consequence of the fact that it was a total derivative which was asympotically real. This work therefore extended previous results of the last year \[\] \[\], \[\], \[\], putting them on a more systematic and general footing and unifying them with other developments. Related recent developments are due to Niedermaier \[\] and Aratyn et. al. \[\]. The soliton solutions could be expressed simply in terms of rational functions of expectation values of the $r$ fields" $\fiz$, where $r=\hbox{rank }\g$, which diagonalise the ad-action of the principal Heisenberg subalgebra of $\gh$. The role of this Heisenberg subalgebra in other, non-relativistic, soliton theories was first established by the Kyoto school \[\], but the affine Toda theory is of particular interest as it possesses relativistic symmetry in space-time with two dimensions and well illustrates \[\] Zamolodchikov’s idea \[\] that integrable theories result from particular breakings of a conformally invariant theory in which the conservation laws are relics of the conformal (or W-) symmetry. Thus it is possible to relate the structure found in affine Toda theory to the concepts of particle physics in an intriguing way and it is this which motivates our interest in precisely this theory. For example, mass formulae and coupling rules for the solitons emerge bearing remarkable similarities to those enjoyed by the quantum particle excitations of the original fields, as elucidated in recent years \[\], \[\], \[\], \[\], \[\], \[\]. This points to a duality symmetry in which the inversion of the coupling constant is accompanied by a replacement of the roots of $\g$ by the corresponding coroots. All this structure is very reminiscent of ideas entertained fifteen years ago in connection with magnetic monopole solitons arising in spontaneously broken gauge theories in four dimensional space time \[\], \[Ø\][D.I. Olive, Magnetic Monopoles and Electromagnetic Duality Conjectures“ in Monopoles in Quantum field Theory”, edited by N.S. Craigie, P. Goddard and W. Nahm (World Scientific 1982) p157-191]{}, \[\], \[\]. The difference here is that the construction of the dual quantum field theory describing the solitons is now a more realistic possibility.*
Our construction of the soliton solutions has the intuitively attractive feature that the soliton of species $i$“ is created” by the Kac-Moody“ group element given by the exponential $$exp\,
Q\fiz,\eqno(1.1)$$ where $ln\,|Q|$ and $ln\,|z|$ relate to the coordinate and rapidity of the soliton respectively while the superscript $i$”, denoting the soliton species, labels one of the $r$ orbits of the $hr$ roots of $\g$ under the action of the Coxeter element of its Weyl group. It will be convenient to take as the representative element of each such orbit the unique one of the form $$\gamma_{(i)}=c(i)\alpha_{(i)}\eqno(1.2)$$ where $\alpha_{(i)}$ is simple and $c(i)=\pm1$ denotes its colour“, black or white, as explained in and . That the exponentials (1.1) are meaningful as Kac-Moody group elements without any normal ordering is a consequence of the fact that $\fiz$ is nilpotent: in an irreducible representation of $\gh$ at level $x$, powers of $\fiz$ higher than $2x/\gamma_{(i)}^2$ vanish while lower powers usually have finite matrix elements. (In saying this we have chosen to normalise the highest root of $\g$ to have length $\sqrt2$). (The exceptions to this finiteness will have a physical interpretation to be discussed beow). The present paper supplies the proof of these properties. At the same time techniques are found for calculating the expectation values of the $F$’s needed to evaluate the soliton solutions explicitly. The techniques used have an intrinsic interest; the principal vertex operator construction is used together with an exploitation of the outer automorphisms of $\gh$ corresponding to symmetries of the extended Dynkin diagram of $\g$. By-products of interest to the physical interpretation of the soliton solutions concern (a) a classical analogue of Dorey’s fusing rule applicable to soliton solutions rather than the quantum excitation particles, (b) insight into generalised breather” solutions, and (c) expressions for the weight lattice coset of the topological charge when $g$ is simply laced. In section 2 we present our notations and recapitulate the equations considered together with our solutions. In section 3 we establish properties of the outer automorphisms of $\gh$ to be used while in section 4 we present the properties of the expectation values of the $F$’s with respect to the highest weight states of the fundamental representations of $\gh$. These form a pseudo-unitary matrix which plays an important role in subsequent work. Since the results sought are representation dependent we have to proceed by considering successively more elaborate irreducible representations of $\gh$. The simplest possibility occurs when $\g$ is simply laced and at level one, being known as the basic“ representation. As the $\g$ weight of its highest weight state vanishes, that state can be thought of as the vacuum. In this representation, the $F$’s can be constructed explicitly in terms of the principal Heisenberg subalgebra via the principal vertex operator construction \[\], \[\] as explained in section 5. It is therefore possible to calculate the operator product expansion of two $F$’s. The result possesses double poles with c-number residues and simple poles whose residues are proportional to elements of the principal Heisenberg subalgebra or a third $F$. The $F$’s so occurring satisfy Dorey’s fusing rule and this is the algebraic origin of this rule as reported previously . It is easy to check that the square of each $F$ vanishes and to normal order” products of $F$’s in order to calculate matrix elements, rather as is done in evaluating dual string scattering amplitudes. Section 6 extends these results to the other irreducible representations of $\gh$ at levels higher than one (when $g$ is simply laced) in which the vertex operator construction no longer applies. Section 7 treats non-simply laced Lie algebras, denoted $g_{\tau}$, as they are obtained in a unique way by the familiar folding procedure from a simply laced $\g$ and an outer automorphism $\th$ \[\], \[\]. The key result is that $\th$ preserves the principal $su(2)$ subalgebra of $\g$. As a consequence, the principal Heisenberg subalgebra of $\g_{\tau}$ is a subalgebra of that of $\g$. This makes it easy to relate the $F$’s for $\gh$ and $\gh_{\tau}$ and hence establish the final versions of the quoted nilpotency and operator product properties. The physical interpretation of these results, (a), (b) and (c), cited four paragraphs above, are discussed in section 8. Sample calculations of soliton solutions are also given for $su(N)$, $so(8)$ and $G_2$. [**2. The Affine Toda Equations and their Soliton Solutions**]{}
even though it is not difficult to generalise virtually all of our arguments to the latter. Before presenting the equations and the solutions of interest, we shall summarise our conventions for such algebras and their representations.
The Chevalley basis of the affine untwisted Kac-Moody algebra $\gh$ is generated by the elements $$\{e_i,f_i,h_i\}\qquad i=0,1\dots r.$$ It has been proven that $$\eqalign{[h_i,h_j]=&0\cr
[h_i,e_j]=&K_{ji}e_j\cr [h_i,f_j]=&-K_{ji}f_j\cr
[e_i,f_j]=&\delta_{ij}h_i\cr}\eqno(2.1)$$ together with the Serre relations $$\eqalign{(\ad e_i)^{1-K_{ji}}e_j=&0\cr (\ad
f_i)^{1-K_{ji}}f_j=&0\cr}\eqno(2.2)$$ are sufficient to completely specify the algebra, given $K_{ji}$, the Cartan matrix of $\gh$, satisfying the usual properties. This is the analogue of Serre’s theorem for a finite-dimensional Lie algebra. It is then convenient to add the element $d'$ with the defining properties $$[d',e_i]=e_i,\qquad[d',h_i]=0\qquad\hbox{ and }\qquad
[d',f_i]=-f_i, \eqno(2.3)$$ and to regard it, together with the $h_i$, as spanning the Cartan subalgebra which therefore has dimension $r+2$. Notice that the adjoint action of $d'$ grades $\gh$; this is generally known as the principal gradation in view of its connection with the principal $so(3)$ subalgebra of $\g$ as seen below. Since the $e_i$ can be regarded as the step operators for the simple roots of $\gh$, the grade of a step operator counted by $d'$ is simply the height of the associated root. We define the positive integers $m_i$ and $n_i$ to be the lowest for which $$\sum_iK_{ji}m_i=0\qquad\hbox{ and
}\qquad\sum_jn_jK_{ji}=0.\eqno(2.4a)$$ It follows that$$n_i=2m_i/a_i^2\eqno(2.4b)$$where $a_i$ denotes the $i$’th simple root of $\gh$ and the long roots are chosen to have length $\sqrt2$. The Coxeter and dual Coxeter numbers of $\g$ are defined respectively as $$h=\sum_{i=0}^rn_i\qquad\hbox{ and }\qquad\tilde
h=\sum_{i=0}^rm_i.\eqno(2.5)$$ The following element of the Cartan subalgebra is central in that it commutes with all of $\gh$:- $$x=\sum_{i=0}^rm_ih_i.\eqno(2.6)$$ We shall want to compare this presentation of $\gh$, which is particularly relevant to the affine Toda theory, to the more familiar one as a central extension of the loop algebra of $\g$: $$[\lambda^m\otimes\gamma_1,
\lambda^n\otimes\gamma_2]=\lambda^{m+n}
\otimes[\gamma_1,\gamma_2]+\delta_{m+n,0}(\gamma_1,\gamma_2)mx,\eqno(2.7)$$ where $\gamma_1,\gamma_2\in\g$ and ( , ) is the Killing form on $\g$. When $\gh$ is viewed this way it is more natural to append an element $d$ with the grading property $$[d,\lambda^m\otimes\gamma]=m\lambda^m\otimes\gamma.$$ This is known as the homogeneous grading and $d$ can be thought of as the Virasoro generator $-L_0$. Relating the two presentations of $\gh$ is straightforward. For $i\neq0$ $$e_i\Leftrightarrow\lambda^0\otimes E^{\alpha_i},\qquad
f_i\Leftrightarrow\lambda^0\otimes E^{-\alpha_i},\qquad
h_i\Leftrightarrow\lambda^0\otimes H^{\alpha_i},\eqno(2.8a)$$ in terms of the Chevalley generators of $\g$ while, for $i=0$, $$e_0\Leftrightarrow \lambda^1\otimes E^{-\psi},\qquad
f_0\Leftrightarrow\lambda^{-1}\otimes E^{\psi}, \qquad
h_0\Leftrightarrow\lambda^0\otimes (H^{-\psi}+x),\eqno(2.8b)$$ where $\psi\equiv-\alpha_0$ denotes the highest root of $\g$. Finally we can now state the relation between $d$ and $d'$ which count the homogeneous and principal grades respectively $$d'=hd +T_0^3.\eqno(2.9)$$ Here $T_0^3$ is defined to be $\lambda^0\otimes T^3$ where $T^3$ is the generator of the principal or maximal $so(3)$ subalgebra of $\g$ lying in the Cartan subalgebra of the latter. Its adjoint action on the step operators of $\g$ counts the height of the corresponding roots. The analogue of $d'$ for the finite dimensional Lie algebra $\g$ must be obtained by somehow modding out $d$. By (2.9), this is achieved by defining the element of the corresponding Lie group $G$, $S=e^{2\pi iT^3/h}$. Conjugation with respect to $S$ then grades $\g$ according to powers of $\omega=exp(2\pi i/h)$. This structure played a crucial role in understanding the properties of the affine Toda particles arising as field quanta , , and is explained further in section 2.4. The construction of solutions to the affine Toda equations will utilise irreducible highest weight representations of $\gh$, particularly the fundamental ones. The corresponding representation spaces are generated by the action of arbitrary products of the $f_i$ on the highest weight“ state $|\Lambda>$, say, which is annihilated by the $e_i$. The representation is precisely characterised by the action of $h_0,
h_1\dots h_r$ on this state, which must be an eigenvector under this action. We write $$h_i|\Lambda>=\Lambda(h_i)|\Lambda>={2\Lambda\cdot a_i\over
a_i^2}|\Lambda>.\eqno(2.10)$$ The scalar product here is deduced from the invariant scalar product on $\gh$. Unitarity requires the $r+1$ quantities $\Lambda(h_i)$ to be non-negative integers. It follows that the eigenvalue of $x$ take the same positive integral value on all states of the irreducible representation, which we denote $V(\Lambda)$. This is known as the level, and is again denoted $x$, by abuse of notation. The number of irreducible representations at each level is finite and at least one. It is convenient to define the fundamental” weights $\Lambda_0,\Lambda_1,\dots \Lambda_r$ of $\gh$ by $\Lambda_i(h_j)=\delta_{ij}$. Since the $r+1$ simple roots $a_i$ of $\gh$ relate to the $r$ simple roots $\alpha_i$ of $\g$ by $$a_0=(-\psi,1,0),\qquad a_i=(\alpha_i,0,0)\quad i=1,2,\dots r,\eqno(2.11)$$ given the Cartan Weyl basis of the Cartan subalgebra of $\gh$, $(H_0^i,x,d)$, we find $$\Lambda_0=(0,1,?)\qquad \Lambda_i=(\lambda_i,m_i,?)\quad i=1,2,\dots
r.\eqno(2.12)$$ The entries denoted by the question mark are undetermined since the roots span a subspace of codimension $1$ but are conventionally taken to vanish. The levels $m_i$ in (2.12) are the integers defined in (2.4). For more details, see \[\] where the same notation is used. Notice that the state $\otimes_j\rlj^{p_j}$ of the representation $\otimes_jV(\Lambda_j)^{p_j}$, where the $p_j$ are positive integers, is automatically annihilated by the $f_i$ and so is a highest weight state. Thus $$\otimes_j\rlj^{p_j}=|\sum_jp_j\Lambda_j>,\eqno(2.13)$$ where the right hand side denotes the highest weight state of $V(\sum_jp_j\Lambda_j>$. Thus, in principle, all irreducible representations of $\gh$ can be found by the decomposition of products of its fundamental representations. In fact it is possible to improve on this by considering products of only those fundamental representations with level one, ie $m_i=1$. These include the vacuum representation, that with highest weight $\Lambda_0$. If $\g$ is simply laced this condition means that $\lambda$ either vanishes or is minimal ,\[\] The subspace of $\gh$ with principal, $d'$ grade unity contains an element of special significance $$\hat E_1=\sum_{i=0}^r\sqrt{m_i}\,\,e_i.\eqno(2.14)$$ As we see in section 3 it is invariant under all the diagram automorphisms of $\gh$. It was found some time ago \[\] that the integrability of the affine Toda equations owed itself to the existence of field dependent zero curvature potentials \[\]. These can be lifted from the loop algebra of $\g$ to $\gh$ at the cost of introducing an extra, innocuous, field \[\]. The constant, vacuum solutions to the affine Toda equations give rise to potentials depending linearly on $\hat E_1$, (2.14), and its conjugate, $\hat E_{-1}=\hat E_1^{\dagger}$. We see from (2.1) and (2.7) that $$[\hat E_1,\hat E_{-1}]=x.$$ The affine Toda Hamiltonian is one member of an infinite hierarchy of integrable systems and consideration of this makes it natural to seek the subalgebra $C(\hat E_1)$ that commutes with $\hat E_1$, modulo the central term $x$. It is not difficult to show that can be expressed as a direct sum of graded subspaces $$C(\hat E_1)=\oplus\sum_{M\in Z\!\!\!Z}C_M(\hat E_1) \quad\hbox{
where }[d',C(\hat E_1)]=MC_M(\hat E_1).$$ The integers $M$ for which dim $C_M(\hat E_1)$ is non-zero are called exponents of $\gh$ of multiplicity dim $C_M(\hat E_1)$. Tables of these exponents can be found in . These exponents possess the symmetry $M\leftrightarrow-M$ and possess period $h$, the Coxeter number of $\g$. Modulo $h$ the exponents of $\gh$ equal the exponents of $\g$. Since the only occasion in which the multiplicity exceed one (and equals two) is when $\g=D_{2n}$ and the exponent equals $n-1 $ mod $2n-2$, we shall permit ourselves the temporary simplifying assumption that $C_M(\hat E_1)$ is spanned by $\em$ when $M$ is an exponent. Using the invariant bilinear form ( , ) on $\gh$ it follows that $$[\em,\en]={Mx\over h}(\em,\en)\delta_{M+N,0},$$ So the $\em$ generate what is known as the principal Heisenberg subalgebra of $\gh$. We suppose $\em^{\dagger}=\hat E_{-M}$ and choose to normalise consistently with (2.14) so that $$[\em,\en]=Mx\delta_{M+N,0}.\eqno(2.15)$$ It is a remarkable fact that, unless $M=0$, the dimension of the principally $M$-graded subspace $\gh_M$ of $\gh$ equals $r+$ dim $C_M(\hat E_1)$. The remaining $r$ dimensions of the $M$ graded subspace are spanned by vectors $\hat F_M^1,\hat F_M^2,\dots \hat F_M^r$ obeying the following commutators with the Heisenberg subalgebra $$[\em,\hat F_N^i]=\gamma_i\cdot q([M])\hat F_{M+N}^i.\eqno(2.16)$$ It is understood that $[M]$ denotes $M$ modulo $h$ and that $q([M])$ and $\gamma_i$ lie in the root space of $\g$. These quantities arise naturally in the construction of the Cartan subalgebra of $\g$ in apposition, a structure which is essentially the analogue for $\g$ of the construction above for $\gh$ as we briefly explain in the next subsection. (More details can be found in Chapter 14 of and in appendix A of .) If we define the generating function for the $\hat F_N^i$ in terms of a formal complex parameter $z$ as $$\fiz=\sum_{N=-\infty}^{\infty}\, z^{-n}\hat F_N^i,\eqno(2.17)$$ we find from (2.12) that $$[\em,\fiz]=\gamma_i\cdot q([M])z^M\fiz.\eqno(2.18)$$ Thus the complete Heisenberg subalgebra of $\gh$ has been ad-diagonalised by the $\fiz$, which, as we mentioned in the introduction, create the solitons of affine Toda theory when exponentiated, (1.1). The construction outlined above has a well established analogue for the finite-dimensional Lie algebra $\g$. Indeed this played a fundamental role in the construction of the local conservation laws of affine Toda theory and in understanding the mass and coupling properties of the particles which are the quantum excitations of the $r$ fields , , . The mathematics of the construction is due to Kostant \[\]. The analogue of (2.14) in $\g$ is $$E^1=\sum_{i=1}^r\sqrt{m_i}\,E^{\alpha_i}+E^{-\psi}.\eqno(2.19)$$ Now it commutes with its conjugate and, in fact, as $E^1$ is in general position“, the subalgebra of $\g$ commuting with it possesses $r$ dimensions and so is abelian. It can therefore be considered as an alternative Cartan subalgebra ${\bf h'}$, say, said to be in apposition”. Conjugation by $S=e^{2\pi iT^3/h}$ furnishes a linear map $\sigma$ on ${\bf h'}$. $\sigma$ can be shown to be an element of the Weyl group, of order $h$, called the Coxeter element. This possesses $r$ eigenvalues of the form $e^{2\pi i\nu/h}$, where $1\leq\nu\leq
h-1$ is one of the exponents of $\g$. Accordingly, if $h_1,h_2,\dots h_r$ denotes an orthonormal basis of ${\bf h'}$, we can express the $S$-graded elements of ${\bf
h'}$ as $$SE^{\nu}S^{-1}=e^{{2\pi i\nu\over h}}S, \qquad E^{\nu}=q(\nu)\cdot
h\qquad\hbox{where}\qquad\sigma\left(q(\nu)\right)=e^{{2\pi i\nu\over
h}}q(\nu),\eqno(2.20)$$ so that the $q(\nu)$ are the eigenvectors of $\sigma$. Because $\sigma$ is unitary and real we can orthonormalise in the following sense $$q(\nu)\cdot q(\nu')^*=\delta_{\nu,\nu'}, \qquad
q(h-\nu)=q(\nu)^*.\eqno(2.21)$$ This is explained in more detail in . There it is also explained how it is possible to define the step operators $F^{\alpha}$ for the roots with respect to ${\bf h'}$ and that $\sigma$ splits them into precisely $r$ orbits each containing $h$ roots. From the definition (2.20), we have the commutator $$[E^{\nu},F^{\alpha}]=\alpha\cdot q(\nu)F^{\alpha}.\eqno(2.22)$$ The modulus of the structure constant here is proportional to the mass of the affine Toda particles. The fact that this is also proportional to the right Perron Frobenius eigenvector $y_j(1)$ of the Cartan matrix of $\g$ follows from the important identity $$\gamma_j\cdot q(\nu)=2i\sin\,{\pi\nu\over h}y_j(\nu)e^{-i\delta_{jB}\pi
i/\nu}.\eqno(2.23)$$ where $\delta_{jB}=1$ if the vertex $j$ is black", and zero otherwise. $\gamma_j$ is defined in (1.2), being the standard representative of the $i$-th orbit. It is possible to affinise $\g$ to form $\gh$ following the construction (2.7) but using the basis just discussed. The resulting quantities $E_m^{\mu}$ and $F_m^{\alpha}$ are graded with respect to $d$, the homogeneous grade, rather than $d'$, the principal grade. Regarding $z$ as a formal parameter, the following formula succinctly converts between the grades $$\sum_{M\in Z\!\!\!Z}z^{-M}\em=\sum_{m\in Z\!\!\!Z}z^{-mh}z^{-T_0^3}
\left(\sum_{\nu}E_n^{\nu}\right)z^{T_0^3}.\eqno(2.24)$$ This is equivalent to the result derived in the appendix of . (See also \[\] )
A similar expression applies to the $\fiz$ $$\fiz=\sum_Mz^{-M}\hat
F_M^i=\sum_{m\in Z\!\!\!Z}z^{-mh}z^{-T_0^3}F_m^{\gamma_i}z^{T_0^3}
-x(T^3,F^{\gamma_i})/h.\eqno(2.25)$$ The necessity for the last, central, term was explained in .
The affine Toda field theory equations associated with $\gh$ comprise a set of $r$ coupled, differential equations satisfied by $r$ scalar fields $\phi$, taken to lie in the root space of $\g$, taking the relativistically invariant form in two space-time dimensions $$\partial^2\phi+{4\mu^2\over\beta}\left(\sum_{i=
1}^rm_i{\alpha_i\over\alpha_i^2}e^{\beta\alpha_i\cdot\phi}
-{\psi\over\psi^2}e^{-\beta\psi\cdot\phi}\right)=0.\eqno(2.26)$$ $\mu$ denotes a real inverse length scale and $\beta$ an imaginary coupling constant. The coefficients are so arranged that $\phi=0$ is a constant solution. The most general solution was found in a formal sense in but we shall be interested in what we claimed to be the solitonic specialisation of this solution. These specialisations are given by $$e^{-\beta\lambda_j\cdot\phi}={\llj e^{-\mu\hat
E_1x^+}g(0)e^{-\mu\hat
E_{-1}x^-}\rlj\over<\Lambda_0|e^{-\mu\hat
E_1x^+}g(o)e^{-\mu\hat
E_{-1}x^-}|\Lambda_0>^{m_j}},\eqno(2.27)$$ for $j=1,2,\dots r$. $g(0)$ was a Kac-Moody group element arising as an integration constant. The choice which described $n$ solitons was a product of $n$ factors of the form (1.1), one for each soliton, with the moduli of the variables $z$ suitably ordered. The time development" operators could be eliminated, using the identity valid for $j=0,1,2,\dots r$, $$\llj e^{-\mu\hat E_1x^+}g(0)e^{-\mu\hat E_{-1}x^-}\rlj=
e^{\mu^2m_jx^+x^-}\llj g(t)\rlj,\eqno(2.28)$$ where $g(t)$ is obtained by multiplying each occurrence of $\fiz$ by the factor $$W(i,z,x^{\mu})=exp\left(-\mu\{\gamma_i\cdot q(1)zx^+-\gamma_i\cdot
q(1)^*z^{-1}x^-\}\right),\eqno(2.29a)$$ $$=exp\left(\pm\mu|\gamma_i\cdot
q(1)|\{e^{\eta}x^+-e^{-\eta}x^-\}\right),\eqno(2.29b)$$ choosing $$z=\mp e^{\eta}|\gamma_i\cdot q(1)|/\gamma_i\cdot
q(1)\eqno(2.30)$$ for the reasons explained in . There $\eta$ was shown to be the rapidity of the soliton created by this $\fiz$. The result (2.29) is a straightforward application of the commutation relations (2.15) and (2.18). The justification of the specialisation was that it was possible to evaluate the energy and momentum of these solutions, finding the expressions appropriate to $n$ moving solitons, given the choice of the variables $z$ explained in , despite the complexity implied by the imaginary coupling constant $\beta$. The remainder of this paper will be devoted to the study of these formulae. It was crucial in the arguments of that products of factors were bona-fide Kac-Moody group elements. This will be justified by developing the properties of the $\fiz$, some of which are Lie algebraic and some of which, such as the nilpotency, are representation dependent.
Let $\dgh$ denote the Dynkin diagram of the affine untwisted Kac-Moody algebra $\gh$. It is well known that this is the same as the extended Dynkin diagram of the finite-dimensional Lie algebra $\g$. $\dgh$ tends to be more symmetrical than $\dg$, the Dynkin diagram of $\g$, and this will be important in what follows as it will mean that $\gh$ has more outer automorphisms than $\g$ has. There will be several ways in which these can be exploited to simplify calculations of matrix elements of the $F$’s and hence the soliton solutions themselves. As the discussion of this section is rather technical, the reader may wish, at a first reading, to pass immediately to section 4. The symmetries of $\dgh$ form a finite group denoted $Aut\dgh$ whose elements can be denoted by permutations of the vertices of $\dgh$ preserving its structure: $$\tau:i\rightarrow\tau(i)\qquad
\tau\in Aut\dgh.\eqno(3.1)$$ We immediately have a representation of $aut\dgh$ of dimension $r+1$: $$D_{ij}(\tau)=\delta_{i\tau(j)},\eqno(3.2)$$ where $D$ is a real, orthogonal matrix. Its character is simply given by $$\chi(\tau)\equiv
Tr(D(\tau))=\hbox{ No. of points of $\dgh$ fixed by } \tau.\eqno(3.3)$$ Given the character table of $Aut\dgh$, (3.3) can be used to deduce the decomposition of $D$ into irreducible representations. Because of its defining property, $D(\tau)$ must commute with the Cartan matrix of $\gh$, denoted $K$: $$[K,D(\tau)]=0.\eqno(3.4a)$$ It also follows from the definitions (2.4) that $$m_{\tau(i)}=m_i\qquad n_{\tau(i)}=n_i\eqno(3.4b)$$ There is a standard way of lifting any non-trivial $\tau\in Aut\dgh$ to an automorphism of $\gh$ which is outer. The key point is that, if $X_i$ denotes $e_i$, $f_i$ or $h_i$, the map $\th$ defined by $$X_i\rightarrow X_{\tau(i)}=X_jD_{ji}(\tau)\equiv \th(X_i)\eqno(3.5)$$ respects the defining relations (2.1) and (2.2) of $\gh$ by virtue of (3.4) and hence can be extended to an automorphism of $\gh$, by the reconstruction theorem. Notice that, by (2.6), (3.4) and (3.5), the level, $x$, is invariant $$\th(x)=x.\eqno(3.6)$$
$\th$ can readily extended to the enveloping algebra by imposing $\th(ab)=\th(a)\th(b)$ and similarly to states of representations of $\gh$. If $|\Lambda>$ denotes a highest weight state, so that it is annihilated by all $r+1$ $e_i$, then, by (3.5), so is $\tau(|\Lambda>)$. Furthermore, if it is one of the $r+1$ fundamental highest weight states so that $$h_i|\Lambda_j>=\delta_{ij}|\Lambda_j>,$$ we find, by (3.5) that $$\th|\Lambda_i>\sim |\Lambda_{\tau(i)}>.\eqno(3.7)$$ As long as $\tau$ is non-trivial there exists an $i$ for which $\tau(i)$ is distinct from it. Thus as $|\Lambda_i>$ and $|\Lambda_{\tau(i)}>$ are highest weight states of inequivalent irreducible representations of $\gh$, we conclude that $\th$ is an outer automorphism of $\gh$. In fact it is known that the group of outer automorphisms of $\gh$, understood as the quotient of the group of all automorphisms by the invariant subgroup of inner automorphisms is a finite group isomorphic to $Aut\dgh$. Because of the way the soliton solutions are expressed in terms of the principal Heisenberg subalgebra $\{\em\}$ and the $F$’s which ad-diagonalise it, we need to determine the action of $\th$ on these quantities. The first comment is that, by (3.5), $\th$ leaves invariant the subspaces of $\gh$ with principal grade $1$, $-1$ and $0$ respectively. As these subspaces generate all of $\gh$ it follows that $\th$ always respects the principal grade, which is counted by the adjoint action of $d'$. Furthermore, by its explicit form, (2.21), $\hat
E_1$ is invariant: $$\th(\hat E_1)=\hat E_1 \qquad\hbox{ for all }\tau\,\in\, Aut\dgh.\eqno(3.8)$$ Since the $\em$ are characterised by their principal grade and their commutation relation with $\hat E_1$, we conclude that $$\th(\em)=\eta(\tau,M)\em, \qquad M=\hbox{exponent of }\gh,\eqno(3.9)$$where $\eta(\tau,M)$ denotes a two dimensional representation of $Aut\dgh$ when $\g=D_{2k}$ and $M=2k-1$ mod $4k-2$ and a one dimensional representation otherwise (that is, a phase). We shall need explicit formulae for $\eta$ and shall calculate it in all cases of interest. Most often it simply equals unity, as in (3.8), but there are crucial cases when it does not. To proceed further, we need to know more about the structure of $Aut\dgh$ and this can be deduced by considering its action on $\g$ rather than $\gh$. That it has such an action follows from the comment that $\dgh$ is also the extended Dynkin diagram of $\g$ defined by the $r$ simple roots augmented by the negative of the highest root. Therefore $$Aut\dgh\subset Aut\Phi(\g),$$ where $\Phi(\g)$ denotes the root system of $\g$. The structure of $Aut\Phi(\g)$ is well understood . If $W(\g)$ denotes the Weyl group of $\g$, it constitutes an invariant subgroup of $Aut\Phi(\g)$ such that $$Aut\Phi(\g)/W(\g)\cong Aut\dg,$$ where $Aut\dg$ itself can be thought of as the subgroup of $Aut\dgh$ respecting the vertex labelled $0$. Let us define $$W_0(\g)=Aut\dgh\cap W(\g),\eqno(3.10)$$ and analyse its structure. We see immediately that $W_0$ is likewise an invariant subgroup of $Aut\dgh$ and that, $$Aut\dgh/W_0(\g)\cong Aut\dg.\eqno(3.11)$$ We can now deduce that any $\tau\,\in\, Aut\dgh$ has a unique decomposition $$\tau=\rho\tau_0,\quad \rho\,\in\, Aut\dg,\quad \tau_0\in
W_0.\eqno(3.12)$$ By (3.11) there exists $\tau_0\,\in\, W_0$ with $\tau_0(0)=\tau(0)$. Hence $\tau\tau_0^{-1}=\rho\,\in\, Aut\dg$. Furthermore, $\tau_0$ is unique, for, if not, and $\tau_0(0)=\tau_0'(0)$, we would have $\tau_0'\tau_0^{-1}\,\in\, W_0\cap Aut\dg$. As this intersection contains only the unit element the result (3.12) follows. We conclude that the elements of $W_0$ are labelled by the tip points" of $\dgh$, namely all those vertices symmetrically related to the vertex labelled $0$. The fundamental $\g$-weights associated to the tip points of $\dgh$ other than $0$ itself are all minimal , and hence (when $0$ is included) are in one-to-one correspondence with the cosets of the weight lattice of $\g$ with respect to its root lattice, and hence with the centre, $Z(\g)$, of the simply connected Lie group whose Lie algebra is $\g$. This suggests that $W_0(\g)$ and $Z(\g)$ are related. In fact it has been checked that they are isomorphic . We can make the isomorphism more explicit once we lift $\tau$ in $Aut\dgh$ to a automorphism $\td$ of the finite dimensional Lie algebra $\g$. The automorphism (3.5) of $\gh$ worked at any level so, in particular, we may choose level zero in which case $\gh$ is realised by the loop algebra $$e_i=E^{\alpha_i}\quad i=1,\dots r\qquad e_0=\lambda E^{\alpha_0},$$ where we have written the step operators for the extended set of roots of $\g$. With similar expressions for the $f_i$, we see that (3.5) now furnishes an automorphism of $\g$ rather than $\gh$. We denote this $\td$ in order to distinguish it from $\th$. The difference is that now $\tau$ in $W_0$ furnishes an inner automorphism of $\g$. We may as well put $\lambda$ equal to unity. We see that $$\eta (\tau,N)=\eta(\tau,\nu),\eqno(3.13)$$ where $N=nh(g)+\nu$, and $\nu$ is an exponent of $\g$ and so taking values between $1$ and $h-1$. Furthermore, by the construction of $E_{\nu}$ in the appendix of \[\] , $$\eta(\tau,\nu)=1\qquad \tau\in W_0.\eqno(3.14)$$ Furthermore $\eta$ only depends on $\tau$ through the coset (3.11). We can now apply $\td$ to $S=exp(2\pi iT^3/h)$, where $T^3=\rho^v\cdot
H=\sum_{\alpha>0}2\alpha\cdot H/\alpha^2$ is the intersection of the principal $so(3)$ with its Cartan subalgebra. Taking scalar products with the simple roots of $\g$ in turn enables us to deduce $$\tau(\rho^v)-\rho^v=-{2h\lambda_{\tau(0)}\over\alpha_{\tau(0)^2}},$$ from which we find $$\td(S)=S \,exp(-4\pi i\lambda_{\tau(0)}\cdot
H/\alpha_{\tau(0)}^2).\eqno(3.15)$$ As the second factor lies in $Z(\g)$ this establishes the isomorphism $$W_0(\g)\cong Z(\g).\eqno(3.16)$$ This establishes the fact that $W_0$ is abelian, and, in fact, it is maximally so in $\dgh$ as any element therein, commuting with all elements of $W_0$, must lie in $W_0$. We shall see how (3.16) affords important information concerning the asymptotic behaviour of soliton solutions in section (8.1). Applying (3.9) and (3.14) to the statement that $\fiz$ ad-diagonalises the $\em$, we deduce that $$\th(\fiz)=\epsilon(\tau,i)\fiz,\qquad \tau\in W_0(\g),\eqno(3.17)$$ where $\epsilon(\tau,i)$ is one of the irreducible (and hence one-dimensionsal) representations of $W_0\cong Z$. In sections 4 and 8 we shall present arguments to the effect that $$\epsilon(\tau,i)=e^{-2\pi
i\lambda_i\cdot\lambda_j}\qquad\hbox{if}\qquad\tau(0)=j,\eqno(3.18)$$ when $\g$ is simply laced and its roots are chosen to have length $\sqrt2$. It is understood that $\lambda_0$ denotes zero.
The preceding work will help us solve two apparently unrelated problems at the same time. The first is the evaluation of expectation values of the $r$ $\fiz$’s with respect to the highest weight states $\rlj$ of the $r+1$ fundamental representations: $$F_{ji}=\llj\fiz\rlj\eqno(4.1)$$ as these play a crucial role in the expressions for soliton solutions. We shall find the coefficient of proportionality between $F_{ji}$ and $F_{0i}$ which is the vacuum expectation value“ of $\fiz$. The second problem is the determination of the action of the automorphism $\th$ on the $\fiz$. When $\tau\in Aut\dg$, this will be useful for constructing the $\fiz$’s for the nonsimply-laced Lie algebras via the folding of a simply-laced Lie algebra , .Notice that (4.1) is independent of $z$ as only the zero (principal) mode of $\fiz$ contributes. This means that we can equally well write $$\fio=\sum_{j=0}^r\,h_jF_{ji}.\eqno(4.2)$$ The matrix $F$ is not square but there is a natural way of defining an extra column with the label $0$ in terms of the level $x$:- $$\hat F^0(z)=x\qquad\hbox{ so }\qquad F_{i0}=m_i.\eqno(4.3)$$ Taking the $\rlj$ expectation value of the equation $$[\ei,[\emi,\fio]]=\cases{|q(1)\cdot\gamma_i|^2\fio&$i\neq0$\cr
\qquad 0&$i=0$\cr}\eqno(4.4)$$ and substituting (2.16) and (4.2) yields $$\sum_kC_{jk}F_{ki}=\cases{ |q(1)\cdot\gamma_i|^2F_{ji}&$
i\neq 0$\cr\qquad 0&$i=0$\cr}\eqno(4.5)$$ where $$C_{jk}=m_jK_{jk}\eqno(4.6)$$ and $K$ is the Cartan matrix of $\gh$. We see that the $i$”’th column of the matrix $F$ furnishes an eigenvector of $C$ corresponding to the eigenvalue $|q(1)\cdot\gamma_i|^2$ or zero. Notice that apart from the eigenvalue zero these eigenvalues are proportional to the squared masses of the Toda particles. The matrix $C$, (4.6), is not symmetric, but it is symmetrisable, that is, equivalent to its transpose. This means that there is a second scalar product with respect to which $C$ is symmetric and its eigenvectors corresponding to distinct eigenvalues orthogonal. Unfortunately the eigenspaces of $C$ are degenerate in general and the reason is simply the symmetry of $\gh$ which implies that $D(\tau)$ commutes with $C$, by (3.4). That is, each eigenspace of $C$ carries a representation of $Aut\dgh$ contained in $D$, (3.2). Because $\th$, (3.5), respects the principal grades of $\gh$ we can take the zero mode of (3.17) and insert (4.2) to find $$\sum_kD_{jk}(\tau)F_{ki}=\epsilon(\tau,i)F_{ji},\qquad\tau\in
W_0(\g),\eqno(4.7)$$ where we define $\epsilon(\tau,0)=1$. Thus, the columns of $F$ are simultaneous eigenvectors of $C$ and $D(\tau)$. Because $Aut\dgh$ is, by definition, the symmetry group of $C$, and because, by the result of section 3, $W_0$ is its maximal abelian subgroup, we can consider $C$ and $W_0$ as furnishing a complete set of commuting observables whose simultaneous eigenvectors are the columns of $F$. These are therefore orthogonal with respect to the second scalar product. Taking them also to be normalised, the matrix $F$ is unitary with respect to this product, and hence invertible. Considering now any $\tau\in
aut\dgh$, we find $$\th(\fio)=\sum_j\fjo d_{ji}(\tau)\quad\hbox{ where }
d(\tau)=F^{-1}D(\tau)F.\eqno(4.8)$$ Thus $d$ is the matrix representing the action of the automorphism on the $\fio$. It enjoys the following properties $$d^{\dagger}(\tau)d(\tau)=1,\eqno(4.9a)$$ $$d_{i0}(\tau)=d_{0i}(\tau)=\delta_{i0},\eqno(4.9b)$$ $$d_{ij}(\tau)\hbox{ is diagonal, }\quad\tau\in W_0.\eqno(4.9c)$$ Unitarity, (4.9a), follows using the $Aut\dgh$ invariance of the second scalar product. (4.9b) follows from unitarity and (3.6) while (4.9c) expresses (3.17). When $\tau\in Aut\dg$, $D(\tau)$ also enjoys property (4.9b). In fact we have verified case by case that, with a suitable choice of labelling of the superscripts of the $\fiz$, $$D(\tau)=d(\tau),\qquad\qquad\tau\in Aut\dg.\eqno(4.10)$$ Unfortunately we do not know a simple proof. We are now in a position to be more specific about the phases $\epsilon(\tau,i)$ in (4.7) and (3.17). Since $W_0\cong Z(\g)$ is abelian, its irreducible representations are all one dimensional. Evidently, by (4.8), the phases $\epsilon(\tau,i) $ constitute the $r+1$ irreps of $W_0$ occurring in the decomposition of $D$, (3.2). There is a subset of these that we know precisely, namely those occurring in the representation of $W_0$ defined by $D$ acting on the tip points. The reason is that when $W_0$ acts on these tip points there are no fixed points so that it follows that this is the regular representation of $W_0$ and that each irreducible representation occurs precisely once. When $\g=G_2$ or $F_4$, $W_0$ is trivial but when $\g=B_r$ or $C_r$, $W_0=Z_2$ and there are of course two irreps. It is more interesting when $\g$ is simply laced. Then the tip points correspond to minimal weights (understanding $\lambda_0=0$) and, if we choose the labelling appropriately, we have $$\epsilon(\tau_i,j)=e^{-2\pi i\lambda_i\cdot\lambda_j}\quad\hbox{where}\quad
\tau_i(0)=i.\eqno(4.11)$$ Later on, in section 8, we shall present an argument for the extension of (4.11) whereby the label $j$ runs over all $r+1$ values, not just the $|W_0|$ values corresponding to minimal weights.
The symmetry group of $\Delta(s\hat u(N))$ is the dihedral group $D_N$ while $W_0$ is its cyclic subgroup $Z_N$. Labelling the vertices of $\Delta (s\hat u(N))$ in the obvious consecutive manner $0,1,2,\dots N-1$, and considering $W_0$ to be generated by $\tau_1(i)=i+1$, we find
$$F_{mn}=e^{2\pi imn/N}.\eqno(4.12)$$ Then $$d_{mn}(\tau_1)=\delta_{mn}e^{-2\pi im/N},$$ so that, if $\tau_j(i)=i+j=\tau_1^j(i)$, $$d_{mn}(\tau_j)=\delta_{mi}e^{-2\pi imj/N}$$ in agreement with (4.11). Furthermore $$c_{mn}\equiv(F^{-1}CF)_{mn}=\delta_{mn}\left(2sin{\pi m\over
N}\right)^2.$$ These yield the appropriate squared masses. The Lie algebra $D_4$ provides an interesting example as $Aut\Delta(\hat
D_4)\cong S_4$ while $W_0\equiv Z_2\otimes Z_2$ so, by (3.11), $Aut\Delta(D_4)\cong S_3$. The eigenvalues of the matrix (4.6) $$C=\left(\matrix{2&0&0&0&-1\cr0&2&0&0&-1\cr
0&0&2&0&-1\cr0&0&0&2&-1\cr-2&-2&-2&-2&4\cr}\right),$$ exhibits an unusually large degeneracy, being 0,2,2,2 and 6. Labelling the rows and columns of this matrix 0,1,2,3 and 4 we have that the three non-trivial elements of the group $W_0$ take the following form when written in permutation notation:- (01)(23), (02)(31) and (03)(12). Acting on the tip points 0,1,2 and 3 of $\Delta(\hat D_4)$ each irreducible representation occurs precisely once whereas the action on the central vertex 4 yields the scalar representation. We then find that the matrix $F$ diagonalising $C$ and $W_0$ takes the form $$F=\left(\matrix{1&1&1&1&1\cr1&1&-1&-1&1\cr1&-1&1&-1&1\cr1&-1&-1&1&1\cr
2&0&0&0&-4\cr}\right).\eqno(4.13)$$ Notice the order chosen for the columns 1,2 and 3 ensuring the entry 1 on the diagonal given that the first line consists of unit entries. With this choice, (4.11) again holds.
Our goal is to evaluate expectation values of products of $F$’s with respect to the fundamental highest weight states $\rlj$, and hence explicit soliton solutions. We have seen that for a single $F$ the desired result is the solution to a problem involving finite matrices of dimension $r+1$, whatever $\g$. The key to the more general problem is the observation that sufficiently large powers of $\fiz$ always vanish. Since the critical power involves the level and so is representation dependent, our strategy will be to build up from the irreducible representations with the simplest structure. These occur when $\g$ is simply laced and at level $1$. The Frenkel-Kac-Segal vertex operator construction \[\] is familiar in this situation and there is a variant (with historical priority) which expresses the $\fiz$ in terms of the principal Heisenberg subalgebra, the $\em$. The highest weights $\Lambda_j$ of these irreducible level one representations, denoted $\rho_j$, correspond to the tip points of $\dgh$, namely those related to the point $0$ by symmetries of $\dgh$ which can be taken to be unique elements of $W_0$. $\Lambda_0$ itself defines basic" representation whose highest weight state is what physicists would call the vacuum. Denoting $$\rho_j(\fiz)=F_{ji}exp\left(\sum_{N>0}{\gin z^N\hat E_{-N}\over
N}\right)exp\left(\sum_{N>0}{-\gin^*z^{-N}\en\over N}\right),\eqno(5.1)$$ where the sums extend over the positive affine exponents, we can verify the correct commutation relations (2.18) with the $\en$ as well as the correct expectation value in the highest weight state $\rlj$ by virtue of (4.1). We also use the fact that $\rlj$ is the ground state of the Fock space built with the principal Heisenberg subalgebra: $$\en\rlj=0\qquad N>0,\eqno(5.2)$$ and that this Fock space carries the irreducible representation $\rho_j$. Using the familiar normal ordering procedures of string theory, whereby the $\en$ with positive suffices are moved to the right of those with negative suffices, we shall establish the crucial formula $$\rho(\fiz)\rho(\fjz)=X_{i,j}(z,\zeta):\rho(\fiz)\rho(\fjz):\eqno(5.3a)$$ where $$X_{i,j}(z,\zeta)=X_{j,i}(\zeta,z)=\prod_{p=1}^h\left(z-\omega^{-p}\zeta\right
)^{\sigma^p(\gamma_i)\cdot\gamma_j}.\eqno(5.3b)$$ Notice that the powers $\sigma^p(\gamma_i)\cdot\gamma_j$ in (5.3b) can only take the values $0,\pm1$ and $\pm2$ and we shall study the consequences of this later.
The proof of (5.3) employs the standard techniques which immediately yield (5.3a) with $$\xij=exp\left(-\sum_{N>0}{\gin^*\gjn\over N}({\zeta\over
z})^N\right),\eqno(5.4)$$ in which the sum converges when $|z|>|\zeta|$. The sum over the positive affine exponents $N=\nu+nh$, where $\nu=[N]$ is one of the $r$ exponents of $\g$, can be written as a double sum over $n$ and $\nu$. Using the identity for a subset of the logarithmic series $$\sum_{n=0}^{\infty}{y^{nh+\nu}\over nh+\nu}=-{1\over
h}\sum_{p=1}^h
\omega^{p\nu}\hbox{ln}(1-y\omega^{-p}),$$ where $\omega$ is the primitive $h$’th root of unity, this becomes $$\xij=exp\left(\sum_{p=1}^h\hbox{ln}(1-\omega^{-p}{\zeta\over
z})\Biggl\{{1\over h}\sum_{\nu}\omega^{p\nu}\gamma_i\cdot
q(\nu)^*\gamma_j\cdot q(\nu)\Biggr\}\right)$$ $$=\prod_{p=1}^h\left(1-\omega^{-p}{\zeta\over
z}\right)^{\sum_{\nu}\sigma^p\gamma_i\cdot q(\nu)^*\gamma_j\cdot q(\nu)/h},$$ where the sum over $\nu$ extends over the $r$ exponents of $\g$. Because the $r$ eigenvectors of the Coxeter element $\sigma$ satisfy the orthogonality property $$q(\nu)^*\cdot q(\mu)=h\delta_{\mu,\nu},$$ we have the completeness relation $$\sum_{\nu}q(\nu)q(\nu)^*=hI.$$ Inserting this in our previous expression for $\xij$ yields the desired result (5.3), on realising that the sum of powers in (5.3b) vanishes as $\sum_p\sigma^p=0$. The symmetry of the factor $X$ cited in (5.3b) is very important but not quite trivial to verify. Use is made of the fact just mentioned as well as the identity $\sum_{p=0}^{h-1}p\gamma_j\cdot\sigma^p\gamma_i=h\gamma_j\cdot\sigma^{-(1+c(i))
/2}\lambda_i\in
hZ\!\!\!Z$, where $c(i)$ denotes the colour" of the vertex $i$ in the notation of \[\] and identity (2.7a) there is used.
The fact that $\xij$ is symmetric (5.3b) means that we can calculate commutators of the modes of the $F$’s in the familiar way, using deformations of contour integrals. Here we shall be content with studying the singularities of (5.3) which are simply double and simple poles. First note that by (5.3) $$\fiz\fiz=0\eqno(5.5)$$ as the contribution of $p=h$ in the product vanishes and the other factors are regular. This is nilpotency property appropriate to unit level for simply laced algebras. The corresponding results at higher levels will be deduced from this.
Double poles can only occur in (5.3) at points $z=\omega^{-p}\zeta$ when the corresponding factor is raised to the power $-2$. this requires $$\sigma^p\gamma_i+\gamma_j=0$$ which is only possible when $-\gamma_i$ and $\gamma_j$ lie on the same Coxeter orbit, that is, $i$ and $j$ are conjugate. The precise value of $p$ in (5.6) was calculated in . The coefficient of the double pole is then a c-number while the coefficient of the associated simple pole lies in the principal Heisenberg subalgebra. The remaining simple poles in (5.3b) can only occur at points $z=\omega^{-p}\zeta$ and then only if the power of the responsible factor equals $-1$, that is if $\sigma^p\gamma_i+\gamma_j$ is a root. As each root has a unique expression $\sigma^q\gamma_k$, we have the condition $$\sigma^p\gamma_i+\gamma_j=\sigma^q\gamma_k.\eqno(5.6)$$ To find the residue of this pole we collect the coefficient of $\hat
E_{-N}$ in the normal ordered product of (5.3b) and find $$(\omega^{-p}\zeta)^N\gamma_i\cdot
q([N])+\zeta^N\gamma_jq([N])=(\omega^{-q}\zeta)^N\gamma_k\cdot
q([N])$$ using (5.6) and the fact that $q$ is an eigenfunction of $\sigma$. A similar calculation applies to the coefficient of $\en$ and the conclusion is that the residue is proportional to $\hat
F^k(\omega^{-q}\zeta)$. This is the basis of our claim that the poles of the OPE of two $F$’s are controlled by Dorey’s fusing rule which therefore has a purely Kac-Moody-Lie algebraic origin. Later we shall see that this statement applies to all representations and that it has a consequence for the fusing " of soliton solutions. Equation (5.3) for the product of two $F$’s can be extended to an arbitrary product, again in the manner familiar from string theory: $$\rho(\hat F^{i(1)}(z_1))\dots \rho(\hat F^{i(k)}(z_k))\dots\rho(\hat
F^{i(n)}(z_n))=\prod_{1\leq p<q\leq n}X_{i(p),i(q)}(z_p,z_q)$$ $$\times
:\rho(\hat F^{i(1)}(z_1))\dots\rho(\hat F^{i(k)}(z_k))\dots\rho(\hat
F^{i(k)}(z_n)):\eqno(5.7)$$ Initially the moduli of the arguments $z_k$ should decrease to the right for convergence but as the right hand side of (5.7) is meromorphic its domain of definition can be extended by analytic continuation. Because the same factors $X$ occur as before, (5.3), we can conclude that the singularities arising are of the same type, that is, either due to the occurrence of conjugate pairs or to the possibility of producing a third soliton by Dorey’s fusing rule. This conclusion is important as it provides the basis for seeing that the products of Kac-Moody group elements occurring in the soliton solutions are well defined and regular except at the special singular points just mentioned. Taking the expectation value of (5.7), with respect to the highest weight state, $\rlj$, of the irrep $\rho$ and using (5.2) yields the matrix element $$\llj\hat F^{i(1)}(z_1)\dots\hat F^{i(k)}(z_k)\dots\hat
F^{i(n)}(z_n)\rlj=\prod_{1\leq p<p\leq
n}X_{i(p),i(q)}(z_p,z_q)\prod_{p=1}^nF_{ji(p)}.\eqno(5.8)$$
When $\g$ is simply laced, the irreducible level $1$ representations of $\gh$ are labelled by their $\g$ weights which are respectively $0$ and the fundamental minimal weights $\lambda_j$ $$|\Lambda_0>=|1,0>,\qquad\hbox{and}\qquad \rlj=|1,\lambda_j>.$$ Each $\lambda_j$ defines a coset of the weight lattice of $\g$ with respect to its root lattice. At higher levels $x>1$, we expect all $\gh$ irreps with $\g$-weight in the same coset as $\lambda_j$ to occur in the decomposition of the representation: $$D^{(1,\lambda_j)}\otimes\overbrace{ D^{(1,0)}\otimes\dots D^{(1,0)}}
^{x-1\rm\,\, times}\eqno(6.1)$$ In general, it is not easy to perform the decomposition of (6.1) into irreducibles but that is not necessary to establish the nilpotency and operator product expansion properties of the $\fiz$ that we are seeking. Corresonding to (6.1) the construction of $\fiz$ at level two in terms of the level one principal vertex operator construction is $$\ffiz=\fiz\otimes 1+1\otimes\fiz,$$ and we see by (5.5), that $$\ffiz^2=2\fiz\otimes\fiz.$$ So$$\ffiz^3=0\qquad\hbox{at level }2.$$ Repeating the construction at level $x$, $$\ffiz=\fiz\otimes 1\otimes\dots1+1\otimes\fiz\otimes\dots 1+\dots
1\otimes1\otimes\dots\fiz, \eqno(6.2)$$ so$$\ffiz^x=x!\,\fiz\otimes\fiz\otimes\dots\fiz,\eqno(6.3)$$ we find the previously announced nilpotency property $$\ffiz^{x+1}=0.\eqno(6.4)$$ We can also apply this style of argument to the operator product expansion. Considering level $2$ for ease of writing, we have $$\ffiz\ffjz=\fiz\fjz\otimes 1+1\otimes\fiz\fjz+\fiz\otimes
\fjz+\fjz\otimes\fiz.$$ Only the first two terms possess the singularities at $z$ in terms of $\zeta$. For example, the residue of the pole at $z=\omega^{-p}\zeta$ is proportional to $$\fkz\otimes 1+1\otimes\fkz=\ffkz.\eqno(6.5)$$ So Dorey’s fusing rule still applies. The residue of the double pole has doubled and so is proportional to the level. We conclude that at any level, $e^{Q\fiz}$ is well defined in the sense of having finite matrix elements between any pair of states in the representation space. Thus, despite its superficial resemblance to a vertex operator, it requires no normal ordering and so constitutes a bona-fide Kac-Moody group element. As we have emphasised, the key point is that the exponential series terminates, by (6.4).
In order to evaluate the solution (2.27) for a single soliton of species $i$, we seek the fundamental expectation values $$\llj\epx e^{Q\fiz}\emx\rlj,\qquad j=0,1,2.\dots r.\eqno(6.6)$$ If $\lambda_j$ is minimal, so $m_j=1$, then, by (2.28), (2.29), (4.1) and (6.4), this reduces to $$\llj\epx(1+Q\fiz)\emx\rlj=e^{\mu^2x^+x^-}\left(1+F_{ji}QW(i)\right).\eqno(6.7
)$$ This suffices when $\g$ is of $A$-type but for the $D$ and $E$-type simply laced algebras we may consider those $\Lambda_k$ corresponding to vertices $k$ of $\dgh$ adjacent to a tip point j (with $m_j=1$). Then $m_k=2$ and it is not difficult to show that $$|\Lambda_k>={1\over\sqrt2}\left(\rlj\otimes
f_j\rlj-f_j\rlj\otimes\rlj\right)$$
$$={1\over\sqrt2}\left(\rlj\otimes\emi\rlj-\emi\rlj\otimes\rlj\right).\eqno(6.8
)$$
The expectation value of $\fiz$ with respect to $|\Lambda_k>$ is still given by (4.1) whereas that of its square is found, using the nilpotency properties (6.4), (6.3) and (6.8) to be given by $$<\Lambda_k|\epx\fiz^2\emx|\Lambda_k>$$ $$=2\llj\epx\fiz\emx\rlj\llj\ei\epx\fiz\emx\emi\rlj$$ $$-2\llj\ei\epx\fiz\emx\rlj\llj\epx\fiz\emx\emi\rlj.$$ The matrix elements occurring here can be deduced by differentiating (6.7) with respect to $x^+$, $x^-$ or both and the result after substitution is simply $$2e^{2\mu^2x^+x^-}\,F_{ji}^2\,W(i)^2.$$ Thus, in this case, the expectation value (6.6) is given by $$e^{2\mu^2x^+x^-}\left(1+F_{ki}QW(i)+F_{ji}^2Q^2W(i)^2\right).\eqno(6.9)$$
Notice that the dependence upon $z$ is wholly contained in the factor $W(i)$ and that the result is clearly finite. Obviously this line of argument could be extended. The results so far are sufficient for the single soliton solutions of affine $A_r$ and $D_4$, as we see in section 8.4.
We now seek to extend the construction of the $\fiz$ and the verification of their properties to those untwisted affine Kac- Moody algebras which are not simply laced. We shall exploit the existence of a natural embedding of each such algebra within a simply laced one ($\g$ say) as the fixed subalgebra ($\gt$) of an element $\tau$ of $Aut\dg$ lifted to $\g$ and so outer“. Before studying the effect on the $\em$ and $\fiz$ we shall recap the basic ideas . We shall call $\tau$ direct” if within each orbit of points given by its action on $\dg$ no two vertices are linked. The Dynkin diagram of $\gt$, $\dgt$, is obtained by folding" $\dg$, that is, by identifying the vertices on each separate orbit of $\tau$. The set of vertices contained in such an orbit containing vertex $i$ is denoted $<i>$. This will also be used to label the vertices of $\dgt$. As $\tau$ preserves the point $0$ the same folding procedure relates the extended Dynkin diagrams of $\g$ and $\gt$. There is a precise correspondence between each of these direct reductions and the list of simply laced Lie algebras. This can be seen from the diagrams and is as follows:- $$A_{2r-1}\rightarrow C_r, \quad E_6\rightarrow F_4,\quad
D_{r+1}\rightarrow B_r \hbox{ and } D_4\rightarrow G_2.\eqno(7.1)$$ Thinking either of $\g$ or $\gh$, we define $$X_{<i>}=\sum_{i\in<i>}X_i\eqno(7.2)$$ where, as in (3.5), $X_i$ denotes $e_i$, $f_i$ or $h_i$. It is then easy to check the commutation relations $$[h_{<i>},h_{<j>}]=0\eqno(7.3a)$$ $$[h_{<j>},e_{<i>}]=K_{<i><j>}e_{<i>},\eqno(7.3b)$$ $$[h_{<j>},f_{<i>}]=-K_{<i><j>}f_{<i>},\eqno(7.3c)$$ $$[e_{<i>},f_{<j>}]=\delta_{ij}h_{<i>},\eqno(7.3d)$$ where $$K_{<i><j>}={1\over|<i>|}\sum_{i\in<i>,j\in<j>}K_{ij}=
\sum_{j\in<j>}K_{ij}.\eqno(7.4)$$ The Serre relation (2.2) can also be checked. Once we have checked that (7.4) indeed defines a Cartan matrix, we shall conclude that it the Cartan matrix of $\gt$ since (7.3) constitute the defining relations. Using the direct property, we see $K_{<i><i>}=2$ while, from the second expression (7.4) we see that when $<i>\neq<j>$, $K_{<i>,<j>}=0. -1,
-2,$ or $-3$, since the number of points in $<j>$, $|<j>|$, can only equal $1, 2$ or $3$, according to the examples above. By the first version of (7.4) we see that $${K_{<i><j>}\over K_{<j><i>}}={|<j>|\over|<i>|}.$$ In an obvious notation, this gives the ratio of squared root lengths $\alpha_{<i>}^2/\alpha_{<j>}^2$. Since $\tau\in Aut\dg$, $|<0>|=1$ and hence $${\alpha_{<i>}^2\over\alpha_{<0>}^2}={1\over|<i>|}\eqno(7.5)$$
By the definitions (2.4) together with the normalisation $m_0=n_0=1$, we deduce $$m_{<i>}=m_i, \quad n_{<i>}=|<i>|n_i \quad\hbox{and}\quad
h(\gt)=h(\g).\eqno(7.6)$$ The preservation of the Coxeter number $h$ in (7.6) is very striking, but in fact the structure goes deeper. The principal $so(3)$ subalgebra of $\g$ plays a crucial role in the construction of the principal Heisenberg subalgebra of $\gh$. We now show that it respected by $\tau$ and hence survives as the principal $so(3)$ subalgebra of $\gt$, $$so(3)\subset\gt\subset\g.\eqno(7.7)$$ The reason is simply that the Weyl vector of $\g$ is $Aut\dg$ invariant. Also the step operator for the lowest root of both $\g$ and $\gt$ is identified. This together with (7.7), again implies the equality of the Coxeter numbers. Turning to $\gh$ and $\bf{\hat g_{\tau}}$ we see that the principal Heisenberg subalgebra of the latter is a subset of that of $\gh$. It follows that the exponents of $\gt$ are a subset of the exponents of $\g$ (with the same true automatically of the affine exponents). Comparing with (3.9) and (3.13), we see that when $\tau\in Aut\dg$ is direct and nontrivial, $\eta(\nu,\tau)=1$ if and only if $\nu$ is an exponent of $\gt$. Looking at the examples (7.1), we note from tables that in all cases the common Coxeter number is even. Furthermore, with the exception of $D_{2k}$, it is always precisely the even exponents of $\g$ which are deleted in order to obtain those of $\gt$. Hence for $\tau$ nontrivial and direct $\in
Aut\dg$ $$\th(\em)=(-1)^{M+1}\em\eqno(7.8)$$ with the exception of $\g=D_{2k}$. Since in this latter case, two exponents equal $2k-1$, the principal Heisenberg subalgebra at this principal grade, $M=2k-1$, mod $4k-2$, must be two dimensional. As only one of these two exponents survive in the reduction to $B_{2k-1}$, we must be able to choose a basis in the two-dimensional space so that
$$\th(\em)=\em,\qquad\th(\em^{\prime})=-\em^{\prime}.\eqno(7.9)$$ $D_4$ is exceptional in that $Aut\Delta({\bf D_r})$ is larger when $r=4$, being $S_3$ the permutation group of the three tip vertices, which we label $1,2$ and $3$. $D_4$ has exponents $1,3,3^{\prime}$ and $5$, while the reductions ${\bf
g_{(12)}}=B_3$ and ${\bf g_{(123)}}=G_2$ have exponents $1,3, 5$ and $1,3$ respectively. Choosing $\tau$ to be the permutation $(12)$, we can define the basis at $M=6n+3$ to be as in (7.9). Since $S_3$ is represented in this two-dimensional subspace and since the permutation $\sigma=(123)$ which is responsible for the reduction to $G_2$ has no invariant subspace, this two dimensional representation has to be the two dimensional irreducible representation of $S_3$. Hence we calculate $$\hat\sigma(\em)=-{1\over 2}\em-{\sqrt3\over2}\em^{\prime},\eqno(7.10a)$$ $$\hat\sigma(\em)^{\prime}={\sqrt3\over2}\em-{1\over2}\em^{\prime}.\eqno(7.10b)$$ This completes what we have to say concerning the actions of the outer automorphisms of affine untwisted Kac-Moody algebras on their principal Heisenberg subalgebras. Given the choice of labelling whereby (4.10) holds, we can elevate it to include all grades so that $$\th(\fiz)=\hat F^j(z) D_{ji}(\tau)=\hat F^{\tau(i)}(z),\qquad\tau\in
Aut\dg\hbox{ and direct}.\eqno(7.11)$$ We now check this when $\tau$ has order two. On the basis of preceding arguments we expect that $$\th(\fiz)=\hat F^{\tau(i)}(z_{\tau})$$ and need only prove that $z_{\tau}=z$. Acting on the $\em$ $\fiz$ commutator (2.18) with $\th$, $$(-1)^{M+1}[\em,\hat F^{\tau(i)}(z_{\tau})]=z^M\gamma_i\cdot q(\mu)\hat
F^{\tau(i)}(z_{\tau}).$$ Now $\gamma_{\tau(i)}\cdot q(\mu)=(-1)^{M+1}\gamma_i\cdot q(\mu)$, using (2.23) and the facts that $x_{\tau(i)}(\mu)=(-1)^{\mu+1}x_i(\mu)$, as shown in FO and $\delta_{\tau(i)B}=\delta_{iB}$ as $\tau$ is direct. Therefore $$[\em,\hat F^{\tau(i)}(z_{\tau})]=z^M\gamma_{\tau(i)}\cdot
q(\mu)\hat
F^{\tau(i)}(z_{\tau}).$$ Hence $z^M=z_{\tau}^M$ for all exponents $M$ of $\gh$. As this includes $M=1$ the result (7.11) follows. It then follows by the preceding discussion that $$\hat F^{<i>}(z)=\sum_{i\in<i>}\fiz,\eqno(7.12)$$ where, again, $<i>$ labels an orbit of $\tau$. If $\tau$ is direct, we know that no two points $i$ and $j$, say, of $<i>$ are linked in $\dg$ and that, as a consequence, $\gamma_i\pm\gamma_j$ cannot be a root. Hence, by the fusing rule, $$[\fiz,\hat F^j(z)]=0,\eqno(7.13)$$ and we can repeat the arguments of section 6 to deduce that the highest non-vanishing power of $\hat F^{<i>}(z)$ is $|<i>|x$, remembering that the level, $x$, is preserved (3.6). By (7.5), this power equals $x\alpha_{<0>}^2/\alpha_{<i>}^2$ as claimed earlier. As this and lower powers have finite matrix elements, we have completed our proof of the stated properties of the powers of $\fiz$.
The spatial jump of any soliton solution is conserved throughout time: $${\partial\over\partial t}\Delta\phi=0\quad\hbox{ where
}\quad\Delta\phi\equiv\phi(t,x=\infty)-\phi(t,x=-\infty)\eqno(8.1)$$ This is because asymptotically the soliton solution must take values in ${2\pi i\over\beta}\Lambda_W(\g)$, where for the purposes of this section we have assumed that $\g$ is simply laced with roots chosen to have length $\sqrt2$. The expression (8.1) is called the topological quantum number but is known to be difficult to calculate, even for single soliton solutions. The reason is that it depends discontinuously on the phase of the parameter $Q$ in (1.1) where $ln|Q|$ signifies the spatial coordinate of the soliton. However we shall now see that, modulo $2\pi i\Lambda_R(\g)/\beta$, it is independent of this phase, depending only on the label $i$ in (1.1) in a way that we shall determine. To do this we shall consider the quantities $exp(-\beta\lambda_j\cdot\Delta\phi)$ which are easily calculable from the general solution. Actually it is sufficient to consider only those values of $j$ for which the fundamental weight $\lambda_j$ is minimal. As the level of the corresponding representation with highest weight state $\rlj$ is unity, the exponential (1.1) terminates after the second term in the solution. So $$e^{-\beta\lambda_j\cdot\phi(x)}={1+Q\llj\fio\rlj W(i,z,x^{\mu})
\over1+Q\llo\fio\rlo W(i,z,x^{\mu})}$$ remembering (2.29). So, if the minus sign is taken in (2.30), so that a soliton rather than an anti-soliton is considered, $$e^{-\beta\lambda_j\cdot\Delta\phi}={\llj\fio\rlj\over\llo\fio\rlo}.$$
The anti-soliton, coming from the alternative sign choice produces the inverse factor. Now consider the unique element of $W_0$ defined by $\tau_j(0)=j$. Putting $i=0$ in (3.7) and using our freedom to choose a phase to be unity, we have $$\th_j\rlo=\rlj$$ which on insertion into the previous expression gives $$e^{-\beta\lambda_j\cdot\Delta\phi}={\llo\th_j^{-1}\fio\th_j\rlo\over\llo\fio\
rlo}=\epsilon(\tau_j,i),\eqno(8.2)$$ using (3.17). Thus the phase $\epsilon$ acquires a direct significance. At least when $\g$ is simply laced, explicit calculations have led us to conjecture that $$\epsilon(\tau_j,i)=e^{-2\pi
i\lambda_i\cdot\lambda_j}\eqno(8.3)$$ Considering a solution describing several solitons of species $i(1),i(2)\dots i(n)$, we find, by similar arguments that $$e^{-\beta\lambda_j\cdot\Delta\phi}=\prod_{k=1}^n\epsilon(\tau_j,i(k)).
\eqno(8.4)$$ We shall find below that this taken together with the possibility of solitons fusing gives support for (8.3).
We saw that $$g=exp(Q_1\fif)exp(Q_2\fit)\eqno(8.5)$$ constitutes a well-defined Kac-Moody group element unless $z_1$ and $z_2$ are related in the exceptional ways discussed in section (5.3). Here we discuss the interpretation of the singularity given by $$z_1=\omega^{-p}z_2\quad\hbox{where $p$ is such that }\quad
\sigma^p\gamma_{i(1)}+\gamma_{i(2)}=\sigma^q\gamma_k,\eqno(8.6)$$ for some $q$ and $k$. As the group element $g$ creates“ two solitons of species $i(1)$ and $i(2)$ whose rapidities, $ln|z_1|$ and $ln|z_2|$ coincide at the singular point (8.6), we may suspect that some sort of bound state may be formed. We shall confirm this and that the bound state is a third soliton, of species $k$, given by Dorey’s fusing rule. For simplicity, first consider the group element (8.5) evaluated at unit level so that $\fiz^2=0$, by (5.5). Also using (5.3), we find $$g=1+Q_1\fif+Q_2\fit+Q_1Q_2X_{i(1),i(2)}(z_1,z_2):\fif\fit:.\eqno(8.7)$$ As the singular point (8.6) is approached, the factor $X$ in the last term acquires a pole while everything else remain finite if the constants $Q_i$ remain fixed. Alternatively, we can suppose $Q_1$ and $Q_2$ tend to zero in such a way that $Q_1Q_2X_{i(1),i(2)}$ remains finite. Then the limit of (8.7) is simply $$g=1+Q^{\prime}\fkz=expQ^{\prime}\fkz,$$ again using (5.5) and letting $\zeta=z_2$ and $Q^{\prime}$ be a new constant. Thus two solitons have fused” to give a third with the selection rule given precisely by Dorey’s fusing rule, originally formulated for the particle excitations of the theory and now seen to extend to the solitons. It is not difficult to show that the above limiting behaviour of the product of two Kac-Moody group elements equalling a third is independent of the level. This result provides strong support for the validity of (8.3). Recall Braden’s result \[\] that if species $i(1)$ and $i(2)$ fuse to form $k$, then if the corresponding fundamental representations are considered, $D^{\lambda_k}$ occurs in the the Clebsch Gordon series of $D^{\lambda_{i(1)}}\otimes D^{\lambda_{i(2)}}$. This implies that $$e^{-2\pi i\lambda_{i(1)}\cdot\lambda_j}e^{-2\pi
i\lambda_{i(2)}\cdot\lambda_j}=e^{-2\pi i\lambda_{k}\cdot\lambda_j}.$$ It follows that if the phases $\epsilon$ for two solitons are given by (8.3) then so is that of any third soliton obtained by fusion, by (8.4). We have just seen how an analytic continuation of the parameters describing the coordinates and rapidities of a solution describing two or more solitons can lead to a new solution describing a lesser number of solitons. This phenomenon, involving fusing“, does not occur in the simplest affine Toda theory, namely that with $\g=su(2)$, Sine Gordon theory, but there is a related phenomenon that does. This is the occurrence of the breather” solution \[\] which can be regarded as the bound state of soliton and antisoliton, oscillating about their common centre of mass. Classically it possesses a continuous mass spectrum extending from zero to the soliton-antisoliton threshold. In the quantum theory, this continuous spectrum is quantised, with the number of states depending on the coupling constant $\beta$. The lowest mass such state (if it exists) is identified with the quantum excitation of the original Sine-Gordon field. We shall now see, by finding the corresponding Kac Moody group element", that such solutions can be formed of any soliton antisoliton pair in affine Toda theory, and that the energy and momentum can be evaluated by the techniques of our previous paper.
If $\delta_i$ denotes the phase of the complex number $\gamma_i\cdot q(1)$, and $$z_i=\epsilon_ie^{i\delta_i}e^{\eta_i},\qquad\epsilon_i=\pm1,$$ it was shown that, when $\eta_i$ is real, the contribution to $\sqrt2$ times the light cone components $P^{\pm}$ of the momenta due to a factor $expQ_i\hat F^i(z_i)$ in the group element $g(0)$, (2.27), was $M_ie^{\pm\eta_i}$. $M_i$ is the mass of the soliton of species $i$ and was calculated explicitly while $\eta_i$ can be interpreted as its rapidity. In fact this argument holds good even if $e^{\eta_i}$ is complex, providing that its real part is positive. Note that the contribution to the momentum is independent of both the complex number $Q_i$ and the sign $\epsilon_i$. Consider now a group element $$g(0)=exp Q_1\hat F^1(e^{i\delta_1}e^{\eta_1})\,\,exp Q_2\hat
F^2(e^{i\delta_2}e^{\eta_2}).\eqno(8.8)$$ If $\eta_1$ and $\eta_2$ are real, this produces a two soliton solution with momentum $$\sqrt 2P^{\pm}=M_1e^{\pm\eta_1}+M_2e^{\pm\eta_2}.$$ However it is possible for this expression to remain real and positive for a complex choice of rapidities $\eta_1$ and $\eta_2$, providing $M_1=M_2=M$, say. Such a choice is given by $$\eta_1=\eta_2^*=\eta+i\theta,\qquad -\pi/2<\theta<\pi/2,\eqno(8.9)$$ when $$\sqrt2P^{\pm}=2Me^{\pm\eta}cos\,\theta.$$ Since the familiar $su(2)$, Sine-Gordon, breather arises this way, (8.8) and (8.9) can be interpreted as giving rise to a generalised breather solution with mass $2Mcos\,\theta$ and rapidity $\eta$.
Usually the equal mass condition is satisfied by choosing species $1$ and $2$ to be antiparticles of each other but in $D_4$ there is another possibility, as, by triality, the $S_3$ symmetry of $\Delta(D_4)$, there exist three solitons species of equal mass, as is seen below in section 8.4. This raises the possibility of breather solutions with non-zero topological charge which, at present, we are unable to exclude. We anticipate that it will be important to evaluate the higher conserved charges for such solutions.
The affine $su(N)$ soliton solutions are particularly easy to evaluate as all $N$ fundamental representations of $s\hat u(N)$ have level one. The single soliton solution of species $I$ is given by $$e^{-\beta\lambda_J\cdot\phi}={1+Qe^{2\pi
iIJ/N}W(I)\over1+QW(I)},\quad J=1,2,\dots N-1.$$ This follows from (2.27), (4.12) and (6.7). If we consider two solitons of species $I$ and $J$, we find, using additionally (5.8), that $e^{-\beta\lambda_J\cdot\phi}$ equals $${1+e^{2\pi iI(1)J/N}W_1+e^{2\pi
iI(2)J/N}W_2+X_{I(1)I(2)}(z_1,z_2)e^{2\pi
i(I(1)+I(2))J/N}W_1W_2\over1+W_1+W_2+X_{I(1)I(2)}(z_1,z_2)W_1W_2},$$ where $J=1,2,\dots N-1$ and $W_q=Q(q)W(I(q))$ for short.
We can also immediately write down the $s\hat o(8)$ solitons. Using, in addition, (4.13) and (6.9), we have for species $i=1,2$ and $3$, corresponding to three of the four tip points of $\Delta(s\hat o(8))$, $$e^{-\beta\lambda_j\cdot\phi}=\cases{1&$j=i$\cr {1-QW(i)\over 1+QW(i)}&$j\not=
i,4$ \cr {1+Q^2W(i)^2\over(1+QW(i))^2}&$j=4$\cr},$$ while, for species $4$, $$e^{-\beta\lambda_j\cdot\phi}=\cases{1&j=1,2,3\cr
{1-4QW(4)+Q^2W(4)^2\over(1+QW(4))^2}&j=4\cr}.$$ Using the additional results of section 7, concerning non simply laced algebras, we can construct the two species of affine $G_2$ soliton solution. By (7.12), these are created by $$\hat F^{<1>}=\hat F^{<2>}=\hat F^{<3>}=\hat F^1+\hat F^2+\hat
F^3\quad\hbox{and}\quad\hat F^{<4>}=\hat F^4,$$ in terms of the quantities for $\hat D_4$. We have retained the convention for labeling the vertices of $\Delta(\hat D_4)$ $0,1,2,3$ and $4$. It follows from their defining properties that the fundamental highest weight states of $\hat G_2$ are $$|\Lambda_{<0>}>=|\Lambda_0>,\qquad |\Lambda_{<1>}>=|\Lambda_1>\hbox{
or }|\Lambda_2>\hbox{ or }|\Lambda_3>.$$ So the $\hat F^{<4>}$ solution for affine $G_2$ is given by the same expression as the affine $D_4$ solution above. For the other soliton solution, we have, for $j=0$ or $1$ $$<\Lambda_{<j>}|\hat
F^{<1>}(z)|\Lambda_{<j>}>=F_{j1}+F_{j2}+F_{j3},$$ $$<\Lambda_{<j>}|\hat
F^{<1>}(z)^2|\Lambda_{<j>}>=2X_{12}(z,z)(F_{j1}F_{j2}+F_{j2}F_{j3}+F_{j3}F_{j1}
),$$ $$<\Lambda_{<j>}|\hat
F^{<1>}(z)^3|\Lambda_{<j>}>=6X_{12}(z,z)^3F_{j1}F_{j2}F_{j3},$$ where the $\hat D_4$ matrix $F_{ji}$ (4.13) enters the right hand side and (5.7) has been used. As $h=6$ for $D_4$ or $G_2$, we can evaluate $X_{12}(z,z)=1/3$ using (5.3b). Inserting these numerical values, (2.27) yields $$e^{-\beta\lambda_{<1>}\cdot\phi}={1-W-{1\over3}W^2+{1\over27}W^3\over1+3W+W^2
+{1\over27}W^3},$$ where $W=QW_{<1>}$. The other component of this solution is more difficult to evaluate since powers of $\hat F$ up to six survive. Combining the previous methods, we find $$e^{-\beta\lambda_{<4>}\cdot\phi}={1+3W^2-{16\over27}W^3+{1\over3}W^4+{1\over2
7^2}W^6\over\left(1+3W+W^2+{1\over27}W^3\right)^2}.$$
All the single soliton solutions for the affine untwisted Toda theories have recently been worked out explicitly using Hirota’s method . The results so obtained for the enumeration of soliton species and for the mass formulae seem to agree with the results of the general arguments of rather than with whose results were incomplete. It is interesting to see that although the general features of our approach eventually emerge, this Hirota method appears comparatively cumbersome.
We have succeeded in our main aim of establishing the conditions under which products of Kac-Moody group elements of the form (1.1) make sense. The key result is the proof of the vanishing, in a representation of level $x$, of all powers of $\fiz$ exceeding $2x/\gamma_i^2$. In the course of the work some new structural features have emerged as well as new questions such as the general proof of (8.3). The extension of Dorey’s fusing rule to solitons, (at least when $\g$ is simply laced), strengthens the evidence for a duality symmetry between the particles which are the quantum excitations of the elementary fields and the solitons. An intriguing difference concerns the fact that the solitons possess an internal degree of freedom due to the different values of the topological charge possible for a soliton of a given species. It is therefore urgent to clarify the structure of this spectrum but old questions persist such as the dependence of the topological charge (8.1) upon the phase of the constant $Q$ in (1.1). But we have been able to show that, modulo $2\pi
i\Lambda_R(\g)/\beta$, it is independent of $Q$. Although we have seen how the breathers of familiar type fit naturally within our formalism, we have not been able to exclude breathers of a new kind.
There are many ways in which we have good reason to believe our arguments can be extended. One concerns the affine Toda theories associated with twisted affine Kac-Moody algebras (rather than just the untwisted ones we have considered). Another concerns the non-Abelian Toda theories corresponding to non-principal embeddings of $SO(3)$ in $\g$. [**Acknowledgements**]{}
DIO wishes to thank Peter Johnson and Marco Kneipp for discussions. JWRU thanks SERC for the award of a studentship while JWRU and DIO thank the Isaac Newton Institute for hospitality during the early stages of this work. NT was partially funded by NSF contract No PHY90-21984, the Alfred Sloan Foundation and the David and Lucile Packard Foundation.
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---
abstract: 'In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the *univalent foundations program*, which is the topic of the current special year at the Institute for Advanced Study.'
author:
- 'Steve Awodey, Álvaro Pelayo and Michael A. Warren'
title: 'Voevodsky’s Univalence Axiom in homotopy type theory'
---
The Institute for Advanced Study in Princeton is hosting a special program during the academic year 2012-2013 on a new research theme that is based on recently discovered connections between homotopy theory, a branch of algebraic topology, and type theory, a branch of mathematical logic and theoretical computer science. In this brief paper our goal is to take a glance at these developments. For those readers who would like to learn more about them, we recommend a number of references throughout.
[**Type theory**]{} was invented by Bertrand Russell [@Russell:1908], but it was first developed as a rigorous formal system by Alonzo Church [@Church:1933cl; @Church:1940tu; @Church:1941tc]. It now has numerous applications in computer science, especially in the theory of programming languages [@Pierce:2002tp]. Per Martin-Löf [@MartinLof:1998tw; @MartinLof:1975tb; @MartinLof:1982bn; @MartinLof:1984tr], among others, developed a generalization of Church’s system which is now usually called dependent, constructive, or simply [**Martin-Löf type theory**]{}; this is the system that we consider here. It was originally intended as a rigorous framework for constructive mathematics.
In type theory objects are classified using a primitive notion of *type*, similar to the data-types used in programming languages. And as in programming languages, these elaborately structured types can be used to express detailed specifications of the objects classified, giving rise to principles of reasoning about them. To take a simple example, the objects of a product type $A\times B$ are known to be of the form $\langle a, b\rangle$, and so one automatically knows how to form them and how to decompose them. This aspect of type theory has led to its extensive use in verifying the correctness of computer programs. Type theories also form the basis of modern computer proof assistants, which are used for formalizing mathematics and verifying the correctness of formalized proofs. For example, the powerful Coq proof assistant [@coq] has recently been used to formalize and verify the correctness of the proof of the celebrated Feit-Thompson Odd-Order theorem [@gonthier].
One problem with understanding type theory from a mathematical point of view, however, has always been that the basic concept of *type* is unlike that of *set* in ways that have been hard to make precise. This difficulty has now been solved by the idea of regarding types, not as strange sets (perhaps constructed without using classical logic), but as spaces, regarded from the perspective of homotopy theory
In [**homotopy theory**]{} one is concerned with spaces and continuous mappings between them, up to homotopy; a *homotopy* between a pair of continuous maps $f \colon X \to Y$ and $g \colon X \to Y$ is a continuous map $H \colon X \times [0, 1] \to Y$ satisfying $H(x, 0) = f (x)$ and $H(x, 1) = g(x)$. The homotopy $H$ may be thought of as a “continuous deformation" of $f$ into $g$. The spaces $X$ and $Y$ are said to be *homotopy equivalent*, $X\simeq Y$, if there are continuous maps going back and forth, the composites of which are homotopical to the respective identity mappings, i.e. if they are isomorphic “up to homotopy". Homotopy equivalent spaces have the same algebraic invariants (e.g. homology, or the fundamental group), and are said to have the same *homotopy type*.
[**Homotopy type theory**]{} is a new field of mathematics which interprets type theory from a homotopical perspective. In homotopy type theory, one regards the types as spaces, or homotopy types, and the logical constructions (such as the product $A\times B$) as homotopy-invariant constructions on spaces. In this way, one is able to manipulate spaces directly, without first having to develop point-set topology or even define the real numbers. Homotopy type theory is connected to several topics of interest in modern algebraic topology, such as $\infty$-groupoids and Quillen model structures (see [@PeWa2012]); we will only mention one simple example below, namely the homotopy groups of spheres.
To briefly explain the homotopical perspective of types, consider the basic concept of type theory, namely that the *term* $a$ is of *type* $A$, which is written: $$a:A.$$ This expression is traditionally thought of as akin to “$a$ is an element of the set $A$." However, in homotopy type theory we think of it instead as “$a$ is a point of the space $A$." Similarly, every term $f : A \to B$ is regarded as a continuous function from the space $A$ to the space $B$.
This perspective clarifies features of type theory which were puzzling from the perspective of types as sets; for instance, that one can have non-trivial types $X$ such that $(X\to X) \cong X + 1$. But the key new idea of the homotopy interpretation is that the logical notion of identity $a = b$ of two objects $a, b: A$ of the same type $A$ can be understood as the existence of a path $p : a \leadsto b$ from point $a$ to point $b$ in the space $A$. This also means that two functions $f, g: A\to B$ are identical just in case they are homotopic, since a homotopy is just a family of paths $p_x: f(x) \leadsto g(x)$ in $B$, one for each $x:A$. In type theory, for every type $A$ there is a (formerly somewhat mysterious) type ${\textnormal{\texttt{Id}}_{A}}$ of identities between objects of $A$; in homotopy type theory, this is just the *path space* $A^I$ of all continuous maps $I\to A$ from the unit interval. (See [@Awodey:2009bz; @Aw2010; @PeWa2012].)
At around the same time that Awodey and Warren advanced the idea of homotopy type theory, Voevodsky showed how to model type theory using Kan simplicial sets, a familiar setting for classical homotopy theory, thus arriving independently at essentially the same idea around 2005. Both were inspired by the prior work of Hofmann and Streicher, who had constructed a model of type theory using groupoids [@HS].
Voevodsky moreover recognized that this simplicial interpretation satisfies a further crucial property, which he termed *univalence*, and which is not usually assumed in type theory. Adding univalence to type theory in the form of a new axiom has far-reaching consequences, many of which are natural, simplifying and compelling. The [**Univalence Axiom**]{} thus further strengthens the homotopical view of type theory, since it holds in the simplicial model, but fails in the view of types as sets.
The basic idea of the Univalence Axiom can be explained as follows. In type theory, one can have a universe ${\mathcal{U}}$, the terms of which are themselves types, $A : {\mathcal{U}}$, etc. Of course, we do not have ${\mathcal{U}}:{\mathcal{U}}$, so only some types are terms of ${\mathcal{U}}$ – call these the *small* types. Like any type, ${\mathcal{U}}$ has an identity type ${\textnormal{\texttt{Id}}_{{\mathcal{U}}}}$, which expresses the identity relation $A = B$ among small types. Thinking of types as spaces, ${\mathcal{U}}$ is a space, the points of which are spaces; to understand its identity type, we must ask, what is a path $p : A \leadsto B$ between spaces in ${\mathcal{U}}$? The Univalence Axiom says that such paths correspond to homotopy equivalences $A\simeq B$, as explained above (the actual notion of equivalence required is slightly different). A bit more precisely, given any (small) types $A$ and $B$, in addition to the type ${\textnormal{\texttt{Id}}_{{\mathcal{U}}}}(A,B)$ of identities between $A$ and $B$ there is the type ${\texttt{Eq}}(A,B)$ of equivalences from $A$ to $B$. Since the identity map on any object is an equivalence, there is a canonical map, $${\textnormal{\texttt{Id}}_{{\mathcal{U}}}}(A,B)\to{\texttt{Eq}}(A,B).$$ The Univalence Axiom states that this map is itself an equivalence. At the risk of oversimplifying, we can state this succinctly as follows:
Univalence Axiom
: $(A = B)\ \simeq\ (A\simeq B)$.
In other words, identity is equivalent to equivalence.
From the homotopical point of view, this says that the universe ${\mathcal{U}}$ is something like a classifying space for (small) homotopy types, which is a practical and natural assumption. From the logical point of view, however, it is revolutionary: it says that isomorphic things can be identified! Mathematicians are of course used to identifying isomorphic structures in practice, but they generally do so with a wink, knowing that the identification is not “officially" justified by foundations. But in this new foundational scheme, not only are such structures formally identified, but the different ways in which such identifications may be made themselves form a structure that one can (and should!) take into account.
Part of the appeal of homotopy type theory with the Univalence Axiom is the many interesting connections it reveals between logic and homotopy. Another remarkable aspect is that it can be carried out in a [**computer proof assistant**]{}, since type theory exhibits such good computational properties (see [@Simpson:2004bt; @Hales:2008ud] on the use of computer proof assistants in general). In practical terms, this means that it is possible to use the powerful, currently available proof assistants based on type theory, like the Coq system, to develop mathematics involving homotopy theory, to verify the correctness of proofs, and even to provide some degree of automation of proofs.
To give just one example, in homotopy type theory one can directly define the $n$-dimensional sphere $S^n$ as a type, with its associated principles of reasoning. Moreover, for any type $A$ one can define the homotopy groups $\pi_n(A)$, again in a very direct way in terms of the identity type ${\textnormal{\texttt{Id}}_{A}}$ explained above. One can then reason directly in type theory, using the principles associated with these constructions, and prove for example that $\pi_n(S^n) = \mathbb{Z}$ for $n\geq 1$ (as has recently been done by G. Brunerie and D. Licata at the Institute for Advanced Study, using the Univalence Axiom in an essential way). Finally, the proof can be formalized in a proof assistant and verified by a computer. In this way, one not only has new methods of proof in classical homotopy theory, but indeed ones which provide associated computational tools.
Voevodsky has christened this combination of homotopy type theory with the Univalence Axiom, implemented on a computer proof assistant, the [**Univalent Foundations**]{} program. It can be regarded as a new foundation for mathematics in general, not just for homotopy theory, as Voevodsky has shown by developing an extensive code library of formalized mathematics in this setting. Moreover, he is promoting more interaction between pure mathematicians and the developers of such proof assistants, as is occurring in the special year on Univalent Foundations at the Institute for Advanced Study.
For those interested in contributing to this new kind of mathematics, it may be encouraging to know that there are many interesting open questions. The most pressing of them is perhaps the “constructivity” of the Univalence Axiom itself, conjectured by Voevodsky in [@Vo2012]. It concerns the effect of adding the Univalence Axiom on the computational behavior of the system of type theory, and thus on the existing proof assistants. Another major direction, of course, is the further formalization of classical results and current mathematical research in the univalent setting. We expect that it will eventually be possible to formalize large amounts of modern mathematics in this setting, and that doing so will give rise to both theoretical insights and good numerical algorithms (extracted from code in a proof assistant).
In this direction, together with Voevodsky, the last two authors are working on an approach to the theory of integrable systems (using the new notion of $p$-adic integrable system as a test case) in the univalent setting. A preliminary treatment is the construction of the $p$-adic numbers is given in [@PeVoWa2012]. One of Voevodsky’s goals (as we understand it) is that in a not too distant future, mathematicians will be able to verify the correctness of their own papers by working within the system of univalent foundations formalized in a proof assistant, and that doing so will become natural even for pure mathematicians (the same way that most mathematicians now typeset their own papers in TeX). We believe that this aspect of the univalent foundations program distinguishes it from other approaches to foundations, by providing a practical utility for the working mathematician.
Our goal in this announcement has been to give a brief and intentionally superficial glimpse of two closely related, recent developments: homotopy type theory and Voevodsky’s univalent foundations program. At present, these subjects are still developing quite rapidly, and the current literature is (with few exceptions) highly specialized and, unfortunately, largely inaccessible to those without prior knowledge of homotopy theory and logic. One exception is the survey article [@PeWa2012], which goes into much greater depth than the present article, while still being intended for a general mathematical readership; it also contains an introduction to the use of the Coq proof assistant in the univalent setting. See also [@Aw2010; @HoTT; @Vo2012].
*Acknowledgements*. We thank the Institute for Advanced Study for the excellent resources which have been made available to the authors during the preparation of this article. We thank Thierry Coquand, Dan Grayson, and Vladimir Voevodsky for useful discussions on the topic of this paper, and the referees for helpful suggestions. Awodey is partly supported by National Science Foundation Grant DMS-1001191 and Air Force OSR Grant 11NL035. Pelayo is partly supported by NSF CAREER Grant DMS-1055897, NSF Grant DMS-0635607, and Spain Ministry of Science Grant Sev-2011-0087. Warren is supported by the Oswald Veblen Fund.
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\
[**Steve Awodey**]{}\
Carnegie Mellon University\
Department of Philosophy\
Pittsburgh, PA 15213 USA\
\
Institute for Advanced Study\
School of Mathematics, Einstein Drive\
Princeton, NJ 08540 USA\
E-mail addresses: awodey@cmu.edu, awodey@math.ias.edu\
\
\
[**Álvaro Pelayo**]{}\
Washington University\
Department of Mathematics\
One Brookings Drive, Campus Box 1146\
St Louis, MO 63130 USA\
\
Institute for Advanced Study\
School of Mathematics, Einstein Drive\
Princeton, NJ 08540 USA\
E-mail addresses: apelayo@math.wustl.edu, apelayo@math.ias.edu\
\
\
[**Michael A. Warren**]{}\
Institute for Advanced Study\
School of Mathematics, Einstein Drive\
Princeton, NJ 08540 USA.\
E-mail address: mwarren@math.ias.edu\
|
---
abstract: |
We have surveyed submillimeter continuum emission from relatively quiescent regions in the Orion molecular cloud to determine how the core mass function in a high mass star forming region compares to the stellar initial mass function. Such studies are important for understanding the evolution of cores to stars, and for comparison to formation processes in high and low mass star forming regions. We used the SHARC II camera on the Caltech Submillimeter Observatory telescope to obtain 350 data having angular resolution of about 9 arcsec, which corresponds to 0.02 pc at the distance of Orion. Further data processing using a deconvolution routine enhances the resolution to about 3 arcsec. Such high angular resolution allows a rare look into individually resolved dense structures in a massive star forming region.
Our analysis combining dust continuum and spectral line data defines a sample of 51 Orion molecular cores with masses ranging from 0.1 to 46 and a mean mass of 9.8 , which is one order of magnitude higher than the value found in typical low mass star forming regions, such as Taurus. The majority of these cores cannot be supported by thermal pressure or turbulence, and are probably supercritical. They are thus likely precursors of protostars.
The core mass function for the Orion quiescent cores can be fitted by a power law with an index equal to -0.85$\pm$0.21. This is significantly flatter than the Salpeter initial mass function and is also flatter than the core mass function found in low and intermediate star forming regions. When compared with other massive star forming regions such as NGC 7538, this slope is flatter than the index derived for samples of cores with masses up to thousands of . Closer inspection, however, indicates slopes in those regions similar to our result if only cores in a similar mass range are considered. Based on the comparison between the mass function of the Orion quiescent cores and those of cores in other regions, we find that the core mass function is flatter in an environment affected by ongoing high mass star formation. Thus, it is likely that environmental processes play a role in shaping the stellar IMF later in the evolution of dense cores and the formation of stars in such regions.
author:
- 'D. Li, T. Velusamy, P. F. Goldsmith, and W. D. Langer'
title: 'Massive Quiescent Cores in Orion. – II. Core Mass Function'
---
INTRODUCTION
============
The formation processes of high mass and low mass stars have long been suggested to be different. High mass star formation (HMSF) may require supercritical conditions in the natal clouds, while low mass star formation (LMSF) mostly occurs in subcritical gas (e.g. Shu, Adams & Lizano 1987). Massive stars may also be formed through stellar mergers (Bonnell & Bate 2002). The recent reports of disks around massive stars (Chini et al. 2004; Patel et al. 2005), however, suggests that massive stars can be formed through disk accretion, the same way as low mass stars. For better understanding massive star formation, it is important to obtain the physical conditions of HMSF regions.
Past observations of molecular clouds that harbor young stars of different masses find clear differences between HMSF regions and LMSF regions. High mass stars form only in GMCs while low mass stars can form in dark clouds as well as in GMCs. High mass stars are predominantly formed in clusters, while low mass stars may form in isolation. The star formation efficiency is generally higher in HMSF regions (e.g. Myers et al. 1986 and Lada & Lada 2003). The gas properties in these regions also differ. In HMSF regions, the density and temperature tend to be higher, and spectral line widths greater indicating higher degree of turbulence. The causal relationship between the presence of young high mass stars, different molecular cloud characteristics, and the possible variation of the star formation processes, however, is not well established.
The objects directly connecting general molecular cloud material and young stars are the so called ’cores’, which are condensations with elevated density and extinction and are likely to be bound by gravity (Ward-Thompson et al. 2006). Cores are potential precursors of protostars. The density and temperature structure of quiescent cores (no IRAS point sources and no association with molecular outflows) provides important constraints for distinguishing between star formation models and determining the initial conditions of star formation. For example, although a singular isothermal sphere leads to an inside–out collapse at constant accretion rate (Shu 1977), the evolution becomes quite different in the non–isothermal case (Foster & Chevalier 1993), or in situations involving nonthermal pressure support in the form of turbulence or magnetic fields (see, for example, Ward-Thompson, Motte & Andre 1999). Each of these models makes different predictions concerning the form of the radial density profiles in cold prestellar cores and their evolutionary successors, presumed to be represented by Class 0 protostars. The need to test these models has motivated extensive studies of the density profiles of prestellar cores using far-infrared and submillimeter imaging observations (see for example, Evans et al. 2001).
Previous detailed studies of dense cores have largely been focused on low mass star forming regions, such as Taurus and Ophiuchus. These regions are within 150 pc of the earth thus allowing high spatial resolution and strong signals. One important observational aspect to consider when tackling this problem is the mass function of dense molecular cores, especially in their relatively quiescent stages before being disrupted by the onset of stellar energy input. The core mass function may bear clues to the relative universality of initial mass function (IMF) of stars, which remains a prominent question in the field of star formation. If the core mass function resembles that of the stellar IMF, it is likely that the star formation processes within each core, including collapse, disk accretion, jet, and outflow, are uniform in terms of the efficiency of mass transfer from the ISM to stars. On the other hand, if the core mass function differs significantly from the stellar IMF, it is likely that environmental factors, such as competitive accretion, shape the resulting IMF. Most of the past studies find a core mass function similar to the stellar IMF (e.g. Motte, Andre & Meri et al. 1998; Testi & Sargent 1998; Young et al. 2006).
Given the importance of quiescent cores in a massive star forming environment, we have aimed to obtain data for a sizable sample of such cores in Orion, which is the closest known GMC. At a distance of about 450 pc, it is possible to detect individual cores through submillimeter imaging. We have also obtained limited amount of spectral line data (Li et al. 2003, paper I), which help to constrain the temperature of these cores and their ambient environment. Adequate spatial resolution and knowledge of the velocity structure (thermal, turbulent, and systematic) are crucial for obtaining an accurate estimate of the core mass and their dynamic state. In this paper, we will focus on the determination of the mass function of these quiescent cores.
Observations and Data Reduction
===============================
We have used the Submillimeter High Angular Resolution Camera (SHARC II, see Dowell et al. 2003), installed on the 10.4 meter telescope of the Caltech Submillimeter Observatory (CSO) to carry out the survey of quiescent Orion cores. The fields which we have mapped are chosen based on the presence of dense gas and the absence of indications of active star formation, such as IRAS sources and outflows. A more detailed discussion of the selection criteria can be found in paper I. The locations of our maps are indicated in Fig. \[fig:coverage\]. SHARC II is a 12 by 32 bolometer array, and each pixel samples a region of 4.85 on the sky. The footprint size of the total array is 2.59 by 0.97. The size of the individual maps in our data set is several times the footprint size, and the data were obtained by repeated scans of the telescope over the region of interest, using a scanning pattern (see below).
Orion is a region with bright, spatially extended emission at submillimeter wavelengths. The normal throw range of the chopping secondary is not enough for moving the reference beam completely off of the emission from the clouds. Chopping against the cloud emission background brings significant uncertainty into the absolute calibration and decreases the capability of detecting sources of larger sizes (e.g. comparable to the chopper throw angle). Our maps are obtained in the non-chopping BOXSCAN mode of SHARC II. By scanning the desired region quickly and repeatedly in a complex pattern, each sky position is covered by every bolometer in the array and with different time variations. Such redundancy enables the sky image to be computed subsequently through software iteration. We have employed the Comprehensive Reduction Utility for SHARC II (CRUSH) developed by A. Kov[á]{}cs at Caltech. This reconstruction has proven to be fairly stable and consistent in recovering both the bright peaks and extended structure in our maps.
During three observing runs from 2003 to 2005 at the CSO, we have obtained 350 maps for 8 fields toward quiescent portions of Orion. The sizes of these fields ranges from 4’$\times$4’ to 8’$\times$8’. About every 60 minutes, we obtained a short scan of a bright calibrator, such as Mars. The scans of the calibrator sources from each day are later reduced using the same CRUSH program to provide the absolute flux scale for our images. The CRUSH program corrects the data for atmospheric absorption utilizing measurements of opacity ($\tau$) provided by a tipper operated at 350 . The fluxes of planets and point sources at the time of observation are calculated by the FLUXES program obtained from JCMT.
After image reconstruction by CRUSH and flux calibration, the typical RMS noise level on the background part of a image (i.e. devoid of cores with peak flux greater than about the 10 $\sigma$ level) is about 0.15 Jy/beam. We further processed our calibrated data using the deconvolution program Hires (Backus et al. 2005) based on Richardson-Lucy iteration procedure (Richardson 1972; Lucy 1974). The deconvolution procedure is based on an idealized beam pattern obtained from repeated observations of Neptune. The deconvolution processing is stable and generally converges within 50 iterations. The edge pixels, where the signal to noise ratio is poor due to insufficient sampling, have been avoided during iteration (see the observed and deconvolved images of ORI7 in Fig. \[fig:ori7-11\] for an example).
The calibrated and deconvolved images are presented in Fig. \[fig:ori7-11\] through Fig. \[fig:ori8\]. The Hires deconvolution conserves the total flux. When expressed in the same units of Jy per (9)$^2$ beam as in the observed images, the peak values of the brightness of dust cores in the deconvolved image tend to be significantly higher than those in the original image. This increase is expected because deconvolution sweeps the flux from the area covered by the extended error beam pattern into the main beam. It also helps recover condensations that are close to a strong source. Both these consequences of using Hires are important for better determination of the mass and the number count of cores. Another direct result of deconvolution is that the size of cores is generally smaller than in the original image. The best possible determination of the core size is important for evaluating the dynamical state of the cores, as will be discussed later.
The detection of cores in our survey is limited both by the brightness of sources and their sizes. If the source is a point source, then the requirement for a positive detection is a high signal to noise ratio, nominally 10$\sigma$, in the one resolution element that has signal. If the source occupies $N$ resolution elements, then a lower signal to noise ratio in each pixel is adequate. To achieve the same statistical significance as that of the detection of the point source, the required signal to noise ratio in each pixel is scaled as $1/\sqrt{N}$. For an isothermal, optically thin dust core, the total continuum flux is linearly proportional to its dust mass. Therefore, we can define a minimum detection mass $M_{det}$ for a core of a certain size to be $$M_{det} = M_{point} \times \sqrt{N} \lc
\label{det}$$ where $M_{point}$ is the minimum mass of a point source (present only in one beam) to be detected. We note that $M_{det}$ scales linearly with the diameter of the core.
In Fig. \[fig:hist\], the minimum detection mass is plotted as a shaded area in the histogram plotted against mass bins. The lower mass boundary at 0.04 of the region of incompleteness is set by a point source of 1.5 Jy, i.e., a 10 $\sigma$ detection of the peak pixel. $M_{det}$ is calculated based on the same assumption of dust properties and cloud distance as used in the calculation of core masses (see § 4) and a representative dust temperature of 17 K. The upper mass boundary of the region of incompleteness, 0.6 , is set by a core of 0.35 pc diameter corresponding to about 2.5 angular size. This limit corresponds to the largest linear dimension of the SHARC II array. Judging by our maps, there exists no core larger than this size with structure so smooth that it is not resolved by SHARC II. Thus, the higher mass boundary represents a robust detection limit, above which the Orion core sample is complete.
Characterizing the Cores
========================
We used a two step method to identify the cores in our data and obtain their physical parameters. First, we employed Clumpfind (Williams et al. 1994) to obtain the intensity peaks. Clumpfind draws contours at specified levels and selects regions that are enclosed by these contours. In a two dimensional bolometer map, it is straightforward to assign a signal to noise ratio to the intensity peaks. We have used the following criteria to define an individual intensity peak – it must be stronger than 10 $\sigma$ based on a noise estimate from empty regions and the valley between two peaks must be lower than one tenth of the stronger peak. The intensity peaks thus obtained are also inspected to exclude isolated bright pixel spikes, which tend to appear toward edges of the maps, where the integration time is smaller than for the rest of the map. The end result is a list of clump candidates, for which the peak intensity pixels are at a significance level better than 10 $\sigma$, and which are well separated from nearby peaks.
To derive the clump parameters from these intensity peaks, we need to make some assumption about the boundary and shape of these structures. The assumption Clumpfind makes is to look for closed contour boundaries. In a region such as Orion where the core density is high and there is underlying diffuse structure, the intensity contours toward the edge of the clump are affected by emission not directly associated with individual clumps. The resulting core boundaries (contours) thus may have sharp corners and other strange shapes. However, incorrectly defining the core boundaries usually has only a small effect on the derived total intensities, which are dominated by the central pixels as defined by our selection criteria. But the sizes of cores can not be well defined this way. Therefore, we make another assumption to help us define a core, which is that the core structure projected onto the sky is a two dimensional Gaussian. Such an assumption enables us to fit the structure around an intensity peak by a well–defined Gaussian clump. We also simultaneously fit a flat background with the two dimensional Gaussian to account for the underlying diffuse structure. The majority of the cores are found to be nearly circular, thus the clump size is defined by the average of the 1/e dimensions of the fitted Gaussians. These fitted Gaussian prove to be good approximations to the two dimensional brightness structures observed. We present the characteristics of the 51 cores found in this study in Table 1.
Calculation of Core Masses
==========================
For an optically thin dust cloud, the dust emission can be described as (Hildebrand 1983) $$S(\nu)=N(a/D^2)Q(\nu)B(\nu,T_d) \lc
\label{fdust}$$ where $S(\nu)$, having units of erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$ (or Jy), is the flux density produced by a cloud at distance $D$. It is the summation of the emission from $N$ spherical grains, with absorption coefficient $Q(\nu)$ and geometric cross section $a$. The formula is obviously true for point sources and is still applicable for extended sources as long as the dust seen in a telescope beam is emitting isotropically.
The value of the parameter $Q(\nu)$ and its dependence on frequency are affected by intrinsic properties of dust grains, such as their size and composition. A simple power law ($Q(\nu)\propto \nu ^\beta$) model has been predicted by theoretical work (Gezari, Joyce & Simon 1973; Andriesse 1974) and has given reasonable fits to observations, but the spectral index $\beta$ varies from 0.6 to 2.8 (Wright 1987; Mathis & Whiffen 1989; Lis & Menten 1998). Moreover, $\beta$ itself may also depend on $\nu$. The general trend is that $\beta$ is smaller for higher frequencies (Hildebrand 1983; Draine & Lee 1984; Martin & Whittet 1990; Gordon 1995).
In better studied GMCs, such as M17, $\beta \approx 2$ is usually a good description of observations (Goldsmith, Bergin & Lis 1997). By combining continuum data at 350 and 1100 , Lis et al. (1998) find evidence that $\beta$ increases as the telescope beam moves away from the Orion Bar, a photon dominated region (PDR), to more quiescent gas further north. A larger $\beta$ in quiescent clouds is consistent with the hypothesis that grains grow in size in such environments. Lis et al. indicate that dust temperature $T_d = 17$ K and $\beta = 2.5$ for regions near ORI1. Their maps do not cover other regions in our survey, which are located south of the Orion Bar.
With the spectral index known, the actual value of $Q$ can be estimated and compared with ‘standard’ values measured at other wavelengths. Extrapolating the 125 emissivity of Hildebrand (1983) with $\beta=2$ gives $Q(350)=1\times10^{-4}$. Extrapolating the 1300 value of Chini et al. (1997) gives $Q(350)=2.2\times10^{-4}$. Direct measurements of part of the Orion molecular cloud by Goldsmith et al. (1997), with an assumed gas to dust ratio equal to 100 and dust temperature $T_d=17$ K, give $Q(350)=4\times10^{-4}$. We use a representative value of $Q(350)=2\times10^{-4}$ in this paper. The range of dust emissivity as discussed above is indicative of the uncertainties in determining the dust mass from dust continuum emission, due to the complexity involved in modeling $Q(\nu)$.
The temperature of dust grains is another issue in deriving the dust mass. In star forming regions, the dust temperature is determined by energy equilibrium between UV/optical absorption and infrared emission. The fact that grains can be of different sizes dictates a distribution in grain temperature. The existence of large, cold grains makes the dust mass derived from dust emission based on single temperature fits an underestimate (Li, Goldsmith, & Xie 1999). In well–shielded regions, the dust temperature will also be affected by gas–dust coupling. For lower densities ($<10^{5}$ ), the dust temperature will be a few degrees lower than the gas temperature, according to the modeling by Goldsmith (2001). At =$10^{6}$ , $T_d \approx T_{gas}$. In our cores, the average gas density estimated based on data presented in Table 1 is generally greater than $10^{6}$ . It is thus reasonable to use the gas temperature as a direct estimate of dust temperature.
We have good measurements of the gas temperature from observations for most of the survey regions, which are at $\sim$ 1 resolution (Paper I). The one $\sigma$ statistical uncertainty of our temperature measurements due to data noise is 0.9 K. The temperature in regions not covered in our ammonia survey (ORI7 and ORI11) can be estimated based on their distances to the Trapezium cluster. Because these regions are already more than 8 pc away from the main ionizing source, the uncertainty in the temperature estimate due to the external UV field (the main heating source) is smaller than $\sim$ 1.5 K (Paper I and Stacey et al. 1993). Without strong embedded sources and external heating from outside (as is the case for most of our surveyed Orion region), the temperature will drop toward the cloud centers. But the temperature decrease is small for most of the cores, as the external UV field for our selected regions is relatively weak compared to that in regions closer to the center of Orion. We will thus take the dust temperature to be equal to the gas temperature in the following discussion. This approximation is accurate for dense regions and an overestimate of $T_d$ for others. Due to this possible higher than true $T_d$, the dust mass we determine may be underestimated.
Assuming a grain radius $r=0.1$ , a grain density $\rho = 3$ g , cloud distance $D=480$ pc, gas to dust ratio $GDR=100$, and $Q(350)=2\times10^{-4}$ we can rewrite Eq. \[fdust\] in units more convenient for this situation $$M_{core} = 2.4 \times 10^{-2} M_\odot \Bigl[\frac{2\times 10^{-4}}{Q(350)}\Bigr]
\> \Bigl[\frac{\lambda}{350 \mu m}\Bigr]^3 \> \Bigl[\frac{D}{480pc}\Bigr]^2\>\Bigl[\frac{GDR}{100}\Bigr]
\> \Bigl[\frac{S(\nu)}{Jy}\Bigr] \> P_f (T_d) \lp
\label{nd}$$ The Planck factor, $P_f(T_d)=e^{h\nu/kT_d}-1$, is plotted in Fig. \[fig:td\]. We also plot the percentage change of the Planck factor if the dust temperature were to decrease by 1 K.
At a temperature of 15 K, a decrease of 1 K in the dust temperature corresponds to a 23% increase in the Planck factor. The fractional change drops to 13% at 20 K. For ORI1, even if the gas and dust temperatures are not closely coupled, the uncertainty produced by using the gas temperature in deriving the dust mass should not be large thanks to the relatively high temperatures. For colder sources, the knowledge of the dust temperature becomes crucial since the Planck factor diverges toward lower $T_d$. The lowest temperature used in our calculation is 12 K, at which the uncertainty in temperature corresponds to about 40% uncertainty in the derived core mass. Other than the dust emissivity, this factor is the largest source of uncertainty in our calculation. The mass of each core is given in Column 6 of Table 1.
The mean core mass of our sample, 9.8 , is about 10 times larger than those found in low mass star forming regions and isolated dark clouds (e.g. Benson & Myers 1989; Young et al. 2006). We believe this difference is not due to the greater distance of Orion than the well studied LMSF regions. The Benson & Myers (1989) study include both isolated cores and a collection of cores in Taurus. The median FWHM core diameter in that sample is 0.14 pc, which is about seven times the spatial resolution of SHARC II used in the present work ($\sim0.02$ pc at the distance of Orion). The BOLOCAM instrument used in Young et al. (2006) has a beam size of about 28, which corresponds to about 0.019 pc at a distance of 130 pc. This is about the same as the spatial resolution of SHARC II for Orion. The average separation of sources in Young et al. (2006) is about 0.09 pc, significantly larger than our resolution. In short, the higher mass of the cores in our sample is unlikely to be a consequence of their being combinations of multiple low mass cores such as those characterizing the Ophiuchus and Taurus regions.
Core Stability
==============
Without spectroscopic data at a spatial resolution that matches that of the submillimeter continuum, we cannot determine the balance between gravity and turbulence and/or thermal support. We can, however, examine the limiting case of the cores being only thermally supported.
For a spherical, self-gravitating, isothermal, and hydrostatically supported core, its density profile can be described by a family of solutions to the Lane-Emden equation (often called Bonnor-Ebert spheres: Ebert 1955 and Bonnor 1956), with the dimensionless radius being $$\xi = r \sqrt{4 \pi G \rho_c/v_s^2} \lc
\label{xi_def}$$ where $\rho_c$ is the central density and $v_s = \sqrt{kT/(\mu m_H)}$ is the sound speed. Each solution is characterized by a single parameter $\xi_{max}$, which is determined by the outer radius and central density. This represents a truncation of the infinite isothermal sphere and the density profile within is thus fixed by the core size and central density (see discussion by Alves, Lada & Lada 2001).
Tafalla et al. (2004) have studied an analytic approximation to the density profile of Bonnor Ebert spheres $$\rho(\xi) = \frac{1}{1+(\xi/2.25)^{2.5}} \lc
\label{xi}$$ and have shown that it is within 10% of the numerical solution given by Chandrasekhar & Wares (1949) for $\xi < 23$. Ebert (1955) and Bonnor (1956) pointed out that when $\xi$ exceeds a critical value of 6.5, the solution becomes unstable. Assuming $\xi_{max} = 6.5$ and using the observed core radius, a critical core mass $M_{BE-CR}$ can be calculated by integrating Eq. \[xi\].
In Fig. \[fig:m2r\], the sizes and mass of the cores are plotted along with the calculated curves based on critical Bonnor-Ebert spheres at various temperatures. The majority of our cores appear to be too massive to be stable Bonnor-Ebert spheres for a kinetic temperature as high as 30 K. The caveat for such a discussion is that the observed size of our cores are not as clearly defined as required by the theory. But the large excess of the observed core mass relative to the Bonnor Ebert critical mass suggest that increasing the core size by as much as a factor of a few would not make a core thermally stable.
To evaluate the importance of turbulence, we can also consider a modified BE sphere taking into account turbulence (cf. Lai et al. 2003). We can define an equivalent temperature $$T_{eq}=\frac{m_H \Delta V^2 }{8\ln(2) k} \lc
\label{teq}$$ where $\Delta V$ is the total full width half maximum (FWHM) line width of the gas. The observed FWHM line width of at 1 scale toward these regions ranges from 0.7 km/s to 1.4 km/s (paper I). We have plotted the critical mass curves based on $T_{eq} (\Delta V = 1$ km/s) and $T_{eq} (\Delta V = 1.5$ km/s) in Fig. \[fig:m2r\]. A turbulent line width of 1.5 km/s or more would in principle offer significant support. However, given the generally quiescent state of such cores and the decrease of turbulence from low density to high density regions (Paper I), one would not expect the turbulence within our cores to exceed the value observed at the larger (1) scale.
We do not have spectroscopic data with spatial resolution matching that of the present submm continuum data. The relevant extant data is from ammonia (Paper I), giving the linewidth at $\sim$50 scale, which is substantially larger than the core sizes. The ammonia linewidth (which is generally two to three times larger than the linewidth found in the cores studied by Benson & Myers 1989) seems to indicate approximate virial equilibrium between the random turbulent motions in these cores and their self–gravity. If the turbulence decreases at smaller scales as suggested in paper I, then it would not be sufficient to stabilize the cores, and they would be candidates for collapse and the onset of star formation.
Finally, we evaluate the importance of magnetic pressure. For a cloud with uniform density and uniform magnetic field, the maximum mass which can be supported by a steady $B$ field alone can be derived through virial theorem (e.g. Spitzer 1978) $$M_\Phi=\Bigl[\frac{5}{9G}\Bigr]^{1/2} Br^2 \lc
\label{mphi}$$ i.e., the magnetic supported mass is proportional to the flux. It is easy to see that if the cloud is conducting thus freezing the magnetic flux, the contraction of a cloud cannot proceed if the starting cloud mass is smaller than $M_\Phi$. Such a uniform cloud combined with isothermal conditions cannot be in pressure equilibrium. Numerical simulations have been carried out to study magnetized clouds under more realistic conditions. The evolution of such a system is complicated, depending on the initial mass, geometry, and the threading of magnetic field. For our purpose of estimating the overall importance of a steady field, we note that simulations for spherical magnetized clouds (Mouschovias & Spitzer 1976; Tomisaka, Ikeuchi & Nakamura 1988) give the critical mass of a cloud in equilibrium to be of the same form as in Eq. \[mphi\], modified by a correction factor $M^\prime_\Phi = c_\Phi M_\Phi$. The correction factor, $c_\Phi$, is shown to be smaller than unity, as the centrally enhanced density profile of a cloud in pressure equilibrium makes the cloud easier to ’squeeze’ from outside. We thus use Eq. \[mphi\] to estimate the maximum mass that can be supported by a steady magnetic field.
The measurement of the magnetic field strength is difficult. There exists one such measurement in the Orion molecular cloud. Using the IRAM 30m telescope, Crutcher et al. (1999) detected the Zeeman effect in the CN 3mm line near Orion BN/KL. The field strength is derived to be either 190 $\mu$G or 360 $\mu$G depending on the fitting scheme. This is much larger than the $B\sim30 \mu$ G measured in dark clouds (e.g. Goodman et al. 1989). Given the proximity of BN/KL to active star formation, the large value of $B$ could be explained by a rapid collapse freezing the magnetic flux into high density regions along the line of sight. It is not clear how different the magnetic field should be in our quiescent cores, but it is expected to be smaller than the Orion KL values measured by Crutcher et al. We thus take a nominal $B=100\>\mu$G in our calculations of $M_\Phi$. The resulting $M_\Phi$ is also plotted in Fig. \[fig:m2r\]. A static B field of this magnitude adds only minimal support and even the higher value of the magnetic field measured by Crutcher et al. (1999) would not be significant.
To determine the dynamical state of these cores more accurately will require higher angular resolution spectroscopic data for the gas as well as better determination of the magnetic field strength and morphology. At this point we can say that the large mass of the Orion cores derived in section 4 suggests that these cores are either collapsing or are supported by strong turbulence.
Core Mass Function
==================
Based on the core masses derived, we present the statistics of the core masses in Fig. \[fig:hist\], left panel. This figure displays the differential mass distribution, in which the data are binned into mass ranges of specified size. When grouped in mass bins of uniform width 5 , the number of cores within each mass bin drops from 26 in the lowest mass bin (0 to 5 ) to 1 in the highest mass bin (40 to 45 ). The incompleteness of the survey is in the lowest mass bin. To get a better look at the distribution of core masses at lower masses, the histogram is also plotted in terms of mass bins having equal logarithmic widths (Fig. \[fig:hist\], right panel).
We would like to be able to represent the distribution of core masses by some simple function, which is generally referred to as the core mass function, or CMF. In the following we show that determining a robust CMF from the core mass distribution is not a trivial undertaking. There are two widely used approaches to fitting an analytic function to the core mass distribution in order to determine the CMF. These are to employ (1) the cumulative mass function, which considers the fraction of cores having mass greater than some specified value, and (2) the differential mass function discussed above. In this paper, CMF refers to the differential core mass function as defined in Eq. \[dndm\] in the Appendix unless specified otherwise.
We present both types of mass functions in Fig. \[fig:cmf\]. The cumulative CMF appears to suggest two power laws: a steep power law with an equivalent index of -2.2 for the larger mass cores and a flatter power law of slope -1.1 for the lower mass ones. The fitting results based on the assumption of two power laws are similar to results from several past studies (e.g. Reid & Wilson 2005 and references therein). However, as we will discuss in detail in the Appendix, fitting multiple power laws to the cumulative mass function is prone to ambiguous interpretation, especially when the power law index of the CMF is about -1.
For the Orion cores, the single power law fits based on the differential CMF yield indices close to -1. One example based on 10 bins is plotted in the right hand panel of Fig. \[fig:cmf\], which has a best fit index of $\alpha$ = -0.85$\pm0.21$. If we change the number of bins, the exact value of fitted index varies between about -0.8 to -0.95. These values are close to -1 and any two power law interpretation of the cumulative CMF must therefore be considered with some caution.
To understand better the consequences of different fits to the cumulative CMF, we have constructed simulated core samples and analyzed them to evaluate different fitting procedures. The cores are randomly generated within the same range of mass as that of the observed Orion core sample and the probability density function according to which a core will have certain mass is based on a power law. In Fig. \[fig:sim\], the cumulative CMF of the Orion cores and a family of simulated core samples are shown together with our CMF determined for the Orion cores. Each simulated sample has 1000 cores and the true mass functions for the samples are power laws with indices ranging from 0 to -2.0. Fig. \[fig:sim\] makes it obvious that the cumulative CMF plotted in log–log space has curvature for power law indices larger (flatter) than -1.5. In the extreme and illustrative case of $\alpha = 0$, the underlying CMF is strictly flat, i.e., the same probability of detecting high mass cores as low mass ones. The cumulative CMF, however, has a clear turnover, as any sampling (or observation) of a probability density function is done in a limited mass range. This should already sound an alarm for any direct power law fit to a cumulative function as a way of modeling the underlying CMF.
When sampled somewhat sparsely, the cumulative CMF can easily be mistaken to be characterized by two regions each having a different power law index, with the low mass end being flatter and the high mass end being steeper. However, an important result here is that the simulated CMF which best agrees with that of the observed cores has a power law index of -0.80. This value is consistent with the result from the fit to the differential CMF given above. The advantage of using the cumulative mass function lies in the fact that the number of data points is equal to the number sources in the sample. That number will obviously be greatly reduced when binning is required, as is the case for the differential function. In the case of the index being close to -1, however, the differential mass function is a much more reliable method than the cumulative mass function to derive the CMF.
In fact, when comparing with other massive regions, such as Orion B (Johnstone et al. 2001), NGC 7538 (Reid & Wilson 2005) and RCW 106 (Mookerjea et al. 2004), a power law index close to -1 can be derived from their core samples when only those cores in a mass range similar to ours are included. The steeper power law indices ($\sim$ -2.3) based on those samples are usually a fit to the higher mass portion of their cores (see the review in Reid & Wilson 2005). Due to poorer angular resolution and sometimes larger distances, the high end of core mass in these samples can be as large as tens of thousands of and some contain water masers or bright infrared sources, which are signs of active star formation. One should use caution when interpreting these dense structures together with resolved cores, especially when multiple power laws are fitted to a cumulative CMF.
From a theoretical viewpoint, Padoan & Nordlund (2002) give an analytic relation between the CMF index and the power spectrum index, Such a relation would predict a stellar IMF–like CMF for core masses larger than 1 with $\beta = 1.74$, consistent with some observations (Miesch & Bally 1994) , but different from others (Brunt & Heyer 2002). For lower mass cores, these authors state that the mass distribution will be flatter as the number of gravitationally unstable cores drops. Simulations (e.g. Gammie et al. 2003, Klessen & Burkert 2001, Tilley & Pudritz 2004) under varying conditions can produce core samples with a CMF consistent with the Salpeter IMF, although this conclusion is not very restrictive and may depend on the evolutionary time of the observation (see discussion by Gammie et al. 2003). A very recent numerical study (Ballesteros-Paredes et al. 2006) finds that the CMF is dependent on the sonic Mach number and that the CMF in a supersonic turbulent flow may have changing slopes as a function of time. Although different in their opinions regarding the direct relationship between IMF and CMF, these studies all relate the CMF to the turbulence conditions in the clouds included in their calculation. In order to compare theoretical predictions of the CMF with observations, it is important to have similar definition and scales of structure while assembling the statistics. The Orion cores are likely to be supercritical (section 5), are quiescent, and are probably precursors of protostars. The flat CMF of Orion cores suggests that evolution and shaping of the mass distribution of dense gas continues after collapse has already started.
Based on consideration of both the cumulative and the differential CMF, we find that the CMF of the Orion cores has a significantly flatter CMF than that of low mass star forming regions. This may reflect the different environment in these regions. In particular, as the Orion CMF is also significantly flatter than the stellar IMF, one would expect environmental effects in the later stages of star formation to shape the IMF, as it cannot be a result of collapsing each core with a similar star formation efficiency.
Conclusion
==========
We have identified 51 dust cores in a 350 submillimeter continuum survey of the quiescent regions of the Orion molecular cloud using the SHARC II camera. The enhanced spatial resolution of our data using the Hires deconvolution tool and our knowledge of the temperature from ammonia mapping (Paper I) enable us to determine relatively accurately the number, size and the total mass of these cores. This Orion dust core sample:
1. is a collection of resolved or nearly resolved cores, with a mean mass of 9.8 which is one order of magnitude higher than that of resolved cores in low mass star forming regions;
2. includes largely thermally unstable cores, which are unlikely to be stabilized by the magnetic field, suggesting that the cores are supported by strong turbulence or are collapsing;
3. has a power law core mass function with index $\alpha =$ -0.85$\pm 0.21$, which is significantly flatter than the stellar IMF and than that found for core samples in low mass star forming regions.
Our comparison of the use of differential and cumulative mass functions to analyze the core mass distribution indicates that the differential approach, while requiring more cores due to the binning involved, is more robust and has better defined statistical uncertainties. The cumulative mass function approach can erroneously suggest multiple power law indices, particularly if the underlying core mass distribution is characterized by a power law index $\simeq$ -1.
Appendix: Differential and Cumulative Function for Power Law Distribution
=========================================================================
Following the convention used for the stellar initial mass function, we can define a power law core mass function (CMF) for cores as $$\frac{dN}{dM} \sim M^{\alpha} \lc
\label{dndm}$$ where the number of cores $dN$ in certain mass range $dM$ is a power law of index $\alpha$. The cumulative CMF is then $$\begin{aligned}
\nonumber N(>M_0) &=& \int_{M_0}^{M_{max}} dN \\
&\sim& \frac{1}{\alpha+1}M_{max}^{\alpha+1}-\frac{1}{\alpha+1}M_{0}^{\alpha+1} \lp
\label{nm}\end{aligned}$$ where $M_{max}$ is the maximum mass for a certain sample.
It is obvious that in an ideal case, the cumulative CMF will also be a power law, but with a flatter slope. In the literature, the cumulative CMF plotted on a log–log scale has been fitted directly by straight lines to give one or multiple power law indices. The indices are then decreased by one to give the CMF power law indices as defined in Eq. \[dndm\].
Such direct fit does not work when the power law index is close to -1. When $\alpha =$ -1, the integral in Eq. \[nm\] results in an log function, $log (M_{max}) - log (M_0)$ , [*not*]{} a power law. Instead of a single slope on a log–log plot, the appearance of the cumulative mass function has curvature, even when the underlying core mass distribution can be characterized by of a single power law.
As illustrated in Fig. \[fig:sim\], the cumulative mass function does behave as described above. Only when $\alpha$ $\leq$ -1.5 does a straight line fit become reasonable. When the number of cores is large enough to allow binning of the data, the differential core mass function is a straightforward representation of the CMF. The analysis of such a CMF would give directly the index of a power law mass function and with relatively well–defined statistical uncertainty relatively. The advantage of using the cumulative function lies in the fact that the number of data points equals the number of cores, and thus no information is lost. A direct power law fit to an arbitrary cumulative function, however, is unreliable. If working with the cumulative mass function is unavoidable, a family of models should be generated from a series of power laws with different indices and the best fit model defined as the one that best reproduces the cumulative CMF of the data. The caveat for this approach is that the uncertainties are not well defined, as the data points are not independent of each other.
This work was supported by the Jet Propulsion Laboratory, California Institute of Technology. This work has made use of NASA’s Astrophysics Data System. Di Li acknowledges the support coming through the Resident Research Associate program of the National Research Council and the NASA Postdoctoral Program. We thank D. Dowell for his valuable help with observing and data processing. Research at the Caltech Submillimeter Observatory is supported by NSF grant AST-0229008.
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![Our maps are in the green rectangles, which are overlaid on a $^{13}$CO 1-0 integrated intensity map reproduced from Bally et al. (1987). The red rectangle indicates, roughly, the coverage provided by the data from Lis et al. (1998). The yellow star indicates the location of the Trapezium cluster.[]{data-label="fig:coverage"}](f1_p.eps)
![350 images of the Orion cores. The left hand panel is the observed image and the right hand panel is the deconvolved image. The coordinates are in J2000. Both images are plotted in units of Jy per 9 beam. The yellow squares indicate the peak positions of cores found by the Clumpfind program. The beam sizes of the observed and deconvolved images are shown as filled circles. The upper panel shows the survey field ORI7, and the lower panel, ORI11. []{data-label="fig:ori7-11"}](f2a_p.eps "fig:"){width="80.00000%"}\
![350 images of the Orion cores. The left hand panel is the observed image and the right hand panel is the deconvolved image. The coordinates are in J2000. Both images are plotted in units of Jy per 9 beam. The yellow squares indicate the peak positions of cores found by the Clumpfind program. The beam sizes of the observed and deconvolved images are shown as filled circles. The upper panel shows the survey field ORI7, and the lower panel, ORI11. []{data-label="fig:ori7-11"}](f2b_p.eps "fig:"){width="80.00000%"}
![Survey fields, ORI1, and ORI2. Units and layout are the same as those for Figure 2. []{data-label="fig:ori1-2"}](f3a_p.eps "fig:"){width="80.00000%"}\
![Survey fields, ORI1, and ORI2. Units and layout are the same as those for Figure 2. []{data-label="fig:ori1-2"}](f3b_p.eps "fig:"){width="80.00000%"}
![Survey fields, ORI4, and ORI5. Units and layout are the same as those for Figure 2.[]{data-label="fig:ori45"}](f4a_p.eps "fig:"){width="80.00000%"}\
![Survey fields, ORI4, and ORI5. Units and layout are the same as those for Figure 2.[]{data-label="fig:ori45"}](f4b_p.eps "fig:"){width="80.00000%"}
![Survey fields, ORI8SE, and ORI8NW. Units and layout are the same as those for Figure 2.[]{data-label="fig:ori8"}](f5a_p.eps "fig:"){width="80.00000%"}\
![Survey fields, ORI8SE, and ORI8NW. Units and layout are the same as those for Figure 2.[]{data-label="fig:ori8"}](f5b_p.eps "fig:"){width="80.00000%"}
|
---
abstract: 'We construct and analyze a model of the relativistic steady-state magnetohydrodynamic (MHD) rarefaction that is induced when a planar symmetric flow (with one ignorable Cartesian coordinate) propagates under a steep drop of the external pressure profile. Using the method of self-similarity we derive a system of ordinary differential equations that describe the flow dynamics. In the specific limit of an initially homogeneous flow we also provide analytical results and accurate scaling laws. We consider that limit as a generalization of the previous Newtonian and hydrodynamic solutions already present in the literature. The model includes magnetic field and bulk flow speed having all components, whose role is explored with a parametric study.'
author:
- Konstantinos Sapountzis
- Nektarios Vlahakis
bibliography:
- 'Kostas.bib'
date: 'Received/Accepted'
title: Rarefaction wave in relativistic steady magnetohydrodynamic flows
---
Introduction
============
When a flow passes an acute vertex of an angle the information of the boundary’s first derivative discontinuity propagates in the flow, and if the velocity of the flow exceed that of the fastest disturbances, i.e. the fast magnetosonic velocity, that propagation is performed in the form of a rarefaction wave; the resulting flow suffers a weak discontinuity[^1]. The importance of the rarefaction waves is significant in a number of phenomena and as such the reader can find relevant studies in various conditions and environments. The Newtonian hydrodynamic case is presented in many textbooks (see for example Landau & Lifschitz[@Landau_fluid]), while a relativistic analytical approach under the same conditions was made by Granik[@Granik_1982].
Moreover, the relativistic and highly magnetized counterpart of the phenomenon is mainly met at the high energy astrophysics where these extreme conditions apply, notably in Gamma-Ray Bursts. Most astrophysical settings are considered as axisymmetric with ignorable azimuthal ($\phi$) coordinate, in which case the study can be done on the so-called poloidal plane. At sufficiently large cylindrical distances from the symmetry axis, axisymmetry can be well approximated by planar symmetry, with the $\hat \phi$ direction replaced by the $\hat y$ direction in a Cartesian system of coordinates. We can continue to use the term “poloidal plane” for the $xz$ plane of this system and split all vector quantities in “poloidal” (i.e., projections on the $xz$ plane) and transverse ($\hat y$) components. The interested reader is referred to [@Tchekhovskoy_Narayan_2010; @Komissarov_Vlahakis_Konigl_2010] for numerical simulations and relevant discussions for axisymmetric and planar-symmetric Gamma-Ray Burst flows, [@Mizuno_2008; @Zenitani_2010] for simulations including an external pressure profile, and [@Kostas_2013] for a semi-analytical planar-symmetric model containing only transverse magnetic field ($B_y$).
In this paper we present a general semi-analytical model which, besides $B_y$ contains a poloidal magnetic component of arbitrary magnitude, and discuss its potential implications. We also study the effect of the initial transverse velocity, and derive accurate scaling laws for the flow physical quantities. Our study is performed in the framework of the planar symmetric, ideal and relativistic steady-state MHD. The procedure is similar to the one followed in [@Kostas_2013], but also to the hydrodynamic approach of [@Landau_fluid] and its relativistic counterpart [@Granik_1982], and it is based on the class of the $r$ self-similar solutions. Beyond its potential astrophysical applications, the theoretical importance of rarefaction is evident, and the aim of the present work is to provide an insight to the relativistic magnetized regime, and thus to serve as a generalization of the already available hydrodynamical solutions.
We use the similarity property to degrade the system of the high nonlinear partial differential equations to a system of ordinary differential ones that are easier to manipulate. In section \[seceq\] we present the full steady-state equations, and in section \[secsmodel\] we apply the self-similarity to obtain the semi-analytical system. In section \[results\] we integrate the resulting system using a simple numerical algorithm for cases where the relative significance of the poloidal magnetic field alters. We also included some models with different initial transverse velocities, as also three models corresponding to the numerical simulations of [@Mizuno_2008], in order to check further the validity of our model. Section \[discussion\] contains the relevant discussion, while in Appendix \[appA\] we derive analytical scaling laws for the interesting case of a cold and homogeneous flow.
Steady-state equations {#seceq}
======================
The system of relativistic MHD equations is expressed by the equations determining the hydrodynamical properties of the flow under the influence of the electromagnetic field (emf), Maxwell’s and Ohm’s laws. The energy-momentum tensor is constructed as the superposition of the matter ($T^{\mu\nu}_{\rm{hy}}$) and the emf ($T^{\mu \nu}_{\rm{EM}}$) tensors $$T^{\mu \nu}=T^{\mu \nu}_{\rm{hy}}+T^{\mu \nu}_{\rm{EM}} \,.$$ The former one is given by $$T^{\mu \nu}_{\rm{hy}}={h}{\rho}u^{\mu} u^{\nu} +p n^{\mu \nu} \,,$$ where $u^{\nu}=\left({\gamma}c,{\gamma}\bm{v}\right)$ the plasma four-velocity, $\bm{v}$ the three-velocity and ${\gamma}=1/(1-v^2/c^2)^{1/2}$ the Lorentz factor. Neglecting gravity and general relativistic effects we chose a Minkowski metric $g^{\mu\nu}=n^{\mu\nu}=\left(-,+,+,+\right)$. The thermodynamical parameters $h$, $p$, and ${\rho}$ are the enthalpy per rest energy, pressure, and matter density of the plasma as measured in the comoving frame. For a gas obeying a polytropic equation of state the following relation applies $$\label{eqenth0}
{h}=1+\frac{{\hat{\Gamma}}}{{\hat{\Gamma}}-1}\frac{p}{{\rho}c^2}$$ with ${\hat{\Gamma}}$ the usual polytropic index.
The components of the $T^{\mu \nu}_{\rm{EM}}$ in analytical form are $$T^{00}_{\rm EM}=\frac{E^2+B^2}{8 \pi} \,, \quad T^{0j}_{\rm EM}=T^{j0}_{\rm EM}=\left(\frac{\bm{E} \times {\bm{B}}}{4 \pi} \right)_j \,,$$ $$T^{jk}_{\rm EM}=-\frac{E_j E_k + B_j B_k}{4 \pi}+\frac{ E^2 + B^2}{8 \pi} \eta^{jk} \,,$$ where Latin indices $i,j=1,2,3$ stand for the spatial coordinates, while Greek for all, and $\bm{E}, \bm{B}$ the electric and magnetic field as measured in the laboratory frame. We can identify $T^{00}_{\rm EM}\,, T^{j0}_{\rm EM}\,, T^{jk}_{\rm EM}$ as emf energy density, energy flux, and magnetic stress contributions, respectively. The full energy-momentum tensor provides the equations of motion in the covariant form $T^{\mu\nu}_{,\nu}=0$, but a straightforward use of all of these equations leads to difficult manipulating forms. Thus it is a common practice to substitute some of them with other equivalent, but simpler ones, as explained below.
At the steady-state limit, the continuity equation $({\rho}u^{\nu})_{\,,\nu}=0$, is written in vector form $$\label{continuity}
{\nabla}\cdot \left( {\gamma}{\rho}\bm{v} \right) =0 \,.$$ The projection of the energy-momentum equation on the proper time direction ($u_\nu T^{\mu\nu}_{,\nu}=0$) provides the entropy conservation $$\label{entropycons}
\bm{v} \cdot {\nabla}{ \left( \frac{p}{{\rho}^{{\hat{\Gamma}}}}\right) } = 0 \,.$$ The polytropic index takes the adiabatic values, $4/3$ and $5/3$ in the limits of ultrarelativistic and nonrelativistic temperatures, respectively.
Maxwell’s equations for the steady-state become $$\begin{aligned}
\label{Maxwell}
{{\nabla}} \cdot {\bm{B}}=0 \,, \quad {\nabla}\cdot {\bm{E}}= \frac{4 \pi}{c} J^0 \,,
\nonumber\\
{\nabla}\times {\bm{B}}= \frac{4 \pi}{c}{\bm{J}}\,, \quad {\nabla}\times {\bm{E}}=0\,,\end{aligned}$$ where $J^{\nu}=\left(J^0,\bm{J}\right)$ the four-current, $J^0/c$ the charge density and $\bm{J}$ the current density. Moreover, at the limit of an infinite electrical conducting plasma the comoving electric field is zero, and Ohm’s law yields $$\label{ohms}
{\bm{E}}=-\frac{\bm{v}}{c} \times\bm{B}\,.$$ We can write explicitly the spatial components of the momentum equation using Maxwell’s equations as $$\label{spatmomentum}
-{\gamma}{\rho}\left({\bm{v}} \cdot {\nabla}\right) \left({h}{\gamma}{\bm{v}} \right)
-{\nabla}p +
\frac{J^0 {\bm{E}}+{\bm{J}} \times {\bm{B}}}{c} =0 \,.$$ Equations (\[continuity\]–\[spatmomentum\]) together with the boundary conditions determine in principle the flow, but the high nonlinear character make this task rather difficult.
![The geometry of a planar symmetric rarefied flow and the coordinate system. The coordinate $y$ is ignorable and the plane $xz$ is the “poloidal” plane. Notice the three regions that in principle exist: the undisturbed plasma (which is in pressure equilibrium with its environment), the rarefied one, and the vacuum. The situation is similar to a supersonic hydrodynamic flow around an acute angle; here the flow is magnetized and it is super-fast magnetosonic. The weak discontinuity, i.e. the rarefaction front, and the contact discontinuity separating the plasma fluid from the void space, are also shown. Angles $\theta$, $\vartheta$ stand for the polar angle and the poloidal field/streamline inclination respectively; both are measured from the $z$-axis clockwise.[]{data-label="geom"}](figures/sketch.eps){width="42.00000%"}
We carry a first partial integration assuming Cartesian coordinates with the axis origin on the boundary discontinuity and planar symmetry along the $\hat y$ direction $\left(\partial / \partial y=0\right)$, see Fig. \[geom\]. From Faraday’s law the electric field is related to an electric potential, $\bm E=-{\nabla}V$. Assuming that the potential is also planar symmetric, $V=V(x,z)$, we find that the transverse electric field vanishes ($E_y$=0). That symmetry in conjunction with Ohm’s law provides $\bm{v}_p \parallel \bm{B}_p$, and we can therefore write the flow velocity in the form $$\label{masintdef}
{\bm{v}}=\frac{{k}}{{\gamma}{\rho}}{\bm{B}} + c{\chi}\hat y \,,$$ where $${k}={\gamma}{\rho}\frac{v_p}{B_p} \,, \quad
{\chi}=\frac{v_y}{c}-\frac{v_p}{c}\frac{B_y}{B_p} \,.$$ Both quantities ${k}$, ${\chi}$ are integrals of motion, i.e. remain constant along a poloidal streamline (or field line). The former integral stands for the ratio of the mass to the magnetic flux, while the second one is investigated later. Furthermore, we introduce the fluxes per unit length in the $\hat y$ direction, ${A}$ and ${\Psi}$, to label the poloidal magnetic field lines and the poloidal streamlines, respectively: $${A}=\int \bm B_p \cdot d\bm s\times\hat y \,, \quad {\Psi}=\int{\gamma}{\rho}\bm v_p \cdot d\bm s\times\hat y \,,$$ where the integration is performed on a line on the polidal plane, starting from a point of the $z$ axis. Reverting the above relationships we obtain $$\bm B_p= {\nabla}{A}\times \hat y \,, \quad \bm v_p=\frac{1}{{\gamma}{\rho}} {\nabla}{\Psi}\times \hat y \,,$$ where the equivalency of whether we use the poloidal magnetic field lines, or the poloidal streamlines is stated explicitly, ${\nabla}{\Psi}={k}{\nabla}{A}$. In this paper we use the magnetic field line notation to project the energy momentum equation parallel ($\hat{b}=\bm{B}_p / B_p$) and perpendicularly ($\hat{n}=-{\nabla}{A}/ \left|{\nabla}{A}\right|$) to the field lines direction; notice that in the limit of the negligible poloidal magnetic field this choice poses some easy-lifted complications of interpretation nature, see the end of next section.
The constancy of ${k}$ is derived by inserting the velocity form (\[masintdef\]) in the continuity equation (\[continuity\]), using the planar symmetry and the zero divergence of $\bm B$ to obtain $\bm B_p \cdot {\nabla}{k}=0 \Leftrightarrow{k}={k}({A}) $. The substitution of the same velocity expression in Ohm’s law (\[ohms\]) yields $$\label{electrictoBp}
{\bm{E}}=-{\chi}{\nabla}{A}\,, \quad E={\chi}B_p \,,$$ which in conjunction with the Faraday’s law provides also the constancy of ${\chi}$. It is useful to compare that integral with the corresponding Ferraro’s isorotation law in axisymmetric flows, which is related to the so-called light cylinder; see for example [@beskin_mhd] for a general analysis. On this cylinder the ratio $E/B_p$, which in the axisymmetric case is a function of the cylindrical distance, becomes unity. In the planar symmetric case no such cylinder exists, and this is reflected to the constancy of the ratio $E/B_p$. This is going to have an important role to the power laws derived later.
Moreover, Eq. (\[entropycons\]) provides the usual polytropic equation $$\label{eqQdef}
Q({A})=p / {\rho}^{{\hat{\Gamma}}} \,,$$ and thus $Q({A})$ integral states the entropy conservation along streamlines. Two more quantities complete the set of the integrals $$\begin{aligned}
{P}={P}({A}) = {h}{\gamma}v_y -\frac{B_y}{4 \pi {k}} \,, \\
\mu=\mu({A})= {h}{\gamma}- \frac{{\chi}B_y}{4 \pi {k}c} \,,\end{aligned}$$ and stand for the total (matter + emf) momentum-to-mass flux ratio, and the total energy-to-mass flux ratio, respectively. No more independent integrals exist, but a useful combination which appears often in the subsequent calculations is ${\chi_A}^2={P}{\chi}/(\mu c)$.[^2]
Besides the five integrals (${k}$, ${\chi}$, $\mu$, ${\chi_A}$ or ${P}$, $Q$) two more equations are needed to fully determine the flow. For convenience we introduce the “Alfvénic” Mach number $$\label{Mdef}
M^2\equiv\frac{\left({\gamma}v_p\right)^2}{B_p^2/(4 \pi {\rho}{h})}=\frac{ 4 \pi {h}{k}^2}{{\rho}} \,,$$ and the magnetization parameter $$\label{sigdef}
\sigma=-\frac{{\chi}B_y}{4 \pi {\gamma}{h}c {k}} \,,$$ i.e., the ratio of the Poynting to mass energy flux. In terms of the above quantities the physical quantities are written as $$\begin{aligned}
\label{primedef}
{\rho}=\frac{4 \pi {h}{k}^2}{M^2} \,,
\quad p=Q{\rho}^{{\hat{\Gamma}}} \,,
\quad {h}= 1+\frac{{\hat{\Gamma}}}{{\hat{\Gamma}}-1}\frac{p}{{\rho}c^2} \,, \quad \\
\label{eqnprimedefB}
{\bm {B}} = {\nabla}{A}\times \hat y -\frac{4 \pi \mu{k}c ({\chi}^2 -{\chi_A}^2)}{{\chi}(M^2 +{\chi}^2-1)} \hat y \,, \quad
{\bm{E}}=-{\chi}{\nabla}{A}\,, \quad \\
{\gamma}=\frac{\mu}{{h}} \frac{M^2+{\chi_A}^2-1}{M^2+{\chi}^2-1}\,, \quad \\
{\gamma}\frac{\bm{v}}{c}=\frac{M^2}{4 \pi c {k}{h}}
{\nabla}{A}\times \hat y
+\frac{{\chi_A}^2\mu}{{\chi}{h}} \frac{M^2+{\chi}^2-{\chi}^2/ {\chi_A}^2}{M^2+{\chi}^2-1}\hat y \,. \quad\end{aligned}$$
An interesting situation arises when the $M^2+{\chi}^2-1$ denominator vanishes corresponding to the so-called Alfvénic surface. The requirement that $B_y$ remains finite at that surface yields ${\chi}^2={\chi_A}^2$. Since these are integrals of motion they remain equal everywhere, meaning that $B_y=0$ and the flow carries no Poynting flux. For this reason magnetized planar symmetric flows cannot be trans-Alfvénic.
The initial conditions determine the integrals of motion, but one seeks for the quantities ${A}$ and $M$ or ${h}$; the last two are related by the expression, using Eqs. (\[primedef\]): $$\label{mach-h}
M^2=4\pi {k}^2\left(\frac{{\hat{\Gamma}}}{{\hat{\Gamma}}-1} \frac{Q}{c^2}\right)^{\frac{1}{{\hat{\Gamma}}-1}} {h}\left({h}-1\right)^{-\frac{1}{{\hat{\Gamma}}-1}} \,.$$
The two remaining equations are the Bernoulli (or wind equation) $$\begin{aligned}
\label{bernoulli1}
\frac{\mu^2}{{h}^2} \frac{\left(M^2+{\chi_A}^2-1\right)^2 - \left({\chi_A}^2/{\chi}\right)^2 \left(M^2+{\chi}^2-{\chi}^2/{\chi_A}^2\right)^2 }{\left(M^2 +{\chi}^2-1\right)^2} \nonumber\\
=1 + \left(\frac{M^2 {\nabla}{A}}{4 \pi c {k}{h}} \right)^2 \,, \qquad\end{aligned}$$ which is obtained by substituting all the quantities in the identity ${\gamma}^2 - ({\gamma}{v_y}/c)^2 =1+({\gamma}{v_p}/c)^2$, and the transfield equation obtained by projecting the momentum equation perpendicular to the magnetic field $$\begin{aligned}
\label{transfield}
M^2 \mid {\nabla}{A}\mid^2 \left[
{\nabla}^2 {A}- {\nabla}{A}\cdot {\nabla}\ln \mid {\nabla}{A}\mid \right]
\nonumber \\
-\frac{{\hat{\Gamma}}-1}{{\hat{\Gamma}}} {\nabla}\left[
16 \pi^2 {k}^2 c^2 \frac{{h}({h}-1)}{M^2}
\right] \cdot {\nabla}{A}\nonumber \\
+{\chi}\frac{d{\chi}}{d{A}} \mid {\nabla}{A}\mid^4
+\left( {\chi}^2-1 \right) {\nabla}^2 {A}\mid {\nabla}{A}\mid^2
\nonumber \\
-\frac{1}{2} {\nabla}\left( \frac{4 \pi {k}\mu c}{{\chi}}
\frac{{\chi}^2 -{\chi_A}^2}{M^2 +{\chi}^2-1}
\right)^2 \cdot {\nabla}{A}=0 \,.\end{aligned}$$
Roughly speaking we can state that the solution of the transfield equation determines the shape of the streamlines, while Bernoulli the energetics along them, but this distinction is not clear neither fruitful. Both equations must be solved simultaneously and a simple inspection shows the difficulties involved. The task of finding an analytical solution in the general case seems impossible, and thus all the efforts are focused on the quest of a solution with specific symmetries suitable to describe the particular problem; in our case this is the self-similar shape of the poloidal streamlines.
The self-similar model {#secsmodel}
======================
In order to induce the similarity property we assume that all the quantities have a dependence of the form $r^{{F}_i}f_i(\theta)$ where $r=\sqrt{x^2+z^2}$ the distance from the corner and $\theta$ the angle measured from the $z$-axis ($x=r\sin\theta$, $z=r\cos\theta$). Our goal is to determine the various exponents ${F}_i$ in such a way that the resulting differential equations will be separable on the variables $r$, $\theta$. The method of similarity is quite familiar and has been used before in a number of studies both in Newtonian [@Blandford_1982], [@Vlahakis_Tsinganos_1998] or in the relativistic context [@Begelman_1992; @Contopoulos_1994; @Vlahakis_2003a].
The substitution of the similarity expressions in Eqs. (\[bernoulli1\], \[transfield\]) is straightforward and by inspection we conclude that our model derives separable equations under the following forms $$\begin{aligned}
{A}=-r^{F}a(\theta) \,, \quad
{k}=r^Y \kappa(\theta) \,, \quad
Q=r^Z q(\theta) \,,
\nonumber \\
M=M(\theta) \,, \quad {h}={h}(\theta) \,, \quad
{\chi_A}^2\,,\mu\,,{\chi}=\mbox{const}\,.\end{aligned}$$ Substituting the above forms into the Bernoulli equation (\[bernoulli1\]), we obtain ${F}-Y-1=0$, while from Eq. (\[mach-h\]) we conclude that $Q$ follows the dependence $Z=-2\left({F}-1\right)\left({\hat{\Gamma}}-1\right)$; notice that both ${k}$, $Q$ are integrals and thus their angular dependence is related to the one of the poloidal flux: $\kappa={k}_0\, a^{1-1/{F}}$, $q = q_0\,a^{-2\left({\hat{\Gamma}}-1\right)\left(1-1/{F}\right)}$, with constant ${k}_0$ and $q_0$. For purely algebraic reasons, we use $f(\theta)\equiv 4 \pi c k_0 /({F}{\chi}^2 a^{1/{F}})$ instead of $a$. Accordingly $f$ is proportional to the radial distance along the same magnetic field line (or streamline): for a line passing through ($r_0$, $\theta_0$) any other point obeys $r/r_0=f/f_0$ with $f_0=f(\theta_0)$. Moreover we introduce one more angle $\vartheta$ that stands for the angle between the poloidal magnetic field line (or streamline) and the $z$-axis: $\tan\vartheta=B_x/B_z$. Using the latter variable, we express the parallel $\hat b \equiv \bm B_p / B_p$ and the perpendicular direction $\hat n\equiv-{\nabla}{A}/ \left|{\nabla}{A}\right|$ to the poloidal magnetic field lines as $$\begin{aligned}
\hat b=\cos\left(\vartheta-\theta\right)\hat r+\sin\left(\vartheta-\theta\right) \hat\theta \,, \nonumber \\
\hat n=\sin\left(\vartheta-\theta\right) \hat r-\cos\left(\vartheta-\theta\right) \hat\theta \,.
\label{unitvecs}\end{aligned}$$
Under these assumptions the expressions for the physical quantities become $$\begin{aligned}
\label{eqnsdens}
{A}=-\left(\frac{4\pi c {k}_0 r}{{F}{\chi}^2 f}\right)^{F}\,, \quad
{\rho}=\frac{4 \pi {h}{k}_0^2}{M^2}
{A}^{2({F}-1)/{F}}
\,, \quad \\
\label{eqnsby}
{\bm B}_p = \frac{-{F}{A}}{r \sin\left(\vartheta-\theta\right)} \hat b \,, \quad
{\bm B}_y=
\frac{{F}A \mu f {\chi}({\chi}^2 -{\chi_A}^2)}{r(M^2+ {\chi}^2-1)} \hat y \,,\quad \\
\bm{E}=\frac{-{F}{A}{\chi}}{r \sin\left(\vartheta-\theta\right)} \hat n \,, \quad \\
\label{eqnsgamma}
{\gamma}=\frac{\mu}{{h}}\frac{M^2 + {\chi_A}^2-1}{M^2+{\chi}^2-1} =\frac{\mu}{{h}\left( {1+\sigma}\right)}\,, \quad \\
\label{eqnsgamvp}
\frac{{\gamma}\bm v_p}{c}=\frac{M^2}{{\chi}^2 f {h}\sin\left(\vartheta-\theta\right)}\hat b \,, \quad \\
\label{eqnsgamvy}
\frac{{\gamma}\bm v_y}{c}=\frac{{\chi_A}^2 \mu}{{h}{\chi}}\frac{M^2+{\chi}^2 - {\chi}^2/ {\chi_A}^2}{M^2+{\chi}^2-1}\hat y \,. \quad\end{aligned}$$
Before proceeding further to the equations involved, we note that the angle $\vartheta$ is related to the derivative of the function $f$: The form of $A\propto (r/f)^F$ implies that ${\nabla}A$ is parallel to $f\hat r - (df/d\theta) \hat \theta$, and since $\hat n=-{\nabla}A / |{\nabla}A| $ a first equation is obtained using Eq. (\[unitvecs\]) $$\label{feq}
\frac{df}{d\theta}=\frac{f}{\tan\left(\vartheta-\theta\right)} \,.$$ The Bernoulli equation (\[bernoulli1\]) is now written as $$\begin{aligned}
\label{bernoulli}
\frac{\mu^2}{{h}^2} \frac{\left(M^2+{\chi_A}^2-1\right)^2 - \left({\chi_A}^4/{\chi}^2 \right) \left(M^2+{\chi}^2-{\chi}^2 / {\chi_A}^2\right)^2 }{\left(M^2 +{\chi}^2-1\right)^2} \nonumber\\
=1 + \left[\frac{M^2}{{\chi}^2 {h}f \sin\left(\vartheta-\theta\right)} \right]^2 \,. \qquad\end{aligned}$$ Besides its algebraic form the differential one is also used at the subsequent calculations: $$\begin{aligned}
\label{eqvartheta}
\frac{1}{\tan\left(\vartheta-\theta\right)} \frac{d\vartheta }{d\theta}
=\frac{{\chi}^4 f^2 {h}\sin^2\left(\vartheta-\theta\right)}{M^4}\frac{d{h}}{d\theta} \nonumber \quad \\
+\left[1-\frac{{\chi}^2 \mu^2 f^2\left({\chi}^2 - {\chi_A}^2 \right)^2 \sin ^2\left(\vartheta-\theta \right)}{\left(M^2+{\chi}^2-1\right)^3}\right]\frac{1}{M^2}\frac{dM^2}{d\theta} \,. \quad\end{aligned}$$ Equation (\[mach-h\]) provides a relationship between $M^2$ and ${h}$ (both are functions of $\theta$ alone; note the $Q$ and ${k}$ are constants along streamlines and their combination $Q k^{2({\hat{\Gamma}}-1)} $ is a global constant), implying $$\begin{aligned}
\label{eqenth}
\frac{d{h}}{d\theta}=-\frac{{h}u_s^2}{M^2}\frac{d M^2}{d\theta}\,, \quad u_s^2=\frac{\left({\hat{\Gamma}}-1\right)\left({h}-1\right)}{{\hat{\Gamma}}-1+\left(2-{\hat{\Gamma}}\right){h}} \,,\end{aligned}$$ where $u_s=c_s/\sqrt{c^2-c_s^2}$ the sound proper velocity (over $c$), with $c_s=\sqrt{{\hat{\Gamma}}p / ({\rho}{h})}$.
Applying the similarity expressions to the transfield equation (\[transfield\]) we find $$\begin{aligned}
({F}-1)\left[
\frac{\mu^2({\chi}^2-{\chi_A}^2)^2}{(M^2 +{\chi}^2-1)^2}+
\frac{2({\hat{\Gamma}}-1) {h}({h}-1){\chi}^2}{{\hat{\Gamma}}M^2}\right]
=
\nonumber \quad \\
\frac{1}{{\chi}^2 {f}^2}\left[\frac{{\chi}^2-1}{\sin^2(\vartheta-\theta)}+M^2\right]
\frac{d\vartheta}{d\theta}
+\frac{({F}-1)({\chi}^2-1)}{{\chi}^2 {f}^2\sin^2(\vartheta-\theta)}
\nonumber \qquad \\
+\frac{\sin(2\vartheta-2\theta)}{2}
\left[
\frac{{\chi}^2{h}}{M^2}\frac{d{h}}{d\theta}
-\frac{\mu^2({\chi}^2-{\chi_A}^2)^2}{(M^2 +{\chi}^2-1)^3}
\frac{dM^2}{d\theta}
\right]
\,, \qquad\end{aligned}$$ which, in combination with Eqs. (\[eqvartheta\], \[eqenth\]) gives an equation for $M$: $$\label{Meq}
\frac{dM^2}{d\theta}=\frac{\left({F}-1\right) M^2}{\tan\left(\vartheta-\theta\right)}\frac{N}{D} \,,$$ $$\begin{aligned}
N=-\frac{M^2\left({\chi}^2-1\right)}{{\chi}^4 f^2 {h}^2 \sin^2\left(\vartheta-\theta\right)}
+\frac{2\left({\hat{\Gamma}}-1\right)\left({h}-1\right)}{{\hat{\Gamma}}{h}}
\nonumber\\
+\frac{\mu^2}{{h}^2{\chi}^2}\frac{M^2\left({\chi}^2-{\chi_A}^2\right)^2}{\left(M^2+{\chi}^2-1\right)^2}
\,, \\
D=\frac{1-M^2-{\chi}^2}{M^2}u_s^2-\frac{\mu^2}{{\chi}^2{h}^2}\frac{M^2 \left({\chi}^2-{\chi_A}^2 \right)^2}{ \left(1-M^2-{\chi}^2\right)^2} \nonumber\\
+\frac{M^2}{ {\chi}^4 f^2 {h}^2}\left[ \frac{{\chi}^2-1}{\sin^2\left(\vartheta-\theta\right)}+M^2\right]
\,.\end{aligned}$$
The overall procedure of integration can be stated as follows. The differential Eqs. (\[feq\], \[eqenth\], \[Meq\]) together with the algebraic Bernoulli Eq. (\[bernoulli\]) and the initial conditions consist the system of equations that fully describe the flow. The initial conditions $f_0,M_0,h_0,\vartheta_0$ and the specific values of ${F}$, ${\hat{\Gamma}}$ determining the evolution of the various quantities along the initial surface $\theta_0$, also provide the integrals ${k},{\chi}, {\chi_A}^2 ,\mu,Q$ and complete the necessary set of parameters. The integration derives the evolution of the quantities along a specific poloidal streamline and then the similarity property ${A}\propto \left(r / f \right)^{F}$ is used to extend this solution to the rest of the flow.
Two remarks are easily obtained by the straightforward inspection of Eq. (\[Meq\]). The rarefaction wave front occurs when the denominator vanishes, since these are the only points where the first derivatives might suffer a discontinuity. In order to give an intuitive interpretation, we write both $N,\, D$ in terms of the physical quantities: $$\begin{aligned}
N=\frac{2p_{\rm total}}{{\rho}{h}c^2}\,, \quad p_{\rm total}=p+\frac{B^2-E^2}{8 \pi} \,, \quad \nonumber \\
\label{denom}
D=\frac{\left(\frac{{\gamma}v_\theta}{c}\right)^4-\left(\frac{{\gamma}v_\theta}{c}\right)^2\left(u_s^2+\frac{B^2-E^2}{4\pi{\rho}{h}c^2}\right)+u_s^2 \frac{B_\theta^2-E_r^2}{4\pi{\rho}{h}c^2}}{\left(\frac{{\gamma}v_\theta}{c}\right)^2} \,. \quad\end{aligned}$$ The nature of the denominator vanishing becomes clear if we use both Eq. (\[eqnprimedefB\]) to rewrite the last term as $B_\theta^2-E_r^2=(1-{\chi}^2)B_\theta^2$. The comparison with the dispersion relations for the magnetosonic disturbances, see Appendix C in [@Vlahakis_2003a], reveals that the denominator vanishes when the $\hat\theta$ component of the flow proper velocity is equal with the comoving fast or slow magnetosonic phase velocity of a wave propagating along the $\hat\theta$ direction. These are the actual singular points of the steady-state flow [@Bogovalov_1994] forming the so-called modified fast/slow surface, or limiting characteristics, and it is already met in a number of approaches (see [@Tsigkanos_1996; @Bogovalov_1997] and references therein).[^3]
In the limit of vanishing poloidal magnetic field ($B_p \to 0$) a complication of interpretation nature enters. In such a case ${A}$ becomes zero. Also $k$, ${\chi}$, $M^2$ become infinite, but their ratio retains a finite value $$\begin{aligned}
\frac{{\chi}^2}{M^2}=\sigma\,, \quad \frac{M^2}{{k}^2}=\frac{4\pi {h}}{{\rho}}\,, \quad {k}{\nabla}{A}= {\nabla}{\Psi}\,.\end{aligned}$$ For that reason, the integration has to be performed for $\sigma$ rather than for $M^2$, as in [@Kostas_2013]. In general, one could use $\sigma$ instead of $M^2$ even when $B_p$ exists, by using the expressions: $$\begin{aligned}
\label{eqnssigM}
\sigma=\frac{{\chi}^2-{\chi_A}^2}{M^2+{\chi_A}^2-1}\,, \\
\frac{d\sigma}{d\theta}=-\frac{\sigma}{M^2+{\chi_A}^2-1}\frac{dM^2}{d\theta}\,, \nonumber\\
\label{seq}
=-\frac{\sigma}{M^2+{\chi_A}^2-1}\frac{\left({F}-1\right) M^2}{\tan\left(\vartheta-\theta\right)}\frac{N}{D} \nonumber\end{aligned}$$ (with the latter substituting Eq. \[Meq\]).
Numerical Results {#results}
=================
Suppose a homogeneous magnetized plasma having magnetic field $B_{p0}\hat z + B_{y0} \hat y$ and bulk velocity $v_{p0}\hat z + v_{y0} \hat y$ fills the space $z<0$, $x<0$, supported by some external pressure on the plane $x=0$, $z<0$, see Fig. \[geom\]. Our goal is to explore how a sudden pressure drop at the origin $x=z=0$ modifies the flow characteristics in the region $z>0$, through the rarefaction wave that propagates as a weak discontinuity inside the body of the flow.
Since we require the initial flow to be homogeneous we fix the parameter ${F}=1$. We chose three different set of initial configurations for cold super-fast magnetosonic flows given in Table \[arrayinitial\]. In relation to the strength of the poloidal magnetic field we include cases in which: (i) the poloidal magnetic field component is negligible ($B_p \ll |B_y|$, LP model), (ii) the poloidal magnetic field is mildly smaller than the transverse one ($B_p < |B_y|$, MP model), (iii) both components are of similar magnitude ($B_p \sim |B_y|$, EP model). Our attempt to increase further the strength of the poloidal magnetic field is restricted by the condition of staying in the super-fast magnetosonic regime. We also include a fourth model (TD) in which the thermal energy is nonnegligible. The implications of the initial transverse velocity are studied in the remaining two models (LP01, LP03). These are the same as the cold flow model (LP) except their initial transverse velocities ($v_{y0} = 0.1, \, 0.3$ respectively).
The results of the integration show that both the poloidal poloidal magnetic field and the initial transverse velocity affect the spatial scale of acceleration as also the rarefaction wave front inclination. Finally, and in order to compare our results with the ones obtained by numerical simulations, we included models (HDB), (MHDA), (MHDB) that stand for the corresponding scenarios simulated in [@Mizuno_2008]. The initial parameters for these models are shown in Table \[arrayinitialMiz\].
Model $\sigma_0$ $-(B_y/B_p)_0$ $v_{y0}$ $M_0$
----------------------- ------------ ---------------- ---------- ---------
low poloidal (LP) $10$ $40000$ $0.0005$ $12000$
mild poloidal (MP) $10$ $158$ $0.0005$ $50$
equal poloidal (EP) $10$ $3.5$ $0.0005$ $1.10$
thermal driven (TD) $0.1$ $0.6$ $0.01$ $2.0$
low poloidal01 (LP01) $10$ $40000$ $0.1$ $12000$
low poloidal03 (LP03) $10$ $40000$ $0.3$ $11300$
: The initial conditions of our models were set at $\theta_0=-\pi/2$. All models represent cold flows ${h}=1$, except (TD) which is actually a thermally dominated one with ${h}_0=10$ and ${\hat{\Gamma}}=4/3$, share the same total energy flux $\mu=1100$, the same initial Lorentz factor ${\gamma}_0=100$, are homogeneous ${F}= 1$, and the poloidal streamlines (and field lines since $\bm B_p \parallel \bm v_p$) are initially parallel to the $z$-axis ($\vartheta_0=0$). \[arrayinitial\]
The initial conditions for all models were specified at $\theta_0=-\pi /2$ (i.e. at $z=0$, $x<0$). The main criterion over which they were selected was the total energetic context $$\mu={\gamma}{h}\left(1+\sigma\right) \,.$$ All models shown in Table \[arrayinitial\] share the same value of $\mu = 1100$. The models of the cited simulations (Table \[arrayinitialMiz\]) do not share this same value; the corresponding fluxes are shown in the relevant column of that Table. The thermally driven models (TD and HDB) have very high enthalpy (${h}_0=10,\, 21.6$) and thus a polytropic index of ${\hat{\Gamma}}=4/3$ was chosen.
The results for the magnetic dominated models (LP, MP, EP) appear in Fig. \[figcoldmods\] The first row of diagrams shows the physical shape of the flow, some streamlines, and the spatial distribution of the Lorentz factor. One must have in mind that if a line attains a specific value of the Lorentz factor at some point, self-similarity will finally ascribe this efficiency and to the rest lines starting from the lines close to the corner to the most exterior ones. So the suitable measure for the efficiency achieved is not the efficiency itself, but the relevant energetic evolution along a streamline as a function of the angle $\theta$ or similarly of the relative distance $r/r_0=f/f_0$, where $r_0$ the initial radial distance that the line originates from the base of the flow ($\theta_0=-\pi/2$) and $f_0$ the value of $f$ at that point.
The energetics are shown in the second row, where we draw the energy fluxes per mass energy flux in the laboratory frame. The Lorentz factor for a cold flow is equal to the inertial energy flux (rest mass energy plus bulk kinetic), while the thermal energy $\left( {h}-1\right) {\gamma}$ is absent. During the rarefaction evolution the Poynting flux is converted efficiently to kinetic, reaching soon to its maximum possible value (${\gamma}_{max} = \mu$); $\left(r/r_0\right)_{95}$ the point where ${\gamma}$ attains $95\%$ to its maximum value. When $B_p$ becomes comparable to $|B_y|$ it has significant impact both on the rarefaction wave front and the spatial scale of acceleration in the last one. The analytical results obtained in Appendix \[appA\] interpret exactly this behavior, a summary of which is shown in Fig. \[thapp\]. The calculated rarefaction front corresponds to the dashed lines in the second row diagrams.
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{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}\
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
An important conclusion of our study appears in the third panel exhibiting the evolution of the integrable quantities along the streamline. The inverse of the Alfvénic Mach number and the magnetization parameter follow a power law decrease ($M^{-2}, \sigma \propto r^{-2/3}$), and only small deviations are noticed in the EP model. The scale of the Alfvénic Mach number is associated to the planar geometry, and is further discussed in the next section as also in Appendix \[appA\] where a formal derivation of the scaling law is given. The extend of the rarefied region, equals to the so-called Prandtl-Meyer angle which is defined as the angle between the initial and the final orientation of the flow ($\theta_{PM} = \vartheta_\infty \sim \theta_\infty$). Its monotonic increase with decreasing magnetic field component ratio ${\zeta}_0$, with ${\zeta}\equiv -B_y / B_p$, is also provided in Appendix \[appA\], Eq. (\[eqnassympPM\]), and demonstrated in Fig. \[thapp\] for low initial transverse velocities. As for the relative magnetic field component strength included in the diagrams we notice that the ratio doesn’t alter much, although in the last case a small difference exists that wouldn’t be significant if it didn’t had serious implications on the derivation of the expression providing $\theta_{PM}$ angle (see in Fig. \[thapp\] how sensitive is the value of $\theta_{PM}$ as a function of ${\zeta}_0$, for not too high values of ${\zeta}_0$).
The physical quantities normalized to their initial values appear in the last row. The decrease observed in all except the transverse velocity, is intuitively expected due to the rarefaction process and the relevant conversion of the Poynting energy. The density decrease follows also the $-2/3$ power law, as Eq. (\[eqnsdens\]) suggests (${\rho}\propto M^{-2}$). Similar behavior follow the magnetic field components, where only small deviations at the low ${\zeta}_0$ cases exist. Some special attention should be given in the transverse velocity evolution that shows a peculiar pattern either of increase (LP), or decrease (MP, EP), explained in the next section.
The results of the last three models (TD, LP01, LP03) appear in Fig. \[figrestmods\]. The main characteristics of the mixed type scenario is the much lengthier spatial scales of acceleration, the latter appearance of the wave front, and the extension of the rarefied region; compare for example with (EP) model. A small bump observed in the thermal energy curve $({h}-1){\gamma}$ is due to the magnetic acceleration of the flow. That acceleration yields an increases of the inertial of the thermal energy rather than of the thermal energy context, see that ${h}$ monotonically decreases in the bottom row. This first phase of acceleration occurs in expense of the Poynting energy and for that reason the Alfvénic Mach number follows the power law scaling mentioned before. That behavior breaks when the thermal energy becomes the leading one, but this region falls out of the diagram.
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}\
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Models (LP01, LP03) are dedicated to the implications of significant initial transverse velocity in contreast to the (LP) model. As seen in Fig. \[figrestmods\] the effects on the rarefaction wave inclination and on the maximum extension of the rarefied area are important, while $M^{-2}$ and $\sigma$ still follow the $-2/3$ power law. Besides its high initial value, the transverse velocity finally declines to small values asymptotically.
Model $\sigma_0$ $B_{p0}$ $-B_{y0}$ $\mu$
------------------------- ------------ ---------- ----------- ---------------------
poloidal field (MHDA) $0$ $21.27$ $0$ $2.83\times 10^{6}$
transverse field (MHDB) $100$ $0$ $149$ $5.38\times 10^{5}$
hydrodynamic flow (HDB) $0$ $0$ $0$ $4.11\times 10^{4}$
: All models share the same initial velocity along the $z$-axis, with ${\gamma}_0=7.089$ ($\vartheta_0=0$, $v_{z0}=0.990$, $v_{y0}=0$). The thermal energy is also common, with ${h}_0=4\times 10^5$ (${\rho}_0=10^{-4}$, $p_0=10$, ${\hat{\Gamma}}=4/3$). \[arrayinitialMiz\]
As a final application we examine the consistency of our steady-state solution with the simulations appearing in [@Mizuno_2008], and their similar ones in [@Zenitani_2010]. Using the same set of initial parameters (Table \[arrayinitialMiz\][^4]), the results obtained (Fig. \[figMiz\]) are in excellent agreement with the simulations. The situation is identical until of course the point, where a contact discontinuity occurs, the left plateau at their diagrams, corresponding to a termination of the rarefaction process. This picture is expected whenever a nonzero external pressure/density exists, since a contact discontinuity between the two fluids, but also a shocked region in the exterior medium are formed, see next section.
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Interpretation of the results - Discussion {#discussion}
==========================================
The main characteristic of the rarefaction model presented above is the significant acceleration of the flow. Depending on the available energy reservoir this acceleration is performed in expense of the Poynting energy (${h}{\gamma}\sigma$, magnetically driven), of the thermal one (${h}$, thermally driven), or both (mixed type). But no mater in what form the leading energy is, in all of the flows the acceleration achieved leads to completely matter-dominated flows (${\gamma}_{max}\sim \mu$).
A phenomenological interpretation for the acceleration spatial scale is based on the magnetization parameter power law ($\sigma \propto r^{-2/3}$) that applies to all cold flows, even if small deviations like in (EP) model exist. Since in all cases the same initial values of $\mu,{\gamma},\sigma$ apply, the relative distances are ascribed to the early appearance of the rarefaction process. In contrast to this, in the thermal driven rarefaction (TD) the acceleration takes place in much greater distances, despite the early appearance of the rarefaction wave front. The lengthier, and thus less efficient action, is associated with the conversion of the thermal energy to bulk kinetic, which follows a much shallower law than the Poynting energy decrease (see Fig. \[figrestmods\]). For the case of a mixed type rarefaction where the poloidal magnetic field is significant, ($B_p>|B_y|$) and ${\chi_A}^2 \sim 1 - \epsilon$ with $\epsilon<1/{h}{\gamma}$. Thus $\sigma \propto 1/M^2 \propto {\rho}/{h}\propto {\rho}^{2-{\hat{\Gamma}}}$, while if significant thermal context exists, then ${h}\propto {\rho}^{{\hat{\Gamma}}-1}$ with ${\hat{\Gamma}}=4/3$ exhibiting the slower scale of the thermal conversion. That result is in agreement with other models where rarefaction was considered in the negligible poloidal magnetic field limit, in both planar [@Kostas_2013] and axisymmetric flows [@Komissarov_Vlahakis_Konigl_2010].
The rarefaction wave front is determined by the vanishing of the denominator of Eq. (\[denom\]). In general that vanishing occurs at the modified fast magnetosonic surface which corresponds to points where ${\gamma}\theta /c$ equals the comoving phase velocity of the fast magnetosonic waves. Thus the lines of $\theta=$const are also the characteristics of our system, exactly as in the hydrodynamic homogeneous rarefaction, forming around the poloidal velocity the Mach cone of the fast magnetosonic velocities; Fig. 2 in [@Kostas_2013] is very instructive. Considering the cone at the axis origin the wave front is the envelope surface of all the fast magnetosonic disturbances emitted from the point of the boundary discontinuity. As such, the presence of a poloidal magnetic component and transverse initial velocity both affect the wave front inclination and $\theta_{RW}$ is no longer given by a simple expression like $\sqrt{\sigma_0}/{\gamma}_0$ as in the $B_p \ll |B_y|$, $v_{y0}\approx 0$ flows, but from the more complicated expression (\[eqnassympPM\]) and Fig. \[thapp\]. For the mixed type scenario (TD), the Mach cone is not obtained by the sonic disturbances expected to propagate in a much narrower opening of $u_s/({\gamma}_0 v_{p0}) \sim 6.5 \times 10^{-3}$, if the flow was purely hydrodynamic, but from the more extended one of the fast magnetosonic disturbances ($c_s<c_f$, where $c_f$ the velocity of the fast magnetosonic disturbances at the comoving frame, which is significantly affected by the presence of $B_p$).
The cold and uniform flow is studied in detail in Appendix \[appA\], where the scaling of $M^2 \propto r^{-2/3}$ is formally derived. Beyond that, the same law can be derived by more intuitive arguments. The invariance of $$\label{eqBcscal}
\left({\gamma}v_p\right)^2 \frac{B^2-E^2}{ B_p^2}=\mbox{const}
\quad \Leftrightarrow \quad \frac{B_{\rm co}}{{\rho}}=\mbox{const} \,,$$ where $B_{\rm co}$ the magnetic field in the comoving frame, is connected to the magnetic flux conservation and is obtained using the integral expressions.[^5] The continuity equation and the flux conservation along the streamline provides ${\rho}{\gamma}v_\theta={k}({\nabla}{A})_r \sim {k}{A}/r$, which is used in conjunction with $D=0\Leftrightarrow ({\gamma}v_\theta)^2 = (B^2-E^2)/(4\pi{\rho})$ (Eq. \[denom\]) to obtain $B^2-E^2 \propto 1/({\rho}r^2)$. Combining this scale with Eq. (\[eqBcscal\]), we derive the density evolution ${\rho}\propto r^{-2/3}$ and by Eq. (\[Mdef\]) the requested $M^2\propto r^{2/3}$ power law.
The transverse velocity evolution induces implications in some of the models considered (TD, EP, LP03), while both cases of an increasing (LP models) or a decreasing profile of $v_y$ exist. The general behavior of both velocity components is obtained in terms of $\sigma$ in the analysis of Appendix \[appA\], see Eqs. (\[uyanal\]), (\[upanal\]), according to which ${\chi_A}^2>1$ leads to the decrease, while ${\chi_A}^2<1$ to the increase of the transverse velocity[^6]. The asymptotic values of the velocity components is obtained by setting $\sigma \to 0$ $$\frac{v_{y\infty}}{c}=\frac{{\chi_A}^2}{{\chi}}\,, \quad \frac{v_{p\infty}}{c}=\frac{v_{r\infty}}{c}=\sqrt{1-\frac{{\chi_A}^4}{{\chi}^2}-\frac{1}{\mu^2}} \,,$$ and simply state that all the plasma momentum in the $y$-direction has been transferred to the matter. The momentum conservation dictates the increase of the matter’s transverse momentum (${\gamma}v_y$), but the acceleration at the poloidal direction leads also to the inertial matter enhancement (${\gamma}$). Weather inertial increase suppresses the transverse velocity one, or not, is not uniquely determined; it depends on the initial conditions in the way that the relationship above determines. The velocity expressions are also useful in the calculation of the magnetic components asymptotic ratio. For that purpose, we apply their value to the ${\chi}$ integral, and after some manipulation we find $$\frac{{\zeta}_\infty}{{\zeta}_0}=\frac{v_{p0}}{v_{p\infty}}\frac{M_0^2+{\chi_A}^2-1}{M_0^2}$$ by which the insignificant alteration of the ${\zeta}$ ratio is concluded.
The evolution of the remaining parameters can also be understood analytically. The transverse magnetic field is easily obtained by Eq. (\[eqnsby\]), which reveals the $B_y \propto r^{-2/3}$ decrease, but at distances where $M^2\gg {\chi}^2$ due to the presence of the ${\chi}^2$ term in the denominator. The ${\zeta}$ evolution indicates then that the poloidal component follows the same evolution $B_p \propto r^{-2/3}$, except for the case where a minor deviation of ${\zeta}$ is observed. It is instructive to compare the scaling of the magnetic field component with the one obtained by the axissymetric MHD steady-state models, $B_p\propto 1/\varpi^2$, $B_\phi\propto 1/\varpi$, see for example [@Vlahakis_2003a]. Besides the differences in the decrease of the two components, in rarefaction the Poynting energy conversion $\mu-{h}{\gamma}\propto |B_y| \propto r^{-2/3}$ is to be compared with the one obtained in the semi-analytical results of [@Vlahakis_2003a] and the numerical ones ([@Komissarov_Vlahakis_Barkov_2009] and references therein) where the conversion is much slower caused by the slow decrease of $\varpi B_\phi$. Thus the magnetic driven rarefaction is to be considered as a powerful and short scaling mechanism to convert Poynting energy to bulk kinetic, and as such its contribution to the high energy astrophysical phenomena might be important.
The magnetization parameter does follow the $-2/3$ power law in a great extend, but as Eq. (\[usfleqM2\]) suggests, this behavior deviates whenever ${\zeta}$ is close to unity and $v_y \sim {\chi}$, e.g. in (EP) model. The deviation has not serious implications in the scalings of most physical quantities, but it affects the calculations for the maximum extend of the rarefied flow ($\theta_{PM}$), see Eq. (\[eqnassympPM\]). The two terms appearing in this expression are of the same order magnitude and thus the accurate numerical integration is unavoidable, especially for low ${\zeta}$ flows. The resulting $\theta_{PM}$ for low initial transverse velocity as a function of ${\zeta}_0$ are shown in Fig. \[thapp\].
Our model describes the rarefaction until its full completion, leading to a completely matter-dominated flow that fills the space up to polar angle $\theta_{PM}$. This will be indeed the end state if the flow is surrounded by void space. If the pressure or density of the environment is nonzero then the expansion of the flow will modify the environment as well, and a contact discontinuity (CD) will be formed. The details of the final state depend on the characteristics of the environment (for example if it is a hydrodynamic super-sonic flow in the $\hat z$ direction a shocked region will be formed and the pressure at the CD will be related to the shock jump conditions). For an initially uniform environment the CD will be conical $\theta=\theta_{CD}$ passing trough the origin. The environment characteristics define the value of the pressure $P_{CD}$ at CD, and thus the rarefaction ceases at some angle $\theta_F$ in which pressure equilibrium is reached $$\label{nonzeroext}
\left[\frac{B^2-E^2}{8\pi} + P\right]_{\theta=\theta_F} = P_{CD}\,.$$ The above equation defines the angle $\theta_F$, after which (and up to $\theta_{CD}$) the flow remains uniform. Our model correctly describes the rarefaction till the flow becomes uniform, so we can use it to find the end state from Eq. (\[nonzeroext\]), and also find $\theta_{CD}=\left[\vartheta\right]_{\theta=\theta_F}$.
If the flow is cold then Eq. (\[nonzeroext\]) can be much simplified. Since the comoving magnetic field scales with the density $\sqrt{B^2-E^2} \propto \rho \propto 1/M^2$, Eq. (\[nonzeroext\]) can be rewritten as $$\label{nonzeroextcold}
\left[\frac{M_0^2}{M^2}\right]_{\theta=\theta_F} = \sqrt{\frac{8\pi P_{CD}}{B_0^2-E_0^2}}\,.$$ If for example the environment is a uniform static medium with pressure 25 times smaller than the initial magnetic pressure of the flow, then at $\theta_F$ the ratio $M_0^2/M^2=0.2$. Using the third row of Fig. \[figcoldmods\], e.g. for model LP, we find $r/r_0\approx 300 $, and from the other diagrams for the same model all the rest physical quantities. The terminal Lorentz factor is $\sim 350 $ (corresponding to efficiency ${\gamma}/\mu \sim 30\%$) and the flow inclination $\vartheta=\theta_{CD}=0.5 \vartheta_\infty=0.16^\circ$.
More complicated environments are beyond the scope of this paper, but they are definitely an interesting application of the model. Possibly the environment itself can also be modeled with an $r$ self-similar model. In this paper we focus on the strongly magnetized cases and highly relativistic velocities, however the model applies to other cases as well, for example to slow magnetosonic weak discontinuities; these will be examined in another connection.
This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.
The cold homogeneous case {#appA}
=========================
We derive here the analytical expressions for the cold (${h}\to 1, u_s \to 0$)) and homogeneous flow (${F}=1$, see Eq. \[eqnsdens\]), that serves as an extension of the hydrodynamic solutions of [@Landau_fluid] and [@Granik_1982]. At that limit, the rarefaction wave front is determined straightforwardly by the vanishing of the denominator of Eq. (\[Meq\]). The qualitative behavior of the solution is easily understood, if we use Eq. (\[denom\]): $$\label{denomcold}
D=0 \Rightarrow \; \left(\frac{{\gamma}v_\theta}{c}\right)^2=\frac{B^2-E^2}{4\pi{\rho}c^2} \,.$$ As we proceed to the integration from the initial surface $\theta_0 = -\pi/2$ to the higher angle ones, the $d/d\theta$ derivatives equal to zero indicating the uniform flow phase. That uniformity breaks at the point $\theta_{RW}$ where the $\hat\theta-$component of the flow proper velocity becomes equal to the fast magnetosonic one yielding the weak discontinuity. From this angle and beyond a (0/0) form arises and the derivatives attain a finite value signaling the initiation of the rarefaction process.
It is easy to derive an analytical expression for the rarefaction wave angle in terms of the initial quantities and not only for the cold limit. For this purpose, we combine the vanishing of $D$ (Eq. \[denom\]) and the Bernoulli equation (\[bernoulli\]) to eliminate $f$ and find $$\begin{aligned}
\label{rwanglenoassump}
\sin ^2\left(\vartheta-\theta\right)= \frac{\sigma ^2 M^2}{{\chi}^2 \left(v_p/c\right)^2}-\frac{{\chi}^2-1}{M^2}
\nonumber \quad \\
- \frac{1-M^2-{\chi}^2}{M^2 \left({\gamma}v_p/c\right)^2}u_s^2 \,, \quad\end{aligned}$$ where use of Eq. (\[eqnsdens\])-(\[eqnsgamvy\]) was also made. We would like to underline that the above expression provides not only the rarefaction wave front angle $\theta_{RW}$, i.e. when we consider the initial values of the quantities, but also relates the appearing quantities at the subsequent rarefaction phase.
A point of special attention for the following calculations is the transverse velocity $v_y$ which, even if it is negligible at the beginning, it is possible to end up with significant values (see for example models EP, TD) affecting the derived asymptotic expressions. Under this perspective two helpful and accurate expressions are: $$\begin{aligned}
\label{usfleqM2}
M^2=\frac{{\chi}}{\sigma}\left({\chi}-\frac{v_y}{c}\right) \,, \\
\label{usfleqxa2}
{\chi_A}^2=\frac{{h}{\gamma}{\chi}}{\mu}\left(\frac{v_y}{c}+\frac{\sigma}{{\chi}}\right) \,.\end{aligned}$$ It is also useful to express the velocity components in terms of the magnetization parameter. Eq. (\[eqnsgamma\]), (\[eqnsgamvy\]), (\[eqnssigM\]) yield $$\begin{aligned}
\label{uyanal}
\frac{v_y}{c}=\frac{{\chi_A}^2+\sigma\left({\chi_A}^2-1\right)}{{\chi}}\,, \quad \\
\label{upanal}
\frac{v_p}{c}=\sqrt{1-\frac{{\chi_A}^4}{{\chi}^2}\left(1+\frac{{\chi_A}^2-1}{{\chi_A}^2}\sigma\right)^2-\frac{{h}^2}{\mu^2}\left(1+\sigma\right)^2} \,. \quad\end{aligned}$$
We now focus on the cold flow limit and we use Eq. (\[usfleqM2\]) in Eq. (\[rwanglenoassump\]) to obtain $$\label{rwanglenoassump2}
\sin^2\left(\vartheta-\theta\right)=\frac{\sigma}{{\chi}}\left[\frac{{\chi}- \frac{v_y}{c}}{\left(v_p /c\right)^2}+\frac{1-{\chi}^2}{{\chi}-\frac{v_y}{c}}\right] \,.$$ The effects of the transverse velocity are important in cases where the integral ${\chi}$ is close to unity, i.e. when the ratio ${\zeta}$ is comparable to the transverse velocity. Assuming the initial values we obtain the rarefaction wave front angle; for a specific ${\gamma}_0$, the angle depends on both $v_{y0},\,{\zeta}_0$ via ${\chi}$. In Fig. \[thapp\] we give the relevant plot as function of ${\zeta}_0$ for two different values of the initial transverse velocity. Notice that at the limit of the negligible poloidal magnetic field ($B_p \to 0, \, {\chi}\to \infty$) the above expression becomes $\sin^2\theta_{RW}=\sigma_0 (c^2-v_{p0}^2)/v_{p0}^2$ in agreement with the results of [@Kostas_2013].
![The calculated angles for a cold flow (${\gamma}_0=100$, $\sigma_0=10$) as a function of ${\zeta}_0$. **Top:** The rarefaction front ($\sin^2\theta_{RW}$) for two different values of the transverse velocity, and the asymptotic expression (${\zeta}_0 \to\infty$). Notice that for improper initial conditions (low ${\zeta}_0$ or high $v_{y0}$) we obtain $\sin^2\theta_{RW}>1$ corresponding to sub-fast magnetosonic flow where the rarefaction wave is impossible. **Bottom:** The resulting extension of the rarefied area ($\tan \theta_{PM}$) for a cold flow of negligible $v_{y0}$.[]{data-label="thapp"}](figures/sin_genK4.eps "fig:"){width="35.00000%"}\
![The calculated angles for a cold flow (${\gamma}_0=100$, $\sigma_0=10$) as a function of ${\zeta}_0$. **Top:** The rarefaction front ($\sin^2\theta_{RW}$) for two different values of the transverse velocity, and the asymptotic expression (${\zeta}_0 \to\infty$). Notice that for improper initial conditions (low ${\zeta}_0$ or high $v_{y0}$) we obtain $\sin^2\theta_{RW}>1$ corresponding to sub-fast magnetosonic flow where the rarefaction wave is impossible. **Bottom:** The resulting extension of the rarefied area ($\tan \theta_{PM}$) for a cold flow of negligible $v_{y0}$.[]{data-label="thapp"}](figures/thPMfigK3.eps "fig:"){width="35.00000%"}
In order to derive the spatial evolution of the integrable quantities we use Eqs. (\[bernoulli\]), (\[denom\]) to eliminate $\vartheta$ this time: $$\begin{aligned}
\label{appassymptotnoassump}
\frac{1}{f^2} = \frac{{\chi}^2 {h}^2 {\gamma}^2 \sigma^2}{M^2}
-{\chi}^4{h}^2\frac{{\chi}^2-1}{M^6}\left({\gamma}\frac{v_p}{c}\right)^2 \quad \nonumber \\
-{\chi}^4{h}^2\frac{1-M^2-{\chi}^2}{M^6}u_s^2 \,. \quad\end{aligned}$$ In the cold limit and by use of Eqs. (\[usfleqM2\]), (\[usfleqxa2\]), (\[uyanal\]), (\[eqnsgamma\]) we obtain a rather simple expression $$\label{appasymM2}
\frac{1}{f^2}=\frac{{\chi}^4}{M^6} \left[ \left(1-{\chi_A}^2\right)^2 \mu^2+{\chi}^2-1 \right] \,,$$ which gives the power law evolution of the Alfvénic Mach number $M^2 \propto f ^{2/3}\propto r ^{2/3}$. It is instructive to compare this result with the one obtained at the negligible poloidal case. In that limit both $M^2$ and ${\chi}^2$ become infinite, but their ratio retains the finite value of $1/\sigma$, see Eq. (\[eqnssigM\]). Thus the same spatial scaling is provided in terms of $\sigma \propto r^{-2/3}$.
We use this simple result to calculate the maximum extend of the rarefied area. For that purpose Eq. (\[feq\]) provides $$\begin{aligned}
\frac{d\theta}{d M^2}=\frac{3}{2}\frac{\tan\left(\vartheta-\theta\right)}{M^2} \,.\end{aligned}$$ The tangent appearing is obtained from the Bernoulli Eq. (\[bernoulli\]) $$\label{eqappsin}
\sin\left(\vartheta-\theta\right)=\frac{M^2 + {\chi}^2-1}{\sqrt{M^2 \ H\left(M^2\right)}} \,,$$ where $H(M^2)$ is a polynomial of $M^2$ $$\begin{aligned}
H\left(M^2\right)=\frac{\left[\mu^2\left({\chi}^2-{\chi_A}^4\right)-{\chi}^2\right] M^4}{\left[\mu^2(1-{\chi_A}^2)^2+{\chi}^2-1\right]{\chi}^2}-2M^2+1-{\chi}^2 \,.\end{aligned}$$ The resulting expression must be calculated numerically $$\begin{aligned}
\label{eqnassympPM}
\theta_{PM}=\theta_{RW}+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber\\
+\displaystyle\frac{3}{2}\int^\infty_{M^2_0} \! \frac{M^2 +{\chi}^2-1}{\sqrt{M^2 H(M^2)-\left(M^2+{\chi}^2-1\right)^2}} \frac{dM^2}{M^2} \,. \quad\end{aligned}$$ In Fig. \[thapp\] we show the obtained $\theta_{PM}$ values as a function of the ${\zeta}_0$ for the same model parameters (${F},\mu,{\gamma}_0,v_{y0}$) like the ones used in (LP, MP, EP) models..
In the limit of negligible poloidal magnetic field and $v_y=0$ we get $H(M^2)=\sigma^{-1} M^2[\mu^2-(\sigma+1)^2]$; the second term is of order $\sim \mu^2/{\gamma}^2$ and thus can be ignored. The transformation $dM^2 /M^2 = -d\sigma /\sigma $ provides $$\begin{aligned}
\theta_{PM}=\theta_{RW}+\displaystyle\frac{3}{2}\int_0^{\sigma_0} \! \frac{1+\sigma}{\mu \sqrt \sigma}\,\, d\sigma \,.\end{aligned}$$ In that limit, $\theta_{RW}\approx -\sqrt\sigma_0/{\gamma}_0$, and the above calculation yields $\theta_{PM}=2\sqrt\sigma / \mu$ which is exactly the result found in [@Kostas_2013].
[^1]: A discontinuity on the derivatives of the flow quantities rather than on the quantities itself, in our case first derivatives discontinuity.
[^2]: The notation was chosen in accordance with the axisymmetric flows in which ${\chi_A}$ is the value of ${\chi}$ at the Alfvénic surface, while the integral ${P}$ corresponds to the angular momentum-to-mass flux ratio.
[^3]: Notice that a similar distinction for the Alfvénic point does not exist, since in the relevant dispersion relation the trigonometric/projecting terms cancel out.
[^4]: Note that the values of “magnetic fields” given in [@Mizuno_2008] are in fact the magnetic field over $\sqrt{4 \pi}$.
[^5]: In the general case of nonnegligible thermal content we find $\left({\gamma}v_p/c\right)^2 (B^2-E^2)/B_p^2={\chi}^2-1+\mu^2(1-{\chi_A}^2)^2/{h}^2$. It is interesting to note that the same relation holds in the axisymmetric case as well, with ${\chi}$ however replaced by $E/B_p=\varpi\Omega/c$, the cylindrical distance in units of the light cylinder radius.
[^6]: In an axisymmetric trans-Alfvénic flow ${\chi_A}^2<1$ always hold since at the Alfvén surface $M^2=1-{\chi_A}^2$. In a planar symmetric flow this limitation does not exist, and thus we included cases with ${\chi_A}^2>1$ in our study.
|
---
abstract: 'Theoretical predictions for the magnetic moments of the physical ${\Delta}$ baryons are extracted from lattice QCD calculations. We utilize finite-range regulated effective field theory that is constructed to have the correct Dirac moment mass dependence in the region where the [*u*]{} and [*d*]{} quark masses are heavy. Of particular interest is the chiral nonanalytic behaviour encountered as the $N\, \pi$ decay channel opens. We find a $\Delta^{++}$ magnetic moment (at the $\Delta$ pole) of $\mu_{\Delta^{++}}=4.99 \pm 0.56\ \mu_N$. This result is within the Particle Data Group range of 3.7–7.5 $\mu_N$ and compares well with the experimental result of Bosshard [*et al.*]{} of $\mu_{\Delta^{++}}=4.52 \pm 0.51 \pm 0.45\ \mu_N$. The interplay between the different pion-loop contributions to the $\Delta^+$ magnetic moment leads to the surprising result that the proton moment may exceed that of the $\Delta^+$, contrary to conventional expectations.'
author:
- 'I.C. Cloet[^1], D.B. Leinweber[^2] and A.W. Thomas[^3]'
title: Delta Baryon Magnetic Moments From Lattice QCD
---
[Introduction]{}
The magnetic moments of the $\Delta$ baryons have already caught the attention of the experimental community and hold the promise of being accurately measured in the foreseeable future. Experimental estimates exist for the ${\Delta^{++}}$, based on the reaction $\pi^+\ p\ \to \pi^+\ \gamma'\ p$. The Particle Data Group [@Hagiwara:pw] cites a range of values, 3.7–7.5 $\mu_N$, for the ${\Delta^{++}}$ magnetic moment, with the two most recent experimental results being $\mu_{\Delta^{++}}=4.52 \pm 0.51 \pm 0.45\ \mu_N$ [@Bosshard] and $\mu_{\Delta^{++}}=6.14 \pm 0.51\ \mu_N$ [@LopezCastro:2000ep]. This gives an idea of the elusive nature of the $\Delta$ magnetic moments. In principle, the $\Delta^+$ magnetic moment can be obtained from the reaction $\gamma\ p\ \to \pi^o\ \gamma'\ p$, as demonstrated at the Mainz microtron [@Kotulla; @Drechsel:2001qu] and Kotulla [*et al.*]{} have recently reported an initial measurement of $\mu_{\Delta^+} = 2.7^{+1.0}_{-1.3} {\rm (stat.)}
\pm 1.5 {\rm (syst.)} \pm 3 {\rm (theor.)}~\mu_N$ [@Kotulla:2002cg]. Recent extrapolations of octet baryon magnetic moments have utilized a Pad$\acute{\rm e}$ approximant [@Leinweber:1998ej; @Hackett-Jones:2000qk; @Leinweber:2001ui; @Cloet:2002eg], an analytic continuation of chiral perturbation theory ($\chi$PT) which incorporates the correct leading nonanalytic (LNA) structure of $\chi$PT. This idea was recently extended to a study of decuplet baryons in the Access Quark Model [@Cloet], where the next-to-leading nonanalytic (NLNA) structure of $\chi$PT was also included. Incorporating the NLNA terms into the extrapolation function contributes little to the octet baryon magnetic moments, however it proves vital for the decuplet. This is because the NLNA terms from $\chi$PT contain information regarding the branch point at the octet-decuplet mass-splitting, $m_{\pi} = \delta$, associated with the $\Delta \to N \pi$ decay channel, which plays a significant role in decuplet magnetic moments. Here we include the NLNA behaviour in a chiral extrapolation of the magnetic moments of the $\Delta$ baryons calculated in lattice QCD. Although the Pad$\acute{\rm e}$ technique provides a well behaved method of extending $\chi$PT to heavy quark masses, it is not so obvious how it could be extended to higher order in a chiral expansion. We have therefore chosen to calculate explicitly the pion loop diagrams which give rise to the LNA and the NLNA behaviour of the $\Delta$ magnetic moments using finite-range regularization [@Young:2002ib; @Donoghue:1998bs]. In this investigation we select the dipole-vertex regulator. We stress that, in the context of the chiral extrapolation of lattice QCD data, the use of a finite-range regulator is the preferred alternative to dimensional regularization as demonstrated in Ref. [@Young:2002ib]. Ideally, the regulator mass should be constrained by lattice QCD results. However, the data available lies at large quark masses and provides little guidance. As such, we explicitly explore the regulator-mass dependence of our results, finding a sensitivity of 3.6% or less in the charged $\Delta$ magnetic moments, for variation of the dipole mass parameter in the range 0.5–1.0 GeV.
[Extrapolations – An Effective Field Theory]{}
To one-loop order the dimensionally-regulated $\chi$PT result for decuplet baryon magnetic moments is [@Banerjee:1995wz; @Correction]
$$\begin{gathered}
\mu_i = a_0 + \sum_{j= \pi, K} \frac{M_N}{32 {\pi}^2 f_j^2}
\Bigl( \alpha_j^i\, \frac{4}{9}\, {\cal H}^2\, {\cal F}(0,m_j,\mu) \\
+ \beta_j^i\, {\cal C}^2\, {\cal F}(-\delta,m_j,\mu) \Bigr)
+ a_2\, m_{\pi}^2\, .
\label{CPT}\end{gathered}$$
The ${\cal F}(\delta,m_j,\mu)$ functions (with $\delta$ the octet-decuplet mass difference) are the nonanalytic components of the meson loop diagrams shown in Fig. 1. These expressions are given explicitly in Ref. [@Banerjee:1995wz]. The constants $a_0$ and $a_2$ multiplying the analytic terms are not specified within $\chi$PT and must be determined by other means. It is well known that dimensionally-regulated $\chi$PT expansions for baryon properties are troubled by a lack of clear convergence [@Hatsuda:tt; @SVW; @Bernard:2002yk; @Thomas:2002sj]. This problem must be addressed if $\chi$PT results are to be applied far beyond the chiral limit, for example in the extrapolation of lattice data where pion masses up to 1 GeV are involved. Indeed, truncated dimensionally-regulated $\chi$PT expansions for magnetic moments exhibit the incorrect behaviour for large $m_{\pi}$. To address these concerns we adopt finite-range regulated $\chi$PT by calculating the meson-loop diagrams using a dipole-vertex regulator. The loop contributions then approach zero naturally as $m_{\pi}$ becomes large. We combine the analytic terms of the chiral expansion into a single term that maintains the chiral expansion to the order we are working, while guaranteeing the correct magnetic moment mass dependence at heavy pion masses.
The extrapolation function which we employ to obtain decuplet baryon magnetic moment predictions from lattice QCD data, thus takes the form $$\begin{gathered}
\mu = \frac{a_0}{1+ c_0 \, m_{\pi}^2} +
\sum_{T'} \chi^{j}_{T'}~ G_{TT'}(m_j, \Lambda) \\
+ \sum_{B} \chi^{j}_{B}~ G_{TB}(m_j, \delta, \Lambda)\, ,
\label{MM}\end{gathered}$$ where the sums over $T'$ and $B$ include the decuplet and octet intermediate states, respectively. The first term encapsulates the pion mass dependence of the photon field coupling directly to the baryon and reflects previous success in the extrapolation of octet baryon magnetic moments [@Leinweber:1998ej; @Hackett-Jones:2000qk; @Cloet:2002eg]. This term ensures the correct behaviour, as a function of quark mass, of the Dirac moments of the quarks at heavy quark masses (as $m_{\pi}^2 \propto m_q$ up to 1 GeV$^2$ – above this range one should use $m_q$ directly). The parameters $a_0$ and $c_0$ are chosen to optimize the fit to lattice data. The functions $G_{TT'}(m_j, \Lambda)$ and $G_{TB}(m_j, \delta, \Lambda)$ are the (heavy baryon) loop contributions to the magnetic form factors, in the limit $q \to 0$, for decuplet-decuplet and decuplet-octet transitions respectively, see Fig. 1. These functions are given by $$\begin{aligned}
G_{TT'}(m_j, \Lambda) &= \lim_{q \to 0}\
\frac{1}{2\pi} \int\ {\rm d}^3 k\ {\cal U}(k)\ {\cal U}(k')\
\frac{(\hat{q}\times \vec{k})^2}{(\omega_k \ \omega_{k'})^2}\, , \nonumber \\
G_{TB}(m_j, \delta, \Lambda) &=
\lim_{q \to 0}\
\frac{1}{2\pi} \int\ {\rm d}^3 k\ {\cal U}(k)\ {\cal U}(k') \nonumber \\
& \hspace{10mm} \frac{(\omega_k + \omega_{k'} - \delta)(\hat{q}\times \vec{k})^2 }
{\omega_k\ \omega_{k'}(\omega_k + \omega_{k'})(\omega_k - \delta)
(\omega_{k'} - \delta)}\, ,
\label{loops}\end{aligned}$$
where $q,~k,~k'$ are the momenta of the photon, incoming meson and outgoing meson, respectively. Note, $\vec{k}' = \vec{k} + \vec{q}$ and $\omega_k=\sqrt{k^2+m_j^2}$. The functions ${\cal U}(k)$ and ${\cal U}(k')$ are used to regulate the loop integrals - we use a dipole with a finite mass parameter $\Lambda$. Therefore
$$\begin{aligned}
{\cal U}(k) &= \left(\frac{\Lambda^{2}}{\Lambda^2+k^2} \right)^2\, , \\
{\cal U}(k') &= \left(\frac{\Lambda^{2}}{\Lambda^2+(k')^2} \right)^2\, .\end{aligned}$$
[lclclclcc]{}
& & &\
Channel & $\chi$ & Channel & $\chi$ & Channel & $\chi$ & Channel & $\chi$ \
\
$\Delta^+$ & 0.265 & $\Delta^0$ & 0.353 & $\Delta^{-}$ & 0.265 & $\Delta^0$ & -0.265 \
$\Sigma^{*+}$ & 0.184 & $\Delta^{++}$ & -0.265 & $\Delta^{+}$ & -0.353 & $n$ & -0.795 \
$p$ & 0.795 & $\Sigma^{*0}$ & 0.123 & $\Sigma^{*-}$ & 0.061 & &\
$\Sigma^{+}$ & 0.552 & $n$ & 0.265 & $p$ & -0.265 & &\
& & $\Sigma^{0}$ & 0.368 & $\Sigma^{-}$ & 0.184 & &\
The masses, $m_j$, of Eq. (\[MM\]) are octet-meson masses associated with the meson cloud and $\chi_{i}$ are model independent constants [@Li:1971vr] that give the magnitude of the loop contribution for each intermediate baryon state. These coefficients are given by [@Banerjee:1995wz] $$\begin{aligned}
\chi^{(\pi)}_{T_i} = \frac{M_N\ {\cal H}^2}{72\, {\pi}^2 \,
(f_{\pi})^2}\ {\alpha^{(\pi)}_{T_i}}\, , \hspace{2.5mm}
\chi^{(K)}_{T_i} = \frac{M_N\ {\cal H}^2}{72\, {\pi}^2\,
(f_{K})^2}\ {\alpha^{(K)}_{T_i}}\, , \nonumber \\
\chi^{(\pi)}_{B_i} = \frac{M_N\ {\cal C}^2}{32\, {\pi}^2\,
(f_{\pi})^2}\ {\beta^{(\pi)}_{B_i}}\, , \hspace{3.0mm}
\chi^{(K)}_{B_i} = \frac{M_N\ {\cal C}^2}{32\, {\pi}^2\,
(f_{K})^2}\ {\beta^{(K)}_{B_i}}\, ,
\label{CC}\end{aligned}$$ where $\cal H$ describes meson couplings to decuplet baryons and $\cal C$ is the octet–decuplet coupling constant. We assign ${\cal H}$ and ${\cal C}$ their $SU(6)$ values of ${\cal H} = -3 D$ and ${\cal C} = -2 D$ [@Butler:1992pn] where the tree level value for $D$ is $0.76$. The decay constants take the values $f_{\pi}=93~\rm MeV$ and $f_{K}= 1.2\, f_{\pi}\rm$ [@Jenkins:1992pi], appropriate to an expansion about the chiral SU(2) limit. Note also that $M_N$ is the nucleon mass and the parameters $\alpha$ and $\beta$ are given in Ref. [@Butler]. The model independent loop coefficients, $\chi$, are summarized in Table \[table:ChiralC5\]. In these calculations the complex mass scheme [@LopezCastro:2000ep; @ElAmiri:xa] is adopted as a method of incorporating the finite life of the $\Delta$ resonances, whilst also retaining electromagnetic gauge invariance. That is, one aims to extract the magnetic moment of the $\Delta$ at the pole position in the complex energy plane, $\delta^{\rm (pole)} \equiv \delta_R - i\, \delta_I = 270 - i\, 50$ MeV [@Hagiwara:pw] for the physical pion mass ($m_{\pi}^{\rm phys}$). Since the value at the position of the $\Delta$ pole is independent of the path chosen, we illustrate extrapolations along the path $\delta_R$ constant and $\delta_I$ given by $$\delta_{I} =
G\ \pi\ (\delta_R^2 - m_{\pi}^2)^{\frac{3}{2}}
\left(\frac{\Lambda^2}{\Lambda^2 + \delta_R^2- m_{\pi}^2} \right)^4\ \Theta(\delta_R - m_{\pi})\, .
\label{complex}$$ The latter is motivated by the usual expression for the $\Delta \to N\, \pi$ self-energy [@Young:2002ib], with $G$ chosen to ensure $\delta_I = 50$ MeV at $m_{\pi}^{\rm phys}$. To relate the kaon and pion masses we utilize the following relations provided by $\chi$PT $$\begin{aligned}
m_K^2 &= {m_K^{(0)}}^2 + \frac{1}{2} m_{\pi}^2\, , \\
m_K^{(0)} &= \sqrt{(m_K^{\mathrm{phys}})^2 -
\frac{1}{2}(m_{\pi}^{\mathrm{phys}})^2}\, .\end{aligned}$$ This allows us to also incorporate kaon loops (at fixed strange quark mass) in extrapolations as a function of $m_{\pi}^2$, to the physical mass regime.
If we expand the first term of Eq. (\[MM\]) as a Taylor series about $m_{\pi} = 0$ to order ${\cal O}(m_{\pi}^4)$, and take the limit as $\Lambda \to \infty$ in the loop integrals, we obtain the traditional $\chi$PT expansion for decuplet magnetic moments, as given in Eq. (\[CPT\]) (where the infinite constants encountered in dimensional regularization simply redefine $a_0$). Further, since the loop contributions approach zero much faster than $1/m_{\pi}^2$ for any reasonable value of $\Lambda$, Eq. (\[MM\]) guarantees the correct mass dependence of the Dirac magnetic moment in the heavy quark mass regime. From previous studies of the nucleon axial form factor [@Guichon:1982zk] it has been consistently demonstrated that $\Lambda$ in the dipole regulator must have a magnitude $<$ 1 GeV. For this investigation we assign $\Lambda = 0.8$ GeV, the optimal value obtained in analyses of state of the art full QCD simulations of the nucleon mass [@Young:2002ib]. In Fig. 2 we plot Eqs. (\[loops\]) (without the chiral coefficient prefactors) with the above value of $\delta$ and a regulator of $\Lambda=0.8$ GeV. Four types of intermediate baryon states are considered. The opening of the $N \, \pi$ decay channel has an interesting effect on the magnetic moment contribution. Fig. 3 presents a plot of the total loop contribution for each $\Delta$ baryon, determined using Eqs. (\[loops\]) and the chiral coefficients given in Table \[table:ChiralC5\].
[Results]{}
------------------ ------------- ------------ --------------- ------------ ------------ ------------
kappa Baryon Mass Pion Mass $\Delta^{++}$ $\Delta^+$ $\Delta^-$ $\Delta^-$
(GeV) (GeV) ($\mu_N$) ($\mu_N$) ($\mu_N$) ($\mu_N$)
\[+0.7ex\]
\[-1.7ex\] 0.152 1.74 (50) 0.964 (12) 2.81 (18) 1.40 (9) 0.000 (00) -1.40 (9)
0.154 1.57 (60) 0.820 (11) 3.19 (28) 1.59 (14) 0.000 (00) -1.59 (14)
0.156 1.39 (80) 0.665 (13) 3.67 (43) 1.83 (21) 0.000 (00) -1.83 (21)
------------------ ------------- ------------ --------------- ------------ ------------ ------------
The lattice QCD results given in Table \[table:Ldata\] are extracted from Ref. [@Leinweber:1992hy]. These lattice calculations employ sequential-source three-point function based techniques [@Comment] utilizing the conserved vector current such that no renormalization is required in relating the lattice results to the continuum. These simulations utilized twenty-eight quenched gauge configurations on a $24 \times 12 \times
12 \times 24$ periodic lattice at $\beta = 5.9$, corresponding to a lattice spacing of 0.128(10) fm. Moments are obtained from the form factors at $0.16~\rm GeV^2$ by assuming equivalent $q^2$ dependence for both the electric and magnetic form factors. This assumption will be tested in future studies incorporating the ideas presented here. Uncertainties are statistical in origin and are estimated by a third-order single-elimination jack-knife analysis [@BE]. In Figs. 4–7 we present fits of the extrapolation function, Eq. (\[MM\]), to the lattice data as a function of $m_{\pi}^2$. The resulting magnetic moment predictions, along with the fit parameters $a_0$ and $c_0$, are summarized in Table \[table:Results\]. In Fig. 4 we show two experimental values for the $\Delta^{++}$ magnetic moment, the result $\mu_{\Delta^{++}}=4.52 \pm 0.50 \pm 0.45\ \mu_N$ is from Ref. [@Bosshard] and $\mu_{\Delta^{++}}=6.14 \pm 0.51\ \mu_N$ is given in Ref. [@LopezCastro:2000ep]. The discrepancy between these two results is a reasonable indication of the current level of systematic error in the experimental determination of the $\Delta^{++}$ magnetic moment. Our prediction of $\mu_{\Delta^{++}}=4.99 \pm 0.56\ \mu_N$ agrees well with the first experimental result, however it lies slightly below the range of the second. We note that the approach used in Ref. [@LopezCastro:2000ep] does not respect unitarity and when this shortcoming is addressed the authors find $\mu_{\Delta^{++}}=6.01 \pm 0.61\ \mu_N$ [@LopezCastro:2000ep], resulting in a somewhat better agreement between our prediction and this experimental result. The result reported here for the $\Delta^+$ magnetic moment of $2.49 \pm 0.27\ \mu_N$ is in agreement with the initial measurement of Kotulla [*et al.*]{} [@Kotulla:2002cg], namely $\mu_{\Delta^+} = 2.7^{+1.0}_{-1.3} {\rm (stat.)} \pm 1.5 {\rm (syst.)} \pm
3 {\rm (theor.)} \mu_N$, however the experimental error at this time is still very large. To address the issue of regulator-mass dependence we vary $\Lambda$ between 0.5 and 1.0 GeV, in each case readjusting $a_0$ and $c_0$ to fit the lattice data and find only a slight regulator-mass dependence of 3.6% or less, for each of the charged $\Delta$ baryons. Ultimately, lattice results at lighter quark masses will constrain $\Lambda$ and therefore reduce this uncertainty. The inclusion of the $\Delta^0$ is more for completeness rather than a firm result, as the lattice data in quenched QCD yields identically zero in this case. By varying the regulator $\Lambda$ between 0.5 and 1.0 GeV we find the moment remains positive, with the order of magnitude of our result unchanged. We await new, unquenched lattice data in order to obtain a stronger prediction for the $\Delta^0$ magnetic moment.
[lcccc]{} Baryon &$a_0$ &$c_0$ &Lattice ($\mu_N$) & Experiment ($\mu_N$)\
\
\[-2.0ex\] $\Delta^{++}$ & 4.87 &0.82 & 4.99 (56) & 3.7–7.5\
$\Delta^{+}$ & 2.44 &0.83 & 2.49 (27) & $2.7^{+1.0}_{-1.3} {\rm (stat.)} \pm 1.5 {\rm (syst.)} \pm 3 {\rm (theor.)}$\
$\Delta^{0}$ & 0.63 &381 & 0.06 (00) &\
$\Delta^{-}$ &-2.40 &0.80 &-2.45 (27) &
The interesting feature of the plots given in Figs. 4–7 is the cusp at $m_{\pi} = \delta$ which indicates the opening of the octet decay channel, $\Delta \to N \pi$. The physics behind the cusp is intuitively revealed by the relation between the derivative with respect to $m_{\pi}^2$ of the magnetic moment and the derivative with respect to the momentum transfer, $q^2$, provided by the pion propagator $1/(q^2+m_{\pi}^2)$ in the heavy baryon limit. Derivatives with respect to $q^2$ are proportional to the magnetic radius in the limit $q^2 \to 0$ $$\langle r_M^2 \rangle = \left. -6\, \frac{d\, G_M(q^2)} {dq^2}\right|_{q^2=0}\, .$$ If we consider for example $\Delta^{++} \to p \pi^+$, with $|j,m_j\rangle = |3/2,3/2\rangle$, the $N\, \pi$ state is in relative P-wave orbital angular momentum with $|l,m_l\rangle = |1,1\rangle$. Thus the pion makes a positive contribution to the magnetic moment. As the opening of the $p\, \pi^+$ decay channel is approached from the heavy quark-mass regime, the range of the pion cloud increases. Just above threshold the pion cloud extends towards infinity as $\Delta E \to 0$, $\Delta E \Delta t \sim \hbar$ and the radius of the magnetic form factor diverges similarly, $ ({\partial}/{\partial q^2}) G_M \propto ({\partial}/{\partial m_{\pi}^2})
G_M \to - \infty$ from above threshold. Below threshold, $G_M$ becomes complex and the magnetic moment of the $\Delta$ is identified with the real part. The imaginary part describes the physics associated with photon-pion coupling in which the pion is subsequently observed as a decay product.
Therefore decuplet to octet transitions enhance the $\Delta$ magnetic moments when the octet decay channel is closed. However, as the decay channel opens this physics serves to suppress the magnetic moment as the chiral limit is approached. These transitions are energetically favourable, making them of paramount importance in determining the physical properties of $\Delta$ baryons. The inclusion of octet–decuplet transitions in octet magnetic moment extrapolations is less important. Significant energy must be borrowed from the vacuum for these transitions to occur. The formulation of effective field theory with a finite-range regulator naturally suppresses transitions from ground state octet baryons to excited state baryons. Physically, the finite size of the meson-cloud source suppresses these high energy contributions. Hence any new curvature associated with octet–decuplet transitions for octet baryons is small [@Leinweber:1998ej]. Indeed, the inclusion of the $p \to \Delta\, \pi$ channels in the proton moment extrapolation of Fig. 5 increases the predicted moment by only 0.05 $\mu_N$ from 2.61 to 2.66 $\mu_N$. For this reason octet to decuplet transitions have been omitted in other chiral extrapolation studies of the octet magnetic moments [@Hackett-Jones:2000qk; @Cloet:2002eg]. In the simplest SU(6) quark model with $m_u = m_d$ the $\Delta^{+}$ and proton moments are equal. However most quark models include spin dependent $q-q$ interactions that enhance the $\Delta^{+}$ magnetic moment relative to the proton. This phenomenology is supported by the lattice results [@Leinweber:1992hy] at large quark masses. Consequently, previous linearly extrapolated lattice predictions [@Leinweber:1992hy], using the same lattice data, suggested that the $\Delta^+$ moment should be greater than that of the proton. We find evidence that supports a different conclusion, with predicted values for the $\Delta^+$ and proton moments lying close at 2.49(27) $\mu_N$ and 2.66(17) $\mu_N$, respectively. The primary reason for this result is the interplay between the three different pion loop contributions; $\Delta^+ \to \Delta^{++}\, \pi^-$, $n\, \pi^+$ and $\Delta^0\, \pi^+$. The transition to a $\Delta^{++}\, \pi^-$ state largely cancels the $n\, \pi^+$ contribution, which dominates the proton magnetic moment. This means that the $\Delta^0\, \pi^+$ transition is the main loop contribution to the $\Delta^+$ magnetic moment, where the $\Delta^+ \to \Delta^0\, \pi^+$ coupling is weak, relative to that of the $p \to n\, \pi^+$ transition. The proton magnetic moment extrapolation is included in Fig. 5, and provides an illustration of the importance of incorporating known chiral physics in any extrapolation to the physical world. Thus, an experimental value for the $\Delta^{+}$ magnetic moment would offer very important insight into the role of spin dependent forces and chiral nonanalytic behaviour in the structure of baryon resonances.
[Conclusion]{}
Finite-range regulated $\chi$PT has been applied to the extrapolation of lattice QCD results for the decuplet baryon magnetic moments. The magnetic moments of the four $\Delta$ baryons at the resonance pole are determined. Experimental values exist only for the $\Delta^{++}$ magnetic moment. A result for the $\Delta^{+}$ should be forthcoming in the near future. The Particle Data Group gives a range of 3.7–7.5 $\mu_N$ for the $\Delta^{++}$ moment, with the two most recent experimental results being $\mu_{\Delta^{++}}=4.52 \pm 0.51 \pm 0.45\ \mu_N$ [@Bosshard] and $\mu_{\Delta^{++}}=6.14 \pm 0.51\ \mu_N$ [@LopezCastro:2000ep]. Our lattice QCD prediction of $\mu_{\Delta^{++}}=4.99 \pm 0.56\ \mu_N$ compares well with the first of these experimental results. An interesting result from this investigation is the prediction that the $\Delta^+$ magnetic moment lies close to and probably below that of the proton moment. Therefore a new high precision experimental measurement of the $\Delta^+$ moment would offer valuable insight into spin dependent forces and chiral nonanalytic behaviour of excited states. The lattice data used in these calculations is now over 10 years old, determined with pion masses greater than 600 MeV. The arrival of new lattice data at lower pion masses is therefore eagerly anticipated, and will help constrain the fit parameters and associated statistical uncertainties. All of this should be forthcoming in the next few years and offers an excellent opportunity to test the predictions of QCD.
We thank Ross Young for interesting and helpful discussions. This work was supported by the Australian Research Council and the University of Adelaide.
[20]{}
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C. Bernard, S. Hashimoto, D. B. Leinweber, P. Lepage, E. Pallante, S. R. Sharpe and H. Wittig, arXiv:hep-lat/0209086. A. W. Thomas, “Chiral extrapolation of hadronic observables”, arXiv:hep-lat/0208023.
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[^1]: icloet@physics.adelaide.edu.au
[^2]: dleinweb@physics.adelaide.edu.au
[^3]: athomas@physics.adelaide.edu.au
|
---
author:
- 'Hiroya <span style="font-variant:small-caps;">Sakurai</span>[^1], Naohito <span style="font-variant:small-caps;">Tsujii</span>$^{1}$, and Eiji <span style="font-variant:small-caps;">Takayama-Muromachi</span>'
title: ' Thermal and Electrical Properties of $\gamma$-Na$_{x}$CoO$_{2}$ ($0.70\leq x\leq 0.78$) '
---
Na$_{x}$CoO$_{2}$ is a quite attractive compound for condensed matter physicists and chemists because of its rich physical properties. For instance, Na$_{0.5}$CoO$_{2}$ shows unusually high thermoelectrical performance[@NaCo2O4N1; @NaCo2O4N2]. In the case of $x=0.35$, superconductivity below $\sim$5 K is induced by water insertion[@NCO]. Na$_{0.75}$CoO$_{2}$ shows a magnetic transition at $T_{\mbox{c}}=22$ K, which is believed to be caused by spin density wave (SDW) formation[@MotohashiSDW; @SugiyamaSDW1]. Moreover, for Na$_{x}$CoO$_{2}$ ($x=0.7$ - 0.75), Sommerfeld constant, $\gamma$, has been estimated from the specific heat to be $\sim$24-30 mJ/Co mol$\cdot$K$^{2}$[@MotohashiSDW; @Ando; @Bruhwiler; @Miyoshi; @Sales], and the heavy fermion behavior has been observed[@Miyoshi].
Despite intensive studies, there are still serious discrepancies in physical property data of Na$_{x}$CoO$_{2}$. The reports on the specific heat are inconsistent with each other, particularly those on low-temperature data: the $C/T$-$T^{2}$ curve ($C$: specific heat, $T$: temperature) in the low-temperature region has been reported to show an upturn[@Ando; @Bruhwiler; @Carretta], linear dependence[@Bayrakci; @Miyoshi], or a downturn[@Sales].
As regards the electrical resistivity, the variation of the data is less than that of the specific heat. When the transition at $T_{\mbox{c}}$ is magnetically observed, the resistivity shows a drop at $T_{\mbox{c}}$, while no anomaly at $T_{\mbox{c}}$ in a transition-free sample is observed. A slight bend in the resistivity around 100 K is seen clearly in the data reported by Shi $et$ $al.$[@Shi] and is observed in most cases[@Bruhwiler; @Sales; @Foo], the origin of which is left unclarified.
The discrepancies in the physical properties including magnetic properties seem to be mainly caused by their strong dependencies on the Na content $x$. Very recently, we prepared $\gamma$-Na$_{x}$CoO$_{2}$ samples by varying $x$ minutely and measured their magnetic properties. We found that the solid-solution range of the system is quite narrow with $0.70\leq x\leq 0.78$, and moreover, its magnetic properties are very sensitive to $x$[@Sakurai]. Most previous studies were carried out only for one of the two end members with $x\sim 0.7$ and $\sim 0.78$. In the present study, we measured the specific heats and the electrical resistivities for the same sets of samples used in the previous magnetic measurements, in order to elucidate their dependences on $x$ and to clarify the origin of the above-mentioned discrepancies.
The powder samples of Na$_{x}$CoO$_{2}$ ($x=0.70$, 0.72, 0.74, 0.76, 0.78, 0.80, and 0.82) were synthesized by the conventional solid state reaction from the stoichiometric mixtures of Na$_{2}$CO$_{3}$ (99.99%) and Co$_{3}$O$_{4}$ (99.9%). The detailed preparation method and the results of chemical analyses of the samples are described elsewhere[@Sakurai]. The specific heat data were collected using a commercial physical property measurement system (PPMS, Quantum Design) for the sintered samples with weights of 14-27 $mg$. The measurements were usually carried out with decreasing temperature, and the magnetic field ($H$) was applied before cooling. The electrical resistivity ($\rho$) was also measured using PPMS by the conventional four-probe method for the sintered samples with typical dimensions of $8\times 3\times 1$ $mm^{3}$ (1-2 $mm$ distance between adjacent probes) with increasing temperature at a rate of 1 K/min, after the samples were cooled down to 1.8 K at the same rate.
The $C/T$-$T^{2}$ curves of Na$_{0.78}$CoO$_{2}$ are shown in Fig. \[Cp\](a). Two anomalies are seen: (i) a sharp peak at $T_{\mbox{c}}=22$ K and (ii) a dull downward bend at $T_{\mbox{k}}\simeq 9$ K. These anomalies correspond to the magnetic transitions[@Sakurai]. The latter transition will be discussed later. The $C/T$-$T^{2}$ curves of Na$_{x}$CoO$_{2}$ with various $x$ values are shown in Fig. \[Cp\](b).
![ (a) $C/T$-$T^{2}$ curves at various magnetic fields. The broken and solid lines are visual guides. The inset shows $C/T$ under 0 Oe. The dotted lines are $C/T = 26.0 + 4.34 \times 10^{-2}T^{2}$ and a visual guide. (b) $C/T$-$T^{2}$ curves of Na$_{x}$CoO$_{2}$ with various $x$ values. The inset shows the $x$-dependences of $\Delta C/T$ (red markers) and $T_{\mbox{c}}$ (blue markers). The solid lines are visual guides. []{data-label="Cp"}](Fig1.eps){width="7cm"}
The peak position and shape at $T_{\mbox{c}}$ measured with increasing temperature were completely the same as those measured with decreasing temperature, and, as seen in Fig. \[Cp\](a), $T_{\mbox{c}}$ is also independent of $H$. Moreover, it does not depend on $x$ as shown in Fig. \[Cp\](b). Only the jump in $C/T$ at $T_{\mbox{c}}$, $\Delta C/T$, decreases with decreasing $x$ below 0.78 as seen in the inset of Fig. \[Cp\](b). These strongly suggest that the phase responsible for the transition in question does not change but only its fraction varies with $x$, which is consistent with the magnetic measurements[@Sakurai]. Namely, for the samples with $x$ above 0.74, a phase separation into Na-rich and Na-poor domains occurs. In addition, as seen in the inset of Fig. \[Cp\](b), the independent behavior of $\Delta C/T$ on $x$ for $x>0.78$ supports the higher limit of the solid-solution range of $x=0.78$[@Sakurai].
The Sommerfeld constants $\gamma$ and Debye temperatures $\Theta$ were first estimated by the function of $$C/T=\gamma+AT^{2}
\label{LinearC}$$ ($A=\frac{12\pi^{4}}{5}\frac{Nk_{B}}{\Theta^{3}}$: $N$, the number of the atom and $k_{B}$, Boltzmann constant) using the data between 26 and 36 K ($700\leq T^{2} \leq 1300$ K$^{2}$). The parameters obtained are shown in Fig. \[Para\] by circular markers. Since the temperature range seems to be high, the same parameters were estimated from the same data by a different function: $$C=\gamma T + 9Nk_{B}(\frac{T}{\Theta})^{3}\int^{\Theta/T}_{0}dx\frac{x^{4}e^{x}}{(e^{x}-1)^{2}} \mbox{ (Debye model), }
\label{Debye}$$ where the fitting parameters are only $\gamma$ and $\Theta$. As shown in Fig. \[Para\], the $\gamma$ and $\Theta$ values obtained from the two different equations are in good agreement with each other, which means that eq. \[LinearC\] is applicable to this temperature range of these compounds. These $\gamma$ values agree well with those reported previously[@MotohashiSDW; @Ando; @Bruhwiler; @Miyoshi; @Sales], and lie on the line of $\gamma = 74.6-62.4x$. This $x$-dependence of $\gamma$ indicates that the density of states (DOS) decreases with increasing Fermi energy, although it is difficult to estimate the influence of the phase separation of the samples with higher $x$ on the $\gamma$ values; it is confirmed that this negative inclination corresponds to that of the energy dependence of DOS above the Fermi energy of Na$_{0.5}$CoO$_{2}$[@Singh].
![ $x$-dependences of $\gamma$ (black markers) and $\Theta$ (red markers). The circular and triangular markers represent the values estimated using eqs. \[LinearC\] and \[Debye\], respectively. The black and red lines are $\gamma =74.6-62.4x$ and $\Theta =373+232x$, respectively. []{data-label="Para"}](Fig2.eps){width="7cm"}
Below about 15 K, the line of eq. \[LinearC\] is located above the experimental data (see the inset of Fig. \[Cp\](a)), which implies that some part of DOS is lost due to the transition at $T_{\mbox{c}}$. This fact seems to be in favor of the SDW formation at $T_{\mbox{c}}$ rather than the typical second-order transition suggested previously[@MotohashiSDW]. However, it should be noted that the SDW state is realized only in a part of a sample as mentioned above and suggested previously[@Sakurai].
The dull anomaly at $T_{\mbox{k}}$, which has been observed by a single crystal[@Sales], is likely due to another weak ferromagnetic transition[@Sakurai]. The transition temperature $T_{\mbox{k}}$ is independent of $H$ as in the case of $T_{\mbox{c}}$. Since the degree of the anomaly seems to change synchronically with $\Delta C/T$ at $T_{\mbox{c}}$, it seems that the domain which undergoes the transition at $T_{\mbox{c}}$ is followed by the transition at $T_{\mbox{k}}$. The transition at $T_{\mbox{k}}$ may be caused by the change in the magnetic structure formed in the first transition at $T_{\mbox{c}}$ with additional lost of DOS. It is, however, necessary to perform microscopic experiments, such as nuclear magnetic resonance (NMR), to elucidate details of the transition.
The $T$-dependence of $\rho$ for $x=0.78$ is shown in Fig. \[rho\](a). Three characteristic features are seen in this log-log plot: (i) a metallic behavior in the entire $T$-range, (ii) a steep decrease in $\rho$ below $T_{\mbox{c}}$, and (iii) bending of the curve at $T_{\mbox{b}}=120$ K. The metallic behavior even below $T_{\mbox{c}}$ and $T_{\mbox{k}}$ is consistent with the existence of the residual $\gamma$ of approximately 10 mJ/Co mol$\cdot$K$^{2}$ for $x=0.78$ at 0 K.
The steep decrease in $\rho$ below $T_{\mbox{c}}$ is, of course, related to the magnetic transition seen in the specific heat and the magnetic susceptibility[@Sakurai]. Indeed, as shown in Fig. \[rho\](b), this anomaly becomes less pronounced with decreasing $x$ in consistent with the specific heat and magnetic susceptibility data. Therefore, this anomaly reflects the intrinsic nature of the domain which undergoes the magnetic transition at $T_{\mbox{c}}$. In the case of a lower $x$ value, a marked decrease in $\rho$ below 40 K, which is seen even for $x=0.78$, seems to continue down to 1.8 K as reported by Miyoshi $et$ $al$.[@Miyoshi], the origin of which is unknown.
The bending-like variation of $\rho$ at $T_{\mbox{b}}$ has been seen in the in-plane resistivity measured by a single crystal[@NaCo2O4N1; @Bruhwiler; @Foo], which means that the bend is intrinsic. To explain this behavior, the data above 30 K were fitted by the equation $\rho=AT^{2}\ln (E_{F}/T)$ ($E_{F}$: Fermi energy), which is based on a two-dimensional (2D) Fermi gas model[@Hodges; @Bloom]. The resulting function seemed to reproduce the data to some extent, but $E_{F}$ was very small, being about 700 K. Then, we calculated the exponents $\alpha$ in $\rho\propto T^{\alpha}$ for the two temperature ranges 40-70 K ($\alpha _{\mbox{mid}}$) and 150-210 K ($\alpha _{\mbox{high}}$) for $x=0.78$ to obtain $\alpha _{\mbox{mid}}=0.371$ and $\alpha _{\mbox{high}}=0.833$. To eliminate the influence of residual resistivity, the residual resistivity, which was estimated to be 1.36 m$\Omega\cdot cm$ from the data below 10 K by a trinomial, was subtracted from the raw data. For the data compensated for the residual resistivity, the exponents become $\alpha _{\mbox{mid}}=0.558$ and $\alpha _{\mbox{high}}=0.992$ (see Fig. \[rho\](a)). Thus, $\rho$ obeys the almost $T$-linear relation in the high temperature range as the high-$T_{\mbox{c}}$ cuprates do. The optical spectra for $x=0.7$ also suggest a phenomenon similar to that of the high-$T_{\mbox{c}}$ cuprates[@Hwang]. On the other hand, the compensated data gave the $T^{1.46}$-dependence for the temperature range below $T_{\mbox{c}}$. From these facts, a typical Fermi liquid behavior was not seen at least for $x=0.78$ in any $T$-range and by any procedure. In cases of $x=0.74$ and 0.76, the $T^{2}$-dependence was seen but only below 5 K, which is consistent with the data obtained by a single crystal[@Miyoshi].
The $\alpha _{\mbox{low}}$, $\alpha _{\mbox{mid}}$ and $\alpha _{\mbox{high}}$ values were determined for every $x$ using the data compensated for the residual resistivities as shown in Fig. \[rhoPara\]. $T_{\mbox{b}}$ was determined from the crossing point of the two lines with $\alpha _{\mbox{mid}}$ and $\alpha _{\mbox{high}}$ and is plotted in Fig. \[rhoPara\]. $\alpha _{\mbox{high}}$ is almost independent of $x$ and is close to unity, indicating the $T$-linear-like behavior in the high $T$-range. On the other hand, $T_{\mbox{b}}$ increases linearly with $x$. Similar tendencies in $\alpha _{\mbox{mid}}$, $\alpha _{\mbox{high}}$, and $T_{\mbox{b}}$ were obtained when we used the raw data. Since no anomaly is seen around $T_{\mbox{b}}$ in the specific heat and the magnetic susceptibility[@Sakurai], the bend is not due to a transition. However, at the present stage, the physical meaning of $T_{\mbox{b}}$ is unclear, although the anomalies at approximately $T_{\mbox{b}}$ have been observed by angle-resolved photoemission spectroscopy and optical measurements[@Hasan; @Wang]. Similar bends have been observed in the resistivity of LiV$_{2}$O$_{4}$ ($T_{\mbox{b}}\sim 300$-400 K[@LiV2O4]) and in-plane resistivities of Sr$_{n+1}$Ru$_{n}$O$_{3n+1}$ ($T_{\mbox{b}}\sim 250$ K for $n=1$[@Sr2RuO4] and $T_{\mbox{b}}\sim 200$ K for $n=2$[@Sr3Ru2O7]). Thus, since these compounds have been expected to have a strong orbital or spin fluctuation, the bend of Na$_{x}$CoO$_{2}$ may be due to some kind of fluctuation, such as an orbital or spin fluctuation.
![ $x$-dependences of $T_{\mbox{b}}$, $\alpha _{\mbox{low}}$, $\alpha _{\mbox{mid}}$ and $\alpha _{\mbox{high}}$ estimated from the data compensated for the residual resistivities from the raw data. The solid lines represent $T_{\mbox{b}}=-35.8+198x$, $\alpha _{\mbox{mid}}=1.41-1.15x$, and $\alpha _{\mbox{high}}=0.610-0.365x$, respectively. []{data-label="rhoPara"}](Fig4R.eps){width="7cm"}
Finally, we will discuss about the physical properties for $x=0.70$. The resistivity of this compound shows a $T^{\alpha}$-dependence with $\alpha _{\mbox{low}}\sim 1$ below 7 K, and the $\alpha _{\mbox{low}}$ value increased up to 1.4 with increasing magnetic field up to 7 T, which is consistent with the results measured by a single crystal[@Li]. Since no magnetic transition is observed in the specific heat and magnetic measurements[@Sakurai], this behavior of the electrical resistivity suggests that this compound is in the vicinity of the magnetic instability which virtually exists below $x=0.70$. This idea seems to be consistent with the enhancement of $C/T$ below 10 K, because $\gamma$ of a nearly ferromagnetic or nearly antiferromagnetic compound is often enhanced by the spin fluctuation[@Moriya]. Since the Wilson ratio $R=\frac{\pi^{2}}{3}(\frac{k_{B}}{\mu _{B}})^{2}\frac{\chi}{\gamma}$ ($\mu _{B}$: Bohr magneton and $\chi$: magnetic susceptibility) is calculated to be $R=2.8$-2.9 for $x=0.70$, using $\chi =1.25$-$1.17\times 10^{-3}$ emu/mol at 26-36 K, the spin fluctuation seems to be ferromagnetic. The band calculation also indicates this ferromagnetic tendency[@Singh].
In summary, we have performed the specific heat and resistivity measurements of Na$_{x}$CoO$_{2}$ ($x=0.70$-0.78) for the same sets of samples used in the previous magnetic measurements[@Sakurai]. In the specific heat for $x=0.78$, two anomalies were seen, corresponding to magnetic transitions at $T_{\mbox{c}}=22$ K and $T_{\mbox{k}}\simeq 9$ K. Both anomalies become less pronounced simultaneously with decreasing $x$, while keeping $T_{\mbox{c}}$ and $T_{\mbox{k}}$ unchanged, and disappear below $x=0.72$. This behavior is consistent with our phase separation model proposed previously. The Sommerfeld constant was estimated to be $\gamma=26$-31 mJ/mol$\cdot$K$^{2}$, which is consistent with the previous reports. The resistivity measurements showed that this system is metallic for both the entire $x$ and $T$ ranges. The steep decrease at $T_{\mbox{c}}$ and the bending-like variation at $T_{\mbox{b}}$ (=120K) were found in the resistivity for $x=0.78$. $T_{\mbox{b}}$ increased slightly with $x$, the origin of which is unclear. From these results, for a higher $x$ value, a phase separation into Na-rich and Na-poor domains occurs as we previously proposed, while, for a lower $x$ value, the system is expected to be in the vicinity of the magnetic instability which virtually exists below $x=0.70$.
Acknowledge {#acknowledge .unnumbered}
===========
Special thanks to S. Takenouchi (NIMS) for chemical analysis. We would like to thank K. Takada, T. Sasaki, A. Tanaka, M. Kohno (NIMS), and K. Ishida (Kyoto University) for fruitful discussion. This study was partially supported by a Grants-in-Aid for Scientific Research (B) from Japan Society for the Promotion of Science (16340111). One of the authors (H.S) is a Research Fellow of the Japan Society for the Promotion of Science.
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[^1]: E-mail: sakurai.hiroya@nims.go.jp
|
---
abstract: 'Debris disks are tenuous, dust-dominated disks commonly observed around stars over a wide range of ages. Those around main sequence stars are analogous to the Solar System’s Kuiper Belt and Zodiacal light. The dust in debris disks is believed to be continuously regenerated, originating primarily with collisions of planetesimals. Observations of debris disks provide insight into the evolution of planetary systems; the composition of dust, comets, and planetesimals outside the Solar System; as well as placing constraints on the orbital architecture and potentially the masses of exoplanets that are not otherwise detectable. This review highlights recent advances in multiwavelength, high-resolution scattered light and thermal imaging that have revealed a complex and intricate diversity of structures in debris disks, and discusses how modeling methods are evolving with the breadth and depth of the available observations. Two rapidly advancing subfields highlighted in this review include observations of atomic and molecular gas around main sequence stars, and variations in emission from debris disks on very short (days to years) timescales, providing evidence of non-steady state collisional evolution particularly in young debris disks.'
author:
- 'A. Meredith Hughes,$^1$ Gaspard Duchêne,$^{2,3}$ Brenda C. Matthews$^{4,5}$'
title: 'Debris Disks: Structure, Composition, and Variability'
---
circumstellar disks, planet formation, extrasolar planetary systems, main sequence stars, planetesimals, circumstellar matter
1. Debris disks are a common phenomenon around main sequence stars, with current detection rates at $\sim 25$% despite the limited sensitivity of instrumentation (for Kuiper Belt analogues, a few $\times$ Solar System levels for even the nearest stars, and orders of magnitude from Solar System levels for exozodis) indicating that this figure is unquestionably a lower limit. Debris disks are more commonly detected around early-type and younger stars, and they frequently show evidence of two dust belts, like the Solar System.
2. High-resolution imaging of structures in outer debris disks reveals a rich diversity of structures including narrow and broad rings, gaps, haloes, wings, warps, clumps, arcs, spiral arms, and eccentricity. There is no obvious trend in disk radial extent with stellar spectral type or age. Most of the observed structures can be explained by the presence of planets, but for most structures there is also an alternative proposed theoretical mechanism that does not require the presence of planets.
3. The combined analysis of a debris disk’s SED with high-resolution (polarized) scattered light images and thermal emission maps is a powerful method to constrain the properties of its dust. Despite some shortcomings in current models, a coherent picture arises in which the grain size distribution is characterized by a power law similar to that predicted from collisional models, albeit with a minimum grain size that is typically a few times larger than the blowout size. Beyond the ubiquitous silicates, constraints on dust composition remain weak. Remarkably, dust in most debris disks appears to be characterized by a nearly universal scattering phase function that also matches that observed for dust populations in the Solar System, possibly because most dust grains share a similar aggregate shape.
4. Many debris disk systems harbor detectable amounts of atomic and/or molecular gas. Absorption spectroscopy of a few key edge-on systems reveals volatile-rich gas in a wide variety of atomic tracers, and there is evidence of an enhanced C/O ratio in at least some systems. Emission spectroscopy reveals that CO gas is common, especially around young A stars. The quantities of molecular gas found in most systems are likely insufficient to strongly affect the dust dynamics or planet formation potential. While the origin of the gas is still a matter of discussion, there is increasing evidence that the sample is not homogeneous. The molecular gas in some systems is clearly second-generation like the dust, while in other systems the disk is likely to be a “hybrid” with second-generation dust coexisting with at least some primordial gas.
5. Detection of time variable features in debris disk SEDs, images and light curves on timescales of days to years provide a window into ongoing dynamical processes in the disks and potentially the planetesimal belts as well.
6. Planet-disk interaction is now directly observable in a handful of systems that contain both a directly imaged planet(ary system) and a spatially resolved debris disk. These rare systems provide valuable opportunities to place dynamical constraints on the masses of planets, and thereby to calibrate models of planetary atmospheres.
<!-- -->
1. [*M star exploration:*]{} Most studies of debris disk structure and composition have so far focused on F, G, K, and A stars. While the incidence of M star debris disks appears to be low, the known disks include some of the most iconic systems including AUMic, and their gas and dust properties are poorly understood. Given then high frequency of terrestrial planets around M dwarfs, understanding their debris disks is a high priority.
2. [*Physics of Debris Disk Morphology:*]{} Multiwavelength imaging is beginning to untangle the underlying physical mechanisms sculpting debris disk structure. Dynamically-induced disk structures from planets or stellar flybys should affect even the largest grains, whereas those caused by the ISM or radiation effects tend to produce the largest effect on the smallest grains. As limits on gas emission improve, gas becomes a less plausible mechanism for sculpting narrow and eccentric rings in many systems. Imaging of structures across the electromagnetic spectrum, particularly comparison of ALMA thermal imaging with high-contrast scattered light imaging, will be key for moving from categorization of structure to conceptualization of the underlying physics.
3. [*The connection between hot, warm and cold dust belts:*]{} While multiple components are often detected in debris disk systems, identifying correlations between them has not yet delivered a fully coherent picture. Understanding the relative importance of in situ dust production and dust migration, as well as the physical mechanisms explaining the latter, remains an open question. Future observations sensitive enough to detect dust at intermediate radii between separate belts will help in clarifying this issue.
4. [*Variability:*]{} While studies of gas absorption variability have a long and fascinating history – revealing the dynamics and composition of falling evaporating bodies – studies of the variability of dust, both directly in emission and indirectly through stellar variability, are in their infancy. Currently there are a handful of known instances, each of which has multiple proposed explanations ranging from planetary dynamics to collisional avalanches. While large variations in integrated infrared light are rare, they are important for understanding stochastic events. Furthermore, the recent detection of fast-moving scattered light features in the dust around AUMic suggests that at least in some cases systems previously assumed to be static might be variable on more subtle spatial and flux contrast scales. Debris disk variability is an area that is wide open for discovery and modeling, particularly in the approaching epoch of [*JWST*]{}.
5. [*Gas statistics and chemistry:*]{} As the number of gas detections in debris disks increases, the opportunity for characterization of atomic and molecular abundances of the gas also grows. Clear opportunities include surveys designed to measure incidence of gas emission in a statistically meaningful way, in addition to detection of molecules other than CO and the corresponding opportunity to characterize the composition of exocometary gas. It would also be beneficial to amass rich data sets comparable to that of $\beta$Pic for a sample of several different objects so that modeling of both the atomic and molecular components can begin to explore the diversity of exosolar gas composition and better distinguish between primordial and secondary gas origins.
DISCLOSURE STATEMENT {#disclosure-statement .unnumbered}
====================
The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The authors wish to thank the following people for providing feedback and commentary on the article: Christine Chen, Kevin Flaherty, Paul Kalas, Grant Kennedy, Sasha Krivov, Luca Matra, Aki Roberge, Kate Su and Ewine van Dishoeck. The authors thank Eugene Chiang and Mark Wyatt for points of clarification. The authors are also grateful to the following people who agreed to share data used in the preparation of the figures in this review: Dan Apai, John Carpenter, Cail Daley, Bill Dent, Carsten Dominik, Jane Greaves, Paul Kalas, Markus Kasper, Mihoko Konishi, Meredith MacGregor, Sebastian Marino, Julien Milli, Johan Olofsson, Glenn Schneider. A.M.H. gratefully acknowledges support from NSF grant AST-1412647, and G.D. from NSF grants AST-1413718 and AST-1616479.
|
---
abstract: 'Descriptive Complexity has been very successful in characterizing complexity classes of decision problems in terms of the properties definable in some logics. However, descriptive complexity for counting complexity classes, such as $\fp$ and $\shp$, has not been systematically studied, and it is not as developed as its decision counterpart. In this paper, we propose a framework based on Weighted Logics to address this issue. Specifically, by focusing on the natural numbers we obtain a logic called Quantitative Second Order Logics (QSO), and show how some of its fragments can be used to capture fundamental counting complexity classes such as $\fp$, $\shp$ and $\fpspace$, among others. We also use QSO to define a hierarchy inside $\shp$, identifying counting complexity classes with good closure and approximation properties, and which admit natural complete problems. Finally, we add recursion to QSO, and show how this extension naturally captures lower counting complexity classes such as $\shl$.'
address: 'Pontificia Universidad Católica de Chile & IMFD Chile'
author:
- Marcelo Arenas
- Martin Muñoz
- Cristian Riveros
bibliography:
- 'biblio.bib'
title: Descriptive Complexity for Counting Complexity Classes
---
Introduction
============
Preliminaries {#sec:preliminaries}
=============
A logic for quantitative functions {#sec:logic}
==================================
Counting under $\qso$ {#sec:complexity}
=====================
Extending $\qso$ to capture classes beyond counting {#sec:extentions}
---------------------------------------------------
Exploring the structure of $\shp$ through $\qso$ {#sec:syntactic}
================================================
Adding recursion to QSO {#sec:beyond}
=======================
Concluding remarks and future work {#sec:conclusions}
==================================
Acknowledgements
================
|
---
author:
- 'Benjamin Arras, Ehsan Azmoodeh, Guillaume Poly and Yvik Swan'
---
In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein’s method. We provide several applications, and discuss comparison with recent similar results on the same topic.
0.3cm [**Keywords**]{}: Stein’s method, Stein discrepancy, Second Wiener chaos, Variance Gamma distribution, 2-Wasserstein distance, Malliavin Calculus
[**MSC 2010**]{}: 60F05, 60G50, 60G15, 60H07
Introduction
============
Background
----------
Stein’s method is a popular and versatile probabilistic toolkit for stochastic approximation. Presented originally in the context of Gaussian CLTs with dependant summands (see [@S72]) it has now been extended to cater for a wide variety of quantitative asymptotic results, see [@Chen-book] for a thorough overview of Gaussian approximation or https://sites.google.com/site/steinsmethod for an up-to-date list of references on non-Gaussian and non-Poisson Stein-type results. To this date one of the most active areas of application of the method is in Gaussian analysis, via Nourdin and Peccati’s so-called Malliavin/Stein calculus on Wiener space, see [@n-pe-1] or Ivan Nourdin’s dedicated webpage https://sites.google.com/site/malliavinstein.
Given two random objects $F, F_{\infty}$, Stein’s method allows to compute fine bounds on quantities of the form $$\sup_{h\in \mathcal{H}} \left|
\E\left[ h(F) \right] - \E \left[ h(F_{\infty}) \right] \right|.$$ The method rests on three pins :
A. a “Stein pair”, i.e. a linear operator and a class of functions $(\mathcal{A}_\infty, \mathcal{F}(\mathcal{A}_\infty))$ such that $ \E \left[ \mathcal{A}_\infty(f(F_{\infty})) \right] = 0$ for all $f \in \mathcal{F}(\mathcal{A}_\infty)$;
B. a contractive inverse operator $\mathcal{A}_\infty^{-1}$ acting on the centered functions $\bar{h} = h - \E h(F_{\infty})$ in $\mathcal{H}$ and contraction information, i.e. tight bounds on $\mathcal{A}_\infty^{-1}(\bar{h})$ and its derivatives;
C. handles on the structure of $F$ (such as $F=F_n=T(X_1, \ldots,
X_n)$ a $U$-statistic, $F = F(X)$ a functional of an isonormal Gaussian process, $F$ a statistic on a random graph, etc.).
Given the conjunction of these three elements one can then apply some form of transfer principle : $$\label{eq:6}
\sup_{h\in \mathcal{H}} \left|
\E\left[ h(F) \right] - \E \left[ h(F_{\infty}) \right] \right|
= \sup_{h\in \mathcal{H}} \left|
\E\left[ \mathcal{A}_\infty\left( \mathcal{A}_\infty^{-1}(\bar h(F)) \right) \right]\right|;$$ remarkably the right-hand-side of the above is often much more amenable to computations than the left-hand-side, even in particularly unfavourable circumstances. This has resulted in Stein’s method delivering several striking successes (see [@B-H-J; @Chen-book; @n-pe-1]) which have led the method to becoming the recognised and acclaimed tool it is today.
In general the identification of a Stein operator is the cornerstone of the method. While historically most practical implementations relied on adhoc arguments, several general tools exist, including Stein’s *density approach* [@S86] and Barbour’s *generator approach* [@B90]. A general theory for Stein operators is available in [@LRS]. In many important cases, these are first order differential operators (see [@dobler]) or difference operators (see [@LS]). Higher order differential operators have recently come into light (see [@g-variance-gamma; @p-r-r]).
Once an operator is $\mathcal{A}_{\infty}$ identified, the task is then to bound the resulting rhs of ; there are many ways to perform this. In this paper we focus on Nourdin and Peccati’s approach to the method. Let $F_{\infty}$ be a standard Gaussian random variable. Then the appropriate operator is $ \mathcal{A}_{\infty}f(x) = f'(x) - xf(x).$ Given a sufficiently regular centered random variable $F$ with finite variance and smooth density, define its Stein kernel $\tau_F(F)$ through the integration by parts formula $$\label{eq:9}
\E[\tau_F(F) \phi'(F) ] = \E \left[ F \phi(F)
\right] \mbox{ for all absolutely continuous } \phi.$$ Then, for $f$ a solution to $f'(x) - xf(x) = h(x) -\E[h(F_{\infty})]$ write $$\begin{aligned}
\E[h(F)] - \E[h(F_{\infty})] & = \E \left[ f_h'(F)
- F f_h(F) \right] = \E \left[ (1-\tau_F(F)) f_h'(F) \right]\end{aligned}$$ so that $$\begin{aligned}
\left| \E[h(F)] - \E[h(F_{\infty})] \right| \le \| f_h'\|
\sqrt{\E \left[ (1-\tau_F(F))^2 \right]}. \end{aligned}$$ At this stage two good things happen : (i) the constant $\sup_{h \in \mathcal{H}}\|f_h'\|$ (which is intrinsically Gaussian and does not depend on the law of $F$) is bounded for wide and relevant classes $\mathcal{H}$; (ii) the quantity $$\label{eq:7}
S(F \, || \, F_{\infty}) = \E \left[ (1-\tau_F(F))^2 \right]$$ (called the *Stein discrepancy*) is tractable, via Malliavin calculus, as soon as $F$ is a sufficiently regular functional of a Gaussian process. These two realizations opened a wide field of applications within the so-called “Malliavin Stein Fourth moment theorems”, see [@n-pe-ptrf; @n-pe-1]. A similar approach holds also if $F_{\infty}$ is centered Gamma, see [@n-pe-2; @a-m-m-p], and more generally if the law of the target random variable $F_{\infty}$ belongs to the family of Variance Gamma distributions, see [@thale] for the method and [@g-variance-gamma] for the bounds on the corresponding solutions. See also [@kusuoka; @viquez] for other generalizations. We stress that in the Gaussian case, Stein’s method provides bounds e.g. in the Total Variation distance, whereas technicalities related to the Gamma and Variance Gamma targets impose that one must deal with smoother distances (i.e. integrated probability measures of the form with $\mathcal{H}$ a class of smooth functions) in such cases.
Purpose of this paper
---------------------
The primary purpose of this paper is to extend Nourdin and Peccati’s “Stein discrepancy analysis” to provide meaningful bounds on $ \mathrm{d}(F, F_{\infty})$ for $\mathrm{d}(.,.)$ some appropriate probability metric and random variables $F_{\infty}$ belonging to the second Wiener chaos, that is $$\label{target-wiener1}
F_{\infty} = \sum_{i=1}^q \alpha_{\infty,i} (N^2_i -1).$$ where $q\ge 2$, $\{N_i\}_{i=1}^{q}$ are i.i.d. $\mathscr{N}(0,1)$ random variables, and the coefficients $\{ \alpha_{\infty,i}\}_{i=1}^{q}$ are distinct.
Such a generalization immediately runs into a series of obstacles which need to be dealt with. We single out three crucial questions : (Q1) what operator $\mathcal{A}_{\infty}$? (Q2) what quantity will play the role of the Stein discrepancy $S(F \, || \, F_{\infty})$? (Q3) what kind of distances $\mathrm{d}(.,.)$ can we tackle through this approach?
In this paper we provide a complete answers to a more general version of (Q1), hereby opening the way for applications of Stein’s method to a wide variety of new target distributions. We use results from [@a-p-p] to answer (Q2) for chaotic random variables. We also provide an answer to (Q3) for $\mathrm{d}(.,.)$ the $p$-Wasserstein distances with $p\le 2$, under specific assumptions on the structure of $F$. Such a result extends the scope of Stein’s method to so far unchartered territories, because aside for the case $p=1$, $p$-Wasserstein distances do not admit a representation of the form .
Overview of the results
-----------------------
In the first part of the paper, Section \[sec:steins-method-second\], we discuss Stein’s method for target distributions of the form . In Section \[benjarras\] we introduce an entirely new Fourier-based approach to prove a Stein-type characterization for a large family of $F_{\infty}$ encompassing those of the form . The operator $\mathcal{A}_{\infty}$ we obtain is a differential operator of order $q$. In Section \[sec:mall-based-appr\] we use recent results from [@a-p-p] to derive a Malliavin-based justification for our $\mathcal{A}_{\infty}$ when $F_{\infty}$ is of the form . We also introduce a new quantity $\Delta(F_n, F_{\infty})$ for which we will provide a heuristic justification in Section \[sec:cattyw-steins-meth\] of the fact that $\Delta(F_n, F_{\infty})$ generalizes the Stein discrepancy $S(F \, || \, F_{\infty})$ in a natural way for chaotic random variables. Finally we argue that quantitative assessments for general targets of the form are out of the scope of the current version of Stein’s method. In the second part of the paper, Section \[sec:preliminaries\], we introduce an entirely new *polynomial* approach to Stein’s method to provide bounds on the Wasserstein-2 distance (and hence the Wasserstein-1 distance) in terms of $\Delta(F_n, F_{\infty})$. Our approach bypasses entirely the need for estimating bounds on solutions of Stein equations. More specifically we provide a tool for providing quantitative assessments on $\mathrm{d}_{W_2}(F_n, F_{\infty})$ in terms of the generalized Stein discrepancy $\Delta(F_n, F_{\infty})$ for $F_{\infty}$ as in and $$F_n = \sum\limits_{i=1}^{\infty} \alpha_{n, i} (N_i^2-1)$$ still with $\left\{ N_i \right\}_{i\ge 1}$ i.i.d. standard Gaussian and now $\left\{ \alpha_{n, i} \right\}_{i\ge1}$ not necessarily distinct real numbers. As mentioned above, the fact that we bound the Wasserstein-2 distance is not anecdotal : this distance (which is useful in many important settings, see [@villani-book]) does not bear a dual representation of the form and is thus entirely out of the scope of the traditional versions of Stein’s method. In Section \[sec:idea-behind-proof\] we provide an intuitive explanation of the proof of our main results. In Section \[s:applications\] we apply our bounds to particular cases and compare them to the only competitor bounds available in the literature which are due to [@thale], wherein only the case with $q=2$ and $\alpha_1 = -\alpha_2$ is covered. Finally in Section \[lem:appendix\] we provide the proof.
Stein’s method for the second Wiener chaos {#sec:steins-method-second}
==========================================
Here we set up Stein’s method for target distributions in the second Wiener chaos of the form $$F_\infty= \alpha_{\infty,1} (N^2_1 -1) + \cdots +
\alpha_{\infty,q}(N^2_q -1)$$ where $\{N_i\}_{i=1}^{q}$ is a family of i.i.d. $\mathscr{N}(0,1)$ random variables.
Overview of known results {#sec:overv-known-results}
-------------------------
In the special case when $\alpha_{\infty,i}=1$ for all $i$, then $F_\infty= \sum_{i=1}^{q} (N^2_i -1) \sim \chi^2_{(q)}$ is a centered chi-squared random variable with $q$ degree of freedom. Pickett [@pickett] has shown that a Stein’s equation for target distribution $F_\infty$ is given by the first order differential equation $$x f'(x) + \frac12 (q-x) f(x) = h(x) - \E[h(F_\infty)].$$ For more recent results in this direction consult [@g-p-r] and references therein.
Another important contribution in our direction is given by Gaunt in [@g-2normal] with $q=2$ and $\alpha_{\infty,1}= - \alpha_{\infty,2}= \frac12$. In this case $$F_\infty \stackrel{\text{law}}{=} N_1 \times N_2$$ where $N_1$ and $N_2$ are two independent $\mathscr{N} (0,1)$ random variables. He has shown that a Stein’s equation for $F_\infty$ can be given by the following second order differential equation
$$x f''(x) + f'(x) - x f(x) = h(x) - \E [h (F_\infty)].$$ It is a well known fact that the density function of the target random variable $N_1 \times N_2$ is expressible in terms of the modified Bessel function of the second kind so that it is given by solution of a known second order differential equation and the Stein operator follows from some form of duality argument.
The more relevant studies of target distributions having a second order Stein’s differential equations include: Variance-Gamma distribution [@g-variance-gamma], Laplace distribution [@p-r-laplace], or a family of probability distributions given by densities $$f_s(x)= \Gamma(s) \sqrt{\frac{2}{s \pi}} \exp(- \frac{x^2}{2s})
U(s-1,\frac12,\frac{x^2}{2s}), \quad x>0, \, s \ge \frac12$$ appearing in preferential attachment random graphs, and $U(a,b;x)$ is the Kummer’s confluent hypergeometric function, see [@p-r-r]. We stress the fact that Stein’s method is [completely open]{} even in the simple case when the target random variable $F_\infty$ has only two non-zero eigenvalues $\alpha_{\infty,1}$ and $\alpha_{\infty,2}$, i.e. $$F_\infty = \alpha_{\infty,1} (N^2_1 -1) + \alpha_{\infty,2} (N^2_2 -1)$$ such that $\vert \alpha_{\infty,1}\vert \neq \vert \alpha_{\infty,2} \vert $. It is worth mentioning that such distributions are beyond the Variance-Gamma class, and are appearing more and more in very recent and delicate limit theorems, see [@b-t] for asymptotic behavior of generalized Rosenblatt process at extreme critical exponents and [@m-p-r-w] for asymptotic nodal length distributions.
Fourier-based approach
----------------------
Before stating the next theorem, we need to introduce some notations. For any $d$-tuple $(\lambda_1,...,\lambda_d)$ of real numbers, we define the symmetric elementary polynomial of order $k\in\{1,...,d\}$ evaluated at $(\lambda_1,...,\lambda_d)$ by: $$\begin{aligned}
e_{k}(\lambda_1,...,\lambda_d)=\sum_{1\leq i_1<i_2<...<i_k\leq d}\lambda_{i_1}...\lambda_{i_k}.\end{aligned}$$ We set, by convention, $e_{0}(\lambda_1,...,\lambda_d)=1$. Moreover, for any $(\mu_1,...,\mu_d)\in\mathbb{R}^*$ and any $k\in\{1,...,d\}$, we denote by $(\lambda/\mu)_k$ the $d-1$ tuple defined by: $$\begin{aligned}
(\frac{\lambda}{\mu})_k=\bigg(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_{k-1}}{\mu_{k-1}},\frac{\lambda_{k+1}}{\mu_{k+1}},...,\frac{\lambda_d}{\mu_d}\bigg).\end{aligned}$$ For any $(\alpha,\mu)\in\mathbb{R}_+^*$, we denote by $\gamma(\alpha,\mu)$ a Gamma law with parameters $(\alpha,\mu)$ whose density is: $$\begin{aligned}
\forall x\in\mathbb{R}_+^*,\ \gamma_{\alpha,\mu}(x)=\dfrac{\mu^\alpha}{\Gamma(\alpha)}x^{\alpha-1}\exp\big(-\mu x\big).\end{aligned}$$
\[benjarras\] Let $d \ge 1$ and $(m_1, \ldots, m_d) \in \mathbb{N}^d$. Let $((\alpha_1,\mu_1),...,(\alpha_d,\mu_d))\in \big(\mathbb{R}_+^*\big)^{2d}$ and $(\lambda_1, \ldots, \lambda_d) \in
\R^{\star}$ and consider: $$F = -\sum_{i=1}^d\lambda_i\dfrac{m_i\alpha_i}{\mu_i}+\sum_{i=1}^{d}\lambda_i\gamma_i\big(m_i\alpha_i,\mu_i\big),$$ where $\{\gamma_i\big(m_i\alpha_i,\mu_i\big)\}$ is a collection of independent gamma random variables with appropriate parameters. Let $Y$ be a real valued random variable such that $\E[|Y|]<+\infty$. Then $Y \stackrel{\text{law}}{=} F$ if and only if $$\begin{aligned}
&\E \bigg[ \big(Y+\sum_{i=1}^d\lambda_i\dfrac{m_i\alpha_i}{\mu_i}\big)(-1)^d\bigg(\prod_{j=1}^d\dfrac{\lambda_j}{\mu_j}\bigg)\phi^{(d)}(Y)+\sum_{l=1}^{d-1}(-1)^l\bigg(Ye_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})\nonumber\\ &+\sum_{k=1}^d\lambda_k\dfrac{m_k\alpha_k}{\mu_k}\left(e_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})-e_{l}((\frac{\lambda}{\mu})_k)\right)\bigg)\phi^{(l)}(Y)+Y\phi(Y) \bigg]=0,\label{eq:2}\end{aligned}$$ for all $\phi\in S(\mathbb{R})$.
$(\Rightarrow)$. Let $F$ be as in the statement of the theorem. We denote by $J_+=\{j\in\{1,...,d\}:\lambda_j>0\}$ and $J_-=\{j\in\{1,...,d\}:\lambda_j<0\}$. Let us compute the characteristic function of $F$. For any $\xi\in \mathbb{R}$, we have: $$\begin{aligned}
\phi_F(\xi)&=\E[\exp(i\xi F)],\\
&=\exp\bigg(-i\xi\sum_{k=1}^d\lambda_k\dfrac{m_k\alpha_k}{\mu_k}\bigg)\prod_{j=1}^d\E\bigg[\exp\bigg(i\xi\lambda_j\gamma_j(m_j\alpha_j,\mu_j)\bigg)\bigg],\\
&=\exp\bigg(-i\xi<m\alpha;\lambda/\mu>\bigg)\prod_{j\in J_+}\bigg(\exp\bigg(\int_0^{+\infty}\bigg(e^{i\xi\lambda_j x}-1\bigg)\bigg(\frac{m_j\alpha_j}{x}e^{-\mu_jx}\bigg)dx\bigg)\bigg)\\
&\times\prod_{j\in J_-}\bigg(\exp\bigg(\int_0^{+\infty}\bigg(e^{i\xi\lambda_j x}-1\bigg)\bigg(\frac{m_j\alpha_j}{x}e^{-x\mu_j}\bigg)dx\bigg)\bigg),\\
&=\exp\bigg(-i\xi<m\alpha;\lambda/\mu>\bigg)\exp\bigg(\int_0^{+\infty}\bigg(e^{i\xi x}-1\bigg)\bigg(\sum_{j\in J_+}\frac{m_j\alpha_j}{x}e^{-\frac{x\mu_j}{\lambda_j}}\bigg)dx\bigg)\\
&\times\exp\bigg(\int_0^{+\infty}\bigg(e^{-i\xi x}-1\bigg)\bigg(\sum_{j\in J_-}\frac{m_j\alpha_j}{x}e^{-\frac{x\mu_j}{(-\lambda_j)}}\bigg)dx\bigg),\end{aligned}$$ where we have used the Lévy-Khintchine representation of the Gamma distribution. We denote $\mu_j/\lambda_j$ by $\nu_j$. Differentiating with respect to $\xi$ together with standard computations, we obtain: $$\begin{aligned}
\prod_{k=1}^d(\nu_k-i\xi)\dfrac{d}{d\xi}\bigg(\phi_F(\xi)\bigg)=\bigg[-i<m\alpha,\lambda/\mu>\prod_{k=1}^d(\nu_k-i\xi)+i\sum_{k=1}^dm_k\alpha_k\prod_{l=1,l\ne k}^d(\nu_l-i\xi)\bigg]\phi_F(\xi).\end{aligned}$$ Let us introduce two differential operators characterized by their symbols in Fourier domain. For smooth enough test functions, $\phi$, we define: $$\begin{aligned}
&\mathcal{A}_{d,\nu}(\phi)(x)=\frac{1}{2\pi}\int_{\mathbb{R}}\mathcal{F}(\phi)(\xi)\bigg(\prod_{k=1}^d(\nu_k-i\xi)\bigg)\exp(ix\xi)d\xi,\\
&\mathcal{B}_{d,m,\nu}(\phi)(x)=\frac{1}{2\pi}\int_{\mathbb{R}}\mathcal{F}(\phi)(\xi)\bigg(\sum_{k=1}^dm_k\alpha_k\prod_{l=1,l\ne k}^d(\nu_l-i\xi)\bigg)\exp(ix\xi)d\xi,\\
&\mathcal{F}(\phi)(\xi)=\int_{\mathbb{R}}\phi(x)\exp(-ix\xi)dx.\end{aligned}$$ Integrating against smooth test functions the differential equation satistifed by the characteristic function $\phi_F$, we have, for the left hand side: $$\begin{aligned}
\int_{\mathbb{R}}\mathcal{F}(\phi)(\xi)\bigg(\prod_{k=1}^d(\nu_k-i\xi)\bigg)\dfrac{d}{d\xi}\bigg(\phi_F(\xi)\bigg)d\xi&=\int_{\mathbb{R}}\mathcal{F}\big(\mathcal{A}_{d,\nu}(\phi)\big)(\xi)\dfrac{d}{d\xi}\bigg(\phi_F(\xi)\bigg)d\xi,\\
&=-\int_{\mathbb{R}}\dfrac{d}{d\xi}\bigg(\mathcal{F}\big(\mathcal{A}_{d,\nu}(\phi)\big)(\xi)\bigg)\phi_F(\xi)d\xi,\\
&=i\int_{\mathbb{R}}\mathcal{F}\big(x\mathcal{A}_{d,\nu}(\phi)\big)(\xi)\phi_F(\xi)d\xi,\end{aligned}$$ where we have used the standard fact $d/d\xi(\mathcal{F}(f)(\xi))=-i\mathcal{F}(xf)(\xi)$. Similarly, for the right hand side, we obtain: $$\begin{aligned}
\operatorname{RHS}&=\int_{\mathbb{R}}\mathcal{F}(\phi)(\xi)\bigg[-i<m\alpha,\lambda/\mu>\prod_{k=1}^d(\nu_k-i\xi)+i\sum_{k=1}^dm_k\alpha_k\prod_{l=1,l\ne k}^d(\nu_l-i\xi)\bigg]\phi_F(\xi)d\xi,\\
&=i\int_{\mathbb{R}}\mathcal{F}\big(-<m\alpha,\lambda/\mu>\mathcal{A}_{d,\nu}(\phi)+\mathcal{B}_{d,m,\nu}(\phi)\big)(\xi)\phi_F(\xi)d\xi.\end{aligned}$$ Thus, $$\begin{aligned}
\int_{\mathbb{R}}\mathcal{F}\big((x+<m\alpha,\lambda/\mu>)\mathcal{A}_{d,\nu}(\phi)-\mathcal{B}_{d,m,\nu}(\phi)\big)(\xi)\phi_F(\xi)d\xi=0\end{aligned}$$ Going back in the space domain, we obtain the following Stein-type characterization formula: $$\begin{aligned}
\E\big[(F+<m\alpha,\lambda/\mu>)\mathcal{A}_{d,\nu}(\phi)(F)-\mathcal{B}_{d,m,\nu}(\phi)(F)\big]=0.\end{aligned}$$ In order to conclude the first half of the proof, we need to compute explicitely the coefficients of the operators $\mathcal{A}_{d,\nu}$ and $\mathcal{B}_{d,m,\nu}$ in the following expansions: $$\begin{aligned}
\mathcal{A}_{d,\nu}=\sum_{k=0}^da_k\dfrac{d^k}{dx^k},\\
\mathcal{B}_{d,m,\nu}=\sum_{k=0}^{d-1}b_k\dfrac{d^k}{dx^k}.\end{aligned}$$ First of all, let us consider the following polynomial in $\mathbb{R}[X]$: $$\begin{aligned}
P(x)=\prod_{j=1}^d(\nu_j-x)=(-1)^d\prod_{j=1}^d(x-\nu_j).\end{aligned}$$ We denote by $p_0,...,p_d$ the coefficients of $\prod_{j=1}^d(X-\nu_j)$ in the basis $\{1,X,...,X^d\}$. Vieta formula readily give: $$\begin{aligned}
\forall k\in\{0,...,d\},\ p_k=(-1)^{d+k}e_{d-k}(\nu_1,...,\nu_d),\end{aligned}$$ It follows that the Fourier symbol of $\mathcal{A}_{d,\nu}$ is given by: $$\begin{aligned}
\prod_{k=1}^d(\nu_k-i\xi)=P(i\xi)=\sum_{k=0}^d(-1)^ke_{d-k}(\nu_1,...\nu_d)(i\xi)^k.\end{aligned}$$ Thus, we have, for $\phi$ smooth enough: $$\begin{aligned}
\mathcal{A}_{d,\nu}(\phi)(x)=\sum_{k=0}^d(-1)^{k}e_{d-k}(\nu_1,...,\nu_d)\phi^{(k)}(x).\end{aligned}$$ Let us proceed similarly for the operator $B_{d,m,\nu}$. We denote by $P_k$ the following polynomial in $\mathbb{R}[X]$ (for any $k\in\{1,...,d\}$): $$\begin{aligned}
P_k(x)=(-1)^{d-1}\prod_{l=1,l\ne k}^d(x-\nu_l).\end{aligned}$$ A similar argument provides the following expression: $$\begin{aligned}
P_k(x)=\sum_{l=0}^{d-1}(-1)^le_{d-1-l}(\underline{\nu}_k)x^l,\end{aligned}$$ where $\underline{\nu}_k=(\nu_1,...,\nu_{k-1},\nu_{k+1},...,\nu_d)$. Thus, the symbol of the differential operator $B_{d,m,\nu}$ is given by: $$\begin{aligned}
\sum_{k=1}^dm_k\alpha_k\prod_{l=1,l\ne k}^d(\nu_l-i\xi)=\sum_{l=0}^{d-1}(-1)^l\bigg(\sum_{k=1}^dm_k\alpha_ke_{d-1-l}(\underline{\nu}_k)\bigg)(i\xi)^l.\end{aligned}$$ Thus, we have: $$\begin{aligned}
B_{d,m,\nu}(\phi)(x)=\sum_{l=0}^{d-1}(-1)^l\bigg(\sum_{k=1}^dm_k\alpha_ke_{d-1-l}(\underline{\nu}_k)\bigg)\phi^{(k)}(x).\end{aligned}$$ Consequently, we obtain: $$\begin{aligned}
&\E\big[(F+<m\alpha,\lambda/\mu>)\sum_{k=0}^d(-1)^{k}e_{d-k}(\nu_1,...,\nu_d)\phi^{(k)}(F)\\
&-\sum_{l=0}^{d-1}(-1)^l\bigg(\sum_{k=1}^dm_k\alpha_ke_{d-1-l}(\underline{\nu}_k)\bigg)\phi^{(k)}(F)\big]=0.\end{aligned}$$ Finally, there is a straightforward relationship between $e_{k}(\nu_1,...,\nu_d)$ and $e_{d-k}(\lambda_1/\mu_1,...,\lambda_d/\mu_d)$. Namely, $$\begin{aligned}
\forall k\in\{0,...,d\},\ e_{k}(\nu_1,...,\nu_d)=\dfrac{\prod_{j=1}^d\mu_j}{\prod_{j=1}^d\lambda_j}e_{d-k}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d}).\end{aligned}$$ Thus, multiplying by $\prod_{j=1}^d\lambda_j/\prod_{j=1}^d\mu_j$, the previous Stein-type characterisation equation, we have: $$\begin{aligned}
&\E\bigg[(F+<m\alpha,\lambda/\mu>)(-1)^d\bigg(\prod_{j=1}^d\frac{\lambda_j}{\mu_j}\bigg)\phi^{(d)}(F)+\sum_{l=1}^{d-1}(-1)^l\bigg(Fe_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})\\
&+\sum_{k=1}^d\lambda_km_k\frac{\alpha_k}{\mu_k}\big(e_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})-e_{l}((\frac{\lambda}{\mu})_k)\big)\bigg)\phi^{(l)}(F)+F\phi(F)\bigg]=0.\end{aligned}$$ $(\Leftarrow)$ Let $Y$ be a real valued random variable such that $\E[|Y|]<+\infty$ and: $$\begin{aligned}
\forall \phi\in S(\mathbb{R}),\ &\E \bigg[ (Y+<m\alpha,\lambda/\mu>)(-1)^d\bigg(\prod_{j=1}^d\frac{\lambda_j}{\mu_j}\bigg)\phi^{(d)}(Y)+\sum_{l=1}^{d-1}(-1)^l\bigg(Ye_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})\\ &+\sum_{k=1}^d\lambda_km_k\frac{\alpha_k}{\mu_k}\big(e_{l}(\frac{\lambda_1}{\mu_1},...,\frac{\lambda_d}{\mu_d})-e_{l}((\frac{\lambda}{\mu})_k)\big)\bigg)\phi^{(l)}(Y)+Y\phi(Y) \bigg]=0.\end{aligned}$$ By the previous step, this implies that: $$\begin{aligned}
&\forall \phi\in S(\mathbb{R}),\ \int_{\mathbb{R}}\mathcal{F}\big((x+<m\alpha,\lambda/\mu>)\mathcal{A}_{d,\nu}(\phi)-\mathcal{B}_{d,m,\nu}(\phi)\big)(\xi)\phi_Y(\xi)d\xi=0,\\
&\Leftrightarrow\ \int_{\mathbb{R}}\mathcal{F}\big(x\mathcal{A}_{d,\nu}(\phi)\big)(\xi)\phi_Y(\xi)d\xi=\int_{\mathbb{R}}\mathcal{F}\big(-<m\alpha,\lambda/\mu>\mathcal{A}_{d,\nu}(\phi)+\mathcal{B}_{d,m,\nu}(\phi)\big)(\xi)\phi_Y(\xi)d\xi,\\
&\Leftrightarrow\ \prod_{k=1}^d(\nu_k-i\xi)\dfrac{d}{d\xi}\bigg(\phi_Y\bigg)(.)=\bigg[-i<m\alpha,\lambda/\mu>\prod_{k=1}^d(\nu_k-i\xi)+i\sum_{k=1}^d\alpha_km_k\prod_{l=1,l\ne k}^d(\nu_l-i\xi)\bigg]\phi_Y(.),\end{aligned}$$ in $S'(\mathbb{R})$. Since $\E[|Y|]<+\infty$, the characteristic function of $Y$ is differentiable on the whole real line so that: $$\begin{aligned}
\forall\xi\in\mathbb{R},\ \dfrac{d}{d\xi}\bigg(\phi_Y\bigg)(\xi)=\bigg[-i<m\alpha,\lambda/\mu>+i\sum_{k=1}^dm_k\alpha_k\dfrac{1}{\nu_k-i\xi}\bigg]\phi_Y(\xi)\end{aligned}$$ Moreover, we have $\phi_Y(0)=1$. Thus, by Cauchy-Lipschitz theorem, we have: $$\begin{aligned}
\forall\xi\in\mathbb{R},\ \phi_Y(\xi)=\phi_F(\xi).\end{aligned}$$ This concludes the proof of the theorem.
Taking $\alpha_k=\mu_k=1/2$ in the previous theorem implies the following straightforward corollary:
\[benjarras2\] Let $d \ge 1$, $q\geq 1$ and $(m_1, \ldots, m_d) \in \mathbb{N}^d$ such that $m_1+...+m_d=q$. Let $(\lambda_1, \ldots, \lambda_d) \in
\R^{\star}$ and consider: $$F = \sum_{i=1}^{m_1}\lambda_1(N_i^2-1)+\sum_{i=m_1+1}^{m_1+m_2}\lambda_2(N_i^2-1)+...+\sum_{i=m_1+\ldots+m_{d-1}+1}^q\lambda_d(N_i^2-1),$$ Let $Y$ be a real valued random variable such that $\E[|Y|]<+\infty$. Then $Y \stackrel{\text{law}}{=} F$ if and only if $$\begin{aligned}
&\E \bigg[ \big(Y+\sum_{i=1}^d\lambda_im_i\big)(-1)^d2^d\bigg(\prod_{j=1}^d\lambda_j\bigg)\phi^{(d)}(Y)+\sum_{l=1}^{d-1}2^l(-1)^l\bigg(Ye_{l}(\lambda_1,...,\lambda_d)\nonumber\\ &+\sum_{k=1}^d\lambda_km_k\left(e_{l}(\lambda_1,...,\lambda_d)-e_{l}((\underline{\lambda}_k)\right)\bigg)\phi^{(l)}(Y)+Y\phi(Y) \bigg]=0,\label{eq:3}\end{aligned}$$ for all $\phi\in S(\mathbb{R})$.
Let $F$ be as in the statement of the theorem. Then, it is sufficient to observe that we have the following equality in law: $$\begin{aligned}
F\stackrel{\text{law}}{=}-\sum_{k=1}^dm_k\lambda_k+\sum_{i=1}^d\lambda_i\gamma_i\big(\frac{m_i}{2},\frac{1}{2}\big).\end{aligned}$$ To end the proof of the corollary, we apply the previous theorem with $\alpha_k=\mu_k=1/2$ for every $k$.
\[ex:eicheltha\][ Let $d=1$, $m_1=q\geq 1$ and $\lambda_1=\lambda>0$. The differential operator reduces to (on smooth test function $\phi$): $$\begin{aligned}
-2\lambda(x+q\lambda)\phi^{(1)}(x)+x\phi(x).\end{aligned}$$ This differential operator is similar to the one characterising the gamma distribution of parameters $(q/2,1/(2\lambda))$. Indeed, we have, for $F\stackrel{\text{law}}{=}\gamma\big(q/2,1/(2\lambda)\big)$, on smooth test function, $\phi$: $$\begin{aligned}
\E\bigg[F\phi^{(1)}\big(F\big)+\big(\frac{q}{2}-\frac{F}{2\lambda}\big)\phi\big(F\big)\bigg]=0\end{aligned}$$ We can move from the first differential operator to the second one by performing a scaling of parameter $-1/(2\lambda)$ and the change of variable $x=y-q\lambda$. ]{}
[ Let $d=2,\ q=2$, $\lambda_1=-\lambda_2=1/2$ and $m_1=m_2=1$. The differential operator reduces to (on smooth test function $\phi$): $$\begin{aligned}
T(\phi)(x)&=4(x+<m,\lambda>)\lambda_1\lambda_2\phi^{(2)}(x)-2\big[xe_1(\lambda_1,\lambda_2)+\lambda_1m_1(e_1(\lambda_1,\lambda_2)-e_1(\lambda_2))\\
&+\lambda_2m_2(e_1(\lambda_1,\lambda_2)-e_1(\lambda_1))\big]\phi^{(1)}(x)+x\phi(x),\\
&=-x\phi^{(2)}(x)-\phi^{(1)}(x)+x\phi(x),\end{aligned}$$ where we have used the fact that $e_1(\lambda_1,\lambda_2)=\lambda_1+\lambda_2=0,\ e_1(\lambda_2)=\lambda_2=-1/2,\ e_1(\lambda_1)=\lambda_1=1/2$. Therefore, up to a minus sign factor, we retrieve the differential operator associated with the random variable: $$\begin{aligned}
F=N_1\times N_2.\end{aligned}$$ ]{}
Malliavin-based approach {#sec:mall-based-appr}
------------------------
In this section, we assume that the random objects we consider do live in the Wiener space. Let $\rm X=\{X(h); \ h \in \HH\}$ stand for an isonormal process over a separable Hilbert space $\HH$. The reader may consult [@n-pe-1 Chapter 2] for a detailed discussion on this topic. The main aim of this section is to use Malliavin calculus on the Wiener space to obtain a Stein characterization for target random variables of the form $(\ref{target-wiener1})$. The following definition includes the iterated Malliavin $\Gamma$-operators that lie at the core of this approach. The notation $\mathbb{D}^{\infty}$ stands for the class of infinitely many times Malliavin differentiable random variables.
\[Def : Gamma\] Let $F\in \mathbb{D}^{\infty}$. The sequence of random variables $\{\Gamma_i(F)\}_{i\geq 0}\subset
\mathbb{D}^\infty$ is recursively defined as follows. Set $\Gamma_0(F) = F$ and, for every $i\geq 1$, $$\Gamma_{i}(F) = \langle DF,-DL^{-1}\Gamma_{i-1}(F)\rangle_{\HH}.$$ For instance, one has that $\Gamma_1(F) = \langle
DF,-DL^{-1}F\rangle_{\HH}= \tau_F(F)$ the Stein kernel of $F$.
For further use, we also recall that (see again [@n-pe-1]) the cumulants of the random element $F$ and the iterated Malliavin $\Gamma$- operators are linked by the relation $$\kappa_{r+1}(F)=r! \E [\Gamma_r(F)] \mbox{ for } r=0,1,\cdots.$$ Following [@n-po-1; @a-p-p], we define two crucial polynomials $P$ and $Q$ as follows: $$\label{polynomialP}
Q(x)=\big( P(x)\big)^{2}=\Big(x \prod_{i=1}^{q}(x - \alpha_{\infty,i} ) \Big)^{2}.$$ Finally, for any random element $F$, we define the following quantity (whose first appearance is in [@a-p-p]) $$\label{eq:Delta}
\Delta(F, F_{\infty}):= \sum_{r=2}^{\text{deg}(Q)} \frac{Q^{(r)}(0)}{r!} \frac{\kappa_{r}(F)}{2^{r-1}(r-1)!}.$$
[@a-p-p Proposition 3.2]\[p:static\] Let $F$ be a centered random variable living in a finite sum of Wiener chaoses. Moreover, assume that
- $\kappa_r (F) = \kappa_r (F_\infty)$, for all $2 \le r \le k+1=\text{deg}(P)$, and
- $$\E \Bigg[ \sum_{r=1}^{k+1} \frac{P^{(r)}(0)}{r! \
2^{r-1}} \Big( \Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)]
\Big) \Bigg]^2 = 0.$$
Then, $F \stackrel{{\rm law}}{=} F_\infty,$ and $F$ belongs to the second Wiener chaos.
In fact item [**(ii)**]{} of Proposition \[p:static\] can be used to derive a Stein equation for $F_\infty$. To this end, set $$\begin{aligned}
& a_l= \frac{P^{(l)}(0)}{l! 2^{l-1}} \quad 1 \le l \le q+1,\\
& b_l= \sum_{r=l}^{q+1} a_r \E[\Gamma_{r-l+1}(F_\infty)] = \sum_{r=l}^{q+1} \frac{a_r}{(r-l+1)!} \kappa_{r-l+2}(F_\infty) \quad 2 \le l \le q+1\end{aligned}$$ Now, we introduce the following differential operator of order $q$ (acting on functions $f \in C^q(\R)$) : $$\label{eq:SME}
\mathcal{A}_\infty f (x):= \sum_{l=2}^{q+1} (b_l - a_{l-1} x ) f^{(q+2-l)}(x) - a_{q+1} x f(x).$$ Then, we have the following result.
\[thm:SMC\] Assume that $F$ is a general centered random variable living in a finite sum of Wiener chaoses (and hence smooth in the sense of Malliavin calculus). Then $$F \stackrel{\text{law}}{=}
F_\infty$$ if and only if $\E \left[ \mathcal{A}_\infty f (F) \right]
=0$ for all mappings $f:\R \to \R$ such that $\E \left[ \vert
\mathcal{A}_\infty f (F) \vert \right] < \infty$, and moreover $\E[f^{(q)}(F)^2]< +\infty$.
Repeatedly using the Malliavin integration by parts formulae [@n-pe-1 Theorem 2.9.1], we obtain for any $2 \le l \le q+2$ that $$\begin{aligned}
\E\left[F f^{(q-l+2)}(F) \right]
= \E\left[ f^{(q)}(F)
\Gamma_{l-2}(F) \right] + \sum_{r=q-l+3}^{q-1} \E\left[
f^{(r)}(F) \right] \E\left[ \Gamma_{r+l-q-2}(F) \right].\label{eq:com1}\end{aligned}$$ For indices $l=2,3$, the second term in the right hand side of $(\ref{eq:com1})$ is understood to be $0$. Summing from $l=2$ up to $l=q+2$, we obtain that
$$\label{com2}
\begin{split}
\sum_{l=2}^{q+2} a_{l-1} \, \E\left[F f^{(q-l+2)}(F) \right] & = \sum_{l=2}^{q+2} a_{l-1} \, \E\left[ f^{(q)}(F) \Gamma_{l-2}(F) \right] \\
& \hskip1cm + \sum_{l=4}^{q+2} a_{l-1} \, \sum_{r=q-l+3}^{q-1} \E\left[ f^{(r)}(F) \right] \E\left[ \Gamma_{r+l-q-2}(F) \right]\\
& = \sum_{l=1}^{q+1} a_{l} \, \E\left[ f^{(q)}(F) \Gamma_{l-1}(F) \right] \\
& \hskip1cm + \sum_{l=3}^{q+1} a_{l} \, \sum_{r=q-l+2}^{q-2} \E\left[ f^{(r)}(F) \right] \E\left[ \Gamma_{r+l-q-1}(F) \right]\\
& = \sum_{l=1}^{q+1} a_{l} \, \E\left[ f^{(q)}(F) \Gamma_{l-1}(F) \right] \\
& \hskip1cm + \sum_{l=2}^{q+1} a_{l} \, \sum_{r=1}^{l-2} \E \left[ f^{(q-r)}(F) \right] \E \left[ \Gamma_{l-r-1}(F) \right].
\end{split}$$
On the other hand, $$\label{eq:com4}
\begin{split}
\sum_{l=2}^{q+1} b_l \, \E \left[ f^{(q+2-l)}(F) \right] &=
\sum_{l=0}^{q-1} b_{l+2} \E \left[ f^{(q-l)} (F) \right]\\
&= \sum_{l=0}^{q-1} \left[ \sum_{r=l+2}^{q+1} a_r \E (
\Gamma_{r-l-1}(F_\infty) ) \right] \E \left[ f^{(q-l)}(F) \right]\\
&= \sum_{r=2}^{q+1} a_r \sum_{l=0}^{r-2} \E \left[
\Gamma_{r-l-1}(F_\infty) \right] \times \E \left[ f^{(q-l)}(F)
\right].
\end{split}$$ Wrapping up, we finally arrive at $$\label{eq:com5}
\begin{split}
\E \left[ \mathcal{A}_\infty f (F) \right] & = - \E \Bigg[ f^{(q)}(F) \times \Big( \sum_{r=1}^{q+1} a_r \left[ \Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)] \right] \Big) \Bigg] \\
& \hskip 1cm + \sum_{r=2}^{q+1} a_r \sum_{l=0}^{r-2} \left\{ \E [
f^{(q-l)}(F) ] \times \Big( \E \left[ \Gamma_{r-l-1}(F_\infty)
\right] - \E \left[ \Gamma_{r-l-1}(F) \right] \Big) \right\}\\
& = - \E \Bigg[ f^{(q)}(F) \times \Big( \sum_{r=1}^{q+1} a_r \left[
\Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)] \right] \Big) \Bigg] \\
&\hskip 1cm + \sum_{r=2}^{q+1} a_r \sum_{l=0}^{r-2} \frac{ \E [ f^{(q-l)}(F) ]}{(r-l-1)!} \times \Big( \kappa_{r-l}(F_\infty) - \kappa_{r-l}(F) \Big).
\end{split}$$ We are now in a position to prove the claim. First we assume that $F \stackrel{\text{law}}{=} F_\infty$. Then obviously $\kappa_{r}(F)=\kappa_{r}(F_\infty)$ for $r=2,\cdots,2q+2$. Following the same arguments as in the proof of [@a-p-p Proposition 3.2], one can infer that, in fact, $F$ belongs to the second Wiener chaos. Hence, according to [@a-p-p Lemma 3.1], and the Cauchy–Schwarz inequality, we obtain that $$\begin{split}
\vert \E \left[ \mathcal{A}_\infty f (F) \right] \vert & \le \sqrt{\E \left[f^{(q)}(F) \right]^2} \times
\sqrt{ \E \Big[ \sum_{r=1}^{q+1} a_r \left( \Gamma_{r-1}(F) -
\E[\Gamma_{r-1}(F)] \right) \Big]^2} \\
& = \sqrt{\E \left[f^{(q)}(F) \right]^2} \times \sqrt{\Delta(F,F_\infty)}\\
&= \sqrt{\E \left[f^{(q)}(F) \right]^2} \times \sqrt{\Delta(F_\infty,F_\infty)}= 0.
\end{split}$$ Conversely, assume that $\E \left[ \mathcal{A}_\infty f (F) \right] =0$ for all the suitable functions $f$. Then relation $(\ref{eq:com5})$ implies that, by choosing appropriate polynomials for function $f$, we have $\kappa_r(F)=\kappa_r(F_\infty)$ for $r=2,\cdots,q+1$. Now, combining this observation together with relation $(\ref{eq:com5})$, we infer that $$\E \left[ \sum_{r=1}^{q+1} a_r \Big( \Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)] \Big) \Big \vert F \right]=0.$$ Using e.g. integrations by parts, the latter equation can be turned into a linear recurrent relation between the cumulants of $F$ of order up to $q+1$. Combining this with the knowledge of the $q+1$ first cumulants characterize all the cumulants of $F$ and hence the distribution $F$. Indeed, all the distributions in the second Wiener chaos are determined by their moments.
[ Consider the special case of only two non-zero distinct eigenvalues $\lambda_1$ and $\lambda_2$, i.e. $$\label{eq:target}
F_\infty=\lambda_1(N^2_1 -1) + \lambda_2 (N^2_2 -1)$$ where $N_1, N_2 \sim \mathscr N (0,1)$ are independent. In this case, the polynomial $P$ takes the form $P(x)=x (x - \alpha)(x - \beta)$. Simple calculations reveal that $P'(0)=\lambda_1 \lambda_2, P''(0)= -2 (\lambda_1 + \lambda_2)$, and $P^{(3)}(0)=3!$. Also, $\kappa_2(F_\infty)=\E [\Gamma_{1}(F_\infty)]=2(\lambda_1^2 + \lambda_2^2)$, and $\kappa_3(F_\infty)= 2 \E [\Gamma_{2}(F_\infty)]= 4 (\lambda_1^3 + \lambda_2^3)$. Then, the Stein equation $(\ref{eq:SME})$ reduces to that $$\label{eq:stein-equation-2}
\mathcal{A}_\infty f (x) = -4 (\lambda_1 \lambda_2 x + (\lambda_1 + \lambda_2) \lambda_1 \lambda_2) f''(x) + 2 \left(\lambda_1^2 + \lambda_2^2+ (\lambda_1 + \lambda_2) x \right) f'(x) - x f(x)$$ We also remark that when $\lambda_1=-\lambda_2=\frac12$, and hence $F_\infty \stackrel{\text{law}}{=} N_1 \times N_2$, the Stein’s equation $(\ref{eq:stein-equation-2})$ coincides with that in [@g-2normal equation (1.9)].]{}
One can show that is a specification of in the particular setting considered in the current Section. The proof of this asserstion is quite technical and we do not include it. To convince the reader, let us inspect the particular case where $d=2$, $q=2$ and $\lambda_1\ne\lambda_2$. Using the previous corollary, the differential operator, $T$, boils down to, (on smooth test function $\phi$): $$\begin{aligned}
T(\phi)(x)=4\lambda_1\lambda_2(x+\lambda_1+\lambda_2)\phi^{(2)}(x)-2\big[x(\lambda_1+\lambda_2)+\lambda_1^2+\lambda_2^2\big]\phi^{(1)}(x)+x\phi(x),\end{aligned}$$ which is, up to a minus sign factor, the differential operator appearing in Equation \[eq:stein-equation-2\] with $\alpha=\lambda_1$ and $\beta=\lambda_2$. In the case of $d$ different eigenvalues whose respective multiplicities are equal to $1$, in order to switch from one operator to the other one, we must use the fundamental Newton-Girard identities linking the elementary symmetric polynomials valued at $(\lambda_1,...,\lambda_d)$ together with the power sums valued at $(\lambda_1,...,\lambda_d)$ (which reduce, up to a multiplicative factor, to the cumulants of $F$).
Cattywampus Stein’s method {#sec:cattyw-steins-meth}
--------------------------
In this section, we first propose a heuristic to clarify why $\Delta(F, F_{\infty})$ can be interpreted as a version of the Malliavin-Stein discrepancy, when $F$ belongs to the second Wiener chaos. Let $\mathcal{H}$ be a family of test functions that at least characterizes the convergence in distribution, in the sense that $${\mathrm{d}}_{\mathcal{H}} (F,F_\infty): = \sup_{h \in \mathcal{H}} \Big \vert \E [h(F)] - \E[h(F_\infty)] \Big \vert \approx 0 \quad \text{if and only if} \quad F \stackrel{\text{law}}{\approx} F_\infty.$$ Let $h \in \mathcal{H}$ and take $\mathcal{A}_\infty$ as in $(\ref{eq:SME})$. Consider the Stein equation $$\label{eq:NHSE}
\mathcal{A}_{\infty}f (x) = h (x) - \E[h(F_{\infty})] \qquad h \in \mathcal{H}.$$
[**\[Stein universality assumption\]**]{} \[assu:1\] For any $h \in \mathcal{H}$, equation $(\ref{eq:NHSE})$ has a unique bounded solution $f_h$ so that $$\label{eq:4}
\Vert f^{(r)}_h \Vert_\infty < \infty \qquad \forall \, r=1,\cdots,q$$ uniformly in $h$.
Now, we take a general centered random variable $F$, smooth in the sense of Malliavin differentiability, for example $F$ belongs to a finite sum of Wiener chaoses. Then, under Assumption $\ref{assu:1}$, the proof of Theorem \[thm:SMC\] reveals that for some general constants $C_1,C_2$, using the Cauchy-Swartz inequality, one has
$$\label{eq:8}
\begin{split}
\sup_{h \in \mathcal{H}} \Big \vert \E [h(F)] - \E[h(F_\infty)] \Big \vert & \le C_1 \E \Big \vert \sum_{r=1}^{q+1} a_r \left( \Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)] \right) \Big\vert \\
& \hskip 1cm + C_2 \sum_{r=2}^{q+1} \big \vert \kappa_r(F) - \kappa_r(F_\infty) \big \vert \\
& \le \sqrt{ \E \Big[ \sum_{r=1}^{q+1} a_r \left( \Gamma_{r-1}(F) - \E[\Gamma_{r-1}(F)] \right) \Big]^2 } \\
& \hskip 1cm + C_2 \sum_{r=2}^{q+1} \big \vert \kappa_r(F) - \kappa_r(F_\infty) \big \vert.
\end{split}$$
However, taking into account [@a-p-p Lemma 3.1] when $F$ itself belongs to the second Wiener chaos, then $$\label{eq:11}
\begin{split}
\sup_{h \in \mathcal{H}} \Big \vert \E [h(F)] - \E[h(F_\infty)] \Big
\vert \le C_1 \sqrt{\Delta(F,F_\infty)} + C_2 \sum_{r=2}^{q+1} \big
\vert \kappa_r(F) - \kappa_r(F_\infty) \big \vert.
\end{split}$$ Starting from , to apply Stein’s method in the second Wiener chaos it suffices, in a sense, to provide the estimates required by Assumption \[assu:1\].
This idea is at the heart e.g. of [@thale] where the authors could make use of the estimates provided by [@g-variance-gamma] to apply the above plan to targets as in Example \[ex:eicheltha\] (and, more generally, to variance-gamma distribution).
There are two major flaws to this approach and, therefore, to the classical take on Stein’s method as adapted to the second Wiener chaos. First the bounds required in can only be obtained by solving $q$-th degree inhomogeneous equations and it is unlikely that a unified approach will allow to deal with all targets of the form in one sweep. Indeed different ranges of $\alpha_i$’s imply very different properties for the corresponding $F_{\infty}$ and hence each application of Stein’s method to these targets will necessitate ad-hoc target specific solutions, which in the current state of knowledge we do not even know to be bounded. Second, the bounds on higher order derivatives of $f_h$ will necessarily depend on smoothness assumptions on the test functions $h$; hence several important distances of the form with non-smooth $h$ cannot be tackled via Stein’s method.
Such a flaw was already noted for multivariate Gaussian approximation, see [@CM], where estimates in total variation distance were realized to be beyond the scope of the classical approaches to the method. Recently an original solution was proposed in [@NPS] wherein a class of random vectors $F$ was identified to which one could apply what we will coin an *information theoretic approach to Stein’s method*; this resulted in a general fourth moment bound on Total Variation distance for multivariate normal approximation for random vectors $F$ satisfying a very particular integrability constraint (see [@NPS Condition (2.53)]).
Bypassing the Stein’s method {#sec:preliminaries}
============================
As mentioned in the conclusion to the previous section, bounding the solutions of higher order Stein’s equations is an extremely hard task. However, if the appoximating sequence has the shape of a weighted sum of i.i.d. random variables, we provide a new strategy to bound the two Wasserstein distance between the sequence and the target. Not only we completely bypass the major difficulty of bounding the stein’s solution but we can deal with distances which cannot be reached by usual Stein’s tools.
Fix $p\geq 1$. Given two probability measures $\nu$ and $\mu$ on the Borel sets of $\R^d$ whose marginals have finite absolute moments of order $p$, define the [*Wasserstein distance*]{} (of order $p$) between $\nu$ and $\mu$ as the quantity $${{\bf \rm W}}_p(\nu,\mu) \, = \, \inf_\pi
\bigg ( \int_{\R^d\times\R^d} |x-y|^p d\pi(x,y)\bigg ) ^{1/p}$$ where the infimum runs over all probability measures $\pi$ on $\R^d\times \R^d$ with marginals $\nu$ and $\mu$.
The Wasserstein metric may be equivalently defined by $${{\bf \rm W}}_p(\nu,\mu) \, = \, \big( \inf \E \Vert X-Y\Vert_{d}^p
\big)^{1/p}$$ where the infimum is taken over all joint distributions of the random variables $X$ and $Y$ with marginals $\mu$ and $\nu$ respectively, and $\Vert \, \Vert_d$ stands for the Euclidean norm on $\R^d$. It is also well-known that convergence with respect to ${\bf \rm W}_p$ is equivalent to the usual weak convergence of measures plus convergence of the first $p$th moments. Also, a direct application of Hölder inequality implies that if $1\le p \le q$ then ${\bf \rm W}_p \le {\bf \rm W}_q$. Relevant information about Wasserstein distances can be found, e.g. in [@villani-book].
Main result {#s:main-result}
-----------
Throughout this section, we assume that $\{ W_k\}_{k \ge 1}$ is a general sequence of independent and identically distributed random variables having [**finite moments**]{} of any order such that $\E[W_1]=0$ and $\E[W^2_1]=1$, and moreover $\kappa_r(W_1) \neq 0$ for all $r=2,\cdots,q+1$. For a given sequence $\{\alpha_{n,k}\}_{n,k\ge 1} \subset \R$, we assume that each element $F_n$ of the approximating sequence is of the form
$$\label{eq:app-general-form}
F_n= \sum_{k\ge1} \alpha_{n,k} \, W_k \qquad n\in \N.$$
Similarly, we assume that the target random variable may be written in the following way $$\label{target-wiener}
F_{\infty}:= \sum_{k=1}^q \alpha_{\infty,k} W_k.$$ where $q \ge 2$ and all the coefficients $\{ \alpha_{\infty,k}\}_{1\le k \le q} \subset \R$ are both [non-zero]{} and [pairwise distinct]{} real numbers. Also, without loss of generality, we assume that we are dealing with normalized random variables, meaning that
$$\label{eq:2moment-condition}
\E[F^2_{n}]=\sum_{k\ge1} \alpha^2_{n,k}=1 \quad \text{and} \quad
\E[F^2_\infty]
=\sum_{k=1}^{q} \alpha^2_{\infty,k}=1, \quad \forall \, n \in \N.$$
For any $n \in \N$, we introduce, in the shape of a lemma, a crucial quantity that can be also written as a finite linear combination of the first $2q+2$ cumulants of the random variable $F_n$ of the approximating sequence.
\[lem:linear-combiniation\] For any $n \in \N$ we have $$\label{eq:Delta-general}
\begin{split}
\Delta(F_n,F_\infty)=\Delta(F_n):&= \sum_{k\ge1} \alpha^2_{n,k} \prod_{r=1}^{q} \left( \alpha_{n,k} - \alpha_{\infty,r} \right)^2\\
&=\sum_{r=2}^{2q+2} \Theta_r \sum_{k\ge1} \alpha^r_{n,k}\\
&=\sum_{r=2}^{2q+2} \frac{\Theta_r}{\kappa_r(W_1)} \kappa_r(F_n).
\end{split}$$ where the coefficients $\Theta_r$ are the coefficients of the polynomial $$\label{eq:polynomial}
Q(x)=(P(x))^2= (x \prod_{i=1}^q (x-\alpha_{\infty,i}))^2.$$
Now, we are ready to sate our main results.
\[thm:main-theorem1\] Let all notations and assumption in above are prevail. Then for some constant $C>0$ depending only on target random variable $F_\infty$ (and hence independent of $n$) so that $$\label{eq:main-estimate-bis}
d_{{\bf \rm{W}}_2} (F_n,F_\infty) \le C \,\left(\sqrt{\Delta(F_n)}+\sum_{r=2}^{q+1}|\kappa_r(F_n)-\kappa_r(F_\infty)|\right) \qquad \forall n\ge1.$$ More precisely, there exists a threshold $U_\infty>0$ and a constant $C>0$ depending only on the target random variable $F_\infty$ (and hence independent of $n$) such that the next bound $$\label{eq:upper-threshold}
\sqrt{\Delta(F_n)} +\sum_{r=2}^{q+1} \vert \kappa_r(F_n) - \kappa_r(F_\infty)\vert \le U_\infty,$$ implies
$$\label{eq:main-estimate}
d_{{\bf \rm{W}}_2} (F_n,F_\infty) \le C \,\sqrt{\Delta(F_n)}.$$
In particular, if $\Delta(F_n) \to 0$ and moreover $\kappa_r(F_n) \to
\kappa_r(F_\infty)$ for all $r=2,\cdots,q+1$, then the threshold requirement $(\ref{eq:upper-threshold})$ takes place and therefore the sequence $F_n$ converges in distribution towards target random variable $F_\infty$ at rate $\sqrt{\Delta(F_n)}$.
\[thm:main-theorem2\] Let all the notations and assumptions of Theorem \[thm:main-theorem1\] prevail. Assume further that $\dim_{\mathbb{Q}} \text{span} \left\{
\alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q}\right \}=q$. Then there exists a constant $C>0$ depending only on the target random variable $F_\infty$ such that
$$\label{eq:main-estimate2}
d_{{\bf \rm{W}}_2} (F_n,F_\infty) \le C \,\sqrt{\Delta(F_n)}.$$
\[rem:span1\][ We remark that the condition $\dim_{\mathbb{Q}} \text{span} \left\{ \alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q}\right \}=q$ implies that if $\sum_{i=1}^{q+1} n_i \alpha^2_{\infty,i}=1$ for some $n_i \in \N_0=\N \cup \{ 0\}$, then $n_1=\cdots=n_{q+1}=1$ according to normalization assumption $(\ref{eq:2moment-condition})$. We will use this useful observation in the proof of Theorem \[thm:main-theorem2\]. ]{}
\[rem:span3\][ In light of Theorem \[thm:main-theorem1\], it appears that if $\dim_{\mathbb{Q}} \text{span} \left\{
\alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q}\right \}\neq q$ then even a standard application of Stein’s method (that is, using Stein equations and hypothetical bounds on the solutions) would require the control of more moments than only the first two in order to bound the Wasserstein-1 distance. ]{}
\[rem:span2\]
According to Theorem \[thm:main-theorem2\] when $\dim_{\mathbb{Q}}
\text{span} \left\{
\alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q}\right \}=q$, one can drop the assumption of the separate convergences of the first $q+1$ cumulants, $\kappa_r(F_n) \to \kappa_r(F_\infty)$ for all $r=2,\cdots,q+1$, in Theorem \[thm:main-theorem1\] and hence the only requirement $\Delta(F_n) \to 0$ implies the convergence in distribution of the sequence $F_n$ towards the target random variable $F_\infty$. For example, let $q=2$ and
1. $\alpha_{\infty,1} \in \mathbb{Q}$. Then, obviously $\alpha_{\infty,2} \in \mathbb{Q}$, and therefore $\dim_{\mathbb{Q}} \text{span}
\{ \alpha^2_{\infty,1},\alpha^2_{\infty,2} \}=1 \neq q=2$. Hence, for convergence in distribution of the sequence $F_n$ towards the target random variable $F_\infty$ in addition to convergence $\Delta(F_n) \to
0$ one needs also the convergence $\kappa_3(F_n) \to
\kappa_3(F_\infty)$, see also Remark \[rem:thale\].
2. $\alpha_{\infty,1}\in \R - \mathbb{Q}$ be a irrational number, and therefore according to normalization assumption $(\ref{eq:2moment-condition})$ the coefficient $\alpha_{\infty,2}$ will be also an irrational number. In this case, we have $\dim_{\mathbb{Q}} \text{span} \left\{ \alpha^2_{\infty,1},\alpha^2_{\infty,2}\right \}=q=2$. Hence, the sole requirement $\Delta(F_n) \to 0$ is enough for convergence in distribution of the sequence $F_n$ towards the target random variable $F_\infty$.
Idea behind the proof {#sec:idea-behind-proof}
---------------------
The main idea leading the proof relies on the following non trivial observation: $F_n=\sum_k \alpha_{n,k} W_k$ converges in distribution towards $F_\infty=\sum_{k=1}^q \alpha_{\infty,k} W_k$ if and only if one can find $q$ coefficients among the sequence $\{\alpha_{n,k}\}_{k\ge 1}$ which are very close to the corresponding terms $\{\alpha_{\infty,k}\}_{1\le k \le q}$ and if the $l^2$-norm of the remaining terms is small. The main difficulty is to quantify this phenomenon by only using the Stein discrepancy $\Delta(F_n,F_\infty)$. To reach this goal, we proceed in several steps which are sketched below:
- : Without loss of generality, we can assume that, for each $n$, the sequence $|\alpha_{n,k}|$ decreases with $k$. Indeed, this will be useful to identify which coefficients have a non-zero limit and which coefficients go to zero.
- : Here we prove several inequalities which, roughly speaking, express the fact that if for some $k$ the coefficient $|\alpha_{n,k}|$ is bounded from below by a fixed constant, then it is necessarily close to one of the limiting coefficients $\alpha_{\infty,k}$.
- : Here we prove that a certain number of the coefficients $\alpha_{n,k}$ (say $\alpha_{n,1},\cdots,\alpha_{n,l_n}$) are all close to one element (say $\alpha_\infty(n,k)$) of the set $\{\alpha_{\infty,1},\cdots,\alpha_{\infty,q}\}$ with possible repetitions. The difficult part consists in showing that $\sum_{k> l_n} \alpha_{n,k}^2$ is small. To do so, it is equivalent to prove that $\sum_{k=1}^{l_n} \alpha_{n,k}^2$ is close to one. Having in mind that $\sum_{k=1}^{l_n} |\alpha_{n,k}-\alpha_\infty(n,k)|$ is small, the latter claim is in turn equivalent to $\sum_{k=1}^{l_n}\alpha_\infty(n,k)^2=1$. We argue by contradiction using a maximality argument.
- : As we said in step 3, there might be repetitions among the coefficients $\alpha_\infty(n,k)$ and they may also vary with $n$. However, if the coefficients $\{\alpha_{\infty,k}^2\}_{1\le k \le q}$ are rationally independent, then there is only one way to chose $\alpha_\infty(n,k)$ in the set $\{\alpha_{\infty,1},\cdots,\alpha_{\infty,q}\}$ such that $\sum_{k=1}^{l_n}\alpha_\infty^2(n,k)=1$. This is the idea behind Theorem \[thm:main-theorem2\]. However, when we do not assume rational independence of the coefficients, we need to use the assumption of the convergence of the cumulants through a Vandermonde argument to proceed.
Applications: second Wiener chaos {#s:applications}
---------------------------------
In this section, we apply our main results in a desirable framework when the approximating sequence $F_n$ are elements of the second Wiener chaos of the isonormal process $\rm X=\{X(h); \ h \in \HH\}$ over a separable Hilbert space $\HH$. We refer the reader to [@n-pe-1] Chapter 2 for a detailed discussion on this topic. Recall that the elements in the second Wiener chaos are random variables having the general form $F=I_2(f)$, with $f \in \HH^{\odot 2}$. Notice that, if $f=h\otimes h$, where $h \in \HH$ is such that $\Vert h \Vert_{\HH}=1$, then using the multiplication formula one has $I_2(f)=\rm X (h)^2 -1 \stackrel{\text{law}}{=} N^2 -1$, where $N \sim \mathscr{N}(0,1)$. To any kernel $f \in \HH^{\odot 2}$, we associate the following *Hilbert-Schmidt* operator $$A_f : \HH \mapsto \HH; \quad g \mapsto f\otimes_1 g.$$ We also write $\{\alpha_{f,j}\}_{j \ge 1}$ and $\{e_{f,j}\}_{j \ge 1}$, respectively, to indicate the (not necessarily distinct) eigenvalues of $A_f$ and the corresponding eigenvectors. The next proposition gathers some relevant properties of the elements of the second Wiener chaos associated to $\rm X$.
\[second-property\] Let $F=I_{2}(f)$, $f \in \HH^{ \odot 2}$, be a generic element of the second Wiener chaos of $\rm X$, and write $\{\alpha_{f,k}\}_{k\geq 1}$ for the set of the eigenvalues of the associated Hilbert-Schmidt operator $A_f$.
1. The following equality holds: $F=\sum_{k\ge 1} \alpha_{f,k} \big( N^2_k -1 \big)$, where $\{N_k\}_{k \ge 1}$ is a sequence of i.i.d. $\mathscr{N}(0,1)$ random variables that are elements of the isonormal process $\rm X$, and the series converges in $L^2$ and almost surely.
2. For any $r\ge 2$, $$\kappa_r(F)= 2^{r-1}(r-1)! \sum_{k \ge 1} \alpha_{f,k}^r.$$
3. For polynomial $Q$ as in $(\ref{eq:polynomial})$ we have $\Delta(F) = \sum_{k\geq 1} Q(\alpha_{f,k})$. In particular $\Delta(F_\infty)=0$.
The next corollary is a direct application of our main findings, namely Theorem \[thm:main-theorem1\] and Theorem \[thm:main-theorem2\] and provides quantitative bounds for the main results in [@n-po-1; @a-p-p].
\[cor:2wiener\] Assume that the normalized sequence $F_n=\sum_{k\ge 1} \alpha_{n,k} \big( N^2_k -1 \big)$ belongs to the second Wiener chaos associated to the isonormal process $\rm X$, and the target random variable $F_\infty$ as in $(\ref{target-wiener})$. Then there exists a constant $C>0$ depending only on target random variable $F_\infty$ (and hence independent of $n$) such that
1. $$d_{{\bf \rm W}_2}(F_n,F_\infty) \le \, C \, \bigg( \sqrt{\Delta(F_n)} + \sum_{r=2}^{q+1} \vert \kappa_r(F_n) - \kappa_r(F_\infty) \vert \bigg).$$
2. if moreover $\dim_{\mathbb{Q}} \text{span} \{ \alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q} \} =q$, then $d_{{\bf \rm W}_2}(F_n,F_\infty) \le \, C \, \sqrt{\Delta(F_n)}$. This implies that the sole convergence $\Delta(F_n) \to \Delta(F_\infty)=0$ is sufficient for convergence in distribution towards the target random variable $F_\infty$.
\[rem:thale\]
The upper bound in Corollary \[cor:2wiener\], part (a) requires the separate convergences of the first $q+1$ cumulants for the convergence in distribution towards the target random variable $F_\infty$ as soon as $\dim_{\mathbb{Q}} \text{span} \{ \alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q} \} < q$. This is in very consistent with a quantitative result in [@thale]. In fact, when $q=2$ and $\alpha_{\infty,1}=- \alpha_{\infty,2}=1/2$, then the target random variable $F_\infty$ $( = N_1 \times N_2$, where $N_1,N_2 \sim \mathscr{N}(0,1)$ are independent and equality holds in law) belongs to the class of [*Variance–Gamma*]{} distributions $VG_c(r,\theta,\sigma)$ with parameters $r=\sigma=1$ and $\theta=0$. Then, [@thale Theorem 5.10, part (a)] reads
$$\label{eq:thale-bound}
d_{{\bf \rm W}_1} (F_n,F_\infty) \le C\, \sqrt{\Delta(F_n) + 1/4 \, \kappa^2_3(F_n)}.$$
Therefore, for the convergence in distribution of the sequence $F_n$ towards the target random variable $F_\infty$ in addition to convergence $\Delta(F_n) \to \Delta(F_\infty)=0$ one needs also the convergence of the third cumulant $\kappa_3(F_n) \to \kappa_3(F_\infty)=0$. Also note that in this case we have $\dim_{\mathbb{Q}} \text{span} \{ \alpha^2_{\infty,1}, \alpha^2_{\infty,2} \} =1 < q=2$.
\[ex:thale\][ Again assume that $q=2$ and $\alpha_{\infty,1}=- \alpha_{\infty,2}=1/2$. Consider the fixed sequence $$F_n=F=\alpha_{\infty,1} (N^2_1 -1) - \alpha_{\infty,2} (N^2_2 -1) \qquad n \ge1.$$ Then $\kappa_{2r}(F_n)=\kappa_{2r}(F_\infty)$ for all $r\ge1$, in particular $\kappa_2(F_n)=\kappa_2(F_\infty)=1$, and $\Delta(F_n)=\Delta(F_\infty)=0$. However, it is easy to see that the sequence $F_n$ does not converges in distribution towards the target random variable $F_\infty$, because $2=\kappa_3(F_n) \nrightarrow \kappa_3(F_\infty)=0$. Again, we would like to stress that in this case we have $\dim_{\mathbb{Q}} \text{span} \{ \alpha^2_{\infty,1}, \alpha^2_{\infty,2} \} =1 < q=2$. Therefore the requirement of separate convergences of the first $q+1$ cumulants is essential in Theorem \[thm:main-theorem1\] as soon as $\dim_{\mathbb{Q}} \text{span} \{ \alpha^2_{\infty,1}, \cdots, \alpha^2_{\infty,q} \} < q$. ]{}
[**\[Bai–Taqqu Theorem, 2015\]**]{} [ We conclude this section with a more ambitious example, providing rates of convergence in a recent result given by [@b-t Theorem 2.4]. We stress that many more examples and situations could be tackled by our method. Let $Z_{\gamma_1,\gamma_2}$ be the random variable defined by: $$\begin{aligned}
Z_{\gamma_1,\gamma_2}=\int_{\mathbb{R}^2}\bigg(\int_0^1(s-x_1)^{\gamma_1}_+(s-x_2)^{\gamma_2}_+ds\bigg)dB_{x_1}dB_{x_2},
\end{aligned}$$ with $\gamma_i\in(-1,-1/2)$ and $\gamma_1+\gamma_2>-3/2$. By Proposition 3.1 of [@b-t], we have the following formula for the cumulants of $Z_{\gamma_1,\gamma_2}$: $$\kappa_m\big(Z_{\gamma_1,\gamma_2}\big) =\frac{1}{2}(m-1)!A(\gamma_1,\gamma_2)^mC_m(\gamma_1,\gamma_2,1,1)$$ where, $$\begin{aligned}
A(\gamma_1,\gamma_2)&=[(\gamma_1+\gamma_2+2)(2(\gamma_1+\gamma_2)+3)]^{\frac{1}{2}}\\
&\times[B(\gamma_1+1,-\gamma_1-\gamma_2-1)B(\gamma_2+1,-\gamma_1-\gamma_2-1)\\
&+B(\gamma_1+1,-2\gamma_1-1)B(\gamma_2+1,-2\gamma_2-1)]^{-\frac{1}{2}},\\
C_m(\gamma_1,\gamma_2,1,1)&=\sum_{\sigma\in\{1,2\}^m}\int_{(0,1)^m}\prod_{j=1}^m[(s_j-s_{j-1})^{\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1}_+B(\gamma_{\sigma'_{j-1}}+1,-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)\\
&+(s_{j-1}-s_j)^{\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1}_+B(\gamma_{\sigma_j}+1,-\gamma_{\sigma_{j}}-\gamma_{\sigma'_{j-1}}-1)]ds_1...ds_m,\\
&B(\alpha,\beta)=\int_0^1u^{\alpha-1}(1-u)^{\beta-1}du.\end{aligned}$$ Let $\rho\in(0,1)$ and $Y_{\rho}$ be the random variable defined by: $$\begin{aligned}
Y_{\rho}=\dfrac{a_{\rho}}{\sqrt{2}}(Z_1^2-1)+\dfrac{b_{\rho}}{\sqrt{2}}(Z_2^2-1),\end{aligned}$$ with $Z_i$ independent standard normal random variables and $a_{\rho}$ and $b_{\rho}$ defined by: $$\begin{aligned}
&a_{\rho}=\dfrac{(\rho+1)^{-1}+(2\sqrt{\rho})^{-1}}{\sqrt{2(\rho+1)^{-2}+(2\rho)^{-1}}},\\
&b_{\rho}=\dfrac{(\rho+1)^{-1}-(2\sqrt{\rho})^{-1}}{\sqrt{2(\rho+1)^{-2}+(2\rho)^{-1}}}.\end{aligned}$$ For simplicity, we assume that $\gamma_1\geq \gamma_2$ and $\gamma_2=(\gamma_1+1/2)/\rho-1/2$. Then [@b-t Theorem 2.4] implies that as $\gamma_1$ tends to $-1/2$: $$\begin{aligned}
\label{eq:bai-taqqu}
Z_{\gamma_1,\gamma_2}\stackrel{\text{law}}{\to} Y_{\rho}.\end{aligned}$$ Note that, in this case, $\gamma_2$ automatically tends to $-1/2$ as well. To prove the previous result, the authors of [@b-t] prove the following convergence result: $$\begin{aligned}
\forall m\geq 2,\ \kappa_m\big(Z_{\gamma_1,\gamma_2}\big)\rightarrow\kappa_m\big(Y_{\rho}\big)=2^{\frac{m}{2}-1}(a_{\rho}^m+b_{\rho}^m)(m-1)!.\end{aligned}$$ Now, using Corollary \[cor:2wiener\] and applying Lemma \[lem:cumasym\], we can present the following quantative bound for convergence $(\ref{eq:bai-taqqu})$, namely as $\gamma_1$ tends to $-1/2$: $${\mathrm{d}}_{W_2} (Z_{\gamma_1,\gamma_2},Y_{\rho}) \le C_\rho \, \sqrt{- \gamma_1 - \frac{1}{2}},$$ where $C_{\rho}$ is some strictly positive constant depending on $\rho$ uniquely. ]{}
In order to apply Corollary \[cor:2wiener\] to obtain an explicit rate for convergence $(\ref{eq:bai-taqqu})$, we need to know at which speed $\kappa_m\big(Z_{\gamma_1,\gamma_2}\big)$ converges towards $\kappa_m\big(Y_{\rho}\big)$. For this purpose, we have the following lemma:
\[lem:cumasym\] Under the above assumptions, for any $m\geq 3$, we have, as $\gamma_1$ tends to $-1/2$: $$\begin{aligned}
\kappa_m\big(Z_{\gamma_1,\gamma_2}\big)=\kappa_m\big(Y_{\rho}\big)+O\big(-\gamma_1-\frac{1}{2}\big)\end{aligned}$$
First of all, we note that, as $\gamma_1$ tends to $-1/2$: $$\begin{aligned}
A(\gamma_1,\gamma_2)&=[(\gamma_1+\frac{1}{\rho}(\gamma_1+\frac{1}{2})+\frac{3}{2})(2\gamma_1+\frac{2}{\rho}(\gamma_1+\frac{1}{2})+2)]^{\frac{1}{2}}\\
&\times[B(\gamma_1+1,-(1+\frac{1}{\rho})(\gamma_1+\frac{1}{2}))B(\frac{1}{\rho}(\gamma_1+\frac{1}{2})+\frac{1}{2},-(1+\frac{1}{\rho})(\gamma_1+\frac{1}{2}))\\
&+B(\gamma_1+1,-2\gamma_1-1)B(\frac{1}{\rho}(\gamma_1+\frac{1}{2})+\frac{1}{2},-\frac{2}{\rho}(\gamma_1+\frac{1}{2}))]^{-\frac{1}{2}},\\
&\approx \dfrac{(-\gamma_1-1/2)}{\sqrt{(1+\frac{1}{\rho})^{-2}+(\frac{4}{\rho})^{-1}}}-C_{\rho}(-3+2\gamma+2\psi\big(\frac{1}{2}\big))(\gamma_1+\frac{1}{2})^2\\
&+o((-\gamma_1-1/2)^2),\end{aligned}$$ where $\gamma$ is the Euler constant, $\psi(.)$ is the Digamma function and $C_{\rho}$ some strictly positive constant depending on $\rho$ uniquely. Note that $-3+2\gamma+2\psi\big(1/2\big)<0$. Moreover, we have: $$\begin{aligned}
\nonumber C_m(\gamma_1,\gamma_2,1,1)&\approx \sum_{\sigma\in\{1,2\}^m}\int_{(0,1)^m}\prod_{j=1}^m\bigg\{\mathbb{I}_{s_j>s_{j-1}}\bigg[(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}-\log(s_j-s_{j-1})+(-\gamma-\psi\big(\frac{1}{2}\big))\\
\nonumber &+o(1) \bigg]+\mathbb{I}_{s_j<s_{j-1}}\bigg[ (-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}-\log(s_{j-1}-s_j)+(-\gamma-\psi\big(\frac{1}{2}\big))+o(1)\bigg]\bigg\}ds_1...ds_m\\
\nonumber &\approx \sum_{\sigma\in\{1,2\}^m}\int_{(0,1)^m}\prod_{j=1}^m\bigg[(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}+\mathbb{I}_{s_j>s_{j-1}}\log((s_j-s_{j-1})^{-1})\\
&+\mathbb{I}_{s_j<s_{j-1}}\log((s_{j-1}-s_j)^{-1})+(-\gamma-\psi\big(\frac{1}{2}\big))+o(1)\bigg]ds_1...ds_m.\label{approxCm}\end{aligned}$$ Note that $-\gamma-\psi\big(\frac{1}{2}\big)>0$. The diverging terms in $C_m(\gamma_1,\gamma_2,1,1)$ are $B(\gamma_{\sigma'_{j-1}}+1,-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)$ and $B(\gamma_{\sigma_j}+1,-\gamma_{\sigma_{j}}-\gamma_{\sigma'_{j-1}}-1)$. At $\sigma$ and $j$ fixed, the only possible values are: $$\begin{aligned}
&B(\gamma_1+1,-\gamma_{1}-\gamma_{2}-1)=B(\gamma_1+1,-(\gamma_1+\frac{1}{2})(1+\frac{1}{\rho})),\\
&\approx -\dfrac{1}{(1+\frac{1}{\rho})(\gamma_1+\frac{1}{2})}+(-\gamma-\psi(\frac{1}{2}))+o(1),\\
&B(\gamma_2+1,-\gamma_{1}-\gamma_{2}-1)=B(\frac{1}{\rho}(\gamma_1+\frac{1}{2})+\frac{1}{2},-(\gamma_1+\frac{1}{2})(1+\frac{1}{\rho})),\\
&\approx -\dfrac{1}{(1+\frac{1}{\rho})(\gamma_1+\frac{1}{2})}+(-\gamma-\psi(\frac{1}{2}))+o(1),\\
&B(\gamma_1+1,-2\gamma_{1}-1)\approx -\dfrac{1}{2(\gamma_1+\frac{1}{2})}+(-\gamma-\psi(\frac{1}{2}))+o(1),\\
&B(\gamma_2+1,-2\gamma_{2}-1)=B(\frac{1}{\rho}(\gamma_1+\frac{1}{2})+\frac{1}{2},-\frac{2}{\rho}(\gamma_1+\frac{1}{2})),\\
&\approx -\dfrac{\rho}{2(\gamma_1+\frac{1}{2})}+(-\gamma-\psi(\frac{1}{2}))+o(1).\end{aligned}$$ Moreover, we have, for $j$ fixed: $$\begin{aligned}
(s_j-s_{j-1})^{\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1}_+&=\mathbb{I}_{s_j>s_{j-1}}(s_j-s_{j-1})^{\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1}\\
&\approx \mathbb{I}_{s_j>s_{j-1}}[1+\log(s_j-s_{j-1})(\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1)
+o((\gamma_{\sigma_j}+\gamma_{\sigma'_{j-1}}+1))].\end{aligned}$$ Developing the product in the right hand side of (\[approxCm\]), we obtain: $$\begin{aligned}
C_m(\gamma_1,\gamma_2,1,1)&\approx \sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}\\
&+(-\gamma-\psi\big(\frac{1}{2}\big))\sum_{\sigma\in\{1,2\}^m}\sum_{j=1}^m\prod_{k=1,\ k\ne j}^m(-\gamma_{\sigma_k}-\gamma_{\sigma'_{k-1}}-1)^{-1}\\
&+\sum_{\sigma\in\{1,2\}^m}\sum_{j=1}^m\prod_{k=1,\ k\ne j}^m(-\gamma_{\sigma_k}-\gamma_{\sigma'_{k-1}}-1)^{-1}\int_{(0,1)^m}\bigg[\mathbb{I}_{s_j>s_{j-1}}\log((s_j-s_{j-1})^{-1})\\
&+\mathbb{I}_{s_j<s_{j-1}}\log((s_{j-1}-s_j)^{-1})\bigg]ds_1...ds_m+o((-\gamma_1-\frac{1}{2})^{-m+1})\\
&\approx \sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}\\
&+(-\gamma-\psi\big(\frac{1}{2}\big)+\frac{3}{2})\sum_{\sigma\in\{1,2\}^m}\sum_{j=1}^m\prod_{k=1,\ k\ne j}^m(-\gamma_{\sigma_k}-\gamma_{\sigma'_{k-1}}-1)^{-1}\\
&+o((-\gamma_1-\frac{1}{2})^{-m+1})\\\end{aligned}$$ This leads to the following asymptotic for the cumulants of $Z_{\gamma_1,\gamma_2}$, $$\begin{aligned}
\kappa_m\big(Z_{\gamma_1,\gamma_2}\big)&\approx \frac{(m-1)!}{2}\bigg[\dfrac{(-\gamma_1-1/2)}{\sqrt{(1+\frac{1}{\rho})^{-2}+(\frac{4}{\rho})^{-1}}}-C_{\rho}(-3+2\gamma+2\psi\big(\frac{1}{2}\big))(\gamma_1+\frac{1}{2})^2\\
&+o((-\gamma_1-1/2)^2)\bigg]^m\bigg[\sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}\\
&+(-\gamma-\psi\big(\frac{1}{2}\big)+\frac{3}{2})\sum_{\sigma\in\{1,2\}^m}\sum_{j=1}^m\prod_{k=1,\ k\ne j}^m(-\gamma_{\sigma_k}-\gamma_{\sigma'_{k-1}}-1)^{-1}\\
&+o((-\gamma_1-\frac{1}{2})^{-m+1})\bigg],\\
&\approx \frac{(m-1)!}{2}\dfrac{(-\gamma_1-1/2)^m}{\bigg(\sqrt{(1+\frac{1}{\rho})^{-2}+(\frac{4}{\rho})^{-1}}\bigg)^m}\sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m(-\gamma_{\sigma_j}-\gamma_{\sigma'_{j-1}}-1)^{-1}\\
&+O((-\gamma_1-\frac{1}{2}))\\
&\approx \kappa_m\big(Y_{\rho}\big)+O((-\gamma_1-\frac{1}{2})),\end{aligned}$$ where we have used similar computations as in the proof of Theorem $2.4$ of [@b-t] for the last equality.
Proof of Theorem \[thm:main-theorem1\]
--------------------------------------
In what follows, we will need the following useful lemma.
\[lem:appendix\] For the vector $\bm{ \alpha_\infty} = (\alpha_{\infty,1}, \cdots,\alpha_{\infty,q}) \in \R^q$ where $\alpha_{\infty,i}$ are non-zero and distinct, we denote $$d(x,\bm{ \alpha_\infty} ):= \min_{i=1,\cdots,q} \, \vert x - \alpha_{\infty,i} \vert, \qquad \forall \, x \in \R.$$ Then, there exists a constant $M$ such that $$d(x, \bm{ \alpha_\infty})^2 \le M \prod_{i=1}^q (x - \alpha_{\infty,i})^2.$$
Consider the function $f:\R - \{ \alpha_{\infty,1},\cdots, \alpha_{\infty,q} \} \to \R$ given by $$f(x):= \frac{\prod_{i=1}^q (x - \alpha_{\infty,i})^2}{d(x, \bm{ \alpha_\infty})^2}.$$ Then, obviously $f$ is a continuous function on $\R - \{ \alpha_{\infty,1},\cdots, \alpha_{\infty,q} \}$ and can be extended to a continuous function on whole real line $\R$ by setting $f(\alpha_{\infty,i}):= \prod_{j \neq i} (\alpha_{\infty,j} -
\alpha_{\infty,i})^2 \neq 0$ at each point $\alpha_{\infty,i}$ for $i=1,\cdots,q$. On the other hand, note that we have $f(x) \to \infty$ as $\vert x \vert$ tends to infinity. Hence, $f$ is bounded from below by a positive constant, say $M$.
We split the proofs of Theorems \[thm:main-theorem1\] and \[thm:main-theorem2\] in several steps. Throughout, $C$ stands for a generic constant that is independent of $n$ but may differ from line to line.
[**Step 1**]{}: (Re-ordering the coefficients) Under the second moment conditions $(\ref{eq:2moment-condition})$, we know that for any fixed $n \ge 1$, we have $\lim_{k \to \infty} \alpha_{n,k}=0$. Therefore, $\max \{ \vert \alpha_{n,k} \vert \, : \, k \ge 1 \}$ is attained in, at least one value, say $\vert \alpha_{n,k_1}\vert$. Similarly, $\max \{ \vert \alpha_{n,k} \vert \, : \, k \neq k_1 \}$ is attained in some value $\vert \alpha_{n,k_2}\vert$. We repeat this procedure by induction and we may build a decreasing sequence $\vert \alpha_{n,k_1}\vert \ge \vert \alpha_{n,k_2}\vert \ge \cdots
\vert \alpha_{n,k_p}\vert \ge \cdots$ such that for all $i \ge 1$, we have $$\vert \alpha_{n,k_i}\vert = \max \{ \vert \alpha_{n,k}\vert \, : \, k\neq k_1,k_2,\cdots,k_{i-1}\}.$$ Also, for all $n,p \ge 1$ we have $1 \ge \sum_{i=1}^{p}\alpha^2_{n,k_i} \ge \sum_{i=1}^{p} \alpha^2_{n,k} \to 1$ as $p$ tends to infinity. Therefore $$\label{eq:re-ordering}
F_n \stackrel{\text{law}}{=} \sum_{i=1}^{\infty} \alpha_{n,k_i} W_i, \qquad \forall \, n \ge 1.$$ To emphasize the maximality property of $\alpha_{n,k_i}$, we denote $\alpha_{n,k_i}$ by $\alpha_{\text{max}}(n,i)$, and in the rest of the proof we assume that for each $n \ge 1$, $F_n$ is given by the right hand side of $(\ref{eq:re-ordering})$.
[**Step 2**]{}: (Bounding the $\alpha_{\text{max}}(n,i)$’s from below) For any $p\ge 1$, we introduce the quantity
$$\label{eq:Delta-p}
\Delta_p(F_n):= \sum_{k=p}^{\infty} \alpha^2_{\text{max}}(n,k) \prod_{r=1}^{q} \left( \alpha_{\text{max}}(n,k) -\alpha_{\infty,r}\right)^2.$$
Next, we observe that $$\begin{aligned}
\Delta_p(F_n)& = \sum_{k=p}^{\infty} Q(\alpha_{\text{max}}(n,k)) =\sum_{r=2}^{2q+2} \Theta_r \sum_{k=p}^{\infty}
\alpha^r_{\text{max}}(n,k)
\nonumber \\
\label{eq:product-estimate}
& = \Theta_2 \Big(1 - \sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k)
\Big) + \sum_{r=3}^{2q+2} \Theta_r \sum_{k=p}^{\infty}
\alpha^r_{\text{max}}(n,k).\end{aligned}$$
Besides, for any $r\ge3$ and the maximality property of the coefficients $\alpha_{\text{max}}(n,k)$ together with the normalization assumption $(\ref{eq:2moment-condition})$ we have the following estimate (which is valid for all $r\ge 3$): $$\left\vert \sum_{k=p}^{\infty} \alpha^r_{\text{max}}(n,k) \right\vert \le \left \vert \alpha^{r-2}_{\text{max}}(n,p) \right \vert \times \left( 1- \sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k) \right).$$ Also note that, since $ \vert \alpha_{\text{max}}(n,p) \vert \le
1$, we always have $ \vert \alpha_{\text{max}}(n,p) \vert^{r-2} \le
\vert \alpha_{\text{max}}(n,p) \vert$ (still for all $r\ge 3$). We may deduce $$\Big \vert \Delta_p(F_n) - \Theta_2 \Big(1 - \sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k) \Big)
\Big \vert \le \vert \alpha_{\text{max}}(n,p) \vert \Big(1 -
\sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k) \Big) \sum_{r=3}^{2q+2}
\vert \Theta_r \vert,$$ leading in turn to the lower bound $$\label{eq:general-lower-estimate}
\Big \vert \alpha_{\text{max}}(n,p) \Big \vert \ge \frac{\vert \Theta_2 \vert}{\sum_{r=3}^{2q+2} \vert \Theta_r \vert} - \frac{\Delta_p(F_n)}{ \Big(1 - \sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k) \Big)
\times \sum_{r=3}^{2q+2} \vert \Theta_r \vert}.$$ Note that in the right hand side of , the first summand depends only on the limiting law. In order to deal with the second summand, we need control on $1 - \sum_{k=1}^{p-1} \alpha^2_{\text{max}}(n,k)$. To this end we introduce the following useful quantities: $$\label{eq:vartheta}
\vartheta = \min \Big\{ 1 - \sum_{i=1}^q n_i \alpha^2_{\infty,i} \, \Big\vert \, (n_1,\cdots,n_q) \in \N_0^q, \quad \text{and} \quad 1 - \sum_{i=1}^q n_i \alpha^2_{\infty,i} > 0 \Big\}.$$ $$\label{eq:varkappa}
\varkappa=\max \Big\{ 1 - \sum_{i=1}^q n_i \alpha^2_{\infty,i} \, \Big\vert \, (n_1,\cdots,n_q) \in \N_0^q, \quad \text{and} \quad 1 - \sum_{i=1}^q n_i \alpha^2_{\infty,i} < 0 \Big\}.$$ Note that, for any vector $(n_1,\cdots,n_q) \in \N_0^q$ such that $ 1
- \sum_{i=1}^q n_i \alpha^2_{\infty,i} > 0$ we have $$\vartheta < 1 - \sum_{i=1}^q n_i \alpha^2_{\infty,i} \le 1 - \alpha^2_{\text{min}}(\infty) \left( \sum_{i=1}^q n_i \right)$$ with $\alpha^2_{\text{min}}(\infty)= \min \{ \alpha^2_{\infty,i} \,
: \, i=1,\cdots,q\}$. This leads to the important upper bound estimate $$\label{eq:repetition-upper-estimate}
\sum_{i=1}^q n_i \le \frac{1- \vartheta}{\alpha^2_{\text{min}}(\infty)},$$ which is finite because our assumption on the coefficients of the target random variable $F_\infty$ implies that $\alpha^2_{\text{min}}(\infty) \neq 0$. Finally we set $$\label{eq:L-uuper-estimate}
L:= \lfloor \frac{1-\vartheta}{\alpha^2_{\text{min}}(\infty)} \rfloor
\mbox{ and }\varrho:= \min \{ \vartheta,\vert \varkappa \vert \}.$$
[**Step 3**]{}: (Induction procedure) We now aim to use Step $2$ to control the distance between the eigenvalues $\alpha_{\text{max}}(n,p)$ and the eigenvalues $\alpha_{\infty,p}$ of the target random variable $F_\infty$ in terms of $\Delta(F_n)$. Taking into account the assumption $\Delta(F_n) \to 0$ as $n$ tends to infinity, one can find an $N_0 \in \N$ such that $$\label{eq:Delta_n-estimate}
\Delta(F_n) \le \min \{ \frac{\vert \Theta_2 \vert}{2}, \frac{\varrho
\vert \Theta_2 \vert}{4} \} \, \text{ and } \, \sqrt{L+1} \times
\sqrt{M \times \left( \frac{2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert \Theta_2 \vert} \right)^2 \times \Delta(F_n)} < \frac{\varrho}{2}.$$ Next, for any $n \ge N_0$ we define $$\mathscr{A}_n:= \Big \{ 1 \le p \le L : \text{ for any } 1 \le l \le p \text{ we have } \vert \alpha_{\text{max}}(n,l) \vert \ge \frac{\vert \Theta_2 \vert}{2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert} \Big\}.$$ The collection $\mathscr{A}_n$ is not empty. In fact, using the estimate $(\ref{eq:general-lower-estimate})$ with $p=1$, one can immediately get $$\label{eq:alpha1-estimate}
\Big \vert \alpha_{\text{max}}(n,1) \Big\vert \ge \frac{\vert \Theta_2 \vert}{\sum_{r=3}^{2q+2} \vert \Theta_r \vert} - \frac{\Delta_1(F_n)}{ \sum_{r=3}^{2q+2} \vert \Theta_r \vert} \ge \frac{\vert \Theta_2 \vert}{2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}$$ because, for $n\ge N_0$, we know that $\Delta_1(F_n) = \Delta(F_n)\le \frac{\vert \Theta_2 \vert}{2}$ thanks to the first inequality in $(\ref{eq:Delta_n-estimate})$. Note that for any $p \ge 1$, we have $\Delta_p(F_n) \le \Delta(F_n) \to 0$ as $n$ tends to infinity. On the other hand, $\mathscr{A}_n$ is set bounded by $L$, and therefore has a maximal element which we denote by $L_n$. By the definitions of $\Delta(F_n)$ and of the set $\mathscr{A}_n$ we infer that $$\sum_{k=1}^{L_n} \prod_{r=1}^{q} \left( \alpha_{\text{max}}(n,k) -
\alpha_{\infty,r} \right)^2 \le \left( \frac{ 2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert \Theta_2 \vert} \right)^2 \Delta(F_n).$$ Then, in virtue of Lemma \[lem:appendix\] this reads $$\label{eq:d-estimate}
\sum_{k=1}^{L_n} d(\alpha_{\text{max}} (n,k), \bm{ \alpha_\infty} )^2
\le M \times \left( \frac{ 2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert
\Theta_2 \vert} \right)^2 \Delta(F_n).$$ On the other hand, for any $1 \le k \le L_n$ there exists some $\alpha_{\infty}(n,k) \in \{ \alpha_{\infty,1}, \cdots,
\alpha_{\infty,q} \}$ realizing the minimum in definition of $d(\alpha_{\text{max}} (n,k), \bm{ \alpha_\infty})$. Here, one has to note that the coefficients $\alpha_\infty(n,k)$ in general can be repeated. So, we can rewrite $(\ref{eq:d-estimate})$ as $$\label{eq:after-d-estimate}
\sum_{k=1}^{L_n} \left \vert \alpha_{\text{max}} (n,k) - \alpha_{\infty}(n,k) \right \vert^2 \le M \times \left( \frac{ 2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert
\Theta_2 \vert} \right)^2 \Delta(F_n),$$ which gives us part of the control we seek. It still remains to show that the remainder is well-behaved. To this end we will show that $\forall n \ge
N_0$ there exists $l_n \in \left\{ 1, \ldots, L+1 \right\}$ such that $$\label{eq:after-d-estimate2b}
\sum_{k=1}^{\ell_n} \left \vert \alpha_{\text{max}} (n,k) -
\alpha_{\infty}(n,k) \right \vert^2 \le M \times \left( \frac{ 2
\sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert
\Theta_2 \vert} \right)^2 \Delta(F_n)$$ and $$\label{eq:after-d-estimate2a}
\sum_{k=\ell_n+1}^{\infty} \alpha^2_{\text{max}} (n,k) \le C \sqrt{\Delta(F_n)}.$$ First, taking into account the estimate $(\ref{eq:after-d-estimate})$, one can infer that $$\begin{aligned}
\left\vert (1- \sum_{k=1}^{L_n} \alpha^2_{\text{max}} (n,k) ) \right. &\left.- (1 -
\sum_{k=1}^{L_n} \alpha^2_{\infty}(n,k)) \right\vert \nonumber\\
& = \left\vert \sum_{k=1}^{L_n} \left( \alpha^2_{\infty}(n,k) - \alpha^2_{\text{max}} (n,k) \right) \right\vert\nonumber \\
& \le 2 \sum_{k=1}^{L_n} \vert \alpha_{\text{max}} (n,k) - \alpha_{\infty}(n,k) \vert\nonumber \\
&\le \sqrt{L_n} \times \sqrt{ M \times \left( \frac{ 2
\sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert \Theta_2 \vert}
\right)^2 \times \Delta(F_n)}\nonumber\\
& \le \sqrt{L} \times \sqrt{ M \times \left( \frac{ 2
\sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert \Theta_2 \vert}
\right)^2 \times \Delta(F_n)}. \label{eq:variance-estimate}\end{aligned}$$ In order to conclude we now seek for an index $l_n$ such that $$\label{eq:sum=0}
\sum_{k=1}^{\ell_n} \alpha^2_{\infty} (n,k) = 1.$$ Given $n \ge N_0$ we have three possibilities.
- If $1- \sum_{k=1}^{L_n} \alpha^2_\infty(n,k) =0$ then we can take $\ell_n=L_n$ and, by and , we are done.
- If $1- \sum_{k=1}^{L_n} \alpha^2_\infty(n,k) >0$ then $1- \sum_{k=1}^{L+1}
\alpha^2_{\infty}(n,k) =0$ and we can take $\ell_n=L+1$. Indeed we necessarily have $1- \sum_{k=1}^{L_n} \alpha^2_\infty(n,k) > \varrho$ by definition of $\varrho$. Using the second inequality in $(\ref{eq:Delta_n-estimate})$ as well as the estimate given in $(\ref{eq:variance-estimate})$ one can infer that $$\label{eq:1}
1 - \sum_{k=1}^{L_n} \alpha^2_{\text{max}} (n,k) \ge \frac{\varrho}{2}.$$ Now, using estimate $(\ref{eq:general-lower-estimate})$ with $p=L_n +1$ together with the first estimate in $(\ref{eq:Delta_n-estimate})$ we obtain that $$\label{eq:last-alpha-estimate}
\vert \alpha_{\text{max}} (n,L_n +1) \vert \ge \frac{\vert \Theta_2 \vert}{2 \sum_{r=3}^{2q+2} \vert \Theta_r \vert}.$$ If $L_n$ were strictly less than $L$, then it would contradict the fact that $L_n$ is the maximal element of the set $\mathscr{A}_n$, and therefore $L_n=L$. Now, following exactly the same lines as in the beginning of this step and using $(\ref{eq:last-alpha-estimate})$ one can infer that
1. $\sum_{k=1}^{L+1} \left \vert \alpha_{\text{max}} (n,k) -
\alpha_{\infty}(n,k) \right\vert^2 \le M \times \left( \frac{ 2
\sum_{r=3}^{2q+2} \vert \Theta_r \vert}{\vert \Theta_2 \vert}
\right)^2\times \Delta(F_n).$
2. $ \left\vert (1- \sum_{k=1}^{L+1} \alpha^2_{\text{max}} (n,k) ) - (1 - \sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k)) \right\vert < \frac{\varrho}{2}.$
Now, we are left to show that $1- \sum_{k=1}^{L+1}
\alpha^2_{\infty}(n,k) =0$. First, note that according to definition of $L$, if $1- \sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k) \neq 0$, then we have to have that $1- \sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k)
<0$. Now, again using definition of $\varkappa$ and $\varrho$, this implies that $1- \sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k) \le -
\varrho$. Now, taking into account $(ii)$ and the fact that $1-
\sum_{k=1}^{L+1} \alpha^2_{\text{max}} (n,k) \ge 0$, we arrive to $$-\frac{\varrho}{2}\le1 - \sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k) \le -
\varrho.$$ That is obviously a contradiction and therefore $1-
\sum_{k=1}^{L+1} \alpha^2_{\infty}(n,k) =0$. Finally, employing the same estimates as in $(\ref{eq:variance-estimate})$ we get $$\begin{aligned}
\label{intermediary-tail}
\sum_{k \ge \ell_n +1} \alpha^2_{\text{max}}(n,k) &= \Big \vert 1 - \sum_{k \le \ell_n} \alpha^2_{\text{max}}(n,k) \Big \vert \\
&= \Big \vert \big(1 - \sum_{k \le \ell_n} \alpha^2_{\text{max}}(n,k)
\big) - \big( 1 - \sum_{k \le \ell_n} \alpha^2_{\infty}(n,k) \big)
\Big\vert \\
& \le C \Big( \sum_{k \le \ell_n} \vert \alpha_{\text{max}}(n,k) - \alpha_{\infty}(n,k) \vert^2 \Big)^{\frac{1}{2}}\\
& \le C \, \sqrt{ \Delta(F_n)}.\end{aligned}$$
- The case $1-
\sum_{k=1}^{L_n} \alpha^2_\infty(n,k) <0$ can be also discussed in the same way, this time leading to some $\ell_n < L$.
The square root in is clearly not sharp and we need to remove it. To do this, observe that an obvious consequence of is $$\alpha^2_{\text{max}}(n,k)\le C \sqrt{\Delta(F_n)} \qquad \forall \,
k \, \ge \ell_n+1 .$$ Thus, if $\Delta(F_n)$ is small enough in the sense that $C \sqrt{\Delta(F_n)} \le U_\infty < \min_{1 \le i \le q+1}
\alpha_{\infty,i}$, since $\alpha_{\infty,i} \neq
0$ by assumption, it follows trivially that we may find a universal positive constant $C$ such that for any $n \in \N$ and for any $k\ge \ell_n+1$ it holds that $$\prod_{i=1}^q(\alpha_{\text{max}}(n,k)-\alpha_{\infty,i})^2>C.$$ Hence $$\begin{aligned}
\Delta(F_n) \ge \Delta_{\ell_n+1}(F_n) \ge C \sum_{k=\ell_n+1}^\infty \alpha^2_{\text{max}}(n,k),\end{aligned}$$ leading to the final estimate $$\label{eq:variance-tail-estimate}
\sum_{k=\ell_n+1}^\infty \alpha^2_{\text{max}}(n,k) \le \frac{1}{C} \, \Delta(F_n).$$
**Step 4:** (An algebraic argument) We have showed that $\forall n \ge N_0$ there exists $l_n \in \left\{ 1, \ldots, L+1 \right\}$ such that $$\label{eq:after-d-estimate2b}
\sum_{k=1}^{\ell_n} \alpha^2_{\infty}(n, k)=1 \mbox{ and }\sum_{k=1}^{\ell_n} \left \vert \alpha_{\text{max}} (n,k) -
\alpha_{\infty}(n,k) \right \vert^2 +\sum_{k=\ell_n+1}^{\infty}
\alpha^2_{\text{max}} (n,k) \le C \Delta(F_n).$$ For every $n$, let $\nu(n,k), k=1, \ldots, q$ stand for the multiplicity of the coefficient $\alpha_{\infty,k}$ realizing the minimum in the definition of the $d$-distance (the numbers $\nu(n,k)$ can, a priori, be equal or take value zero). Clearly $\sum_{k=1}^{q} \nu(n,k) \alpha^2_{\infty,k}=1$. In order to reap the conclusion in 2-Wasserstein distance, we are only left to show that, in fact, we have that $\nu(n,k)=1$ for all $1\le k \le q$ and $\ell_n =q$.
In this case $\dim_{\mathbb{Q}} \text{span} \left\{
\alpha^2_{\infty,1},\cdots,\alpha^2_{\infty,q}\right \}=q$. Then, condition $\sum_{k=1}^{q} \nu(n,k) \alpha^2_{\infty,k}=1$ necessarily implies $\nu(n,k)=1$ for all $k=1, \ldots, q$ and thus $\ell_n=q$ (recall Remark \[rem:span1\]).
For any $r=2,\cdots,q+1$, one can write $$\begin{split}
\kappa_r(F_n) & = \kappa_r(W_1) \sum_{k=1}^{\ell_n} \alpha^r_{\text{max}}(n,k) + \kappa_r(W_1) \sum_{k=\ell_n+1}^{\infty} \alpha^r_{\text{max}}(n,k).
\end{split}$$ Therefore, according to $(\ref{eq:variance-tail-estimate})$, we obtain $$\label{eq:step4-1}
\Big\vert \kappa_r(F_n) - \kappa_r(W_1) \sum_{k=1}^{\ell_n} \alpha^r_{\text{max}}(n,k) \Big\vert \le C \, \Delta(F_n).$$\[eq:step4-2\] Moreover, from the Cauchy-Schwarz inequality and , we have $$\label{eq:step4-3}
\Big \vert \sum_{k=1}^{\ell_n} \alpha^r_{\text{max}}(n,k) - \sum_{k=1}^{q}\nu(n,k) \alpha^r_{\infty,k} \Big \vert \le C \, \sqrt{\Delta(F_n)}$$ for any $r=2,\cdots,q+1$. Combining estimates $(\ref{eq:step4-1})$ and $(\ref{eq:step4-3})$ together, we arrive to $$\label{eq:step4-4}
\Big\vert \kappa_r(F_n) - \kappa_r(W_1) \sum_{k=1}^{q}\nu(n,k) \alpha^r_{\infty,k} \Big \vert \le C \, \sqrt{\Delta(F_n)}.$$ Now, we introduce the following $q \times q$ so-called Vandermonde matrix which is invertible because we assumed that the coefficients $\alpha_{\infty,k}$ are pairwise distinct $$\mathbb{M}=
\begin{bmatrix}
\alpha^2_{\infty,1} & \alpha^2_{\infty,2} & \cdots & \alpha^2_{\infty,q} \\
\cdots & \cdots & \\
\alpha^{q+1}_{\infty,1} & \alpha^{q+1}_{\infty,2} &\cdots &\alpha^{q+1}_{\infty,q}
\end{bmatrix}.$$ Set ${\bf V}_n:= (\nu(n,1),\cdots,\nu(n,q))^t$ ($t$ stands for transposition) and consider the vector $$\Xi=\left( \frac{\kappa_2(F_\infty)}{\kappa_2(W_1)}, \cdots, \frac{\kappa_{q+1}(F_\infty)}{\kappa_{q+1}(W_1)}\right)^t.$$ Note that by our assumption $\kappa_r(W_1) \neq 0$ for all $2 \le r \le q+1$. Now, inequality $(\ref{eq:step4-4})$ entails that $$\label{determineV-1}
\Vert \mathbb{M} {\bf V}_n -\Xi \Vert_{\infty} \le C\left( \sqrt{\Delta(F_n)}+\sum_{r=2}^{q+1}\vert \kappa_r(F_\infty)-\kappa_r(F_n)\vert\right),$$ which, by inversion, implies that $$\label{eq:determineV-2}
\Vert {\bf V}_n - \mathbb{M}^{-1}\Xi \Vert_{\infty} \le C\left( \sqrt{\Delta(F_n)}+\sum_{r=2}^{q+1}\vert \kappa_r(F_\infty)-\kappa_r(F_n)\vert\right).$$
If the right hand side of the estimate $(\ref{eq:determineV-2})$ is strictly less than $1$, say $\frac{1}{3}$, then because ${\bf V}_n$ is a vector of integer numbers, we necessarily have $${\bf V}_n = \lfloor \mathbb{M}^{-1}\Xi \rfloor$$ where $\lfloor \cdot \rfloor$ denotes the standard integer part. In order to conclude the proof, it remains to show that $ \lfloor \mathbb{M}^{-1}\Xi \rfloor$ is the vector $(1,\cdots,1)^t$ and $\ell_n=q$. Note that $ \lfloor \mathbb{M}^{-1}\Xi \rfloor$ does not depend on the approximating sequence $F_n$ but only on the target random variable $F_\infty$. So we might place ourselves in the obvious situation where the approximating sequence $F_n$ is nothing else than the target itself. In this case ${\bf V}_n=(1,\cdots,1)^t$ and necessarily
$$\lfloor \mathbb{M}^{-1}\Xi \rfloor=(1,\cdots,1)^t.$$
\[rem:true-rate\][ Here, we would like to strongly highlight that in order to show that all the multiplicity numbers $\nu(n,k)=1$ for all $k=1,\cdots,q$, one needs that the all distances $\vert \kappa_r(F_n) - \kappa_r(F_\infty)\vert$ are very small (in the above sense) for all $r=2,\cdots,q+1$. However, after doing that taking into account the estimate $\eqref{eq:step4-4}$, one can immediately observe that the true convergence rate is given by $\sqrt{\Delta(F_n)}$. ]{}
Acknowledgments {#acknowledgments .unnumbered}
===============
BA’s research is supported by a Welcome Grant from the Université de Liège. YS gratefully acknowledges support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy).
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(B. Arras and Y. Swan) <span style="font-variant:small-caps;">Mathematics department, Université de Liège, Liège, Belgium</span>
(E. Azmoodeh) <span style="font-variant:small-caps;">Department of Mathematics and Statistics, University of Vaasa, Finland</span>
(G. Poly) <span style="font-variant:small-caps;">Institut de Recherche Mathématiques de Rennes, Université de Rennes 1, Rennes, France</span>
*E-mail address*, B. Arras [barras@ulg.ac.be ]{}
*E-mail address*, E. Azmoodeh [ehsan.azmoodeh@uva.fi ]{}
*E-mail address*, G. Poly [guillaume.poly@univ-rennes1.fr ]{}
*E-mail address*, Y. Swan [yswan@ulg.ac.be ]{}
|
---
author:
- Florian Griese
- Tobias Knopp
- Cordula Gruettner
- Florian Thieben
- Knut Müller
- Sonja Loges
- Peter Ludewig
- Nadine Gdaniec
bibliography:
- 'sample.bib'
title: Simultaneous Magnetic Particle Imaging and Navigation of large superparamagnetic nanoparticles in bifurcation flow experiments
---
Introduction
============
Magnetic Particle Imaging (MPI) uses non-linear magnetization characteristics to spatially resolve the distribution of superparamagnetic nanoparticles in 3D at high temporal resolution.[@gleich_tomographic_2005; @knopp_online_2016] The highest sensitivity is achieved with particles in the magnetic core size range of to .[@ferguson_magnetic_2015; @ferguson_optimization_2009] Due to different signal characteristics for different particle types, it is even possible to distinguish between different particle types with the multi-contrast approach introduced by Rahmer et al.[@Rahmer2015] Lately, MPI has been utilized to track balloon catheters for stenosis clearing[@Salamon2016a; @Herz2018] and to navigate magnetically coated catheters through difficult bifurcations[@Rahmer2017]. To this end, a soft-magnetic sphere has been attached to the catheter. With both possibilities it is feasible to simultaneously image the catheter’s position with the coated magnetic particles and steer the catheter in any direction by applying a magnetic force on the sphere.[@Rahmer2017] It has already been shown that micron-sized devices with soft-magnetic spheres attached can be tracked by the imaging capabilities, and moved to target areas by the force capabilities, of MPI.[@Nothnagel2016] A similar principle is applied with rotational magnetic fields to selectively control helical micro-devices. In terms of application, these screw-like devices might be used to unscrew a radioactive source out of its shielding close to a target region e.g. for local cancer treatment.[@Rahmereaal2845] The same principle has been demonstrated on a human scale for multiple screws by Rahmer et al.[@rahmer_remote_2018] Bakenecker et al.[@BAKENECKER2018] have also utilized rotational focus fields to control the actuation of magnetically coated swimmers and have moved them through bifurcations while simultaneously imaging and magnetically navigating them. The ability to move particles with MPI for targeted drug delivery has also been presented. [@le_real-time_2017; @mahmood_novel_2015; @zhang_development_2017; @kuboyabu_usefulness_2016; @bente_biohybrid_2018] Griese et al.[@griese_imaging_2018] have demonstrated how to simultaneously move and image micron-sized magnetic particles with the Magnetic Particle Imaging/Navigation (MPIN) method with a temporal resolution of . Micron-sized particles have been used to target endothelial cells of the central nervous system to identify over-expressed surface proteins such as ICAM-1. The over-expression caused by neurological diseases can be imaged by Molecular Magnetic Resonance Imaging.[@gauberti_molecular_2018] The micron-sized particles do not cross the blood brain barrier and the enhanced permeability retention effect occurring with smaller particles is insignificant.
The ability to navigate and image magnetic particles with MPIN qualifies for the following potential application scenarios. In the case of targeted drug delivery, magnetic particles can be attached to therapeutic substances (e.g. fibrinolytic medications) and be injected into an organism. In the case of acute stroke, the MPIN method could navigate and concentrate the medicated particles locally in the vascular territory of the vessel occlusion (see Fig. \[fig:CarotisStenosis\]), thus enhancing the disintegration of the blood clot. A normal injection with thrombolytic medications without magnetic particles can result in ineffective dissolving of the blood clot because the blood flows mainly through the unblocked branch of the bifurcation between external carotid artery and internal carotid artery. Normally, the blood clot is then cleared using a catheter in an invasive procedure. Therefore, the usage of therapeutic functionalized magnetic particles could help reduce the dose of the medication or even make an invasive intervention superfluous.
![A blood clot causing a stenosis in internal carotid artery. The blood flow at the bifurcation of the internal and external carotid artery goes mainly through the unblocked external artery making it hard for therapeutic substances to take effect.[]{data-label="fig:CarotisStenosis"}](CarotisStenosis.png){width="10.0cm"}
In this work, we investigate two different types of particles with different magnetic core diameters and determine which type satisfies the best compromise in terms of magnetic manipulability and best imaging performance for MPI. To quantify the navigation characteristics of the particles outside the MPI system, we analyze the magnetophoretic mobility. Furthermore, we perform measurements with a Magnetic Particle Spectrometer (MPS) to identify the particles’ imaging characteristics. Finally, we use inflow bifurcation experiments to investigate the flow velocity, up to which it is possible to navigate the particles to one side of the bifurcation, even with a stenosis in one branch of the bifurcation phantom.
Theory
======
Magnetic Navigation of Particles
--------------------------------
The magnetic force on an isotropic suspension of magnetic particles in a magnetic gradient field can be determined to be $$\begin{aligned}
\label{eq:ForceZ1}
{\boldsymbol{F}}_{m}=({\boldsymbol{m}}\cdot \nabla){\boldsymbol{B}}=\frac{V_m \Delta \chi}{\mu_0}({\boldsymbol{B}} \cdot \nabla){\boldsymbol{B}}=V_m \Delta \chi\nabla(\frac{\lVert{{\boldsymbol{B}}}\rVert^2}{2\mu_0})=\frac{4}{3}\pi r_m^3 \Delta \chi\nabla(\frac{\lVert {{\boldsymbol{B}}}\rVert^2}{2\mu_0}) \end{aligned}$$ with the dipole moment ${\boldsymbol{m}}$, the magnetic flux density ${\boldsymbol{B}}$, the vacuum permeability $\mu_0$, the difference in magnetic susceptibility $\Delta \chi$, the volume $V_m$, and $r_m$ the magnetic core radius of the particles.[@krishnan_fundamentals_2016]\[p.592\]
The magnitude of the magnetic selection field of an MPI field-free point scanner is linear, increasing with distance to the field-free point (FFP). In one direction, e.g. the $y$-direction, this results in $$B_y=\mu_0 H_y = \mu_0 G_y y$$ with the gradient strength $G_y = \frac{\partial H_y}{\partial y}$, the magnetic field ${\boldsymbol{H}}$, and assuming the origin of coordinates to be the field-free point. Thus, the magnetic force induced by the gradient field of an MPI scanner on a magnetic particle[@Nothnagel2016] in the $y$-direction can be expressed as $$\begin{aligned}
\label{eq:ForceZ2}
{\boldsymbol{F}}_{m}=\frac{4}{3}\pi r_m^3 \Delta \chi\nabla(\frac{\lVert{{\boldsymbol{B}}}\rVert^2}{2\mu_0})=\frac{2\mu_0}{3}\pi r_m^3 \Delta \chi\nabla(G_y^2y^2)=\frac{4\mu_0}{3}\pi r_m^3 \Delta \chi G_y^2y,\end{aligned}$$ with $y$ being the distance to the FFP. Opposed to the magnetic force, a drag force acts on a magnetic particle if the particle is moving within a liquid such as water or blood. The drag force, for simplicity given here in the $y$-direction, is defined by $$\begin{aligned}
\label{eq:dragforce}
F_{d,y}(v)=6\pi\eta_{water} r_{h} \Delta v_{m,y}\end{aligned}$$ with $\eta_{water}$ being the fluid viscosity, $r_{h}$ being the hydrodynamic radius of the particles, and $\Delta v_{m,y}$ being the velocity difference of the particles to the flowing liquid.
Taking a look at the motion of particles in the horizontal plane in the case of balancing drag force and magnetic force $F_{m,y}=F_{d,y}$, one can obtain the equilibrium difference velocity of the particles and the fluid using $$\begin{aligned}
\Delta v=\frac{\Delta \chi r_{m}^{3}}{9 \mu_0 \eta r_{h}}\nabla B_{0}^2=\xi\nabla B_{0}^2=\xi\nabla (\mu_{0}H(y))^2=\xi\frac{\partial}{\partial y} (\mu_{0}^2(G_{y} y)^2)= \xi\mu_{0}^2G_{y}^2 y
\label{eq:velocityDifference}\end{aligned}$$ with the magnetophoretic mobility[@krishnan_fundamentals_2016]\[p.593\],[@kara_e._mccloskey_magnetic_2003],[@zhou_magnetic_2016] defined as $$\begin{aligned}
\xi=\frac{\Delta v}{\nabla B_0^2}.\end{aligned}$$ For the described magnetic carrier, according to equation , this results in $$\begin{aligned}
\xi=\frac{\Delta \chi r_{m}^{3}}{9 \mu_{0} \eta r_{h}}\end{aligned}$$ with unit \[$\frac{m^2 s^3 A^2 }{kg^2}$\]. The magnetophoretic mobility can be calculated based on explicit knowledge of characteristic parameters of the particles or based on their velocity in a magnetic gradient field inside a fluid. Following the paths of individual cells in a clearly defined magnetic gradient field is challenging but it is possible to determine indirect measures of the magnetophoretic mobility. For our purpose, a relative measure for the mobility is sufficient. For a given set of particles, the particles should be ordered according to their navigation capabilities.
### Navigation characterization using separation apparatus
In magnetic cell separation procedures an apparatus is used to separate cells from a surrounding medium with the help of magnetic fields. The duration required for the separation in these procedures is inversely proportional to the magnetophoretic mobility. Thus, by comparing these separation times for different particles, a classification regarding their suitability for navigation is possible. The procedure is described in the following.
The work of Andreu et al.[@andreu_simple_2011] describes an apparatus for magnetic separation in detail and derives an analytical model for the separation procedure and the required separation times. The separation apparatus is composed of a radial magnetic gradient as used by De Las Cuevas et al. and Benelmekki et al. [@cuevas; @benelmekki] A homogeneous dispersion of magnetic particles is placed inside a cylindrical cavity of radius $L$. A magnetic gradient field is imposed on the cavity with a uniform magnetic gradient pointing towards the walls of the vessel. The magnetic gradient enforces the particles to move radially towards the vessel wall, which is the final stage of the process, resulting in an inhomogeneous dispersion of particles. The magnetic particles are accumulated at the vessel wall while the center of the cavity is filled with the remaining liquid. The separation process is completed by removing the liquid from the center of the cavity. A light source and a detector are additionally used to determine the turbidity of the solution over time. The turbidity $T$ at time $t$ can be described by $$T = T_\infty+\frac{T_0-T_{\infty}}{1+(\frac{t}{t_{50}})^2}$$ for mono-sized Langevin particles with the initial turbidity $T_0$, the final turbidity $T_\infty$, and the half separation time $t_{50}$.[@witte_particle_2017] According to Andreu et al.[@andreu_simple_2011] the half separation time for mono-sized Langevin particles is given by $$\label{eq:HalfTime}
t_{50}=\left(1-\frac{1}{\sqrt{2}}\right)\frac{L}{v_s},$$ with the saturation velocity $v_s$ of the particles at magnetic saturation. In equation it is assumed that the field strength at the boundary is much higher than the required magnetic field to drive particles into magnetic saturation with saturation magnetization $M_s$. By replacing one $B$ in equation with $B=\frac{\mu_0 M}{\Delta \chi}$ and $M=M_s$, the velocity at saturation $v_s$ is described by $$\label{eq:saturationVelocity}
v_s= \xi \frac{M_s \mu_0}{\Delta \chi}\nabla B_0= \xi \frac{M_s \mu_0^2}{\Delta \chi} G_y$$ containing the magnetophoretic mobility $\xi$. From equation and equation it can be seen that the half separation time $t_{50}$ and the magnetophoretic mobility are inversely proportional to each other. Thus, comparing particles based on the half-separation time gives a hint of the magnetic navigation capabilities relative to each other.
Simultaneous Magnetic Particle Imaging/Navigation Principle
-----------------------------------------------------------
The preclinical MPI scanner considered in this work (Bruker Biospin MRI GmbH, Ettlingen, Germany) has one pair of focus field coils for each direction $x$, $y$, and $z$.[@knopp_joint_2015] These focus fields are originally designed to perform multi-patch MPI to enlarge the field of view.[@gleich_fast_2010; @rahmer_results_2011] Currently, the MPI scanner used in this work is capable of switching between two focus field positions with a frequency of . The distance from the FFP determines the magnetic force as described by equation . Thus, the focus fields can be used in the MPIN method to induce a magnetic force on the particles by appropriately choosing the position of the FFP. The principle of switching between imaging mode and navigation mode can be seen in Fig. \[fig:FPPImagingForce\].
![(Left) Illustration of the FFP position during imaging mode where the magnetic particles (indicated by a black dot) are within the Field of View (FoV). (Right) Visualization of the FFP position during the magnetic force mode where the magnetic particles are not within the FoV but, due to the large distance from the FFP, the particles are moved by the magnetic force. Imaging mode and magnetic force mode cannot be performed in parallel and have to be executed in an alternating manner. []{data-label="fig:FPPImagingForce"}](FFPImagingForceSwitchBetter.pdf){width="16.0cm"}
Since the magnetic force acts only during the navigation time span, equation has to be extended with a switching function $s(t)$ to the following equation $$\begin{aligned}
\label{eq:magneticForceSwitching}
F_{m,y}(y,t)=\frac{4}{3}\pi\mu_0 r_m^3 \Delta \chi G_y^2y s(t) \quad \text{with} ~~ s(t)=\begin{cases}
1 & \text{if} \quad t \in [T(i-1)\varphi+Ti\zeta,T i(\varphi+\zeta)) \\
0 & \text{else}
\end{cases} ~ i=1,\dots,n ~~ \text{and} ~ T=\SI{0.021}{\second}\end{aligned}$$ with the number of imaging drive field cycles (DF-cycles) $\zeta$ and navigation DF-cycles $\varphi$. $\mu_0$ is the vacuum permeability, $G_y$ is the gradient in the $y$-direction, $r_m$ is the magnetic core radius of the particles, $T$ is the time span of one acquisition cycle and $y$ is the distance to the FFP. Equation assumes that the object is within the FFP region during the imaging mode and no force acts on the object.
Methods and Material
====================
For quasi-simultaneous Magnetic Particle Imaging and Magnetic Particle Navigation (MPN), the particles need to be suitable for both applications. The final experiments are performed with an MPI scanner, but to gain insights into the properties of the particles with respect to their imaging and navigation suitability, experiments are performed in advance. For magnetic particle imaging, the particles are analyzed using a magnetic particle spectrometer. For the navigation capabilities, the separation time describing the magnetophoretic mobility is determined using a magnetic separation apparatus.
Magnetic Particles
------------------
Two types of particles with different compositions and magnetic core diameters are used for the experiments. The first type is Dynabeads MyOne (ThermoFisher) with a hydrodynamic diameter of about as a reference. These particles consist of iron oxide embedded in a polystyrene matrix and are not suitable for *in vivo* applications. The second particle type is customized nanomag/synomag-D particles (micromod Partikeltechnologie GmbH) with hydrodynamic diameters in the range of . These biocompatible particles consist of iron oxide cores and a dextran shell. Synomag-D particles have excellent properties as tracers for MPI,[@gruettner2019] but a very slow magnetic mobility. In contrast, nanomag-D particles with diameters in the range of have a high magnetic mobility.[@henstock_remotely_2014] To combine the imaging properties of synomag-D with the high magneto-mobility of nanomag-D particles different amounts of synomag-D are embedded in the iron oxide multi-cores of nanomag and finally coated with dextran. An overview of the used particles is given in Table \[tab:Particles\].
------------------- ----- --------- ----------------- ---------------- --------------------
Sample name Lot Surface Hydrodynamic Polydispersity Percentage of iron
diameter \[nm\] index (PDI) from synomag \[%\]
nanomag-D 308 dextran 405 0.12 -
nanomag/synomag-D 332 dextran 649 0.14 44
nanomag/synomag-D 333 dextran 698 0.24 61
nanomag/synomag-D 334 dextran 641 0.32 40
Dynabeads MyOne 017 COOH 1048 0.13 -
------------------- ----- --------- ----------------- ---------------- --------------------
: Particles and their relevant properties used for the experiments in this work.[]{data-label="tab:Particles"}
Magnetic Particle Spectrometry
------------------------------
The MPS measurements are performed with a custom MPS designed similarly to the device outlined by Biederer et al.[@biederer2009magnetization]. The particles are excited with a sinusoidal signal with a frequency of and an amplitude of . The measurements are performed with background subtraction and 1000 averages.
Num $c_0$ $c_1$ $c_2$ $c_3$ $c_4$ $c_5$ $c_6$ $c_7$ $c_8$ $c_9$ $c_{10}$
---------------------------------------------------- -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ----------
Dilution factor 0 1 2 4 8 16 32 64 128 256 512
Iron concentration \[$\si{\milli\mol\per\liter}$\] 179.05 89.52 44.76 22.38 11.19 5.59 2.79 1.39 0.69 0.35 0.18
: Iron concentrations of the dilution series used for MPS measurements.[]{data-label="tab:serialdilution"}
In order to analyze the spectra of the particles, a dilution series (summarized in Table \[tab:serialdilution\]) of all particle types is prepared. The analyzed volume is for each concentration within an Eppendorf tube. In a second measurement each particle sample at the highest concentration is magnetized with a conventional magnet. The particles remain at the bottom of the tube while the rest of the liquid becomes transparent. In a third measurement series the particles are stirred up with a vortexer. In a fourth measurement cycle the whole process of magnetization and vortexing is repeated twice. With these second, third and fourth measurements it should be ensured that the particles have no remaining remanence caused by the force mode. The particles would then have a different behavior in the imaging mode. Finally, a background measurement containing noise is subtracted from all measurements.
Magnetic Mobility of Particles
------------------------------
The magnetic mobility, in terms of its half separation time, is measured with a Q 100 ml device (SEPMAG Technologies). For these experiments of the particles with iron concentration of are placed inside the cavity. During the measurement the opaque suspension becomes more and more transparent until all particles have been driven towards the walls. A light source illuminates the cavity from one side to the other and the detector on the other side monitors the magnetophoresis process while the suspension homogeneity is measured over time. Afterwards, the half separation time $t_{50}$ is determined by using the Qualitance software by SEPMAG, which fits a sigmoidal curve to the measured values. A decreasing value of the half separation time represents an increase of the magnetophoretic mobility. In addition, the size distribution of the particle suspension before and after the magnetic separation is determined by Dynamic Light Scattering (Nano-S 90. Malvern Panalytical Ltd.).
Magnetic Particle Navigation in Flow within Bifurcation Junction
----------------------------------------------------------------
The particles are analyzed in an MPS system to determine their imaging capabilities and their usability for navigation has been determined in a magnetic separation device. Therefore, all subsequent experiments are performed using only the particles best suited for simultaneous imaging and navigation from the insight gained in the aforementioned experiments.
For manipulating particles in flow through a bifurcation junction using MPN, three different bifurcation phantoms are designed and 3D printed by a Forms 2 stereolithography printer from Formlabs. All bifurcation vessels have a quadratic cross-section of $A=a^2=\SI{12.544}{\square\milli\meter}$ with a side length of $a=\SI{3.544}{\milli\meter}=d_{v}$. This cross-section corresponds to the same cross-section as a circular vessel with a diameter of $d=\SI{4}{\milli\meter}$. The first bifurcation vessel splits into two equally-sized branches with a crossing angle of as seen in Fig. \[fig:DynamicBifurcation\](a). The second bifurcation phantom constitutes a 60% stenosis in the right branch and has a circular cross-section of $A_{60}=\SI{7.52}{\milli\meter}^2$ with a diameter of . The third bifurcation phantom simulates a 100% stenosis in the right branch and has a glued circular cross-section of $A_{100}=\SI{3.76}{\milli\meter}^2$ with a diameter of which constitutes a full blockage for the particles. Both also split with a crossing angle of . The 60% stenosis phantom is shown in Fig. \[fig:DynamicBifurcation\](b). All three phantoms have a centric catheter mount in the inbound flowing branch seen in Fig. \[fig:DynamicBifurcation\](c) to ensure that the particles are injected centrically.
at (0,0) [![a) 3D CAD sketch of the bifurcation junction phantom. b) 3D CAD sketch of the bifurcation junction with 60% stenosis in the right side of the phantom. c) The centric catheter mount is built centrically at the bottom of the inbound tube of every phantom to ensure centric injection of particles.[]{data-label="fig:DynamicBifurcation"}](FlowBifurcationNoStenosisTopView.png "fig:"){width="0.9\linewidth"}]{}; (-2.3,-3.3) – (0,-3.3) node\[midway,above\][y]{}; (-2.3,-3.3) – (-2.3, 0) node\[midway,left\][x]{};
a)
at (0,0.0) [![a) 3D CAD sketch of the bifurcation junction phantom. b) 3D CAD sketch of the bifurcation junction with 60% stenosis in the right side of the phantom. c) The centric catheter mount is built centrically at the bottom of the inbound tube of every phantom to ensure centric injection of particles.[]{data-label="fig:DynamicBifurcation"}](FlowBifurcation60StenosisTopView.png "fig:"){width="1.0\linewidth"}]{}; at (0.0,2.5) [Stenosis]{}; (0.7,1) – (0.7,-1) node\[midway,right\]; (0.9,1.6) – (1.5,2.3) node\[midway,right,below,yshift=-2.5mm\];
b)
at (0,0.0) [![a) 3D CAD sketch of the bifurcation junction phantom. b) 3D CAD sketch of the bifurcation junction with 60% stenosis in the right side of the phantom. c) The centric catheter mount is built centrically at the bottom of the inbound tube of every phantom to ensure centric injection of particles.[]{data-label="fig:DynamicBifurcation"}](CentricCatheterMount.png "fig:"){width="0.9\linewidth"}]{}; at (0.0,-2.7) [Centric Catheter Mount]{};
c)
Additionally for the dynamic flow experiments, a phantom mount is built, positioning the phantom in the center of the FoV as seen in Fig. \[fig:Exp3Real\](a). A mirror at a angle is fixed above the phantom to make the volume flow visible from the outside by looking through the scanner bore. A conventional light source illuminates the bore and the mirror to improve the captured optical images taken by a video camera. The bifurcation phantom is placed horizontally in the x$y$-plane to avoid the influence of the gravity force. The center of the bifurcation is placed off-center in the $y$-direction and in the $x$-direction. The dynamic flow bifurcation phantom is connected via three hoses to three flow sensors from BioTech with a measurement range of . Both outbound hoses from the bifurcation branches are then connected back to one hose and back to the pump. The flow sensors are controlled by an Arduino which logs the flow values triggered by the MPI scanner at measurement start. The inbound hose connected to the entrance of the bifurcation phantom has a bypass enabling catheter insertion. This catheter, Abbott Armada 14 with a diameter of , is fixed in the bottom middle of the 3D-printed phantom as seen in Fig. \[fig:DynamicBifurcation\](c). It ensures an unbiased particle flow at the time of the injection. The whole setup can be seen in Fig. \[fig:Exp3Real\](a) and Fig. \[fig:Exp3Real\](b).
![a) Bifurcation flow measurement setup with Arduino attached to flow sensors, bifurcation phantom visible inside the mirror placed inside the bore and light source. b) Schematic image of navigation of particles in bifurcation flow phantom.[]{data-label="fig:Exp3Real"}](Setup.png){width="1.0\linewidth"}
a)
![a) Bifurcation flow measurement setup with Arduino attached to flow sensors, bifurcation phantom visible inside the mirror placed inside the bore and light source. b) Schematic image of navigation of particles in bifurcation flow phantom.[]{data-label="fig:Exp3Real"}](FlowBifurcationNoStenosis.pdf){width="1.0\linewidth"}
b)
The preclinical MPI scanner used generates a maximum gradient of in the $z$-direction ( in the $x$- and $y$-directions). The FFP offset can be set to a maximum of $\SI{-42}{\milli\tesla}$ to $\SI{42}{\milli\tesla}$ ($\SI{-17.5}{\milli\meter}$ to $\SI{17.5}{\milli\meter}$) in the $z$-direction and $\SI{-17}{\milli\tesla}$ to $\SI{17}{\milli\tesla}$ ($\SI{-14.1}{\milli\meter}$ to $\SI{14.1}{\milli\meter}$) in the $x$- and $y$-directions. This leads to a maximum force on the particles of $1.18\cdot 10^{-12}$ N in the $z$-direction and $2.96\cdot 10^{-13}$ N in the $y$-direction, by assuming a particle diameter of . In our case the FFP in force mode is adjusted away in the opposed $y$-direction, making a total distance from bifurcation center to the FFP of ($4.64 \cdot 10^{-13}$ N). In the $x$-direction the FFP is also set away from the MPI FoV center, making a total distance between FFP and bifurcation center of . For these experiments we only use the MPN (DF-cycles $\varphi=20$) which means no imaging (DF-cycles $\zeta=0$) is done and the force acts all the time on the particles.
The following flow experiments with the bifurcation phantom are conducted with five different flow rates from in the inflowing branch. For each flow velocity the experiment is performed twice. First, the magnetic forces are turned off to gain ground truth flow behavior and exclude any biased flow direction. This ensures that the particles flow equally through both branches. Secondly, in the actual experiment the magnetic forces are turned on to investigate the particles’ flow characteristics influenced by the magnetic field. To eliminate a bias to one side of the bifurcation branch and to show that the method works in both directions of the bifurcation, the experiment is conducted for both branches of the bifurcation for the flow rate .
-------------------- -------------- --------------- -------------
Degree of stenosis Flow rate Flow rate Time \[ms\]
inbound \[\] outbound \[\]
100% 1.36 0.68 322
0%, 60%, 100% 2.72 1.36 161
0%, 60% 5.45 2.72 80
0%, 60% 6.87 3.40 63
0%, 60% 8.18 4.09 53
0% 10.22 5.11 42
-------------------- -------------- --------------- -------------
: Volumetric flow rates used in the measurements and the length of time the particles are influenced by the magnetic force.[]{data-label="tab:Sensorflowrate"}
The measured flow values from the sensors coincide with the theory. The flow in both branches is the same and their sum corresponds to the flow values inbound to the bifurcation. The five investigated flow velocities are given in Table \[tab:Sensorflowrate\]. The distance from the location where particles are released to the center of the bifurcation is $l_v$ = . The particles are ejected with a catheter fixed centrically inside the vessel phantom. The time span from particle release point to the center of the bifurcation is also given in the final column of Table \[tab:Sensorflowrate\].
Magnetic Particle Imaging
-------------------------
MPI with simultaneous navigation application is performed using the preclinical MPI scanner, which is also used in the navigation only experiments. Imaging of static particles without navigation application is performed in advance to check the imaging capabilities of the particles in the MPI scanner. Two system matrices are acquired on a grid of positions covering a volume of and delta samples of filled with of Dynabeads MyOne and nanomag/synomag-D 333. The gradient strength is set to and the drive-field amplitude to resulting in a FoV of . For the imaging measurement the delta samples of are used respectively. The imaging sequence is performed with the same imaging parameters to enable reconstruction using a regularized iterative Kaczmarz algorithm. Before reconstruction, the data from 100 DF-cycles are averaged and frequencies with an SNR above two are selected from the frequency range of , resulting in 423 frequency components. The relative regularization parameter is set to 0.001 and three iterations are performed. Afterwards, the full width at half maximum (FWHM) is determined in the $x$- and $z$-directions within the image.
Simultaneous Magnetic Particle Imaging and Navigation of Phantom with Flow in Bifurcation Junction
--------------------------------------------------------------------------------------------------
The simultaneous MPIN bifurcation experiments within flow are only performed for flow velocity and only with the bifurcation phantom with 100% stenosis. The position of the imaging FoV is adjusted by FF$_{\text{I,xyz}}$= with the help of the focus fields to cover the phantom branch with the simulated stenosis. Due to field deviations the system matrix acquired for the previous imaging experiments cannot be used for the imaging part of the experiments with simultaneous navigation application. Thus, a system matrix with the same imaging parameters used in the previous experiments is acquired at the shifted focus field position. The focus field position for navigation is set to FF$_{\text{F,xyz}}$=. For simultaneous imaging and navigation the number of imaging DF-cycles is set to $\zeta=1$ while the number of navigation DF-cycles is set to $\varphi=20$. The FFP then switches between FF$_{\text{I,xyz}}$ and FF$_{\text{F,xyz}}$ with the ratio of 20:1 taking snapshots at the 100% stenosis. Overall, this results in a temporal resolution for imaging of per image.
Results
=======
Magnetic Particle Spectrometer
------------------------------
In Fig. \[fig:TimeLinearity\](a) the spectra of Dynabeads MyOne and four different nanomag/synomag-D particle types in a concentration of are shown. The batches (332, 308) only generate up to 28 harmonics and the signal decrease is not linear. The spectra of the Dynabeads MyOne, and the nanomag/synomag-D batches (333, 334) generate up to 35 harmonics above noise level. Batch 333 has the strongest signal of all batches but also the steepest linear constant decline of the signal strength as seen in Fig. \[fig:TimeLinearity\](a). Batch 334 has lower signal in the beginning harmonics but its steepness is lower and has a kink at the 18$^{th}$ harmonic where the curvature of the signal strength becomes even less steep. For further investigations, only the results of nanomag/synomag-D batch 333 are shown because batch 333 provides the best result of all batches in the spectrum and is therefore the most promising candidate for imaging abilities.
![(a) Spectra of Dynabeads MyOne, nanomag/synomag-D batches 332,333,334,308. (b) Linear signal strength decrease for various frequencies dependent on the concentration of beads. (c) Linear signal strength decline for various frequencies dependent on the concentration of nanomag/synomag-D batch 333.[]{data-label="fig:TimeLinearity"}](SpectrumComparison.pdf){width="1.0\linewidth"}
\(a) Spectra
![(a) Spectra of Dynabeads MyOne, nanomag/synomag-D batches 332,333,334,308. (b) Linear signal strength decrease for various frequencies dependent on the concentration of beads. (c) Linear signal strength decline for various frequencies dependent on the concentration of nanomag/synomag-D batch 333.[]{data-label="fig:TimeLinearity"}](BeadsMyOneLinearityFreq.pdf){width="1.0\linewidth"}
\(b) Dynabeads MyOne
![(a) Spectra of Dynabeads MyOne, nanomag/synomag-D batches 332,333,334,308. (b) Linear signal strength decrease for various frequencies dependent on the concentration of beads. (c) Linear signal strength decline for various frequencies dependent on the concentration of nanomag/synomag-D batch 333.[]{data-label="fig:TimeLinearity"}](SynomagForce2LinearityFreq.pdf){width="1.0\linewidth"}
\(c) nanomag/synomag-D 333.
In Fig. \[fig:TimeLinearity\](b) and Fig. \[fig:TimeLinearity\](c) the linearity within a frequency over different concentrations is investigated for Dynabeads MyOne and nanomag/synomag-D batch 333. It shows that the signal strength in all frequencies is linear, with coefficients of determination between 0.971 and 0.997.
Since the particles should be used for magnetic navigation applications, their imaging characteristics are investigated for any effects of magnetization of the particles. In Fig. \[fig:Magnetization\](a) the time signal of beads at concentration is shown for different stages. The light blue line presents the signal of non-magnetized particles, while in the dark blue line the particles have been magnetized and settled at the bottom of the tube. The results in the green line are from particles that have been magnetized and vertexed once. The results in the red line are from particles that went through this procedure twice. All four time signals seem similar with only very small deviations. The same four lines are presented in Fig. \[fig:Magnetization\](b) for the nanomag/synomag-D batch 333 particles. Here, all graphs also appear very similar with only very small discrepancies. The magnetization attempt and vertexing do not seem to influence the particles’ imaging behavior.
![(a) Time signal for Dynabeads MyOne in four different stages: non-magnetized, magnetized, magnetized and vertex, 2x magnetized and 2x vertex. (b) Time signal for nanomag/synomag-D 333 in four different stages: non-magnetized, magnetized, magnetized and vertex, 2x magnetized and 2x vertex. []{data-label="fig:Magnetization"}](BeadsMyOneLinearityTimeMag.pdf){width="1.0\linewidth"}
\(a) Dynabeads MyOne
![(a) Time signal for Dynabeads MyOne in four different stages: non-magnetized, magnetized, magnetized and vertex, 2x magnetized and 2x vertex. (b) Time signal for nanomag/synomag-D 333 in four different stages: non-magnetized, magnetized, magnetized and vertex, 2x magnetized and 2x vertex. []{data-label="fig:Magnetization"}](SynomagForce2LinearityTimeMag.pdf){width="1.0\linewidth"}
\(b) nanomag/synomag-D 333.
Magnetic Mobility of Particles
------------------------------
The suspension homogeneity during the separation process for the different particles is shown in Fig. \[fig:MagnetophoreticCurve\]. The homogeneity has been normalized with the initial homogeneity. The relevant navigation parameters calculated based on these time curves (the half separation time values), as well as the size distribution of the particle suspension before and after the magnetic separation, and the polydispersity index (PdI) are given in Table \[tab:MagnetophoreticResults\].
![During the magnetic separation the homogeneity of the suspension is measured over time. The suspension homogeneity is plotted for all particle batches, while water is used as a blank value for comparison.[]{data-label="fig:MagnetophoreticCurve"}](MagnetophoreticResultsV2.pdf){width="0.5\linewidth"}
The half separation times (, ) of nanomag/synomag-D batches (333, 334) are within the same order as the half separation of the Dynabeads MyOne at . In contrast, the half separation time of nanomag/synomag-D (332) particles is much shorter at , whereas the half separation time of plain nanomag-D is much longer at .
Sample name Lot $t_{50}$\[s\] $Z_\text{average}$\[nm\] / PdI before separation $Z_\text{average}$\[nm\] / PdI after separation
---------------------- ----- --------------- -------------------------------------------------- -------------------------------------------------
nanomag-D 308 194.4 405.4 / 0.118 505.0 / 0.229
nanomag/synomag-D 332 57.0 649.2 / 0.143 679.2 / 0.171
nanomag/synomag-D 333 75.0 698.2 / 0.240 1104.0 / 0.297
nanomag/synomag-D 334 83.0 641.3 / 0.325 770.2 / 0.481
Dynabeads MyOne COOH 017 79.0 1048 / 0.126 1105.0 / 0.280
: Half-separation times $t_{50}$ of the different suspensions of iron oxide particles with water as a blank value and size distribution before and after separation. []{data-label="tab:MagnetophoreticResults"}
The size distribution is influenced by the separation process resulting in an increase of $Z_{\text{average}}$. The increase is at its maximum after the magnetic separation for all particles except for nanomag/synomag-D 333 particles. For these particles the difference is before and after the magnetic separation process.
Navigation Experiments in Flow
------------------------------
The results are shown in videos. An overview of the flow measurement results can be found in Table \[tab:Exp3\] with video reference, flow parameters for inbound and outbound bifurcation branches, left or right bifurcation and description.
----------- --------------------- ------------- -------------------------------------------------------------------------------------------------------
Video ref Flow rate inbound / Bifurcation Description of results
outbound \[ml/s\] side
v1.1.0R 2.72 / 1.36 right Particles flow only through the right branch
v1.1.1R 2.72 / 1.36 right Control No Force: Particles flow through both branches equally
v1.1.0L 2.72 / 1.36 left Particles flow only through the left branch
v1.1.1L 2.72 / 1.36 left Control No Force: Particles flow through both branches equally
v1.2.0L 5.45 / 2.72 left Particles flow only through the left branch
v.1.2.1L 5.45 / 2.72 left Control No Force: Particles flow through both branches equally
v1.3.0L 6.87 / 3.40 left Particles flow only through the left branch
v1.3.1L 6.87 / 3.40 left Control No Force: Particles flow through both branches equally
v1.4.0L 8.18 / 4.09 left Particles flow mainly through the left branch, depending on injection pressure
v1.4.1L 8.18 / 4.09 left Control No Force: Particles flow through both branches equally
v1.5.0L 10.22 / 5.11 left Effect still visible but, due to the high flow velocity, not all particles are navigated to the left.
----------- --------------------- ------------- -------------------------------------------------------------------------------------------------------
video broken
In general, it can be observed that magnetic forces from the FFP steer the particles to one side of the bifurcation for flow velocities up to in the inbound branch. The method works for both sides of the bifurcation depending on the position of the FFP. The control measurement shows that the particles flow equally through both branches and a bias for one direction is negligibly low. At the flow velocity inbound, the majority of particles steer to the left branch, while the remaining particles travel too quickly through the navigation window and flow with the current in the right branch. 1[-2.15]{} 1[-1.70]{}
at (0,0) [![a) First time point in the flow measurement with magnetic forces and flow velocity where particles are injected and flow through the right side of the bifurcation. (b) Second time point where particles are injected and pushed to the right side and remain motionless at the side, as long as the magnetic fields are turned on. c) Control measurement where forces are turned off and particles flow equally through both bifurcation branches.[]{data-label="fig:ResultExp3Time1_2"}](image-056cut.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1,1) – (1,0) node\[midway,left\][x]{}; (1,1) – (0.0,1) node\[midway,below\][y]{}; (0.1,-1.6) – node\[midway,right\][$F_x$]{} (0.1,-0.9);
at (0,0) [![a) First time point in the flow measurement with magnetic forces and flow velocity where particles are injected and flow through the right side of the bifurcation. (b) Second time point where particles are injected and pushed to the right side and remain motionless at the side, as long as the magnetic fields are turned on. c) Control measurement where forces are turned off and particles flow equally through both bifurcation branches.[]{data-label="fig:ResultExp3Time1_2"}](image-765cutStopped.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1,1) – (1,0) node\[midway,left\][x]{}; (1,1) – (0.0,1) node\[midway,below\][y]{}; (0.1,-1.6) – node\[midway,right\][$F_x$]{} (0.1,-0.9);
at (0,-0.075) [![a) First time point in the flow measurement with magnetic forces and flow velocity where particles are injected and flow through the right side of the bifurcation. (b) Second time point where particles are injected and pushed to the right side and remain motionless at the side, as long as the magnetic fields are turned on. c) Control measurement where forces are turned off and particles flow equally through both bifurcation branches.[]{data-label="fig:ResultExp3Time1_2"}](image-287cutControl.png "fig:"){width="1.0\linewidth"}]{}; at (-1,1) [Force $F_y$=0]{}; (1,1) – (1,0) node\[midway,left\][x]{}; (1,1) – (0.0,1) node\[midway,below\][y]{};
In Fig. \[fig:ResultExp3Time1\_2\](a) and Fig. \[fig:ResultExp3Time1\_2\](b) examples of two time points during the injection of particles are shown for the flow velocity inbound. The particles are always injected centrically in the phantom and the particles are pushed to the right side and enter the right branch as seen in Fig. \[fig:ResultExp3Time1\_2\](a). At the second time point the particles also move to the right side and enter the right branch but they remain on the right side despite the flow pushing them forward. These particles remain motionless on the right side, as depicted in Fig. \[fig:ResultExp3Time1\_2\](b), as long as the magnetic field sequence is active. After the magnetic fields are turned off the particles are released by the outbound flow. In the control measurement with force $F_y=0$, seen in Fig. \[fig:ResultExp3Time1\_2\](c), the particles are interfused with water and they are distributed equally towards both sides of the bifurcation. 2[-2.15]{} 2[-1.65]{}
at (0,0) [![(a) The particles are entirely pushed to the left side of the bifurcation at a flow rate of . (b) When the magnetic fields are turned off in the control measurement the particles distribute equally to both branches at a flow velocity of .[]{data-label="fig:ResultExp3Time3_4"}](image-172cut.png "fig:"){width="1.0\linewidth"}]{}; (1.5,1) – (0.5,1) node\[midway,below\][Force $F_y$]{}; (2,2) – (2,0) node\[midway,left\][x]{}; (2,2) – (0.0,2) node\[midway,below\][y]{}; (0.0,-1.5) – node\[midway,right\][$F_x$]{} (0.0,-0.8);
at (0,-0.075) [![(a) The particles are entirely pushed to the left side of the bifurcation at a flow rate of . (b) When the magnetic fields are turned off in the control measurement the particles distribute equally to both branches at a flow velocity of .[]{data-label="fig:ResultExp3Time3_4"}](image-285Control.png "fig:"){width="1.0\linewidth"}]{}; at (1,1) [Force $F_y$=0]{}; (2,2) – (2,0) node\[midway,left\][x]{}; (2,2) – (0.0,2) node\[midway,below\][y]{};
The highest flow velocity in our experiments with successful navigation of particles towards one of the bifurcation branches is determined to be . This is seen in Fig. \[fig:ResultExp3Time3\_4\](a) for a tube phantom with a cross-section of $A=\SI{12.544}{\milli\meter}^2$ and an inbound length of $l=\SI{35}{\milli\meter}$. In the control measurement shown in Fig. \[fig:ResultExp3Time3\_4\](b) with no force acting on the particles, the particles are distributed equally to both sides of the bifurcation. The particles can only be stopped at the side of the inbound tube at the velocity . In contrast, this effect does not occur at higher flow velocities.
### Flow analysis with bifurcation and 60%-100% stenosis
Further flow experiments are conducted using a bifurcation phantom with a 60% stenosis in the right branch. The experiments are performed using different flow velocities. For all velocities up to it is possible to navigate the particles to the right branch. However, all particles flow through the left branch during the control measurement, where the high flow velocity in the left side is caused by the 60% stenosis. Only at velocity , the particles flow through both, the left and right branch. The video references of the 60% stenosis measurements and their control measurements for different velocities can be found in Table \[tab:Exp3Stenosis\].
----------- -------------------- ------------- -----------------------------------------------------------------
Video ref Flow rate inbound/ Bifurcation Description of results
outbound \[ml/s\]
v2.1.0R 2.72 / 1.36 right Particles flow only through stenosis in right branch
v2.1.1R 2.72 / 1.36 right Control: Particles flow mainly through left branch
v2.2.0R 5.45 / 2.72 right Particles flow only through stenosis in right branch
v2.2.1R 5.45 / 2.72 right Control: Particles flow mainly through left branch
v2.3.0R 6.87 / 3.40 right Particles flow mostly through stenosis in right branch
v2.3.1R 6.87 / 3.40 right Control: Particles flow mainly through left branch
v2.4.0R 8.18 / 4.09 right Particles flow through left branch and stenosis in right branch
v2.4.1R 8.18 / 4.09 right Control: Particles flow mainly through left branch
----------- -------------------- ------------- -----------------------------------------------------------------
: Video results for 60% stenosis measurements and their controls for different flow velocities.[]{data-label="tab:Exp3Stenosis"}
3[-2.15]{} 3[-1.70]{}
at (0,0) [![(a) The particles are moved into the right branch containing the 60% stenosis at flow velocity . In the control measurement at velocity most of the particles flow through the left side and only a small amount enter the stenosis.[]{data-label="fig:ResultExp3Stenosis60"}](image-279Velo3Cut.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1.5,0.8) node\[above\][Stenosis]{} – (1.4,-0.2); (3,3) – (3,0) node\[midway,left\][x]{}; (3,3) – (0.0,3) node\[midway,below\][y]{};
at (0,-0.075) [![(a) The particles are moved into the right branch containing the 60% stenosis at flow velocity . In the control measurement at velocity most of the particles flow through the left side and only a small amount enter the stenosis.[]{data-label="fig:ResultExp3Stenosis60"}](image-058Velo3ControlCut.png "fig:"){width="1.0\linewidth"}]{}; at (-0.75,1) [Force $F_y$=0]{}; (1.5,0.8) node\[above\][Stenosis]{} – (1.4,-0.2); (3,3) – (3,0) node\[midway,left\][x]{}; (3,3) – (0.0,3) node\[midway,below\][y]{};
In Fig. \[fig:ResultExp3Stenosis60\](a) a snapshot of the 60% stenosis measurement shows for velocity that the force of the magnetic field pushes the particles through the bifurcation branch of the stenosis, although the flow velocity is reduced in that branch. The control measurement with no magnetic force active underlines that more particles flow through the left side with no stenosis, as seen in Fig. \[fig:ResultExp3Stenosis60\](b), due to a higher flow velocity in that bifurcation branch.
4[-2.15]{} 4[-1.80]{} 4[0.24]{}
at (0,0) [![(a) and b) Most of the particles accumulate in the 100% stenosis during the flow measurements with velocity . c) and d) At a higher velocity of only a small amount of particles end up in the 100% stenosis while most of the particles are bent towards the left side.[]{data-label="fig:ResultExp3Stenosis100"}](image-113.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1.5,0.8) node\[above\][Stenosis]{} – (1.4,-0.2); (4,4) – (4,0) node\[midway,left\][x]{}; (4,4) – (0.0,4) node\[midway,below\][y]{};
at (0,0) [![(a) and b) Most of the particles accumulate in the 100% stenosis during the flow measurements with velocity . c) and d) At a higher velocity of only a small amount of particles end up in the 100% stenosis while most of the particles are bent towards the left side.[]{data-label="fig:ResultExp3Stenosis100"}](image-1147.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1.5,0.8) node\[above\][Stenosis]{} – (1.4,-0.2); (4,4) – (4,0) node\[midway,left\][x]{}; (4,4) – (0.0,4) node\[midway,below\][y]{};
at (0,0) [![(a) and b) Most of the particles accumulate in the 100% stenosis during the flow measurements with velocity . c) and d) At a higher velocity of only a small amount of particles end up in the 100% stenosis while most of the particles are bent towards the left side.[]{data-label="fig:ResultExp3Stenosis100"}](image-524.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1.5,0.2) node\[above\][Stenosis]{} – (1.4,-0.4); (-2.15,-1.8) – (-2.15,0) node\[midway,left\][x]{}; (-2.15,-1.8) – (0.0,-1.8) node\[midway,below\][y]{};
at (0,0) [![(a) and b) Most of the particles accumulate in the 100% stenosis during the flow measurements with velocity . c) and d) At a higher velocity of only a small amount of particles end up in the 100% stenosis while most of the particles are bent towards the left side.[]{data-label="fig:ResultExp3Stenosis100"}](image-744.png "fig:"){width="1.0\linewidth"}]{}; (-1.5,1) – (-0.5,1) node\[midway,below\][Force $F_y$]{}; (1.5,0.2) node\[above\][Stenosis]{} – (1.4,-0.4); (-2.15,-1.8) – (-2.15,0) node\[midway,left\][x]{}; (-2.15,-1.8) – (0.0,-1.8) node\[midway,below\][y]{};
Furthermore, the results of the experiments performed using the bifurcation phantom with a 100% stenosis in the right branch are shown in Fig. \[fig:ResultExp3Stenosis100\]. The flow velocity is for the images in (a and b) and for the images in (c and d). In the case of the flow velocity the magnetic force of the focus field is strong enough to move the particles inside the right branch into the stenosis, although the flow velocity inside the right branch is zero due to full blockage. In the inbound tube of the phantom the particles are pushed to the far-right side but as soon as they reach the entrance of the right branch they slightly bend towards the left branch. Most of the particles still end up in the stenosis as seen in Fig. \[fig:ResultExp3Stenosis100\](a,b). The slight bend towards the left is clearly visible in Fig. \[fig:ResultExp3Stenosis100\](c,d) at flow velocity . Here, the flow dynamics bend most of the particles away from the entrance of the right branch to the left side, even though the particles are pushed to the far right in the inbound tube. Only a small amount of particles end up at the stenosis due to turbulent whirls. The video references of the 100% stenosis measurements for the two different velocities can be found in Table \[tab:Exp3Stenosis100\].
----------- -------------------- ------------- -------------------------------------------------------------------------------------
Video ref Flow rate inbound/ Bifurcation Description of results
outbound \[ml/s\]
v3.0.0R 1.36 / 1.36 right Large amount of particles flow to 100% stenosis in right branch
v3.1.0R 2.72 / 2.72 right Particles flow through left branch. Only a small amount end up in the 100% stenosis
----------- -------------------- ------------- -------------------------------------------------------------------------------------
: Videos results for 100% stenosis measurements for flow velocities and .[]{data-label="tab:Exp3Stenosis100"}
Magnetic Particle Imaging
-------------------------
Imaging experiments without force application are performed using the most promising particles to evaluate their imaging capabilities. For these experiments, we show the reconstructed images and calculate the spatial resolution in terms of the full width at half maximum (FWHM) of the point-shaped samples in the reconstructed image. A reconstructed image in the $xz$-plane of a static delta sample filled with nanomag/synomag-D 333 is shown in Fig. \[fig:DeltaSample\](a). In order to determine the FWHM of the sample, profiles in the $x$- and $z$-directions are generated, as shown in Fig. \[fig:DeltaSample\](a) (above and left of the image). The FWHM in the $x$-direction is and in the $z$-direction is indicating a better spatial resolution in the $z$-direction compared to the $x$-direction. For the beads, the reconstructed image in the $xz$-plane is given in Fig. \[fig:DeltaSample\](b) and their profiles in the $x$-direction and $z$-direction (above and left of the image) result in a FWHM of and . Because the gradient strength in the $x$-direction and $y$-direction and DF-amplitudes are the same, the spatial resolution in the $y$-direction is assumed to be the same as in the $x$-direction.
(image2) at (0,0) [![(a) Image of a delta sample filled with beads in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left). b) Image of delta sample containing nanomag/synomag-D 333 in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left).[]{data-label="fig:DeltaSample"}](ImgDeltaSampleBeads_xz.png "fig:"){width="0.5\linewidth"}]{}; (profz) [![(a) Image of a delta sample filled with beads in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left). b) Image of delta sample containing nanomag/synomag-D 333 in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left).[]{data-label="fig:DeltaSample"}](DeltaSampleBeadsZ.pdf "fig:"){width="0.35\linewidth"}]{}; ; (-2.2,0.1)–(2.2,0.1); (0.1,-1.25)–(0.1,1.25);
a)
(image1) at (0,0) [![(a) Image of a delta sample filled with beads in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left). b) Image of delta sample containing nanomag/synomag-D 333 in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left).[]{data-label="fig:DeltaSample"}](ImgDeltaSamplesynomag_xz.png "fig:"){width="0.5\linewidth"}]{}; (profz) [![(a) Image of a delta sample filled with beads in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left). b) Image of delta sample containing nanomag/synomag-D 333 in the $xz$-plane with profile in the $x$-direction (above) and profile in the $z$-direction (left).[]{data-label="fig:DeltaSample"}](DeltaSamplesynomagZ.pdf "fig:"){width="0.35\linewidth"}]{}; ; (-2.2,-0.4)–(2.2,-0.4); (0.1,-1.25)–(0.1,1.25);
b)
Magnetic Particle Imaging and Navigation in Flow
------------------------------------------------
In this bifurcation experiment with 100% stenosis at velocity the imaging and navigation mode of MPI is successfully used to navigate the particles towards the stenosis while the distribution of the particles is captured with the imaging mode. With a ratio between $\zeta=20$ and $\varphi=1$ for force and imaging mode, the induced magnetic force acts sufficiently long enough to maneuver the particles towards the stenosis, although no force is acting on the particles during the short time of the imaging mode.
at (0,0) [![(a) Most of the particles are moved into the right side towards the 100% stenosis at flow velocity while the MPI imaging mode shows the distribution of the particles. (b) In the control measurement afterwards where no forces, where applied all particles flow through the left side. []{data-label="fig:ResultExp3MPIFStenosis100"}](FlowMPIFimage-1108.png "fig:"){width="1.0\linewidth"}]{}; (-4.5,1) – (-3.5,1) node\[midway,below\][Force $F_y$]{}; (-1.5,0.8) node\[above\][Stenosis]{} – (-1.4,-0.2); (-1.0/2,-3) – (-1.0/2,0) node\[midway,left\][x]{}; (-1.0/2,-3) – (-1.0/4,-3) node\[midway,below\][y]{};
\(a) 100% stenosis, flow rate , MPI image
at (0,-0.05) [![(a) Most of the particles are moved into the right side towards the 100% stenosis at flow velocity while the MPI imaging mode shows the distribution of the particles. (b) In the control measurement afterwards where no forces, where applied all particles flow through the left side. []{data-label="fig:ResultExp3MPIFStenosis100"}](FlowBifuStenosis100Velo0Controlimage-100.png "fig:"){width="1.0\linewidth"}]{}; at (-1.25,1) [$F_y$=0]{}; (2.5,0.8) node\[above\][Stenosis]{} – (2.4,-0.2); (-1.0/2,-3) – (-1.0/2,0) node\[midway,left\][x]{}; (-1.0/2,-3) – (0.0,-3) node\[midway,below\][y]{};
\(b) $F_y$=0, 100% stenosis, flow rate
The particles are driven towards the right branch of the bifurcation, while the intensities within the MPI image increase as the particles enter the right branch and are pushed to the stenosis, as seen in Fig. \[fig:ResultExp3MPIFStenosis100\](a). In the control measurement afterwards, shown in Fig. \[fig:ResultExp3MPIFStenosis100\](b), the particles are flowing completely through the left branch of the bifurcation. The references of the videos can be found in Table \[tab:Exp3Stenosis100MPI\].
----------- -------------------- ------------- -----------------------------------------------------------------
Video ref Flow rate inbound/ Bifurcation Description of results
outbound \[ml/s\]
v4.0.0R 1.36 / 1.36 right Large amount of particles flow to 100% stenosis in right branch
v4.0.1R 1.36 / 1.36 right Control: With no force, all particles flow through left branch
----------- -------------------- ------------- -----------------------------------------------------------------
: Video results for 100% stenosis measurements for flow velocities while taking snapshots within imaging mode.[]{data-label="tab:Exp3Stenosis100MPI"}
Discussion
==========
The results from the MPS measurements show that nanomag/synomag-D 333 particles provide the best compromise between imaging capabilities and magnetophoretic mobility of all investigated nanomag/synomag-D particles. The spectrum of nanomag/synomag-D 333 particles indicates a sufficient MPI imaging performance with about 35 harmonics above the noise level. The performance of the Dynabeads MyOne particles is slightly inferior but they generate sufficient harmonics for MPI imaging. The imaging characteristics, in terms of the spectra of both particle types, are not influenced by magnetizing or vertexing them. Thus, both particle types do not show a remaining remanence. For the nanomag/synomag-D 333 particles the intensities of the first five frequencies show a strong linearity dependent on the particle concentration. The regression lines have coefficients of determination of $R=0.996$ to $R=0.997$. This strong linearity is inevitable for MPI applications since the reconstruction principle requires linearity. This criterion is also met by the DynaBeads MyOne particles where intensities of the frequency signals indicate a linear tendency to the iron concentration, but their coefficients of determination for the regression lines are smaller at $R=0.971$ to $R=0.986$. The similar half separation times of the DynaBeads and nanomag/synomag-D batches (333,334) particles indicate an equal magnetophoretic mobility for both particle types. The nanomag/synomag-D 333 batch is therefore suitable for magnetic navigation and has shown the most promising imaging characteristics for MPI by generating the largest number of harmonics above the noise level. Additionally, the investigated polystyrene Dynabeads MyOne particles are not biodegradable, thus are not suitable for human use. The nanomag/synomag-D particles do have a biocompatible coating layer and are therefore much more suited to being resolved by the liver. Due to the better performance in imaging characteristics, similar half time separation and biocompatibility, the nanomag/synomag-D 333 particles are used for further MPN bifurcation flow and MPIN flow bifurcation experiments.
The results of the Magnetic Particle Navigation experiments demonstrate that nanomag/synomag-D 333 particles can be actuated to one side of the bifurcation junction within a liquid medium flowing with a velocity of up to (). The rectangular cross-section $A_{\text{exp}}=(\SI{3.544}{\milli\meter})^2=\SI{12.5}{\square\milli\meter}$ of the tube corresponds to a cross-section of circular vessel tube with diameter of . In the literature [@ford_characterization_2005] the flow rate () within the internal carotid artery with circular diameter of about [@krejza_carotid_2006] is stated. Thus, it is promising to investigate Magnetic Particle Navigation with MPI under realistic conditions within blood circulation.
The effect of the magnetic navigation could be slightly stronger in a real heart since the peristaltic pump generates a constant volume flow whereas the heart pumps with a frequency of about . During diastole the flow volume is noticeably lower, which would give the magnetic force more time to move the particles in the desired direction. With this in mind, it might be possible to move particles from the aorta into the common carotid artery.
For flow velocity () the results show that it is even possible to stop particles by pressing them against the side wall of the phantom. The friction at the side wall and the reduced flow velocity profile close to the side wall are probably responsible for this effect. These findings indicate a promising prospect for targeted drug delivery applications in small arteries where the effect of the drug could be enhanced. The effect could be further investigated by trying to stop particles explicitly with the magnetic force against the flow direction.
Furthermore, the flow bifurcation experiment with a stenosis of 60% show that the nanomag/synomag-D 333 particles can be pushed to one side of a bifurcation junction up to a flow velocity of (). The 60% stenosis increases the flow resistance in its bifurcation branch and a larger amount of the flow volume prefers to flow through the clear side. These circumstances make it more challenging to move the particles through the stenosis. If the stenosis in one branch is a full blockage of 100%, it is still possible to magnetically force the particles to the stenosis at a maximal flow velocity of (). With these findings it might be possible to resolve a stenosis caused by blood clots in a medium-sized artery by using drug-loaded particles to clear blockages. Under these conditions a drug normally injected intravenously would never reach the stenosis. Thus, even a small amount of particles loaded with tissue plasminogen activator (tPA) reaching the blood clot at the stenosis would be highly benefical.
Both particle types show that their MPI signal is sufficient enough to reconstruct an image. But the spatial resolutions, in terms of FWHM in the $x$-direction at and in the $z$-direction for Dynabeads MyOne particles, are not as good as dedicated imaging particles. Rahmer et al.[@rahmer_analysis_2012] achieved spatial resolutions in the sub-millimeter range. For nanomag/synomag-D 333 particles the spatial resolutions in terms of FWHM in the $x$-direction at and in the $z$-direction, are similar to the Dynabeads MyOne particles.
Finally, the results of the Magnetic Particle Imaging and Navigation experiments demonstrate that it is possible to control and image particles through a bifurcation at a flow velocity of () quasi-simultaneously. The particles are navigated through a bifurcation towards the stenosis branch even when 100% blocked.
As seen in video v4.0.0R, it is challenging to maneuver the particles to towards the 100% stenosis since the total blockage creates a high resistance. At the boundary surface of the bifurcation turbulances occur and the negative pressure caused by the flowing liquid pulls the particles out of the stenosis back into the flowing current. With a ratio of 20 to 1 between navigation and imaging, an imaging rate of per image can be achieved to identify the position of the particles.
In terms of improvement, the limiting flow velocity of () could be increased by stronger focus fields to induce a greater force on the particles. Such an adaption is not easily possible with the commerical system used in this work. But with a custom built human scanner[@Rahmer2017SciRob; @graeser_human-sized_2019] it would be feasible to introduce stronger additional focus fields for human brain applications with MPIN.
Conclusion
==========
In this work, we have determined that the nanomag/synomag-D 333 particles provide the best compromise between magnetic manipulability and imaging performance for MPI. With these particles we have further demonstrated the feasibility to maneuver particles within a volume flow of () towards one side of a bifurcation by using the MPN method of an MPI scanner. Additionally, the MPIN method has been successfully used to navigate particles towards a 100% stenosis within a bifurcation, while imaging the particles’ distribution in the stenosis every at a flow velocity of (). In the future, magnetic particles combined with a tissue plasminogen activator (tPA) might be used to resolve blood clots in hard-to-reach positions while using MPIN to monitor the liquidation of the stenosis.
Acknowledgements
================
F.G., N.G., F.T. and T.K. thankfully acknowledge the financial support of the German Research Foundation (DFG, grant number KN 1108/2-1) and the Federal Ministry of Education and Research (BMBF, grant number 05M16GKA). This work was also supported by the BMBF under the frame of EuroNanoMed III (grant number: 13XP5060B, T.K, P.L.).
|
---
abstract: 'Planar channeling of 855 MeV electrons and positrons in straight and bent tungsten (110) crystal is simulated by means of the [<span style="font-variant:small-caps;">MBN Explorer</span>]{}software package. The results of simulations for a broad range of bending radii are analyzed in terms of the channel acceptance, dechanneling length, and spectral distribution of the emitted radiation. Comparison of the results with predictions of other theories as well as with the data for (110) oriented diamond, silicon and germanium crystals is carried out.'
author:
- 'H. Shen'
- 'Q. Zhao'
- 'F.S. Zhang [^1]'
- 'Gennady B. Sushko'
- 'Andrei V. Korol [^2]'
- 'Andrey V. Solov’yov [^3]'
title: ' Channeling and Radiation of 855 MeV Electrons and Positrons in Straight and Bent Tungsten (110) Crystals'
---
Introduction \[Introduction\]
=============================
The basic effect of the channeling process in a straight crystal is in an anomalously large distance which a charged projectile can penetrate moving along a crystallographic direction (either planar or axial) experiencing collective action of the electrostatic field of the crystal atoms [@Lindhard1965]. The channeling can also occur in a bent crystal provided the bending radius $R$ is large enough in comparison with the critical one $R_{\rm c}$ [@Tsyganov1976].
A particle trapped into the channel oscillates in the transverse direction while propagating along a crystallographic direction. The channeling oscillations give rise to a specific type of radiation, – the channeling radiation (ChR) [@ChRad:Kumakhov1976]. Its intensity depends on the type and the energy of a projectile as well as on the type of a crystal and a crystallographic direction. The emission of ChR by an ultra-relativistic projectile in a straight crystal is well studied (see, for example, Refs. [@Andersen_ChanRadReview_1983; @BakEtal1985; @Baier; @BazylevZhevago; @RelCha; @Kumakhov2; @Uggerhoj1993; @Uggerhoj_RPM2005] and references therein).
The motion of a channeling particle in a bent crystal contains two components: the channeling oscillations and circular motion along the bent centerline. The latter motion gives rise to the synchrotron-type radiation (SR) [@Jackson]. Therefore, the total spectrum of radiation formed by an [ultra-relativistic]{} projectile in a bent crystal bears the features of both the ChR and SR. Various aspects of radiation formed in crystals bent with the constant radius $R$ were discussed in Refs. [@ArutyunovEtAl_NP_1991; @Bashmakov1981; @KaplinVorobev1978; @SolovyovSchaeferGreiner1996; @Taratin_Review_1998; @TaratinVorobiev1988; @TaratinVorobiev1989] although with various degree of detail and numerical analysis. A quantitative analysis of the emission spectra based on accurate simulation of the channeling process was carried out for silicon and diamond crystals [@BentSilicon_2014; @Sushko_EtAl_NIMB_v355_p39_2015; @Polozkov_VKI_Sushko_AK_AS_SPB_Diamond_2015].
The condition of stable channeling in a bent crystal, $R \gg R_{\mathrm{c}}$, [@Tsyganov1976] implies that the bending radius exceeds greatly the (typical) curvature radius of the channeling oscillations. Therefore, the SR modifies mainly the soft-photon part of the spectrum. This part of the spectrum is especially interesting in connection with the concept of a crystalline undulator (CU) which implies propagation of ultra-relativistic projectiles along periodically bent crystallographic planes [@KSG1998; @KSG_review_1999]. By means of CU it is feasible to produce undulator-like radiation in the hundreds of keV up to the MeV photon energy range. The intensity and characteristic frequencies of the radiation can be varied by changing the type of channeling particles, the beam energy, the crystal type and the parameters of periodic bending (see recent review [@ChannelingBook2014] for more details). The range of bending amplitudes, $a$, and period, $\lambda_{\rm u}$, within which the operation of CU is feasible for projectiles of the energies ${{\varepsilon}}\simeq 10^{-1}-10^1$ GeV are: $a\simeq10^0-10^1$ Å, $\lambda_{\rm u} \simeq 10^0-10^2$ $\mu$m. Even more exciting (although much more challenging and distant) is the possibility of generating a stimulated emission of the free-electron laser (FEL) type by means of CU [@KSG_review_1999; @ChannelingBook2014]. This novel source of light can generate, at least in theory, the stimulated emission in the photon energy range $10^2-10^3$ keV (i.e. hard X and gamma range) which are not achievable in conventional FELs.
Up to now, several experiments have been (and planned to be) carried out to measure the channeling parameters and the radiation emitted by electrons and positrons in periodically bent crystalline structures prepared by several different technologies [@ChannelingBook2014]. The recent attempts include experiments with 195–855 MeV electron beam at the Mainz Microtron (MAMI) facility [@Backe_EtAl_2008; @Backe_EtAl_2011; @Backe_EtAl_2013; @Backe_EtAl_PRL2014; @BackeLauth_Dyson2016; @BackeLauth:NIMB-v355-p24-2015] carried out with CUs manufactured in Aarhus University (Denmark) using the molecular beam epitaxy technology to produce strained-layer Si$_{1-x}$Ge$_{x}$ superlattices with varying germanium content [@MikkelsenUggerhoj2000]. The CUs of this type are planned to be used in the coming experiments at SLAC with 10-35 GeV electron and positron FACET beam [@Uggerhoj2016]. A set of experiments was performed with few GeV positrons at CERN [@Connell_Dyson2016] with the CU based on synthetic diamond doped with boron [@DiamondBoron]. Other related experiments include investigations of the radiation of sub-GeV electrons in a bent silicon crystal [@Backe_EtAl_PRL_115_025504_2015] and of the effectiveness of deflection of multi-GeV electrons by a thin Si crystal [@Wienands_EtAl_PRL_v114_074801_2015].
The technologies, available currently for preparing periodically bent crystals, do not immediately allow for lowering the values of bending period down to tens of microns range or even smaller keeping, simultaneously, the bending amplitude in the range of several angstroms. These ranges of $a$ and $\lambda_{\rm u}$ are most favourable to achieve high intensity of radiation in a CU [@ChannelingBook2014]. One of the potential options to lower the bending period is related to using crystals heavier than diamond ($Z=6$) and silicon ($Z=14$) to propagate ultra-relativistic electrons and positrons. In heavier crystals, both the depth, $\Delta U\propto Z^{2/3}$, of the interplanar potential and its the maximum gradient, $U^{\prime}_{\max}\propto Z^{2/3}$, attain larger values, resulting in the enhancement of the critical channeling angle $\Theta_{\rm L}\propto (\Delta U)^{1/2}$ [@Lindhard1965] and reduction of the critical radius $R_{\rm c} \propto 1/U^{\prime}_{\max}$ [@Tsyganov1976].
From this end, the tungsten crystal ($Z=74$) is a good candidate for the study. This crystal was used in channeling experiments with both heavy [@Kovalenko_EtAl:JINR_RC-v72-p9-1995; @BiryukovChesnokovKotovBook] and light [@Backe_EtAl:SPIE-6634-2007; @Yoshida_EtAl:PRL_v80_p1437_1998] ultra-relativistic projectiles. In Ref. [@Kovalenko_EtAl:JINR_RC-v72-p9-1995] it was noted that the straight tungsten crystals show high structure perfection. This feature is also of a great importance for successful experimental realization of the CU idea [@Imperfectness2008]. In the cited paper the comparison was carried out of the (110) planar channels in tungsten vs. silicon. In particular, the critical radius in W(110) was estimated as $R_{\mathrm{c}}=0.16$ cm which is seven times smaller than that in Si(110). This allows one, at least in theory, to consider periodic bending with $\lambda_{\rm u}\lesssim 10$ $\mu$m. Indeed, matching the maximum curvature of periodic bending $4\pi^2a/\lambda_{\rm u}^2$ to $R_{\mathrm{c}}^{-1}$ one estimates $\lambda_{\rm u} \sim 2.5$ $\mu$m for $a=1$ Å.
Therefore, it is desirable to carry out a quantitative analysis of both the channeling process and the emission of radiation of high-energy [*light*]{} projectiles in straight and bent tungsten crystal. To the best of our knowledge, the simulations of these processes have been restricted to the the straight oriented crystal and were carried out within model approaches. In Refs. [@Azadegan_EtAl:JPConfSer-v517-p012039-2014; @AzadeganWagner:NIMB-v517-p012039-2014] the continuous potential model was applied to construct the trajectories, whereas the dechanneling phenomenon was considered within the framework of the Fokker-Plank equation. The model of binary collisions was exploited in Ref. [@Efremov_EtAl:RussPhysJ-v50-p1237-2007] to investigate the scattering angle of 0.5 GeV electrons and positrons in the process of axial channeling. No results have been presented for bent tungsten crystal.
In this paper we present the results of simulation of the channeling process and of the radiation emission for ${{\varepsilon}}=855$ MeV electrons and positrons in oriented W(110) crystal. The calculations were performed for both straight and bent crystals. In the latter case the bending radius was varied down to 0.04 cm. The projectile energy chosen corresponds to that used in the ongoing experiments with bent crystals and CUs at MAMI [@BackeLauth_Dyson2016; @Backe_EtAl_PRL2014; @BackeLauth:NIMB-v355-p24-2015] carried out with the electron beam. However, from the view point of future experiments it is instructive to present a comparative analysis of the electron vs. positron channeling. Therefore, both light projectiles are considered in the paper.
As in our recent studies, three-dimensional simulations of the propagation of ultra-relativistic projectiles through the crystal were performed by using the [<span style="font-variant:small-caps;">MBN Explorer</span>]{}package [@MBN_ExplorerPaper; @MBN_ExplorerSite]. The package was originally developed as a universal computer program to allow investigation of structure and dynamics of molecular systems of different origin on spatial scales ranging from nanometers and beyond. The general and universal design of the [<span style="font-variant:small-caps;">MBN Explorer</span>]{}code made it possible to expand its basic functionality with introducing a module that treats classical relativistic equations of motion and generates the crystalline environment dynamically in the course of particle propagation [@ChanModuleMBN_2013]. A variety of interatomic potentials implemented in [<span style="font-variant:small-caps;">MBN Explorer</span>]{}support rigorous simulations of various media. The software package can be regarded as a powerful numerical tool to uncover the dynamics of relativistic projectiles in crystals, amorphous bodies, as well as in biological environments. Its efficiency and reliability has already been benchmarked for the channeling of ultra-relativistic projectiles (within the sub-GeV to tens of GeV energy range) in straight, bent and periodically bent crystals [@ChanModuleMBN_2013; @BentSilicon_2013; @Sub_GeV_2013; @BentSilicon_2014; @SushkoThesis2015; @Multi_GeV_2014; @Sushko_AK_AS_SPB_SASP_2015; @Sushko_EtAl_NIMB_v355_p39_2015; @Si110-SASP-855MeV_2016]. In these papers verification of the code against available experimental data and predictions of other theoretical models was carried out.
The description of the simulation procedure is sketched in Sect. \[Methodology\]. The results of calculations are presented and discussed in Sect. \[Results\].
Methodology \[Methodology\]
===========================
Propagation of an ultra-relativistic projectile of the charge $q$ and mass $m$ through a crystalline medium can be described in terms of classical relativistic dynamics. This framework implies integration of the following two coupled equations of motion: $$\begin{aligned}
\partial {{\bf r}}/ \partial t = {{\bf v}},
\qquad
\partial {{\bf p}}/ \partial t = - q \, \partial U/\partial {{\bf r}}\label{Methodology:eq.01} \end{aligned}$$ where $U=U({{\bf r}})$ is the electrostatic potential due to the crystal constituents, ${{\bf r}}(t), {{\bf v}}(t)$, and ${{\bf p}}(t) = m\gamma{{\bf v}}(t)$ stand, respectively, for the coordinate, velocity and momentum of the particle at instant $t$, $\gamma = \left(1-v^2/c^2\right)^{-1/2} = {{\varepsilon}}/mc^2$ is the relativistic Lorentz factor, ${{\varepsilon}}$ is the particle’s energy, and $c$ is the speed of light.
In [<span style="font-variant:small-caps;">MBN Explorer</span>]{}, the differential equations (\[Methodology:eq.01\]) are integrated using the forth-order Runge-Kutta scheme with variable time step. At each integration step, the potential $U=U({\bf r})$ is calculated as the sum of atomic potentials $U_{\mathrm{at}}$ due to the atoms located inside the sphere of the cut-off radius $\rho_{\max}$ with the center at the instant location of the projectile. The value $\rho_{\max}$ is chosen large enough to ensure negligible contribution to the sum from the distant atoms located at $r>\rho_{\max}$. The search for such atoms is facilitated by using the linked cell algorithm implemented in [<span style="font-variant:small-caps;">MBN Explorer</span>]{}[@MBN_ExplorerPaper]. The algorithm implies (i) a subdivision of the sample into cubic cells of a smaller size, and (ii) an assignment of each atom to a certain cell. As a result, the total number of computational operations is reduced considerably.
To simulate the motion along a particular crystallographic plane with the Miller indices $(klm)$ the following algorithm is used [@ChanModuleMBN_2013].
As a first step, a crystalline lattice is generated inside the rectangular simulation box of the size $L_x\times L_y \times L_z$. The $z$-axis is oriented along the beam direction and is parallel to the $(klm)$ plane, the $y$ axis is perpendicular to the plane. The position vectors of the nodes ${{\bf R}}_j^{(0)}$ ($j=1,2,\dots, N$) within the simulation box are generated in accordance with the type of the Bravais cell of the crystal and using the pre-defined values of the lattice vectors.
[<span style="font-variant:small-caps;">MBN Explorer</span>]{}contains several build-in options which allow further modification of generated crystalline structures. The options relevant to modeling linear and bent crystals are as follows [@SushkoThesis2015].
- *Rotation* of the sample around a specified axis. This option allows one to simulate the crystalline structure oriented along any desired crystallographic direction. In particular, this option allows one to choose the direction of the $z$-axis well away from major crystallographic axes, thus avoiding the axial channeling (when not desired).
- Displacement of the nodes in the transverse direction $y$ with respect to a specified $z$-axis: $y \to y + R(1-\cos\phi)$ where $\phi = \sin(z/R)$. As a result, one obtains the crystalline structure *bent with constant radius* $R$ in the $(yz)$ plane. For values of $R$ much larger than the crystal thickness $L$ (along the $z$ direction), the displacement acquires the form: $$y \to y + {z^2 \over 2R} \,.
\label{Methodology:eq.02}$$
- The nodes determine positions of the atoms in an ideal crystal. More realistic structure includes the probability of the atoms to be displaced from their equilibrium positions (the nodes) due to *thermal vibrations* corresponding to the given temperature $T$. Thus, for each atom, the Cartesian components of the displacement are selected randomly by means of the normal distribution corresponding to fixed root-mean-square amplitude $u_{T}$ of thermal vibrations. The values of $u_T$ for a number of crystals are summarized in [@Gemmel]. For tungsten one finds $u_T=0.05$ Å.
Let us note that by introducing unrealistically large value of $u_T$ (for example, exceeding the lattice constants) it is possible to consider large random displacements. As a result, the *amorphous medium* can be generated.
- *Periodic harmonic displacement of the nodes* can be performed by means of the transformation ${{\bf r}}\to {{\bf r}}+ {{\bf a}}\sin({{\bf k}}\cdot{{\bf r}}+\varphi)$. Here, the vector ${{\bf a}}$ and its modulus, $a$, determine the direction and the amplitude of the displacement, the wave-vector ${{\bf k}}$ determines the axis along which the displacement to be propagated, and $\lambda_{\rm }=2\pi/k$ defines the wave-length of the periodic bending. The parameter $\varphi$ allows one to change the phase-shift of the harmonic bending.
In a special case ${{\bf a}}\perp {{\bf k}}$, this options provides simulation of linearly polarized periodically bent crystalline structure which is an important element of a crystalline undulator.
In addition to the aforementioned options, [<span style="font-variant:small-caps;">MBN Explorer</span>]{}allows one to model periodically bent crystalline structures by generating periodic (harmonic) displacement of the nodes, to construct binary structures (for example, Si$_{1-x}$Ge$_x$ superlattices) by introducing random or regular substitution of atoms in the initial structure with atoms of another type. Also, the simulation box can be cut along specified faces, thus allowing tailoring the generated crystalline sample to achieve the desired form of the sample.
Trajectory of a particle entering the initially constructed crystal at the instant $t=0$ is calculated by integrating equations (\[Methodology:eq.01\]). Initial transverse coordinates, $(x_0, y_0)$, and velocities, $(v_{x,0}, v_{y,0})$, are generated randomly accounting for the conditions at the crystal entrance (i.e., the crystal orientation, beam emittance and energy distribution of the particles). A particular feature of [<span style="font-variant:small-caps;">MBN Explorer</span>]{}is in simulating the crystalline environment “on the fly”, i.e. in the course of propagating the projectile. This is achieved by introducing a dynamic simulation box which shifts following the particle (see Ref. [@ChanModuleMBN_2013] for the details).
Taking into account randomness in sampling the incoming projectiles and in positions of the lattice atoms due to the thermal fluctuations, one concludes that each simulated trajectory corresponds to a unique crystalline environment. Thus, all simulated trajectories are statistically independent and can be analyzed further to quantify the channeling process as well as the emitted radiation.
The averaged spectral distribution of the energy emitted within the cone $\theta\leq \theta_{0}$ with respect to the incident beam (i.e. along the $z$ axis) is computed as follows $$\begin{aligned}
\left\langle{{{\rm d}}E(\theta\leq\theta_{0}) \over {{\rm d}}{{\omega}}} \right\rangle
=
{1 \over N_0}
\sum_{n=1}^{N_0}
\int\limits_{0}^{2\pi}
{{\rm d}}\phi
\int\limits_{0}^{\theta_{0}}
\theta {{\rm d}}\theta\,
{{{\rm d}}^2 E_n \over {{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}}.
\label{Methodology:eq.03}\end{aligned}$$ Here, ${{\omega}}$ stands for the frequency of radiation, $\Omega$ is the solid angle corresponding to the emission angles $\theta$ and $\phi$. The sum is carried over the simulated trajectories of the total number $N_0$, and ${{\rm d}}^2 E_n/{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}$ is the energy per unit frequency and unit solid angle emitted by the projectile moving along the $n$th trajectory. The resulting spectrum accounts for all mechanisms of the radiation formation: (a) channeling radiation (ChR) due to the channeling segments, (b) coherent and incoherent bremsstrahlung (BrS) due to the over-barrier motion. In addition to these, the motion along the arc in a bent crystal results in the synchrotron-type radiation.
The numerical procedures implemented in [<span style="font-variant:small-caps;">MBN Explorer</span>]{}to calculate the distributions ${{\rm d}}^2 E_n/{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}$ [@ChanModuleMBN_2013] are based on the quasi-classical formalism due to Baier and Katkov [@Baier]. A remarkable feature of this method is that it combines classical description of the motion in an external field with the quantum corrections due to the radiative recoil quantified by the ratio $\hbar{{\omega}}/{{\varepsilon}}$. In the limit $\hbar {{\omega}}/{{\varepsilon}}\ll 1$ one can use the classical description of the radiative process which is adequate to describe the emission spectra by electrons and positrons of the sub-GeV energy range (see, for example, [@ChannelingBook2014] and references therein). The corrections lead to strong modifications of the radiation spectra of multi-GeV projectiles channeling in crystalline undulators [@Multi_GeV_2014; @Sushko_AK_AS_SPB_SASP_2015] and in bent crystals [@Sushko_EtAl_NIMB_v355_p39_2015].
Results and Discussion \[Results\]
==================================
Using the algorithm outlined above, classical trajectories were simulated for ${{\varepsilon}}=855$ MeV electrons and positrons incident along the (110) crystallographic plane in straight and bent tungsten crystals of the thickness $L_1=75$ $\mu$m along the incident beam direction. In a bent crystal, the channeling condition [@Tsyganov1976] implies that the centrifugal force $F_{\rm cf}= pv/R \approx {{\varepsilon}}/R$ is smaller than the maximum interplanar force $F_{\max}$. It is convenient to quantify this statement by introducing the dimensionless bending parameter $C$: $$C = {F_{\rm cf} \over F_{\max}} = {{{\varepsilon}}\over R F_{\max}} = {R_{\rm c} \over R}.
\label{Results:eq.01}$$ The case $C = 0$ ($R=\infty$) characterizes the straight crystal whereas $C = 1$ corresponds to Tsyganov’s critical (minimum) radius $R_{\rm c}={{\varepsilon}}/F_{\max}$ [@Tsyganov1976]. Within the framework of the continuous interplanar potential model [@Lindhard1965], one calculates $F_{\max}=42.9$ GeV/cm for W(110) at room temperature by means of the formula for the continuous potential derived in Ref. [@Erginsoy_PRL_v15_360_1965; @AppletonEtAl_PR_v161_330_1967] based on the Molière approximation for the atomic potential [@Moliere]. Hence, $R_{\rm c}\approx0.02$ cm for a 855 MeV projectile.
In the calculations presented below the bending radius was varied from 2 down to 0.0432 cm. For each type of the projectiles and for each value of bending radius (including the case of the straight crystal), the numbers $N_0$ of the simulated trajectories were sufficiently large (approximately $5000$), thus enabling a reliable statistical quantification of the channeling process. In Sect. \[Trajectories\] below we define and describe the quantities obtained. The emission spectra are discussed in Sect. \[Spectra\].
Statistical Analysis of Trajectories \[Trajectories\]
-----------------------------------------------------
Fig. \[Figure.01\] shows simulated trajectories of 855 MeV electrons (two black curves labeled “${{\rm e}}_1$” and “${{\rm e}}_2$”) and of positrons (two green curves “${\rm p}_1$” and “${\rm p}_2$”) propagating in straight W(110) crystal. These selected trajectories as well as other notations and features presented in the figure we use in the text below as a reference material when explaining various quantitative and qualitative characteristics of the particles motion.
![ Selected simulated trajectories of 855 MeV electrons (curves “${{\rm e}}_1$” and “${{\rm e}}_2$”) and of positrons (curves “${\rm p}_1$” and “${\rm p}_2$”) propagating in straight W(110). The trajectories illustrate the channeling and the over-barrier motion as well as the dechanneling and rechanneling effects. The $z$-axis of the reference frame is directed along the incoming projectiles, the $(xz)$-plane is parallel to the (110) crystallographic planes (dashed lines) and the $y$-axis is perpendicular to the planes. The W(110) interplanar distance is $d=2.238$ Å. In the electron case, presented are the accepted (“${{\rm e}}_1$”) and non-accepted (“${{\rm e}}_2$”) trajectories. For the accepted trajectory the characteristic lengths of the channeling motion are indicated: the initial channeling segment, $z_{{\rm ch}0}$, and the segments in the bulk, $z_{{\rm ch}1}, \dots, z_{{\rm ch}6}$. The non-accepted trajectory corresponds to $z_{{\rm ch}0} = 0$. The positron trajectories both refer to the accepted type. []{data-label="Figure.01"}](figure01.eps){width="13.0cm"}
To start with, we explain the geometrical parameters used in the simulations. Dashed horizontal lines in Fig. \[Figure.01\] mark the cross section of the (110) crystallographic planes separated by the distance $d=2.238$ Å. Thus, the $y$-axis is aligned with the $\langle 110 \rangle$ crystallographic axis. The horizontal $z$-axis corresponds to the direction of the incoming beam which was considered ideally collimated in the current simulations. To avoid the axial channeling, the $z$-axis was chosen along the $[10, -10, 1]$ crystallographic direction. A projectile enters the crystal at $z=0$ and exits at $z=L$. The crystal is considered infinitely large in the $x$ and $y$ directions. In the simulations, the integration of the equations of motion (\[Methodology:eq.01\]) produces a 3D trajectory. The black and green curves in the figure represent the projections of the corresponding 3D trajectories on the $(yz)$ plane.
The trajectories shown in Fig. \[Figure.01\] illustrate a variety of features which characterize the motion of a charged projectile in an oriented crystal: the channeling mode, the over-barrier motion, the dechanneling and the rechanneling processes, rare events of hard collisions etc.
The channeling motion is more pronounced and regular for positrons than for electrons. Positively charged projectiles tend to move in between the atomic planes, i.e. in the region with lower volume density of the crystal constituents. As a result, they tend to stay in the channeling mode much longer than negatively charged particles, which move in the vicinity of the atomic chains being attracted by positively charged nuclei. On the other hand, although the electrons have higher rate of the dechanneling (i.e., leaving the channeling mode of motion due to the collisions), the inverse phenomenon, the re-channeling, is also more frequent for them [@Kostyuk-AK-AS-WG_855-Si]. Therefore, as it is seen in the figure, quite often the electron trajectory consists of several channeling segments separated by the intervals of the over-barrier motion. The nearly harmonic pattern of the channeling oscillations for positrons as well as strong anharmonicity in the electron channeling oscillations are well-known phenomena (see, e.g., [@BakEtal1985]). These effects can be explained qualitatively within the framework of the continuous potential model by comparing the profiles of the interplanar potentials for the two types of projectiles (nearly harmonic for positrons and extremely non-harmonic for electrons).
Apart from providing the possibility of illustrative comparison, the simulated trajectories allow one to quantify the channeling process in terms of several parameters and functional dependencies which can be generated on the basis of statistical analysis of the trajectories [@ChanModuleMBN_2013; @ChannelingBook2014; @BentSilicon_2013; @Sub_GeV_2013; @BentSilicon_2014; @Sushko_EtAl_NIMB_v355_p39_2015; @Sushko_AK_AS_SPB_SASP_2015; @Si110-SASP-855MeV_2016].
A randomization of the “entrance conditions” for the projectiles, as explained in Sect. \[Methodology\], leads to different chain of scattering events the different projectiles at the entrance to the bulk. As a result, not all the simulated trajectories start with the channeling segments. In Fig. \[Figure.01\], trajectory “${{\rm e}}_2$” refers to the non-accepted projectile while three other trajectories correspond to the accepted ones. A commonly used parameter to quantify this feature is [*acceptance*]{} defined as the ratio ${{\cal A}}= N_{\rm acc}/N_0$ of the number $N_{\rm acc}$ of particles captured into the channeling mode at the entrance of the crystal (the accepted particles) to the total number $N_0$ of the incident particles. The non-accepted particles experience unrestricted over-barrier motion at the entrance but can rechannel somewhere in the bulk.
![ Acceptance versus bending parameter $C$ (see Eq. (\[Results:eq.01\])) for 855 MeV electrons (open symbols) and positrons (filled symbols). The results of current simulations for W(110) are compared with the data generated previously by means of [<span style="font-variant:small-caps;">MBN Explorer</span>]{}: diamond(110) from Ref. [@Sushko_AK_AS_SPB_SASP_2015], Si(110) from Refs. [@BentSilicon_2014; @BentSilicon_2013], and Ge(110) from Ref. [@SushkoThesis2015]. []{data-label="Figure.02"}](figure02.eps){width="13.0cm"}
The acceptance as a function of the bending parameter $C$ is presented in Fig. \[Figure.02\]. The results of the current set of simulations for W(110) are plotted together with the data obtained earlier for other oriented crystals, as indicated in the caption. For each crystal, the corresponding values of bending radius $R$ are calculated from Eq. (\[Results:eq.01\]) using the following values of the (110) interplanar field $F_{\max}$ at room temerature: 7.0, 5.7, 10.0, and 42.9 GeV/cm for diamond, silicon, germanium, and tungsten, respectively. To be noted is (i) the monotonous decrease of ${{\cal A}}$ with $C$, (ii) larger acceptances for positrons than for electrons for given $C$ and crystal, and (iii) non-monotonous behaviour of ${{\cal A}}$ with respect to the charge number $Z$ of the crystal atom (in the positron case). Feature (i) is readily explained using the concept of the continuous interplanar potential and its modification in the bent channel (decrease of the effective potential due to the centrifugal term which increases with $C$) [@BakEtal1985; @Ellison:NP_B-v206-p205-1982; @BiryukovChesnokovKotovBook; @Tabrizi_AK_AS_WG:JPG]. Feature (ii) follows from the fact that in an oriented crystal a projectile electron, being attracted to an atomic chain (plane), gains the transverse energy via the collisions with the crystal constituents and, thus, switches to the over-barrier motion much faster than a positron. Feature (iii) reflects the resulting impact of the two opposite tendencies. First, both the average value of the interplanar field and the interplanar spacing increase with $Z$, thus acting towards the increase of the acceptance. The opposite tendency, is the increase of the (average) scattering angles in collisional events of the projectile with heavier atoms. For electrons, which tend to move in the vicinity of the atomic chains, the latter tendency outpowers the former one. For positrons, the two curves presented (for $Z=6$ and $Z=74$) indicate that dependence ${{\cal A}}$ on $Z$ is not so strong as for electrons. We note that in Ref. [@Kovalenko_EtAl:JINR_RC-v72-p9-1995] the acceptance of W(110) was estimated to be slightly higher than for Si(110).
Within the framework of the continuous potential approximation [@Lindhard1965], the criterion for distinguishing between the channeling and non-channeling motions is introduced straightforwardly by matching the value of the transverse energy, ${{\varepsilon}}_{\perp}$, of the projectile with the depth, $\Delta U$, of the interplanar potential well. The channeling mode corresponds to ${{\varepsilon}}_{\perp} < \Delta U$. In the simulations based on solving the equations of motion (\[Methodology:eq.01\]) with the potential built up as an exact sum of atomic potentials, another criterion is required to select the channeling segments. Following [@ChanModuleMBN_2013], we assume the channeling to occur when a projectile, while moving in the same channel, changes the sign of the transverse velocity $v_y$ at least two times.
An accepted projectile, stays in the channeling mode of motion over some interval $z_{{\rm ch},0}$ until an event of the dechanneling (if it happens). The initial channeling segment $z_{{\rm ch},0}$ is explicitly indicated for trajectory “${{\rm e}}_1$” in Fig. \[Figure.01\]. Trajectory “${{\rm e}}_2$” corresponds to the non-accepted particle ($z_{{\rm ch},0}=0$). Both positron trajectories stand for the accepted particles, and in case of “${\rm p}_2$” the dechanneling does not occur within the indicated length of the crystal. To quantify the dechanneling effect for the accepted particles, one can introduce [*the penetration depth*]{} $L_{\mathrm{p1}}$ [@ChanModuleMBN_2013] defined as the arithmetic mean of the initial channeling segments $z_{{\rm ch},0}$ calculated with respect to all accepted trajectories. For sufficiently thick crystals ($L \gg L_{\mathrm{p1}}$), the penetration depth approaches the so-called dechanneling length ${{L_{\rm d}}}$ which characterizes the fraction of the channeling particles at large distances $z$ from the entrance in terms of the exponential decay, $\propto \exp(-z/{{L_{\rm d}}})$. (see, e.g., Ref. [@BiryukovChesnokovKotovBook]). We note here that the concept of the exponential decay has been widely exploited to estimate the de-channeling lengths for various ultra-relativistic projectiles in the straight and bent crystals [@Backe_EtAl_2008; @BogdanovDabagov2012; @Scandale_EtAl_2012; @BiryukovChesnokovKotovBook; @PhysRevLett.112.135503; @Wienands_EtAl_PRL_v114_074801_2015].
Random scattering of the projectiles on the crystal atoms can result in the [*rechanneling*]{}, i.e. the process of capturing the over-barrier particles into the channeling mode of motion. In a sufficiently long crystal, the projectiles can experience dechanneling and rechanneling several times in the course of propagation. These multiple events can be quantified by introducing the penetration length $L_{\mathrm p2}$ and the total channeling length ${{L_{\rm ch}}}$. These quantities characterize the channeling process in the whole crystal. In explaining the meaning of the depth $L_{\mathrm p2}$ we refer to the electron trajectories in Fig. \[Figure.01\]. Trajectory “${{\rm e}}_1$” has a non-zero initial channeling segment, $z_{{\rm ch},0}$, as well as several secondary channeling segments, marked as $z_{{\rm ch}1}, \dots, z_{{\rm ch}6}$, which appear due to the rechanneling process. So, altogether this trajectory contains seven channeling segments. Trajectory “${{\rm e}}_2$” exhibit only secondary channeling segments (not marked explicitly) of the total number ten. The penetration length $L_{\rm p2}$ is calculated as the arithmetic mean of all channeling segments (initial and secondary) with respect to the total number of channeling segments in all simulated trajectories. Finally, the total channeling length ${{L_{\rm ch}}}$ per particle is calculated by averaging the sums $z_{{\rm ch}0} + z_{{\rm ch}1} + z_{{\rm ch}2} + \dots$, calculated for each trajectory, over all trajectories.
To conclude the description of the lengths introduced above, we note that $L_{\rm p1}$ characterizes the distance covered by accepted particles moving in the channeling mode. Generally speaking, it depends on the beam emittance at the crystal entrance. The second penetration depth, $L_{\rm p2}$, accounts for the rechanneling events, which occur, on average, for the incident angles not greater than Lindhard’s critical angle $\Theta_{\rm L}$ [@Lindhard1965]. Hence, for sufficiently long crystals $L_{\rm p2}$ mimics the initial penetration depth of the beam with a non-zero emittance equal approximately to $\Theta_{\rm L}=(2\Delta U_0/{{\varepsilon}})^{1/2} \approx 0.56$ mrad. The latter estimate was obtained for a ${{\varepsilon}}=855$ MeV projectile using the value $\Delta U_0 = 132.2$ eV for the interplanar potential well in straight W(110) channel at room temperature.
The results on the acceptance and the characteristic lengths are presented in Table \[Table:ep-A-Lp12-Lch\]. All the data refer to zero emittance beams entering a $L=75$ $\mu$m W(110) crystal. The straight crystal corresponds to infinitely large bending radius, $R=\infty$. Statistical uncertainties due to the finite numbers $N_0$ of the simulated trajectories correspond to the $99.9 \%$ confidence interval.
---------- --------- -------------- --------------- ---------------- ------------------ -------------- --------------- ---------------- ------------------
$R$ $C$ ${{\cal A}}$ $L_{\rm p1}$ $L_{\rm p2}$ ${{L_{\rm ch}}}$ ${{\cal A}}$ $L_{\rm p1}$ $L_{\rm p2}$ ${{L_{\rm ch}}}$
$\infty$ $0.00$ $0.56$ $3.33\pm0.13$ $3.43\pm 0.05$ $12.7 \pm 0.5$ $0.97$ $69.4\pm 1.0$ $59.0\pm 1.6 $ $68.1 \pm 1.0 $
$2.0$ $0.01$ $0.53$ $3.19\pm0.12$ $3.38\pm 0.05$ $8.85 \pm 0.37$
$0.864$ $0.023$ $0.50$ $3.16\pm0.11$ $3.29\pm 0.07$ $4.13 \pm 0.18$ $0.95$ $68.8\pm 0.9$ $66.7 \pm 1.0$ $65.8 \pm 0.9$
$0.432$ $0.046$ $0.42$ $2.90\pm0.14$ $2.94\pm 0.11$ $1.94 \pm 0.14$ $0.94$ $68.3\pm 0.9$ $67.4\pm 1.0 $ $64.4 \pm 0.9$
$0.288$ $0.069$ $0.39$ $2.64\pm0.13$ $2.66\pm 0.11$ $1.32 \pm 0.11$ $0.92$ $66.4\pm 1.0$ $65.9\pm 1.0 $ $61.4 \pm 1.0$
$0.173$ $0.116$ $0.28$ $2.23\pm0.12$ $2.24\pm 0.11$ $0.71 \pm 0.08$ $0.86$ $61.9\pm 1.4$ $61.6\pm 1.4 $ $53.1 \pm 1.4$
$0.115$ $0.174$ $0.21$ $1.79\pm0.09$ $1.80\pm 0.09$ $0.40 \pm 0.05$ $0.72$ $55.2\pm 1.8$ $55.0\pm 1.8 $ $39.8 \pm 1.7$
$0.086$ $0.231$ $0.16$ $1.58\pm0.09$ $1.58\pm 0.09$ $0.26 \pm 0.04$ $0.61$ $49.8\pm 2.2$ $49.5 \pm 2.2$ $30.2 \pm 1.9 $
$0.069$ $0.289$ $0.12$ $1.34\pm0.09$ $1.34\pm 0.09$ $0.17 \pm 0.0$ $0.47$ $45.6\pm 2.5$ $45.5 \pm 2.5$ $21.5 \pm 2.0 $
$0.058$ $0.347$ $0.11$ $1.25\pm0.09$ $1.25\pm 0.09$ $0.14 \pm 0.03$ $0.35$ $38.2\pm 2.8$ $38.1 \pm 2.8$ $13.3 \pm 1.7$
$0.043$ $0.463$ $0.07$ $1.03\pm0.08$ $1.03\pm 0.08$ $0.07 \pm 0.02$ $0.23$ $18.5\pm 1.7$ $18.4 \pm 1.7$ $4.2 \pm 0.7$
---------- --------- -------------- --------------- ---------------- ------------------ -------------- --------------- ---------------- ------------------
: Acceptance ${{\cal A}}$, bending parameter $C$, penetration lengths $L_{\rm p1,2}$ and total channeling ${{L_{\rm ch}}}$ length (all in $\mu$m) for 855 MeV electrons and positrons in straight ($R=\infty$) and bent ($R<\infty$, in cm) W(110) crystal. \[Table:ep-A-Lp12-Lch\]
We start discussion of the data presented in Table \[Table:ep-A-Lp12-Lch\] with the penetration lengths of electrons.
The length $L=75$ $\mu$m of the crystal exceeds greatly the quoted values of $L_{\rm p1}$ and $L_{\rm p2}$. Therefore, these quantities can be associated with the dechanneling lengths of the ideally collimated electron beam (the penetration depth $L_{\rm p1}$) and of the beam with emittance of approximately $\Theta_{\rm L}$ (the depth $L_{\rm p2}$). For a collimated electron beam, to [*estimate*]{} the dechanneling length in straight W(110) channel one can use its relationship [@Baier] to the radiation length $L_{\rm r}$ in the corresponding amorphous medium. This relationship can be written in the following form, which is convenient for a quick estimate of ${{L_{\rm d}}}$ (see Eq. (6.4) in Ref. [@ChannelingBook2014]): $${{L_{\rm d}}}\, \mbox{[$\mu$m]}
= 0.089\times \Delta U_0 \mbox{[eV]}\, {{\varepsilon}}\mbox{[GeV]}\, L_{\rm r} \mbox{[cm]}.
\label{Results:eq.02}$$ For a 855 MeV electron in W(110) ($\Delta U_0 = 132.2$ eV, $L_{\rm r} =0.35$ cm) this formula results in ${{L_{\rm d}}}= 3.5$ $\mu$m which correlates with values of $L_{\rm p1}$ and $L_{\rm p2}$ presented in the table for the straight channel ($R=\infty$). We note that all these values are higher than the dechanneling length which follows from the consideration presented in Ref. [@Azadegan_EtAl:JPConfSer-v517-p012039-2014] where electron dechanneling process in straight W(110) was analyzed in terms of the Fokker-Planck equation. As a result of the analysis, the authors provide the following dependence of ${{L_{\rm d}}}$ (in $\mu$m) on the electron beam energy ${{\varepsilon}}$ (in GeV): ${{L_{\rm d}}}=2.78{{\varepsilon}}$. For a 855 MeV projectile this results in ${{L_{\rm d}}}=2.4$ $\mu$m.
It is instructive to compare the estimates ${{L_{\rm d}}}$ as they follow from Eq. (\[Results:eq.02\]) with the results for $L_{\rm p1}$ obtained previously for 855 MeV electrons by means of [<span style="font-variant:small-caps;">MBN Explorer</span>]{}for other oriented crystals. The simulated data are: $L_{\rm p1}=12.01 \pm 0.40$ $\mu$m for diamond(110) [@Sushko_AK_AS_SPB_SASP_2015], $11.72 \pm 0.30$ $\mu$m for Si(110) [@BentSilicon_2014; @BentSilicon_2013], and $6.57 \pm 0.30$ $\mu$m for Ge(110) [@SushkoThesis2015]. For the same crystals, the estimated values are ${{L_{\rm d}}}=18.1, 15.7,$ and 6.90 $\mu$m, respectively. Hence, the observation is that Eq. (\[Results:eq.02\]) overestimates the dechanneling length for low-$Z$ crystals (diamond and silicon) but provides good results for medium- and high-$Z$ ones (germanium and tungsten).
The penetration lengths $L_{\rm p1,2}$ (as well as the dechanneling length) decrease with the increase in the bending parameter $C$. Within the framework of the continuous potential model this feature can be explained in terms of the depth of the potential well which in a bent channel, $\Delta U_C$, is smaller than in a straight one, $\Delta U_0$, due to the centrifugal term (see, e.g., Refs. [@Ellison:NP_B-v206-p205-1982; @Tabrizi_AK_AS_WG:JPG; @BiryukovChesnokovKotovBook]).
It is instructive to compare the values of $L_{\rm p1,2}$ with the total channeling length ${{L_{\rm ch}}}$ of an electron. In the straight channel, ${{L_{\rm ch}}}$ exceeds $L_{\rm p1,2}$ approximately by a factor of four. Thus, on average, an electron trajectory contains four channeling segments when propagating through a 75 $\mu$m thick W(110) oriented crystal. For an accepted particle, one of these is the initial channeling segment whereas other three are due to the rechanneling. For a non-accepted particle, all four segments appear as a result of the rechanneling events. These figures allow one to estimate the [*rechanneling*]{} length, $L_{\rm rech} \approx 20$ $\mu$m, i.e. the average length of a segment within which a projectile moves in the over-barrier mode. The table shows that as $C$ increases, the decrease rate of ${{L_{\rm ch}}}$ is much larger than that of $L_{\rm p1,2}$. This is a clear indication that in a bent crystal the rechanneling events are much rarer than in a straight one. Starting with some bending parameter $\widetilde{C}$ the rechanneling events virtually cease to occur. To estimate $\widetilde{C}$ one can compare the value of $N_{\rm acc} L_{\rm p1}/N_0
={{\cal A}}L_{\rm p1}$, i.e. the initial channeling segments averaged over all trajectories, including the non-accepted ones, with ${{L_{\rm ch}}}$. For $C\geq \widetilde{C}$ one obtains ${{\cal A}}L_{\rm p1}\approx {{L_{\rm ch}}}$. The electron data presented in Table \[Table:ep-A-Lp12-Lch\] suggests that this approximate equality becomes valid starting with $C\approx 0.1$. Hence, in W(110) channel bent with radius $R\approx R_{\rm c}/10$ or smaller the rechanneling event virtually do not happen (the corresponding rechanneling lengths become infinitely large).
![ Channeling fractions $\xi_{\rm ch0}(z)$ (solid curves) and $\xi_{\rm ch}(z)$ (dashed curves in the left panel, symbols in the right panel) calculated for 855 MeV electrons (left panel, note the log scale of the horizontal axis) and positrons (right panel) in straight ($C=0$) and bent ($C>0$) W(110) channels. Thick (red) line in the left graph shows the dependence $\xi_{\rm ch}(z)\propto z^{-1/2}$, see explanation in the text. []{data-label="Figure.03"}](figure03a.eps "fig:"){width="7.5cm"} ![ Channeling fractions $\xi_{\rm ch0}(z)$ (solid curves) and $\xi_{\rm ch}(z)$ (dashed curves in the left panel, symbols in the right panel) calculated for 855 MeV electrons (left panel, note the log scale of the horizontal axis) and positrons (right panel) in straight ($C=0$) and bent ($C>0$) W(110) channels. Thick (red) line in the left graph shows the dependence $\xi_{\rm ch}(z)\propto z^{-1/2}$, see explanation in the text. []{data-label="Figure.03"}](figure03b.eps "fig:"){width="7.5cm"}
To further quantify the impact of the rechanneling effect basing on the information which can be extracted from the simulated trajectories, one can compute the [*channeling fractions*]{} $\xi_{\rm ch0}(z)=N_{\rm ch0}(z)/N_0$ and $\xi_{\rm ch}(z)=N_{\rm ch}(z)/N_0$ [@ChanModuleMBN_2013; @BentSilicon_2013]. Here $N_{\rm ch0}(z)$ stands for the number of electrons (of the total number $N_0$) which propagate in the same channel where they were accepted up to the distance $z$ where they dechannel. The quantity $N_{\rm ch}(z)$ is the number of particles which are in the channeling mode irrespective of the channel which guides their motion at the distance $z$. With increasing $z$ the fraction $\xi_{\rm ch0}(z)$ decreases as the accepted electrons leave the entrance channel. In the contrast, the fraction $\xi_{\rm ch}(z)$ can increase with $z$ when the electrons, including those not accepted at the entrance, can be captured in the channeling mode in the course of the [rechanneling]{}. These dependencies for a 855 MeV electron channeling in straight and bent W(110) channels are presented in Fig. \[Figure.03\] left.
A striking difference in the behaviour of the two fractions as functions of the penetration distance $z$ is mostly pronounced for the straight channel. Away from the entrance point, the fraction $\xi_{\rm ch0}(z)$ (solid curve) follows approximately the exponential decay law, $\xi_{\rm ch0}(z) \propto \exp\left(-z/L_{\rm p1}\right)$ (not shown in the figure). At large distances, the fraction $\xi_{\mathrm ch}(z)$ (dashed curve) , which accounts for the rechanneling process, decreases much slower following the power law, $\xi_{\rm ch}(z)\propto z^{-1/2}$ [@Kostyuk-AK-AS-WG_855-Si]. This dependence is shown in the figure with thick (red) dashed line. As the bending curvature increases, $C\propto 1/R$, the rechanneling events become rarer, and the difference between two fractions decreases. For $C\gtrsim 0.1$ both curves virtually coincide.
Let us turn to the data on positron channeling presented in Table \[Table:ep-A-Lp12-Lch\] and Fig. \[Figure.03\] right. In contrast to the electron case, the crystal length $L=75$ $\mu$m is much smaller than the positron dechanneling length in straight W(110) channel, ${{L_{\rm d}}}\approx 445$ $\mu$m, which can be obtained by means of the formula [@Dechan01; @ChannelingBook2014; @BiryukovChesnokovKotovBook]: $${{L_{\rm d}}}= \gamma {256 \over 9\pi^2} {{{a_{\rm TF}}}\over r_0} {d\over \Lambda}.
\label{Results:eq.03}$$ Here $r_0=2.8\times 10^{-13}$ cm is the electron classical radius, $\gamma={{\varepsilon}}/mc^2$ is the Lorentz relativistic factor, ${{a_{\rm TF}}}$ is the Thomas-Fermi atomic radius (equal to 0.112 Å for a tungsten atom). The quantity $\Lambda$ stands for a so-called ‘Coulomb logarithm’, which characterizes the ionization losses of an ultra-relativistic positron in an amorphous medium (see e.g. [@Landau4]): $\Lambda = \ln\sqrt(2\gamma)^{1/2} mc^2/I - 23/24$ with $I$ being the mean atomic ionization potential ($I\approx 770$ eV for a tungsten atom).
To estimate the positron dechanneling length in a bent crystal, one can multiply Eq. (\[Results:eq.03\]) by the factor $(1-C)^2$ which exactly accounts for the relative change of the depth of the potential well in a bent channel within the framework of the harmonic approximation for the continuous interplanar potential [@BiryukovChesnokovKotovBook].
For all values of $C$ indicated in Table \[Table:ep-A-Lp12-Lch\], the estimated dechanneling length exceeds the crystal length $L$. Hence, the values of $L_{\rm p1}$ presented in the table can be considered only as a lower bound of the positron dechanneling length. For small bending parameters, $C\lesssim 0.1$, the penetration length is just below $L$, thus signaling that most of the accepted projectiles traverse the whole crystal moving in the channeling mode (trajectory “p2” in Fig. \[Figure.01\] is an exemplary one). Finally, for all $C$, the total channeling length ${{L_{\rm ch}}}$ is very close to the ${{\cal A}}L_{\rm p1}$. Therefore, in contrast to the electron channeling, the rechanneling effect does not play any significant role even in the case of the straight crystal. This statement is illustrated further by Fig. \[Figure.03\]right where the fractions $\xi_{\rm ch0}(z)$ (solid curves) and $\xi_{\rm ch}(z)$ (symbols) are practically indistinguishable for $C>0$ and are very close for $C=0$.
Radiation Spectra \[Spectra\]
-----------------------------
The simulated trajectories were also used to compute spectral distribution $\left\langle {{\rm d}}E/\hbar{{\rm d}}{{\omega}}\right\rangle$ of the emitted radiation, Eq. (\[Methodology:eq.03\]). For each trajectory, the distribution ${{\rm d}}^2 E_n/{{\rm d}}{{\omega}}\, {{\rm d}}{{\Omega}}$ was calculated accounting only for the initial part of the trajectory with $z \leq 10$ $\mu$m. Hence, the spectra discussed below refer to the $L=10$ $\mu$m thick crystals. The integration over the emission angle $\theta$ was performed for two values of the radiation apertures: $\theta_{0} = 0.24$ mrad and $\theta_{0} = 8$ mrad. The first value is close to that used in the experiments with the $855$ MeV electron beam at MAMI [@Backe_EtAl_2008; @Backe_EtAl_2011; @Backe_EtAl_PRL2014; @Backe_EtAl_PRL_115_025504_2015], and is much smaller than the natural emission angle for the beam energy, $\gamma^{-1} \approx 0.6$ mrad. Therefore, the corresponding spectra refer to a nearly forward emission. In contrast, the second aperture exceeds the $\gamma^{-1}$ by order of magnitude, thus providing the emission cone $\theta \leq \theta_{0}$ to collect almost all the radiation from the relativistic projectiles. The latter situation corresponds to the conditions at the experimental setup at SLAC [@Wienands_EtAl_PRL_v114_074801_2015].
\[ht\] ![ Spectral distribution of radiation emitted by 855 MeV electrons (left) and positrons (right) in straight (thick solid line, $C=0$) and bent (solid lines, $C>0$) W(110) crystals. The dashed lines show the simulated spectra for amorphous tungsten. The upper and lower plots correspond to the aperture values $\theta_0 = 0.24$ and $8$ mrad, respectively. All spectra refer to the crystal thickness $L=10~\mu$m.[]{data-label="Figure.04"}](figure04a.eps "fig:"){width="7.5cm"} ![ Spectral distribution of radiation emitted by 855 MeV electrons (left) and positrons (right) in straight (thick solid line, $C=0$) and bent (solid lines, $C>0$) W(110) crystals. The dashed lines show the simulated spectra for amorphous tungsten. The upper and lower plots correspond to the aperture values $\theta_0 = 0.24$ and $8$ mrad, respectively. All spectra refer to the crystal thickness $L=10~\mu$m.[]{data-label="Figure.04"}](figure04b.eps "fig:"){width="7.5cm"}\
![ Spectral distribution of radiation emitted by 855 MeV electrons (left) and positrons (right) in straight (thick solid line, $C=0$) and bent (solid lines, $C>0$) W(110) crystals. The dashed lines show the simulated spectra for amorphous tungsten. The upper and lower plots correspond to the aperture values $\theta_0 = 0.24$ and $8$ mrad, respectively. All spectra refer to the crystal thickness $L=10~\mu$m.[]{data-label="Figure.04"}](figure04c.eps "fig:"){width="7.5cm"} ![ Spectral distribution of radiation emitted by 855 MeV electrons (left) and positrons (right) in straight (thick solid line, $C=0$) and bent (solid lines, $C>0$) W(110) crystals. The dashed lines show the simulated spectra for amorphous tungsten. The upper and lower plots correspond to the aperture values $\theta_0 = 0.24$ and $8$ mrad, respectively. All spectra refer to the crystal thickness $L=10~\mu$m.[]{data-label="Figure.04"}](figure04d.eps "fig:"){width="7.5cm"}
The plots in Fig. \[Figure.04\] present the spectra emitted by electrons (left plots) and positrons (right plots) within the apertures $\theta_{0}=0.24$ and $8$ mrad (top and bottom plots, respectively). In each plot, the thick solid (red) curve shows the spectrum in a straight W(110) channel (the bending parameter $C=0$), while other solid curves correspond to the bent channels ($C>0$, the corresponding values of the bending radius one finds in Table \[Table:ep-A-Lp12-Lch\]). The dashed curves represent the spectra calculated from the simulated trajectories in amorphous tungsten.
There are several features to be mentioned when comparing the emission spectra calculated for two different apertures for each type of the projectile as well as when comparing the electron and positron spectra.
To start with, we discuss the emission spectra from electrons. In the case of a straight channel, the powerful maximum at $\hbar{{\omega}}_{\max}\approx 10$ MeV seen for both apertures (although more pronounced for $\theta_0 = 0.24$ mrad) is mainly due to the ChR. In the maximum, the intensity emitted in the oriented W(110) crystal exceeds noticeably the intensity $\langle {{\rm d}}E\rangle_{\rm am}$ of the incoherent BrS emitted in amorphous tungsten (the dashed curves). For bent crystals, the maximum of the intensity decreases with the increase of the bending parameter $C$. Comparing the curves in the left-top and left-bottom panels of Fig. \[Figure.04\] one notices that, as $C$ increases, the ratio $\eta=\langle {{\rm d}}E\rangle_{C>0}/\langle {{\rm d}}E\rangle_{C=0}$ of the maximum intensities in the bent and straight channels decreases faster for the smaller aperture. For example, for $C=0.023$ the ratio equals to 0.7 for $\theta_0 = 0.24$ mrad but $\eta=0.9$ for $\theta_0 = 8$ mrad. This feature can be explain as follows. In the vicinity of the maximum, the main contribution to the ChR intensity emitted within the cone $\theta\leq \theta_0$ comes from those channeling segments for which the angle of inclination to the cone axis does not exceed $\theta_0$. In a comparatively thin crystal, when $L<L_{\rm rech}$ [@footnote:01], the rechanneling events are rare, so that the total channeling segment is determined by the initial penetration depth $L_{\rm p1}$. In a straight crystal, the whole initial channeling segment is aligned with the emission cone so that the intensity is proportional to $L_{\rm p1}$. In a bent channel, the depth $L_{\rm p1}$ should be compared to the scale $R\theta_0$. The whole initial channeling segment can be considered to be aligned with the cone axis if $L_{\rm p1} < R\theta_0$. In the opposite case, $L_{\rm p1} > R\theta_0$, the emission within the cone $\theta\leq \theta_0$ occurs effectively only from the part of the segment [@BentSilicon_2014; @Polozkov_VKI_Sushko_AK_AS_SPB_Diamond_2015]. As a result, the maximum values of the intensities in the straight and bent crystals can be estimated as follows: $$\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C=0}
=
a {{\cal A}}(0) L_{\rm p1}(0)\,,
\qquad
\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C>0}
=
a {{\cal A}}(C) \min\left\{L_{\rm p1}(C),R\theta_0\right\}\,.
\label{Spectra:eq.01}$$ Here the coefficient $a$ depends on the aperture, ${{\cal A}}(C)$ and $L_{\rm p1}(C)$ stand for the acceptance and penetration depth corresponding for a given bending parameter, see Table \[Table:ep-A-Lp12-Lch\].
For the larger aperture, $\min\left\{L_{\rm p1}(C),R\theta_0\right\} = L_{\rm p1}(C)$ for all $C$ indicated. Therefore, the ratio of the intensities is estimated as $\eta_{l} \sim
\Bigl({{\cal A}}(C)/{{\cal A}}(0)\Bigr)\,\Bigl(L_{\rm p1}(C)/L_{\rm p1}(0)\Bigr)$. For the smaller aperture, $L_{\rm p1}(C) \leq R\theta_0$ only for $C\leq 0.015$ (which corresponds to $R\geq 1.3$ cm). For larger bending parameters one evaluates $\eta_{s} \sim \Bigl({{\cal A}}(C)/{{\cal A}}(0)\Bigr)\,\Bigl(R\theta_0/L_{\rm p1}(0)\Bigr)
< \eta_{l}$.
To conclude the discussion of the electron spectra, let us comment on their behaviour in the photon energy range well away from the maximum. It is seen that in most cases, the spectral distribution of radiation formed in the crystalline medium approaches from above that in amorphous tungsten. The excess over $\langle {{\rm d}}E\rangle_{\rm am}$ is due to the emission of the coherent BrS by over-barrier particles which move along quasi-periodic trajectories, traversing the crystallographic planes under the angle greater than Lindhard’s critical angle (see Refs. [@Andersen1980; @Ter-Mikaelian2001; @Baier; @AkhiezerShulga1982; @BazylevZhevago; @Sorensen1996; @Uggerhoj_RPM2005] for the reviews on the coherent BrS). The only exception from this scenario are the spectral dependencies calculated for the smaller aperture in the straight ($C=0$) and nearly straight ($C=0.01$) crystals. In these cases, the amorphous background radiation is more intensive in the photon energy range $\hbar{{\omega}}\gtrsim 50$ MeV. The explanation is as follows. In a straight crystal, the coherent BrS is emitted by over-barrier projectiles which move under the angles $\Theta>\Theta_{\rm L} = 0.56$ mrad with respect to the $z$-axis. These projectiles mostly radiate within the cone $\gamma^{-1}$ along the instant velocity so that only part of this radiation is emitted in the narrow cone $R\theta_0 =0.24$ mrad in the forward (with respect to the incident beam) direction. In a bent crystal, a particle can become an over-barrier one, still moving along the initial $z$ direction, penetrating into the crystal at the distance $\gtrsim R\Theta_{\rm L}$. Hence, for larger values of the bending parameter the intensity of the coherent BrS emitted in the forward direction increases.
It is a well-established fact that channeling oscillations of electrons have a strong anharmonic character which is a direct consequence of a strong deviation of the electron interplanar potential from a harmonic shape (see, e.g., [@BakEtal1985]). The period of oscillations varies with the amplitude, as it is illustrated by two simulated electron trajectories presented in Fig. \[Figure.01\]. As a result, the spectral distribution of ChR exhibit a rather broad maximum.
In contrast, the channeling trajectories of positrons demonstrate nearly harmonic oscillations between the neighboring planes. This is also in accordance with a well-known result established within the framework of the continuum model of channeling [@Gemmel]. Indeed, for a positively charged projectile the interplanar potential can be approximated by parabola in most part of the channel thus leading to close resemblance between the channeling motion of positrons and the undulator motion. As a result, for each value of the emission angle $\theta$ the spectral distribution of ChR in a straight crystal reveals a set of narrow and equally spaced peaks (harmonics). The harmonic frequencies, ${{\omega}}_n$, of ChR of positrons can be estimated from $$\begin{aligned}
{{\omega}}_{n}
=
{2\gamma^2\, {{\Omega}}_{\rm ch} \over 1 + \gamma^2 \theta^2 + K_{\rm ch}^2/2}
\, n ,
\quad
n=1,2,3,\dots .
\label{Spectra:eq.02}\end{aligned}$$ Here, ${{\Omega}}_{\rm ch}$ and $K_{\rm ch}$ are the frequency and the undulator parameter of the channeling oscillations. The maximum value of the latter can be estimated as $2\pi\gamma(d/2)/\lambda_{\rm ch}$ with $d/2$ and $\lambda_{\rm ch}$ being, respectively, the maximum possible amplitude and the period of the oscillations. Within the framework of harmonic approximation for the interplanar potential, one derives ${{\Omega}}_{\rm ch}=2d/c\Theta_{\rm L}$ and $K_{\rm ch} \leq \gamma\Theta_{\rm L}$. For a 855 MeV positron channeling in straight W(110) crystal this estimate produces $K_{\rm ch} \lesssim 0.9$.
It is known from the theory of undulator radiation (see, e.g., [@AlferovBashmakovCherenkov1989]) that for $K\sim 1$ the emission spectrum contains few harmonics the intensities of which rapidly decrease with the harmonic number $n$. This feature is explicitly seen in the spectral distributions calculated from the simulated trajectories of positrons propagating in [straight]{} W(110), see the thick solid curves in the right plots in Fig. \[Figure.04\]. The well-defined peaks (more pronounced for the smaller aperture) correspond to the harmonics of the ChR. For both apertures, the most powerful first peak is located at $\approx 4$ MeV. This value corresponds to the energy of the first (or, fundamental) harmonic emitted in the forward direction which one obtains from Eq. (\[Spectra:eq.02\]) by setting $n=1$, $\theta=0$ and using the aforementioned estimates for ${{\Omega}}_{\rm ch}$ and $K_{\rm ch}$. The intensities of the emission into higher harmonics (the peaks with $n$ up to 5 are seen located at $\hbar{{\omega}}_n\approx 4n$ MeV) rapidly decrease with $n$. This harmonic-like structure of the spectral distribution of ChR of positrons is clearly distinguishable from smooth curve which characterizes the electron spectrum of ChR.
The data presented in Table \[Table:ep-A-Lp12-Lch\] for the straight channel allows one to compare the maximum intensities of ChR of positrons and electrons in the straight $L=10$ $\mu$m thick crystal. For electrons, the peak intensity can be estimated directly from the first equation in (\[Spectra:eq.01\]). For positrons, the penetration depth $L_{\rm p1}(0)$ exceeds greatly the crystal thickness, therefore, to estimate the peak intensity one substitutes $L_{\rm p1}(0)$ with $L$. As a result, one obtains the following estimate for the ratio of the intensities: $\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C=0}^{\rm pos}/
\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C=0}^{\rm el}
= {{\cal A}}_{\rm pos}(0) L /({{\cal A}}(0) L_{\rm p1}(0))_{\rm el} \approx 5$ which correlates with the value which can be calculated by comparing the peak intensities in the top plots in Fig. \[Figure.04\].
As the bending curvature increases, the spectral distributions of radiation emitted by positrons become modified following two different scenarios.
The first one manifests itself as the decrease in the peak intensities with the increase of $C$. This feature is much more pronounced for the smaller aperture, see top-right plot in Fig. \[Figure.04\]. For example, the intensity of the first harmonic peak for $C=0.026$ is six times less than that in the straight crystal. This decrease rate is much larger than in the case of electron channeling where the corresponding drop is on the level of 30 per cent. Similar effect is seen for the higher harmonics as well as for larger values of $C$.
The explanation is as follows. The data on the penetration depth $L_{\rm p1}$ for positrons, presented in Table \[Table:ep-A-Lp12-Lch\], indicate that $L_{\rm p1}>L=10$ $\mu$m for all considered values of $C$. Hence, on average, all accepted particles propagate through the crystal moving in the channeling mode. As mentioned, in a straight crystal, the peak intensity can be estimated from the first equation in (\[Spectra:eq.01\]) where one substitutes $L_{\rm p1}(0)$ with the crystal length $L$. However, even for the $C$ value as small as 0.023, the emission into the nearly forward cone $\theta_0=0.24$ mrad occurs from the much shorter initial part of the channeling trajectory of the length $R\theta_0\approx 2.1$ $\mu$m. Hence, the estimate of the intensities ratio reads: $\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C=0.026} /
\left\langle {{\rm d}}E(\theta\leq \theta_0)\right\rangle_{C=0}
= {{\cal A}}(0.23) R\theta_0 / {{\cal A}}(0) L \approx 1/5$, which correlates with the data from top-right plot in Fig. \[Figure.04\].
For the larger aperture, $\theta_0=8$ mrad, the scale $R\theta_0$ exceeds the crystal length $L$ for $C$ up to $0.16$. Hence for $C\leq 0.16$ the ratio of the intensities is mainly determined by the ratio of the acceptances, ${{\cal A}}(C)/{{\cal A}}(0)$ which drops off at a much lower rate. This tendency one sees when comparing the peak intensities of the spectral distributions shown in bottom-right plot in Fig. \[Figure.04\].
\[ht\] ![ Same as in bottom-right plot in Fig. \[Figure.04\] but for smaller photon energy range and for several representative values of the bending parameter $C$. The peaks at $\hbar{{\omega}}\leq 1$ MeV are due to the synchrotron radiation (SR, dashed curves). See also explanation in the text. []{data-label="Figure.05"}](figure05.eps "fig:"){width="10cm"}
The second scenario of the spectral intensity modification with the increase of the bending parameter $C$ is due to the contribution of the synchrotron radiation (SR) to the emission spectrum. This feature is most pronounced for the larger aperture, see bottom-right plot in Fig. \[Figure.04\].
Positrons, which channel along the planes in a bent crystal experience two types of motion: the channeling oscillations and the translation along the centerline of the bent channel. The latter motion gives rise to the synchrotron-type radiation, i.e. the one, which is emitted by a charge moving along a circle (or, its part) (see, e.g., [@Jackson]). Therefore, the total spectrum contains the features of both ChR and SR. It was shown in Ref.[@Bashmakov1981], that in the case of planar channeling the crystal bending noticeably affects the spectrum only if the condition $\Theta_{\rm L}(C) \gamma \leq 1$ is met ($\Theta_{\rm L}(C)$ is the critical angle in the bent channel). In this case, the total spectrum preserves the features of ChR if $L_{\rm c} \gg \lambda$ (here, $L_{\rm c} = R/\gamma$ is the formation (coherence) length of the radiation emitted from an arc of the circle of the radius $R$, and $\lambda$ is the radiation wavelength), but becomes of the quasi-synchrotron type in the opposite limit [@Bashmakov1981; @Taratin_Review_1998]. The peculiarities which appear in spectral and spectra-angular distributions of the emitted radiation due to the interference of the two mechanisms of radiation were analyzed in Refs.[@SolovyovSchaeferGreiner1996; @ArutyunovEtAl_NP_1991; @TaratinVorobiev1988; @TaratinVorobiev1989] by analytical and numerical means. In Ref. [@BentSilicon_2014] the results of numerical simulations of the emission spectra by 855 MeV electrons were reported for straight and uniformly bent Si(110) crystal. The influence of the detector aperture on the form of the spectral distribution of the emitted radiation in bent S(110) was explored in [@Sub_GeV_2013]. The results of simulation of planar channeling as well as the calculated spectral distribution of the emitted radiation were reported in [@Sushko_EtAl_NIMB_v355_p39_2015] for 3…20 GeV electrons and positrons in bent Si(111).
The curves in bottom-right plot in Fig. \[Figure.04\] show that for $C>0$ the SR manifests itself as an additional structure in the low-energy part of the spectrum ($\hbar{{\omega}}\lesssim 1$ MeV). The intensity of the SR peak increases with the bending parameter up to $C\sim L/\theta_0$, and then it decreases due to the reason discussed above: the larger curvature is, the smaller is the part of the trajectory which contributes to the radiation cone $\theta_0$. To clearer visualize the relationship of the additional structure and the SR, we present Fig. \[Figure.05\] where the spectra calculated from the simulated trajectories for several representative values of $C$ (solid curves) are plotted in a narrower range of photon energies. Also plotted are the spectral distributions of SR (dashed curves) emitted by 855 MeV positrons moving along the circular arc of the length $L=10$ $\mu$m and of the bending radius $R$ corresponding to the indicated value of $C$ (see Table \[Table:ep-A-Lp12-Lch\]). These dependencies were scaled to match the simulated distributions in the maxima of the SR.
Conclusions
===========
By means of the channeling module [@ChanModuleMBN_2013] of the [<span style="font-variant:small-caps;">MBN Explorer</span>]{}package [@MBN_ExplorerPaper; @MBN_ExplorerSite], we have performed the simulations of classical trajectories of 855 MeV electrons and positrons in $L=75$ $\mu$m thick oriented straight and bent single tungsten crystals. The energy considered is sufficiently large to disregard quantum effects when describing the interaction of the projectile with the constituent atoms. Therefore, the relativistic classical mechanics framework chosen in the paper is fully adequate. The trajectories were analyzed to quantify the channeling process by calculating various parameters (acceptance, penetration length, total channeling length) as functions of the curvature. The obtained results were compared to the data available.
The simulated trajectories were used as the input data for numerical analysis of the intensity of the emitted radiation in 10 $\mu$m thick crystals. In the case of straight crystals the channeling radiation appears atop the incoherent bremsstrahlung background. In a bent channel, the spectrum is enriched by the synchrotron radiation due to the circular motion of the projectile along the bent centerline. In all cases, the spectral distribution of radiation was calculated for two extreme values of the detector aperture $\theta_0$ as compared to the natural angle $\gamma^{-1}$ of the emission cone by an ultra-relativistic projectile. The smaller aperture, $\theta_0 = 0.24$ mrad, being much less than $\gamma^{-1}$ (equal to 0.60 mrad for a 855 electron and/or positron), characterizes the radiation spectrum emitted in the (nearly) forward direction. This aperture is routinely used in the channeling experiments with electrons at the Mainz Microtron facility [@Backe_EtAl_2008; @Backe_EtAl_PRL_115_025504_2015]. The larger aperture, $\theta_0 = 8$ mrad, accounts for nearly all emitted radiation in the case of straight and moderately bent crystals.
The obtained and presented results are of interest in connection with the ongoing experiments with crystalline undulators at MAMI [@Backe_EtAl_2011; @BackeLauth_Dyson2016; @Backe_EtAl_PRL2014] carried out with electron beams as well as with possible experiments by means of the positron beam [@Backe_EtAl_2011a]. In these experiments the silicon-based crystalline undulators were exposed (or discussed to be exposed) to the beam. However, of current interest is the search for other crystalline structures capable to effectively steer ultra-relativistic light projectiles along periodically bent crystallographic planes [@ChannelingBook2014]. Therefore, the results presented in the current paper can serve as benchmark data for further theoretical and experimental efforts in studying channeling and radiation formed in tungsten-based crystalline undulators.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11025524 and 11161130520, the National Basic Research Program of China under Grant No. 2010CB832903, and by the European Commission (the PEARL Project within the H2020-MSCA-RISE-2015 call, GA 690991). Constructive suggestions by Alexey Verkhovtsev are gratefully acknowledged.
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[^1]: Email: fszhang@bnu.edu.cn
[^2]: korol@mbnexplorer.com; On leave from State Maritime University, St. Petersburg, Russia
[^3]: solovyov@mbnresearch.com; On leave from Ioffe Physical-Technical Institute, St. Petersburg, Russia
|
---
abstract: 'In this note we study Morita contexts and Galois extensions for corings. For a coring $\mathcal{C}$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\mathcal{M}^{\mathcal{C}}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called *cleft* $C$*-Galois extensions* over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) $H$-Galois extensions (e.g. [@Doi94], [@DM92]).'
author:
- |
**Jawad Y. Abuhlail**[^1]\
Mathematics Department, Birzeit University\
Birzeit - Palestine
title: 'Morita Contexts for Corings and Equivalences[^2]'
---
Introduction {#introduction .unnumbered}
============
Let $\mathcal{C}$ be a coring over a not necessarily commutative ring $A$ and assume $A$ to be a right $\mathcal{C}$-comodule through $\varrho
_{A}:A\longrightarrow A\otimes _{A}\mathcal{C}\simeq \mathcal{C},$ $a\mapsto
\mathbf{x}a$ for some group-like element $\mathbf{x}\in \mathcal{C}$ (see [@Brz02 Lemma 5.1]). In the first section we study from the viewpoint of Morita theory the relationship between $A$ and its subring of coinvariants $B:=A^{co\mathcal{C}}:=\{b\in A\mid \varrho (b)=b\mathbf{x}\}.$ We consider the $A$-ring $^{\ast }\mathcal{C}:=\mathrm{Hom}_{A-}(\mathcal{C}%
,A)$ and its left ideal $Q:=\{q\in $ $^{\ast }\mathcal{C}\mid \sum
c_{1}q(c_{2})=q(c)\mathbf{x}$ for all $c\in \mathcal{C}\}$ and show that $B$ and $^{\ast }\mathcal{C}$ are connected via a Morita context using $%
_{B}A_{^{\ast }\mathcal{C}}$ and $_{^{\ast }\mathcal{C}}Q_{B}$ as connecting bimodules. Our Morita context is in fact a generalization of Doi’s Morita context presented in [@Doi94].
In the second section we introduce the weak (resp. the strong) structure theorem for $\mathcal{M}^{\mathcal{C}}.$ For the case $_{A}\mathcal{C}$ is locally projective, in the sense of B. Zimmermann-Huignes, we characterize $%
A$ being a generator (a progenerator) in the category of right $\mathcal{C}$-comodules by $\mathcal{M}^{\mathcal{C}}$ satisfying the weak (resp. the strong) structure theorem. Here the notion of Galois corings introduced by T. Brzeziński [@Brz02] plays an important role. The results and proofs are essentially module theoretic and similar to those of [@MZ97] for the catgeory $\mathcal{M}(H)_{A}^{C}$ of Doi-Koppinen modules corresponding to a right-right Doi-Koppinen structure $(H,A,C)$ (see also [@MSTW01] for the case $C=H$).
The notion of a $C$-Galois extension $A$ of a ring $B$ was introduced by T. Brzeziński and S. Majid in [@BM98] and is related to the so called entwining structures introduced in the same paper. In the third section we give equivalent conditions for a $C$-Galois extension $A/B$ to be cleft. Our results generalize results of [@Brz99] from the case of a base field to the case of a commutative ground ring. In the special case $\varrho (a)=\sum
a_{\psi }\otimes x^{\psi },$ for some group-like element $x\in C,$ we get a complete generalization of [@DM92 Theorem 1.5 ] (and ).
With $A$ we denote a not necessarily commutative ring with $1_{A}\neq 0_{A}$ and with $\mathcal{M}_{A}$ (resp. $_{A}\mathcal{M},$ $_{A}\mathcal{M}_{A}$) the category of *unital* right $A$-modules (resp. left $A$-modules, $A$-bimodules). For every right $A$-module $W$ we denote by $\mathrm{Gen}%
(W_{A}) $ (resp. $\sigma \lbrack W_{A}]$) the class of $W$-generated (resp. $%
W$-subgenerated) right $A$-modules. For the well developed theory of categories of type $\sigma \lbrack W]$ the reader is referred to [@Wis88 Section 15].
An $A$-module $W$ is called **locally projective** (in the sense of B. Zimmermann-Huignes [@Z-H76]), if for every diagram $$\xymatrix{0 \ar[r] & F \ar@{.>}[dr]_{g' \circ \iota} \ar[r]^{\iota} & W
\ar[dr]^{g} \ar@{.>}[d]^{g'} & & \\ & & L \ar[r]_{\pi} & N \ar[r] & 0}$$ with exact rows and $F$ f.g.: for every $A$-linear map $g:W\longrightarrow
N, $ there exists an $A$-linear map $g^{\prime }:W\longrightarrow L$, such that the entstanding parallelogram is commutative. Note that every projective $A$-module is locally projective. By [@Z-H76 Theorem 2.1] a left $A$-module $W$ is locally projective, iff for every right $A$-module $M$ the following map is injective $$\alpha _{M}^{W}:M\otimes _{A}W\longrightarrow \mathrm{Hom}_{-A}(^{\ast }W,M),%
\text{ }m\otimes _{A}w\mapsto \lbrack f\mapsto mf(w)].$$ It’s easy then to see that every locally projective $A$-module is flat and $%
A $-cogenerated.
Let $\mathcal{C}$ be an $A$-coring. We consider the canonical $A$-bimodule $%
^{\ast }\mathcal{C}:=\mathrm{Hom}_{A-}(\mathcal{C},A)$ as an $A$-ring with the canonical $A$-bimodule structure, multiplication $(f\cdot g)(c):=\sum
g(c_{1}f(c_{2}))$ and unity $\varepsilon _{\mathcal{C}}.$ If $_{A}\mathcal{C}
$ is locally projective, then we have an isomorphism of categories $\mathcal{%
M}^{\mathcal{C}}\simeq \sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}]$ (in particular $\mathcal{M}^{\mathcal{C}}\subseteq \mathcal{M}_{^{\ast }%
\mathcal{C}}$ is a full subcategory) and we have a left exact functor $%
\mathrm{Rat}^{\mathcal{C}}(-):\mathcal{M}_{^{\ast }\mathcal{C}}\rightarrow
\mathcal{M}^{\mathcal{C}}$ assigning to every right $^{\ast }\mathcal{C}$-module its maximum $\mathcal{C}$*-rational* $^{\ast }\mathcal{C}$-submodule, which turns to be a right $\mathcal{C}$-comodule. Moreover $%
\mathcal{M}^{\mathcal{C}}=\mathcal{M}_{^{\ast }\mathcal{C}}$ iff $_{A}%
\mathcal{C}$ is f.g. and projective. For more investigation of the $\mathcal{%
C}$-rational $^{\ast }\mathcal{C}$-modules see [@Abu03].
After this paper was finished, it turned out that some results in this paper were discovered independently by S. Caenepeel, J. Vercruysse and S. Wang [@CVW].
Morita Contexts
===============
In this section we fix the following: $\mathcal{C}$ is an $A$-coring with group-like element $\mathbf{x}\ $and $A$ is a right $\mathcal{C}$-comodule with structure map $$\varrho _{A}:A\longrightarrow A\otimes _{A}\mathcal{C}\simeq \mathcal{C},%
\text{ }a\mapsto \mathbf{x}a$$ (e.g. [@Brz02 Lemma 5.1]), hence $A\in \mathcal{M}_{^{\ast }\mathcal{C}%
} $ with $a\leftharpoonup g:=\sum a_{<0>}g(a_{<1>})=g(\mathbf{x}a)$ for all $%
a\in A$ and $g\in $ $^{\ast }\mathcal{C}.$ For $M\in \mathcal{M}_{^{\ast }%
\mathcal{C}}$ put $$M^{\mathbf{x}}:=\{m\in M\mid mg=mg(\mathbf{x})\text{ for all }g\in \text{ }%
^{\ast }\mathcal{C}\}.$$ In particular $A^{\mathbf{x}}:=\{a\in A\mid a\leftharpoonup g=ag(\mathbf{x})$ for all $g\in $ $^{\ast }\mathcal{C}\}\subset A$ is a subring. For $M\in
\mathcal{M}^{\mathcal{C}}$ we set $$M^{co\mathcal{C}}:=\{m\in M\mid \varrho (m)=m\otimes _{A}\mathbf{x}%
\}\subseteq M^{\mathbf{x}}.$$ Obviously $B:=A^{co\mathcal{C}}=\{b\in A\mid b\mathbf{x}=\mathbf{x}%
b\}\subseteq A^{\mathbf{x}}$ is a subring and $\varrho _{A}$ is $(B,A)$-bilinear. For $M\in \mathcal{M}^{\mathcal{C}}$ we have $M^{co\mathcal{C}%
}\in \mathcal{M}_{B}.$ Moreover we set $$Q:=\{q\in \text{ }^{\ast }\mathcal{C}\mid \sum c_{1}q(c_{2})=q(c)\mathbf{x}%
\text{ for all }c\in \mathcal{C}\}\subseteq (^{\ast }\mathcal{C})^{\mathbf{x}%
}.$$
\[Psi-pr\]
1. For every right $^{\ast }\mathcal{C}$-module $M$ we have an isomorphism of right $B$-modules $$\omega _{M}:\mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,M)\longrightarrow M^{%
\mathbf{x}},\text{ }f\mapsto f(1_{A})$$ with inverse $m\mapsto \lbrack a\mapsto ma].$
2. Let $_{A}\mathcal{C}$ be locally projective. If $M\in \mathcal{M}^{%
\mathcal{C}},$ then $M^{co\mathcal{C}}=M^{\mathbf{x}}\simeq \mathrm{Hom}%
_{-^{\ast }\mathcal{C}}(A,M)=\mathrm{Hom}^{\mathcal{C}}(A,M).$ Hence $$\Psi _{M}:M^{co\mathcal{C}}\otimes _{B}A\longrightarrow M,\text{ }m\otimes
_{B}a\mapsto ma$$ is surjective *(*resp. injective, bijective*)*, iff $$\Psi _{M}^{\prime }:\mathrm{Hom}^{\mathcal{C}}(A,M)\otimes
_{B}A\longrightarrow M,\text{ }f\otimes _{B}a\mapsto f(a)$$ is surjective *(*resp. injective, bijective*)*.
3. We have $\mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,^{\ast }\mathcal{C}%
)\simeq (^{\ast }\mathcal{C})^{\mathbf{x}}.$ If moreover $_{A}\mathcal{C}$ is $A$-cogenerated *(*resp. locally projective and $^{\Box }\mathcal{C}%
:=\mathrm{Rat}^{\mathcal{C}}(^{\ast }\mathcal{C}_{^{\ast }\mathcal{C}})$*)*, then $Q=(^{\ast }\mathcal{C})^{\mathbf{x}}$ *(*resp. $%
Q=(^{\Box }\mathcal{C})^{co\mathcal{C}}$*)*.
4. For every $M\in \mathcal{M}_{^{\ast }\mathcal{C}}$ *(*resp. $%
M\in \mathcal{M}^{\mathcal{C}}$*)* and all $m\in M,$ $q\in Q$ we have $%
mq\in M^{\mathbf{x}}$ *(*resp. $mq\in M^{co\mathcal{C}}$*)*.
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1. Obvious.
2. Trivial.
3. Considering $^{\ast }\mathcal{C}$ as a right $^{\ast }\mathcal{C}$-module via right multiplication we get $\mathrm{Hom}_{-^{\ast }\mathcal{C}%
}(A,^{\ast }\mathcal{C})\simeq (^{\ast }\mathcal{C})^{\mathbf{x}}$ by (1).If $q\in $ $(^{\ast }\mathcal{C})^{\mathbf{x}},$ then we have for all $g\in $ $^{\ast }\mathcal{C}$ and $c\in \mathcal{C}:$$$g(\sum c_{1}q(c_{2}))=\sum g(c_{1}q(c_{2}))=(q\cdot g)(c)=(qg(\mathbf{x}%
))(c)=q(c)g(\mathbf{x})=g(q(c)\mathbf{x}),$$ i.e. $\sum c_{1}q(c_{2})-q(c)\mathbf{x}\in \mathrm{\func{Re}}(\mathcal{C},A)$ $:=\bigcap \{\mathrm{Ke}(g)\mid g\in \mathrm{Hom}_{A-}(\mathcal{C},A)\}.$ If $_{A}\mathcal{C}$ is $A$-cogenerated, then $\mathrm{\func{Re}}(\mathcal{C}%
,A)=0,$ hence $Q=(^{\ast }\mathcal{C})^{\mathbf{x}}.$
Assume $_{A}\mathcal{C}$ to be locally projective. Then we have for all $%
q\in Q,g\in $ $^{\ast }\mathcal{C}$ and $c\in \mathcal{C}:$$$(q\cdot g)(c)=\sum g(c_{1}q(c_{2}))=g(q(c)\mathbf{x})=q(c)g(\mathbf{x})=(qg(%
\mathbf{x}))(c),$$ hence $q\in $ $^{\Box }\mathcal{C},$ with $\varrho (q)=q\otimes _{A}\mathbf{x%
},$ i.e. $q\in $ $(^{\Box }\mathcal{C})^{co\mathcal{C}}.$ On the other hand, if $q\in (^{\Box }\mathcal{C})^{co\mathcal{C}},$ then for all $g\in $ $%
^{\ast }\mathcal{C}$ we have $q\cdot g=qg(\mathbf{x}),$ i.e. $q\in (^{\ast }%
\mathcal{C})^{\mathbf{x}}=Q.$
4. Let $M\in \mathcal{M}_{^{\ast }\mathcal{C}}.$ Then we have for all $%
q\in Q,$ $g\in $ $^{\ast }\mathcal{C}$ and $m\in M:$$$(mq)g=m(q\cdot g)=m(qg(\mathbf{x}))=(mq)g(\mathbf{x}),$$ i.e. $mq\in M^{\mathbf{x}}.$ If $M\in \mathcal{M}^{\mathcal{C}},$ then we have for all $m\in M$ and $q\in Q:$$$\begin{tabular}{lllll}
$\varrho _{M}(mq)$ & $=$ & $\varrho _{M}(\sum m_{<0>}q(m_{<1>}))$ & & \\
& $=$ & $\sum m_{<0><0>}\otimes _{A}m_{<0><1>}q(m_{<1>})$ & & \\
& $=$ & $\sum m_{<0>}\otimes _{A}m_{<1>1}q(m_{<1>2})$ & & \\
& $=$ & $\sum m_{<0>}\otimes _{A}q(m_{<1>})\mathbf{x}$ & & \\
& $=$ & $\sum m_{<0>}q(m_{<1>})\otimes _{A}\mathbf{x}$ & & \\
& $=$ & $\sum mq\otimes _{A}\mathbf{x},$ & &
\end{tabular}$$ i.e. $mq\in M^{co\mathcal{C}}.\blacksquare $
\[bim\]
1. With the canonical actions $A$ is a $(B,^{\ast }\mathcal{C})$-bimodule.
2. $Q$ is a $(^{\ast }\mathcal{C},B)$-bimodule.
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1. By assumption $A\in \mathcal{M}^{\mathcal{C}}\subseteq \mathcal{M}%
_{^{\ast }\mathcal{C}}.$ For all $b\in B,$ $a\in A$ and $g\in $ $^{\ast }%
\mathcal{C}$ we have $$b(a\leftharpoonup g)=bg(\mathbf{x}a)=g(b(\mathbf{x}a))=g(\mathbf{x}%
(ba))=(ba)\leftharpoonup g.$$
2. For all $a\in A,$ $q\in Q$ and $c\in \mathcal{C}$ we have $$\sum c_{1}(aq)(c_{2})=\sum c_{1}q(c_{2}a)=\sum (ca)_{1}q((ca)_{2})=q(ca)%
\mathbf{x}=(aq)(c)\mathbf{x}.$$
For all $q\in Q,$ $b\in B$ and $c\in \mathcal{C}$ we have $$\sum c_{1}(qb)(c_{2})=\sum c_{1}q(c_{2})b=q(c)\mathbf{x}b=q(c)b\mathbf{x}%
=(qb)(c)\mathbf{x}.$$ On the other hand we have for all $q\in Q,$ $g\in $ $^{\ast }\mathcal{C}$ and $c\in \mathcal{C}:$$$\begin{tabular}{lllll}
$\sum c_{1}(g\cdot q)(c_{2})$ & $=$ & $\sum c_{1}q(c_{21}g(c_{22}))$ & $=$ &
$\sum c_{11}q(c_{12}g(c_{2}))$ \\
& $=$ & $\sum c_{11}(g(c_{2})q)(c_{12})$ & $=$ & $\sum (g(c_{2})q)(c_{1})%
\mathbf{x}$ \\
& $=$ & $\sum q(c_{1}g(c_{2}))\mathbf{x}$ & $=$ & $(g\cdot q)(c)\mathbf{x}.$%
\end{tabular}$$
Moreover we have for all $b\in B,$ $q\in Q,$ $g\in $ $^{\ast }\mathcal{C}$ and $c\in \mathcal{C}:$$$\begin{tabular}{lllll}
$((g\cdot q)b)(c)$ & $=$ & $(g\cdot q)(c)b$ & $=$ & $\sum q(c_{1}g(c_{2}))b$
\\
& $=$ & $\sum (qb)(c_{1}g(c_{2}))$ & $=$ & $(g\cdot qb)(c).\blacksquare $%
\end{tabular}$$
\[Mc\]Keep the notation above fixed.
1. $(A^{\mathbf{x}},^{\ast }\mathcal{C},A,(^{\ast }\mathcal{C})^{\mathbf{%
x}},\widetilde{F},\widetilde{G})$ is a Morita context derived form $%
A_{^{\ast }\mathcal{C}},$ where $$\begin{tabular}{llllllll}
$\widetilde{F}$ & $:$ & $(^{\ast }\mathcal{C})^{\mathbf{x}}\otimes _{A^{%
\mathbf{x}}}A$ & $\longrightarrow $ & $^{\ast }\mathcal{C},$ & $q\otimes
_{A^{\mathbf{x}}}a$ & $\mapsto $ & $qa,$ \\
$\widetilde{G}$ & $:$ & $A\otimes _{^{\ast }\mathcal{C}}(^{\ast }\mathcal{C}%
)^{\mathbf{x}}$ & $\longrightarrow $ & $A^{\mathbf{x}},$ & $a\otimes
_{^{\ast }\mathcal{C}}q$ & $\mapsto $ & $a\leftharpoonup q.$%
\end{tabular}$$
2. $(B,^{\ast }\mathcal{C},A,Q,F,G)$ is a Morita context, where $$\begin{tabular}{llllllll}
$F$ & $:$ & $Q\otimes _{B}A$ & $\longrightarrow $ & $^{\ast }\mathcal{C},$ &
$q\otimes _{B}a$ & $\mapsto $ & $qa,$ \\
$G$ & $:$ & $A\otimes _{^{\ast }\mathcal{C}}Q$ & $\longrightarrow $ & $B,$ &
$a\otimes _{^{\ast }\mathcal{C}}q$ & $\mapsto $ & $a\leftharpoonup q.$%
\end{tabular}$$
If moreover $_{A}\mathcal{C}$ is locally projective, then the two Morita contexts coincide.
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1. By Lemma \[Psi-pr\] we have $\mathrm{End}(A_{^{\ast }\mathcal{C}%
})\simeq A^{\mathbf{x}},$ $(^{\ast }\mathcal{C})^{\mathbf{x}}\simeq \mathrm{%
Hom}_{-^{\ast }\mathcal{C}}(A,^{\ast }\mathcal{C})$ and the result follows by [@Fai81 Proposition 12.6].
2. By Lemma \[bim\] $A$ is a $(B,^{\ast }\mathcal{C})$-bimodule and $Q$ is a $(^{\ast }\mathcal{C},B)$-bimodule. For all $q\in Q,g\in $ $^{\ast }%
\mathcal{C},$ $a\in A$ and $c\in \mathcal{C}$ we have $$F(g\cdot q\otimes _{B}a)(c)=\sum q(c_{2}g(c_{1}))a=(g\cdot qa)(c)=(g\cdot
F(q\otimes _{B}a))(c)$$ and $$\begin{tabular}{lllll}
$F(q\otimes _{B}a\leftharpoonup g)(c)$ & $=$ & $q(c)(a\leftharpoonup g)$ & $%
= $ & $q(c)g(\mathbf{x}a)$ \\
& $=$ & $g(q(c)\mathbf{x}a)$ & $=$ & $\sum g(c_{1}q(c_{2})a)$ \\
& $=$ & $\sum g(c_{1}(qa)(c_{2}))$ & $=$ & $(F(q\otimes _{B}a)\cdot g)(c),$%
\end{tabular}$$ hence $F$ is $^{\ast }\mathcal{C}$-bilinear. Note that by Lemma \[Psi-pr\] $G$ is well defined and is obviously $B$-bilinear. Moreover we have for all $%
a,\widetilde{a}\in A$ and $q,\widetilde{q}\in Q$ the following associativity relations: $$\begin{tabular}{lllll}
$(F(q\otimes _{B}a)\cdot \widetilde{q})(c)$ & $=$ & $\sum \widetilde{q}%
(c_{1}q(c_{2})a)$ & $=$ & $\widetilde{q}(q(c)\mathbf{x}a)$ \\
& $=$ & $q(c)\widetilde{q}(\mathbf{x}a)$ & $=$ & $(qG(a\otimes _{^{\ast }%
\mathcal{C}}\widetilde{q}))(c),$ \\
$G(a\otimes _{^{\ast }\mathcal{C}}q)\widetilde{a}$ & $=$ & $q(\mathbf{x}a)%
\widetilde{a}$ & $=$ & $(q\widetilde{a})(\mathbf{x}a)$ \\
& $=$ & $F(q\otimes _{B}\widetilde{a})(\mathbf{x}a)$ & $=$ & $%
a\leftharpoonup F(q\otimes _{B}\widetilde{a}).$%
\end{tabular}$$ If $_{A}\mathcal{C}$ is locally projective, then $A^{\mathbf{x}}=A^{co%
\mathcal{C}},$ $(^{\ast }\mathcal{C})^{\mathbf{x}}=Q$ by Lemma \[Psi-pr\] and the two contexts coincide.$\blacksquare $
[@Brz02 Definition 5.3] An $A$-coring $\mathcal{C}$ is said to be **Galois**, if there exists an $A$-coring isomorphism $\chi :A\otimes
_{B}A\longrightarrow \mathcal{C}$ such that $\chi (1_{A}\otimes _{B}1_{A})=%
\mathbf{x}.$ Recall that $A\otimes _{B}A$ is an $A$-coring with the canonical $A$-bimodule structure, comultiplication $$\Delta :A\otimes _{B}A\longrightarrow (A\otimes _{B}A)\otimes _{A}(A\otimes
_{B}A),\text{ }\widetilde{a}\otimes _{B}a\mapsto (\widetilde{a}\otimes
_{B}1_{A})\otimes _{A}(1_{A}\otimes _{B}a)$$ and counity $\varepsilon _{A\otimes _{B}A}:A\otimes _{B}A\longrightarrow A,$ $\widetilde{a}\otimes _{B}a\mapsto \widetilde{a}a.$
Consider the functors $$(-)^{co\mathcal{C}}:\mathcal{M}^{\mathcal{C}}\longrightarrow \mathcal{M}_{B}%
\text{ and }-\otimes _{B}A:\mathcal{M}_{B}\longrightarrow \mathcal{M}^{%
\mathcal{C}}.$$ By [@Brz02 Proposition 5.2] $(-\otimes _{B}A,(-)^{co\mathcal{C}})$ is an adjoint pair of covariant functors, where the adjunctions are given by $$\Phi _{N}:N\longrightarrow (N\otimes _{B}A)^{co\mathcal{C}},\text{ }n\mapsto
n\otimes _{B}1_{A} \label{str}$$ and $$\Psi _{M}:M^{co\mathcal{C}}\otimes _{B}A\longrightarrow M,\text{ }m\otimes
_{B}a\mapsto ma. \label{wk}$$ If $\Psi _{M}$ is an isomorphism for all $M\in \mathcal{M}^{\mathcal{C}},$ then we say $\mathcal{M}^{\mathcal{C}}$ satisfies the **weak structure theorem.** If in addition $\Phi _{N}$ is an isomorphism for all $N\in
\mathcal{M}_{B},$ then we say $\mathcal{M}^{\mathcal{C}}$ satisfies the **strong structure theorem** (in this case $(-)^{co\mathcal{C}}$ and $%
-\otimes _{B}A$ give an equivalence of categories $\mathcal{M}^{\mathcal{C}%
}\simeq \mathcal{M}_{B}$).
Let $W\in \mathcal{M}_{A}$ and consider the canonical right $\mathcal{C}$-comodule $W\otimes _{A}\mathcal{C}.$ Then $W\simeq (W\otimes _{A}\mathcal{C}%
)^{co\mathcal{C}}$ via $w\mapsto w\otimes _{A}\mathbf{x}$ with inverse $%
w\otimes _{A}c\mapsto w\varepsilon _{\mathcal{C}}(c)$ and we define $$\beta _{W}:=\Psi _{W\otimes _{A}\mathcal{C}}:W\otimes _{B}A\longrightarrow
W\otimes _{A}\mathcal{C},\text{ }w\otimes _{B}a\mapsto w\otimes _{A}\mathbf{x%
}a. \label{bw}$$ In particular we have for $W=A$ the morphism of $A$-corings $$\beta :=\Psi _{A\otimes _{A}\mathcal{C}}:A\otimes _{B}A\longrightarrow
A\otimes _{A}\mathcal{C}\simeq \mathcal{C},\text{ }\widetilde{a}\otimes
_{B}a\mapsto \widetilde{a}\mathbf{x}a. \label{bet}$$ If $\beta $ is bijective, then $\mathcal{C}$ is a Galois $A$-coring and we call the ring extension $A/B$ $\mathcal{C}$-**Galois**.
\[surj\]For the Morita context $(B,^{\ast }\mathcal{C},A,Q,F,G)$ the following statements are equivalent:
1. $G:A\otimes _{^{\ast }\mathcal{C}}Q\longrightarrow B$ is surjective *(*bijective and $B=A^{\mathbf{x}}$*)*;
2. there exists $\widehat{q}\in Q,$ such that $\widehat{q}(\mathbf{x}%
)=1_{A};$
3. for every right $^{\ast }\mathcal{C}$-module $M$ we have a $B$-module isomorphism $M\otimes _{^{\ast }\mathcal{C}}Q\simeq M^{\mathbf{x}}.$
4. for every right $\mathcal{C}$-comodule $M$ we have $M\otimes _{^{\ast
}\mathcal{C}}Q\simeq M^{\mathbf{co}\mathcal{C}}$ as $B$-modules.
If moreover $_{A}\mathcal{C}$ is locally projective, then *(1)-(4)*are moreover equivalent to:
5. $A_{^{\ast }\mathcal{C}}$ is *(*f.g.*)* projective.
\(1) $\Rightarrow $ (2). Assume $G$ to be surjective. Then there exist $%
a_{1},...,a_{k}$ and $q_{1},...,q_{k}\in Q,$ such that $G(\sum%
\limits_{i=1}^{k}a_{i}\otimes _{^{\ast }\mathcal{C}}q_{i})=1_{A}.$ Set $%
\widehat{q}:=\sum\limits_{i=1}^{k}a_{i}q_{i}\in Q.$ Then we have $$\widehat{q}(\mathbf{x})=(\sum\limits_{i=1}^{k}a_{i}q_{i})(\mathbf{x}%
)=\sum\limits_{i=1}^{k}q_{i}(\mathbf{x}a_{i})=\sum\limits_{i=1}^{k}(a_{i}%
\leftharpoonup q_{i})=G(\sum\limits_{i=1}^{k}a_{i}\otimes _{^{\ast }\mathcal{%
C}}q_{i})=1_{A}.$$
(2) $\Rightarrow $ (3). Consider the $B$-module morphism $$\xi _{M}:M\otimes _{^{\ast }\mathcal{C}}Q\longrightarrow M^{\mathbf{x}},%
\text{ }m\otimes _{^{\ast }\mathcal{C}}q\mapsto mq.$$ Let $\widehat{q}\in Q$ with $\widehat{q}(\mathbf{x})=1_{A}$ and define $%
\widetilde{\xi }_{M}:M^{\mathbf{x}}\longrightarrow M\otimes _{^{\ast }%
\mathcal{C}}Q,$ $m\mapsto m\otimes _{^{\ast }\mathcal{C}}\widehat{q}.$ For every $n\in M^{\mathbf{x}}$ we have $$(\xi _{M}\circ \widetilde{\xi }_{M})(n)=\xi _{M}(n\otimes _{^{\ast }\mathcal{%
C}}\widehat{q})=n\leftharpoonup \widehat{q}=n\widehat{q}(\mathbf{x})=n.$$ On the other hand we have for all $m\in M$ and $q\in Q:$ $$\begin{tabular}{lllllll}
$(\widetilde{\xi }_{M}\circ \xi _{M})(m\otimes _{^{\ast }\mathcal{C}}q)$ & $%
= $ & $\widetilde{\xi }_{M}(m\leftharpoonup q)$ & $=$ & $m\leftharpoonup
q\otimes _{^{\ast }\mathcal{C}}\widehat{q}$ & $=$ & $m\otimes _{^{\ast }%
\mathcal{C}}q\cdot \widehat{q}$ \\
& $=$ & $m\otimes _{^{\ast }\mathcal{C}}q\widehat{q}(\mathbf{x})$ & $=$ & $%
m\otimes _{^{\ast }\mathcal{C}}q,$ & &
\end{tabular}$$ i.e. $\xi _{M}$ is bijective with inverse $\widetilde{\xi }_{M}.$
(3) $\Rightarrow $ (4). Let $M\in \mathcal{M}^{\mathcal{C}}.$ By Lemma \[Psi-pr\] we have $\xi _{M}(M\otimes _{^{\ast }\mathcal{C}}Q)\subseteq M^{co%
\mathcal{C}}\subseteq M^{\mathbf{x}}.$ By assumption $\xi _{M}:A\otimes
_{^{\ast }\mathcal{C}}Q\longrightarrow M^{\mathbf{x}}$ is bijective. Hence $%
M^{\mathbf{x}}=M^{co\mathcal{C}}$ and $M\otimes _{^{\ast }\mathcal{C}}Q%
\overset{\xi _{M}}{\simeq }M^{co\mathcal{C}}.$
(4) $\Rightarrow $ (1) We are done since $G=\xi _{A}.$
Assume $_{A}\mathcal{C}$ to be locally projective.
Then $B\simeq \mathrm{End}(A_{^{\ast }\mathcal{C}}),$ $Q\simeq \mathrm{Hom}%
_{-^{\ast }\mathcal{C}}(A,^{\ast }\mathcal{C})$ and we get (1) $%
\Longleftrightarrow $ (5) by [@Fai81 Corollary 12.8].$\blacksquare $
\[co=x\]For the Morita context $(B,^{\ast }\mathcal{C},A,Q,F,G)$ assume there exists $\widehat{q}\in Q$ with $\widehat{q}(\mathbf{x})=1_{A}$ *(*equivalently $G:Q\otimes _{B}A\longrightarrow $ $^{\ast }\mathcal{C}$ is surjective*)*. Then:
1. For every $N\in \mathcal{M}_{B},$ $\Phi _{N}$ is an isomorphism.
2. $B$ is a left $B$-direct summand of $A.$
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1. Let $N\in \mathcal{M}_{B}.$ Then we have by Theorem \[surj\] the isomorphisms $G:A\otimes _{^{\ast }\mathcal{C}}Q\longrightarrow B$ and $\xi
_{N\otimes _{B}A}:(N\otimes _{B}A)\otimes _{^{\ast }\mathcal{C}%
}Q\longrightarrow (N\otimes _{B}A)^{co\mathcal{C}}.$ Moreover $\Phi _{N}$ is given by the canonical isomorphisms $$N\simeq N\otimes _{B}B\simeq N\otimes _{B}(A\otimes _{^{\ast }\mathcal{C}%
}Q)\simeq (N\otimes _{B}A)\otimes _{^{\ast }\mathcal{C}}Q\simeq (N\otimes
_{B}A)^{co\mathcal{C}}.$$
2. The map $\mathrm{tr}_{A}:A\longrightarrow B,$ $a\mapsto
a\leftharpoonup \widehat{q}$ is left $B$-linear with $\mathrm{tr}_{A}(b)=b$ for all $b\in B.\blacksquare $
\[general\]For the Morita context $(B,^{\ast }\mathcal{C},A,Q,F,G)$ assume there exists $\widehat{q}\in Q$ with $\widehat{q}(\mathbf{x})=1_{A}$ *(*equivalently $G:Q\otimes _{B}A\longrightarrow B$ is surjective*)*. Then:
1. $_{B}A$ and $Q_{B}$ are generators.
2. $A_{^{\ast }\mathcal{C}}$ and $_{^{\ast }\mathcal{C}}Q$ are f.g. and projective.
3. $F:Q\otimes _{B}A\longrightarrow $ $^{\ast }\mathcal{C}$ induces bimodule isomorphisms $$A\simeq \mathrm{Hom}_{^{\ast }\mathcal{C}-}(Q,^{\ast }\mathcal{C})\text{ and
}Q\simeq \mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,^{\ast }\mathcal{C}).$$
4. The bimodule structures above induce ring isomorphisms $$B\simeq \mathrm{End}(A_{^{\ast }\mathcal{C}})\text{ and }B\simeq \mathrm{End}%
(_{^{\ast }\mathcal{C}}Q)^{op}.$$
The result follows by standard argument of Morita Theory (e.g. [@Fai81 Proposition 12.7]).$\blacksquare $
\[f-G\]Consider the Morita context $(B,^{\ast }\mathcal{C},A,Q,F,G)$ and assume that $F:Q\otimes _{^{\ast }\mathcal{C}}A\longrightarrow $ $^{\ast }%
\mathcal{C}$ is surjective. Then:
1. $A_{^{\ast }\mathcal{C}}$ is a generator, $Q\simeq \mathrm{Hom}%
_{B-}(A,B)$ as bimodules and $^{\ast }\mathcal{C}\simeq \mathrm{End}(Q_{B}).$
2. $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem *(*in particular $A/B$ is $\mathcal{C}$-Galois*)*.
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1. The result$\;$follows by standard argument of Morita Theory (e.g. [@Fai81 Proposition 12.7]).
2. By assumption $\varepsilon _{\mathcal{C}}=F(\sum%
\limits_{i=1}^{k}q_{i}\otimes _{B}a_{i})$ for some $\{(q_{i},a_{i})%
\}_{i=1}^{k}\subseteq Q\times A.$ In this case $\Psi _{M}:M^{co\mathcal{C}%
}\otimes _{B}A\longrightarrow M$ is bijective with inverse $\widetilde{\Psi }%
_{M}:M\longrightarrow M^{co\mathcal{C}}\otimes _{B}A,$ $m\mapsto
\sum\limits_{i=1}^{k}mq_{i}\otimes _{B}a_{i}.$ In fact, we have for all $%
m\in M,$ $n\in M^{co\mathcal{C}}$ and $a\in A:$$$\begin{tabular}{lllll}
$(\Psi _{M}\circ \widetilde{\Psi }_{M})(m)$ & $=$ & $\sum%
\limits_{i=1}^{k}(mq_{i})a_{i}$ & $=$ & $\sum%
\limits_{i=1}^{k}(m_{<0>}q_{i}(m_{<1>})a_{i}$ \\
& $=$ & $\sum\limits_{i=1}^{k}m_{<0>}(q_{i}a_{i})(m_{<1>})$ & $=$ & $%
\sum\limits_{i=1}^{k}m_{<0>}\varepsilon _{\mathcal{C}}(m_{<1>})$ \\
& $=$ & $m$ & &
\end{tabular}$$ and $$\begin{tabular}{lllll}
$(\widetilde{\Psi }_{M}\circ \Psi _{M})(n\otimes _{B}a)$ & $=$ & $%
\sum\limits_{i=1}^{k}(na)q_{i}\otimes _{B}a_{i}$ & $=$ & $%
\sum\limits_{i=1}^{k}nq_{i}(\mathbf{x}a)\otimes _{B}a_{i}$ \\
& $=$ & $\sum\limits_{i=1}^{k}n\otimes _{B}q_{i}(\mathbf{x}a)a_{i}$ & $=$ & $%
\sum\limits_{i=1}^{k}n\otimes _{B}(q_{i}a_{i})(\mathbf{x}a)$ \\
& $=$ & $n\otimes _{B}\varepsilon _{\mathcal{C}}(\mathbf{x}a)$ & $=$ & $%
n\otimes _{B}a.\blacksquare $%
\end{tabular}$$
\[C -finite\]For the Morita context $(B,^{\ast }\mathcal{C},A,Q,F,G)$ the following are equivalent:
1. $F:Q\otimes _{B}A\longrightarrow $ $^{\ast }\mathcal{C}$ is surjective *(*bijective*)*;
2. (a) $Q_{B}$ is f.g. and projective;
\(b) $\Omega :A\longrightarrow \mathrm{Hom}_{-B}(Q,B),$ $a\mapsto \lbrack
q\mapsto a\leftharpoonup q]$ is a bimodule isomorphism;
\(c) $_{^{\ast }\mathcal{C}}Q$ is faithful.
If $_{A}\mathcal{C}$ is $A$-cogenerated, then *(1)* $\&$ *(2)* are moreover equivalent to:
3. (a)$\;_{B}A$ is f.g. and projective;
(b) $\Lambda :$ $^{\ast }\mathcal{C}\longrightarrow \mathrm{End}%
(_{B}A)^{op},$ $g\mapsto \lbrack a\mapsto a\leftharpoonup g]$ is a ring isomorphism.
4. $A_{^{\ast }\mathcal{C}}$ is a generator.
If moreover $_{A}\mathcal{C}$ is f.g. and projective, then *(1)-(4)* are equivalent to:
5. $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem.
The implications (1) $\Rightarrow $ (2), (3), (4) follow without any finiteness conditions on $\mathcal{C}$ by standard argument of Morita Theory (e.g. [@Fai81 Proposition 12.7]). Note that $_{^{\ast }\mathcal{C}}Q$ is faithful by the embedding $^{\ast }\mathcal{C}\hookrightarrow \mathrm{End}%
(Q_{B})$ (see Proposition \[f-G\] (1)).
(2) $\Rightarrow $ (1). Let $\{(q_{i},p_{i})\}_{i=1}^{k}\subset Q\times
\mathrm{Hom}_{-B}(Q,B)$ be a dual basis for $Q_{B}.$ By (b) there exist $%
a_{1},...,a_{k}\in A,$ such that $\Omega (a_{i})=q_{i}$ for $i=1,...,k.$ For every $q\in Q$ we have then $(\sum\limits_{i=1}^{k}q_{i}a_{i})\cdot
q=\sum\limits_{i=1}^{k}q_{i}(a_{i}\leftharpoonup
q)=\sum\limits_{i=1}^{k}q_{i}p_{i}(q)=q,$ hence $\sum%
\limits_{i=1}^{k}q_{i}a_{i}=\varepsilon _{\mathcal{C}}$ by (c) and the $%
^{\ast }\mathcal{C}$-bilinear morphism $F:Q\otimes _{^{\ast }\mathcal{C}%
}A\longrightarrow $ $^{\ast }\mathcal{C}$ is surjective.
Assume $_{A}\mathcal{C}$ to be $A$-cogenerated.
\(3) $\Rightarrow $ (1). Let $\{(a_{i},p_{i})\}_{i=1}^{k}\subset A\times
\mathrm{Hom}_{B-}(A,B)$ be a dual basis of $_{B}A.$ By (b), there exist $%
g_{1},...,g_{k}\in $ $^{\ast }\mathcal{C},$ such that $\Lambda (g_{i})=p_{i}$ for $i=1,...,k.$ **Claim**: $g_{1},...,g_{k}\in Q.$ For all $f\in $ $%
^{\ast }\mathcal{C}$ and $i=1,...,k$ we have $$\begin{tabular}{lllll}
$\Lambda (g_{i}\cdot f)(a)$ & $=$ & $a\leftharpoonup (g_{i}\cdot f)$ & $=$ &
$(a\leftharpoonup g_{i})\leftharpoonup f)$ \\
& $=$ & $p_{i}(a)\leftharpoonup f$ & $=$ & $f(\mathbf{x}p_{i}(a))$ \\
& $=$ & $f(p_{i}(a)\mathbf{x})$ & $=$ & $p_{i}(a)f(\mathbf{x})$ \\
& $=$ & $(p_{i}f(\mathbf{x}))(a)$ & $=$ & $\Lambda (g_{i}f(\mathbf{x}))(a),$%
\end{tabular}$$ hence $g_{i}\cdot f=g_{i}f(\mathbf{x}),$ i.e. $g_{i}\in $ $(^{\ast }\mathcal{%
C})^{\mathbf{x}}=Q$ (by Lemma \[Psi-pr\] (2)). Moreover for every $a\in A$ we have: $\Lambda
(\sum\limits_{i=1}^{k}g_{i}a_{i})(a)=\sum\limits_{i=1}^{k}a\leftharpoonup
g_{i}a_{i}=\sum\limits_{i=1}^{k}p_{i}(a)a_{i}=a,$ i.e. $\sum%
\limits_{i=1}^{k}g_{i}a_{i}=\varepsilon _{\mathcal{C}}$ and the $^{\ast }%
\mathcal{C}$-bilinear morphism $F$ is surjective.
(4) $\Rightarrow $ (1). Since $Q\simeq \mathrm{Hom}_{-\text{ }^{\ast }%
\mathcal{C}}(A,^{\ast }\mathcal{C}),$ we have $\func{Im}(F)=\mathrm{tr}%
(A,^{\ast }\mathcal{C}):=\sum \{\func{Im}(h):h\in \mathrm{Hom}_{-\text{ }%
^{\ast }\mathcal{C}}(A,^{\ast }\mathcal{C})\},$ hence $\func{Im}(F)=$ $%
^{\ast }\mathcal{C}$ iff $A_{^{\ast }\mathcal{C}}$ is a generator (e.g. [@Wis88 Page 154]).
Assume $_{A}\mathcal{C}$ to be f.g. and projective.
(1) $\Rightarrow $ (5) follows without any finiteness conditions on $%
\mathcal{C}$ by Proposition \[f-G\] (2).
\(5) $\Rightarrow $ (1). Since $_{A}\mathcal{C}$ is f.g. and projective, we have $\mathcal{M}^{\mathcal{C}}\simeq \mathcal{M}_{^{\ast }\mathcal{C}}$ (e.g. [@Brz02 Lemma 4.3]), hence $^{\ast }\mathcal{C}\in \mathcal{M}^{%
\mathcal{C}},$ $Q=(^{\ast }\mathcal{C})^{co\mathcal{C}}$ and $F=\Psi
_{^{\ast }\mathcal{C}}.\blacksquare $
Galois Extensions and Equivalences
==================================
The notation of the first section remains fixed. For every $M\in \mathcal{M}%
^{\mathcal{C}}$ we have the $\mathcal{C}$-colinear morphism $$\Psi _{M}^{\prime }:\mathrm{Hom}^{\mathcal{C}}(A,M)\otimes
_{B}A\longrightarrow M,\text{ }f\otimes _{B}a\mapsto f(a).$$
In this section we characterize $A$ being a generator (resp. a progenerator) in $\mathcal{M}^{\mathcal{C}}$ under the assumption that $_{A}\mathcal{C}$ is locally projective. Our approach is similar to that of [@MZ97] and our results generalize those obtained there for the special case of the category of Doi-Koppinen modules $\mathcal{M}(H)_{A}^{C}.$
\[A-Gen\]Assume $_{A}\mathcal{C}$ to be locally projective. If $_{B}A$ is flat and $A/B$ is $\mathcal{C}$-Galois, then:
1. $A$ is a subgenerator in $\mathcal{M}^{\mathcal{C}},$ i.e. $\sigma
\lbrack A_{^{\ast }\mathcal{C}}]=\sigma \lbrack \mathcal{C}_{^{\ast }%
\mathcal{C}}].$
2. for each $M\in \mathcal{M}^{\mathcal{C}},$ $\Psi _{M}^{\prime }$ is injective.
3. for every $A$-generated $M\in \mathcal{M}^{\mathcal{C}},$ $\Psi
_{M}^{\prime }$ is an isomorphism.
Assume $_{A}\mathcal{C}$ to be locally projective.
1. Since $A/B$ is $\mathcal{C}$-Galois, $\beta ^{\prime }:=\Psi _{%
\mathcal{C}}^{\prime }$ is an isomorphism, hence $\mathcal{C}$ is $A$-generated. Consequently $\sigma \lbrack A_{^{\ast }\mathcal{C}}]\subseteq
\sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}]\subseteq \sigma \lbrack
A_{^{\ast }\mathcal{C}}],$ i.e. $\sigma \lbrack A_{^{\ast }\mathcal{C}%
}]=\sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}].$
2. With slight modifications, the proof of [@MZ97 Lemma 3.22] applies.
3. If $M\in \mathcal{M}^{\mathcal{C}}$ is $A$-generated, then $\Psi
_{M}^{\prime }$ is surjective, hence bijective by (2).$\blacksquare $
The following result is a generalization of (which in turn generalizes [@DT89 Theorem 2.11]):
\[suf-con\]Assume $A/B$ to be $\mathcal{C}$-Galois.
1. If $_{B}A$ is flat, then $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem.
2. Assume there exists $\widehat{q}\in Q,$ such that $\widehat{q}(%
\mathbf{x})=1_{A}.$ If $_{B}A$ is flat, or for all $b\in B$ and $c\in
\mathcal{C}$ we have $\widehat{q}(cb)=q(c)b,$ then $\mathcal{M}^{\mathcal{C}%
} $ satisfies the strong structure theorem.
<!-- -->
1. The proof is the first part of the proof of [@Brz02 Theorem 5.6].
2. By assumption and Corollary \[co=x\], $\Phi _{N}$ is an isomorphism for all $N\in \mathcal{M}_{B}.$ If $_{B}A$ is flat, then $\mathcal{M}^{%
\mathcal{C}}$ satisfies the weak structure theorem by (1). On the other hand, if for all $b\in B$ and $c\in \mathcal{C}$ we have $\widehat{q}%
(cb)=q(c)b,$ then an analog argument to that in the proof of [@Brz99 Proposition 3.13] shows that $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem.$\blacksquare $
\[gen\]Assume $_{A}\mathcal{C}$ to be locally projective. Then the following are equivalent:
1. $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem;
2. $_{B}A$ is flat and $A/B$ is $\mathcal{C}$-Galois;
3. $_{B}A$ is flat and $\beta ^{\prime }:=\Psi _{\mathcal{C}}^{\prime }$ is an isomorphism;
4. $_{B}A$ is flat and for every $A$-generated $M\in \mathcal{M}^{%
\mathcal{C}},$ $\Psi _{M}^{\prime }$ is bijective;
5. for every $M\in \mathcal{M}^{\mathcal{C}}=\sigma \lbrack \mathcal{C}%
_{^{\ast }\mathcal{C}}],$ the $\mathcal{C}$-colinear morphism $\Psi
_{M}^{\prime }$ is bijective;
6. $\sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}]=\mathrm{Gen}%
(A_{^{\ast }\mathcal{C}});$
7. $_{B}A$ is flat, $\sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}%
}]=\sigma \lbrack A_{^{\ast }\mathcal{C}}]$ and $\mathrm{Hom}_{-^{\ast }%
\mathcal{C}}(A,-):\mathrm{Gen}(A_{^{\ast }\mathcal{C}})\longrightarrow
\mathcal{M}_{B}$ is full faithful;
8. $\mathrm{Hom}^{\mathcal{C}}(A,-):\mathcal{M}^{\mathcal{C}%
}\longrightarrow \mathcal{M}_{B}$ is faithful;
9. $A$ is a generator in $\mathcal{M}^{\mathcal{C}}.$
\(1) $\Longleftrightarrow $ (5) $\&\;$(2) $\Longleftrightarrow $ (3) follow by Lemma \[Psi-pr\]. The equivalences (4) $\Longleftrightarrow $ (5) $%
\Longleftrightarrow $ (6) $\Longleftrightarrow $ (7) follow by [@MZ97 Theorem 2.3]. The equivalence (8) $\Longleftrightarrow $ (9) is evident for any category, and moreover (6) $\Longleftrightarrow $ (9) by the fact that $%
\mathrm{Gen}(A_{^{\ast }\mathcal{C}})\subseteq \sigma \lbrack A_{^{\ast }%
\mathcal{C}}]\subseteq \sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}]=%
\mathcal{M}^{\mathcal{C}}.$ By Lemma \[A-Gen\] we have (3) $\Rightarrow $ (4). Now assuming (1) we conclude that $A/B$ is $\mathcal{C}$-Galois and that $_{B}A$ is flat (since (1) $\Longleftrightarrow $ (5) $%
\Longleftrightarrow $ (7)), hence (1) $\Rightarrow $ (2) follows and we are done.$\blacksquare $
([@MZ97 Definition 2.4]) A left module $P$ over a ring $\mathcal{S}$ is called a **weak generator**, if for any right $\mathcal{S}$-module $%
Y,$ $Y\otimes _{\mathcal{S}}P=0$ implies $Y=0.$ A right module $P$ over a ring $\mathcal{R}$ is called **quasiprogenerator** (resp. **progenerator**), if $P_{\mathcal{R}}$ is f.g. quasiprojective and generates each of its submodules (resp. $P_{\mathcal{R}}$ is f.g., projective and a generator). $P_{\mathcal{R}}$ is called **faithful** (resp. **balanced**), if the canonical morphism $\mathcal{R}\longrightarrow \mathrm{End%
}(_{\mathrm{End}(P_{\mathcal{R}})}P)^{op}$ is injective (resp. surjective).
\[prog\]Assume $_{A}\mathcal{C}$ to be flat. Then the following are equivalent:
1. $\mathcal{M}^{\mathcal{C}}$ satisfies the strong structure theorem;
2. $_{B}A$ is faithfully flat and $A/B$ is $\mathcal{C}$-Galois.
If moreover $_{A}\mathcal{C}$ is locally projective, then *(1)* $\&$ *(2)* are moreover equivalent to:
3. $_{B}A$ is faithfully flat and $\beta ^{\prime }:=\Psi _{\mathcal{C}%
}^{\prime }$ is bijective;
4. $_{B}A$ is faithfully flat and for every $M\in \sigma \lbrack
A_{^{\ast }\mathcal{C}}],$ $\Psi _{M}^{\prime }$ is bijective;
5. $A_{^{\ast }\mathcal{C}}$ is quasiprojective and generates each of its submodules, $_{B}A$ is a weak generator and $\sigma \lbrack \mathcal{C}%
_{^{\ast }\mathcal{C}}]=\sigma \lbrack A_{^{\ast }\mathcal{C}}];$
6. $A_{^{\ast }\mathcal{C}}$ is a quasiprogenerator and $\sigma \lbrack
\mathcal{C}_{^{\ast }\mathcal{C}}]=\sigma \lbrack A_{^{\ast }\mathcal{C}}];$
7. $_{B}A$ is a weak generator, $\Psi _{M}^{\prime }$ is an isomorphism for every $M\in \mathrm{Gen}(A_{^{\ast }\mathcal{C}})$ and $\sigma \lbrack
A_{^{\ast }\mathcal{C}}]=\sigma \lbrack \mathcal{C}_{^{\ast }\mathcal{C}}];$
8. $\mathrm{Hom}^{\mathcal{C}}(A,-):\mathcal{M}^{\mathcal{C}%
}\longrightarrow \mathcal{M}_{B}$ is an equivalence;
9. $A$ is a progenerator in $\mathcal{M}^{\mathcal{C}}.$
(1) $\Longleftrightarrow $(2)$\;$is [@Brz02 Theorem 5.6]. Assume $_{A}%
\mathcal{C}$ to be locally projective. Then (2) $\Longleftrightarrow $ (3)follows by Lemma \[Psi-pr\] and we get (1) $\Longleftrightarrow $ (8) $%
\Longleftrightarrow $ (9) by characterizations of progenerators in categories of type $\sigma \lbrack M]$ (see [@Wis88 18.5, 46.2]). Moreover (4) $\Longleftrightarrow $ (5) $\Longleftrightarrow $ (6) $%
\Longleftrightarrow $ (7) follow from [@MZ97 Theorem 2.5]. Obviously (3) $\Rightarrow $ (4) (note that (3) $\Longleftrightarrow $ (2) $%
\Longleftrightarrow $ (1)). Assume now (4). Then $_{B}A$ is faithfully flat and moreover $\Psi _{\mathcal{C}}^{\prime }$ is bijective, since $\mathcal{C}%
\in \sigma \lbrack A_{^{\ast }\mathcal{C}}]$ by (6), i.e. (4) $\Rightarrow $ (3) and the proof is complete.$\blacksquare $
Assume $_{A}\mathcal{C}$ to be locally projective. Then $\func{Im}%
(F)\subseteq $ $^{\Box }\mathcal{C}.$ In fact we have for all $q\in Q,$ $%
a\in A,$ $g\in $ $^{\ast }\mathcal{C}$ and $c\in C:$ $$((qa)\cdot g)(c)=\sum g(c_{1}q(c_{2})a)=g(q(c)\mathbf{x}a)=q(c)g(\mathbf{x}%
a)=(qg(\mathbf{x}a)(c),$$ hence $qa\in $ $^{\Box }\mathcal{C},$ with $\varrho (qa)=q\otimes _{A}%
\mathbf{x}a.$
Assume $_{A}\mathcal{C}$ to be locally projective and that there exists $%
\widehat{q}\in Q$ with $\widehat{q}(\mathbf{x})=1_{A}$ *(*equivalently $%
G:A\otimes _{^{\ast }\mathcal{C}}Q\longrightarrow $ $B$ is surjective*)*. Then $\mathcal{M}^{\mathcal{C}}$ satisfies the strong equivalence theorem, iff $\func{Im}(F)=$ $^{\Box }\mathcal{C}$ and the following map is surjective for every $M\in \mathcal{M}^{\mathcal{C}}$$$\varpi _{M}:M\otimes _{^{\ast }\mathcal{C}}\text{ }^{\Box }\mathcal{C}%
\longrightarrow M,\text{ }m\otimes _{^{\ast }\mathcal{C}}f\mapsto mf.$$ In this case $Q\otimes _{B}A\overset{F}{\simeq }$ $^{\Box }\mathcal{C}$ and $%
M\otimes _{^{\ast }\mathcal{C}}$ $^{\Box }\mathcal{C}\overset{\varpi _{M}}{%
\simeq }M$ for every $M\in \mathcal{M}^{\mathcal{C}}.$
Consider for every $M\in \mathcal{M}^{\mathcal{C}}$ the commutative diagram $$\xymatrix{M \otimes_{\mathcal{^*{C}}} Q \otimes_B A \ar[rr]^(.55){\xi _{M}
\otimes id_A} \ar[d]_{id_M \otimes F} & & M^{co{\mathcal{C}}} \otimes_B A
\ar[d]^{\Psi_M} \\ M \otimes_{^*{\mathcal{C}}} {}^{\Box }{\mathcal{C}}
\ar[rr]_{\varpi_M} & & M }$$
Assume $\func{Im}(F)=$ $^{\Box }\mathcal{C}$ and $\varpi _{M}$ to be surjective for every $M\in \mathcal{M}^{\mathcal{C}}.$ Then $\Psi _{M}$ is obviously surjective. Let $K=\mathrm{Ke}(\Psi _{M}).$ Since $\Psi _{M}$ is a morphism in $\mathcal{M}^{\mathcal{C}}\simeq \sigma \lbrack \mathcal{C}%
_{^{\ast }\mathcal{C}}]$ we have $K\in \mathcal{M}^{\mathcal{C}},$ hence $%
\Psi _{K}:K^{co\mathcal{C}}\otimes _{B}A\longrightarrow K$ is surjective. By Theorem \[surj\] we have $K\otimes _{^{\ast }\mathcal{C}}Q\overset{\xi _{K}%
}{\simeq }K^{co\mathcal{C}}$ and $A\otimes _{^{\ast }\mathcal{C}}Q\overset{%
\xi _{A}}{\simeq }B,$ hence $$K^{co\mathcal{C}}\simeq K\otimes _{^{\ast }\mathcal{C}}Q=\mathrm{Ke}(\Psi
_{M})\otimes _{^{\ast }\mathcal{C}}Q=\mathrm{Ke}(\Psi _{M}\otimes _{^{\ast }%
\mathcal{C}}id_{Q})=\mathrm{Ke}(id_{M^{co\mathcal{C}}})=0,$$ i.e. $\Psi _{M}$ is bijective. By corollary \[co=x\] $\Phi _{N}$ is bijective for every $N\in \mathcal{M}_{B}.$ Consequently $\mathcal{M}^{%
\mathcal{C}}$ satisfies the strong structure theorem.
On the other hand, assume that $\mathcal{M}^{\mathcal{C}}$ satisfies the strong structure theorem. Note that $F$ is the adjunction of $\Psi _{^{\Box }%
\mathcal{C}},$ hence $Q\otimes _{B}A\overset{F}{\simeq }$ $^{\Box }\mathcal{C%
}$ and consequently $\varpi _{M}$ is also bijective for every $M\in \mathcal{%
M}^{\mathcal{C}}$ by the commutativity of the above diagram.$\blacksquare $
Assume $_{A}\mathcal{C}$ to be locally projective.
1. $\varpi _{A}:A\otimes _{^{\ast }\mathcal{C}}$ $^{\Box }\mathcal{C}%
\longrightarrow A$ is surjective iff there exists $\widehat{g}\in $ $^{\Box }%
\mathcal{C}$ with $\widehat{g}(\mathbf{x})=1_{A}.$ To prove this assume first that $\varpi _{A}$ is surjective. Then there exist $%
\{(a_{i},g_{i})\}_{i=1}^{k}\subset A\times $ $^{\Box }\mathcal{C},$ such that $\sum\limits_{i=1}^{k}a_{i}\leftharpoonup g_{i}=1_{A}.$ Set $\widehat{g}%
:=\sum\limits_{i=1}^{k}a_{i}g_{i}\in $ $^{\Box }\mathcal{C}.$ Then $\widehat{%
g}(\mathbf{x})=(\sum\limits_{i=1}^{k}a_{i}g_{i})(\mathbf{x}%
)=\sum\limits_{i=1}^{k}g_{i}(\mathbf{x}a_{i})=\sum\limits_{i=1}^{k}a_{i}%
\leftharpoonup g_{i}=1_{A}.$ On the other hand, assume there exists $%
\widehat{g}\in $ $^{\Box }\mathcal{C}$ with $\widehat{g}(\mathbf{x})=1_{A}.$ Then for every $a\in A$ we have $1_{A}\leftharpoonup (\widehat{g}a)=(%
\widehat{g}a)(\mathbf{x})=\widehat{g}(\mathbf{x})a=a,$ i.e. $\varpi _{A}$ is surjective.
2. Assume $\varpi _{A}$ to be surjective. If $\Psi _{M}$ is surjective for $M\in \mathcal{M}^{\mathcal{C}},$ then $\varpi _{M}$ is surjective, since $$\varpi _{M}\circ (\Psi _{M}\otimes _{^{\ast }\mathcal{C}}id_{^{\Box }%
\mathcal{C}})=\Psi _{M}\circ (id_{M^{\mathbf{co}\mathcal{C}}}\otimes
_{B}\varpi _{A}).$$
\[fin-gen\]Assume $_{A}\mathcal{C}$ to be f.g. and projective. Then the following are equivalent:
1. $\mathcal{M}^{\mathcal{C}}$ satisfies the weak structure theorem;
2. $_{B}A$ is flat and $A/B$ is $\mathcal{C}$-Galois;
3. $_{B}A$ is flat and $\beta ^{\prime }:=\Psi _{\mathcal{C}}^{\prime }$ is an isomorphism;
4. $_{B}A$ is flat and for every $A$-generated $M\in \mathcal{M}^{%
\mathcal{C}}=\mathcal{M}_{^{\ast }\mathcal{C}},$ the $\mathcal{C}$-colinear morphism $\Psi _{M}^{\prime }$ is bijective;
5. for every $M\in \mathcal{M}^{\mathcal{C}},$ the $\mathcal{C}$-colinear morphism $\Psi _{M}^{\prime }$ is bijective;
6. $_{B}A$ is flat, $\mathcal{M}_{^{\ast }\mathcal{C}}=\sigma \lbrack
A_{^{\ast }\mathcal{C}}]$ and $\mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,-):%
\mathrm{Gen}(A_{^{\ast }\mathcal{C}})\longrightarrow \mathcal{M}_{B}$ is full faithful;
7. $\mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,-):\mathcal{M}_{^{\ast }%
\mathcal{C}}\longrightarrow \mathcal{M}_{B}$ is faithful;
8. $A_{^{\ast }\mathcal{C}}$ is a generator;
9. $F:Q\otimes _{B}A\longrightarrow $ $^{\ast }\mathcal{C}$ is surjective *(*bijective*)*;
10. (a) $Q_{B}$ is f.g. and projective;
\(b) $\Omega :A\longrightarrow \mathrm{Hom}_{-B}(Q,B),$ $a\mapsto \lbrack
q\mapsto a\leftharpoonup q]$ is a bimodule isomorphism;
\(c) $_{^{\ast }\mathcal{C}}Q$ is faithful;
11. (a)$\;_{B}A$ is f.g. and projective;
(b) $\Lambda :$ $^{\ast }\mathcal{C}\longrightarrow \mathrm{End}%
(_{B}A)^{op},$ $g\mapsto \lbrack a\mapsto a\leftharpoonup g]$ is a ring isomorphism.
The result follows by Theorems \[C -finite\], \[gen\] and the fact that in case $_{A}\mathcal{C}$ is f.g. and projective $\mathcal{M}^{\mathcal{C}}=%
\mathcal{M}_{^{\ast }\mathcal{C}}=\sigma \lbrack \mathcal{C}_{^{\ast }%
\mathcal{C}}].\blacksquare $
\[morita\]*(Morita, e.g. [@Fai81 4.1.3, 4.3], [@MZ97 2.6]).* Let $\mathcal{R}$ be a ring, $P$ a right $\mathcal{R}$-module, $\mathcal{%
S}:=\mathrm{End}(P_{\mathcal{R}})$ and $P^{\ast }:=\mathrm{Hom}_{\mathcal{R}%
}(P,\mathcal{R}).$
1. The following are equivalent:
1. $P_{\mathcal{R}}$ is a generator;
2. $_{\mathcal{S}}P$ is f.g. projective and $\mathcal{R}\simeq \mathrm{%
End}(_{\mathcal{S}}P)^{op}$ canonically.
2. The following are equivalent:
1. $P_{\mathcal{R}}$ is a faithful quasiprogenerator and $_{\mathcal{S}%
}P $ is finitely generated;
2. $P_{\mathcal{R}}$ is a progenerator;
3. $_{\mathcal{S}}P$ is a progenerator and $P_{\mathcal{R}}$ is faithfully balanced;
4. $P_{\mathcal{R}}$ and $_{\mathcal{S}}P$ are generators;
5. $P_{\mathcal{R}}$ and $_{\mathcal{S}}P$ are f.g. and projective;
6. $\mathrm{Hom}_{-\mathcal{R}}(P,-):\mathcal{M}_{\mathcal{R}%
}\longrightarrow \mathcal{M}_{\mathcal{S}}$ is an equivalence with inverse $%
\mathrm{Hom}_{-\mathcal{S}}(P^{\ast },-);$
7. $-\otimes _{\mathcal{R}}P^{\ast }:\mathcal{M}_{\mathcal{R}%
}\longrightarrow \mathcal{M}_{\mathcal{S}}$ is an equivalence with inverse $%
-\otimes _{\mathcal{S}}P.$
As a consequence of Theorems \[prog\] and \[morita\] we get
\[fin-prog\]Assume $_{A}\mathcal{C}$ to be f.g. and projective. Then the following are equivalent:
1. $\mathcal{M}^{\mathcal{C}}$ satisfies the strong structure theorem;
2. $_{B}A$ is faithfully flat and $A/B$ is $\mathcal{C}$-Galois;
3. $_{B}A$ is faithfully flat and $\beta ^{\prime }:=\Psi _{\mathcal{C}%
}^{\prime }$ is bijective;
4. $_{B}A$ is faithfully flat and for every $M\in \sigma \lbrack
A_{^{\ast }\mathcal{C}}],$ the map $\Psi _{M}^{\prime }$ is bijective;
5. $A_{^{\ast }\mathcal{C}}$ is quasiprojective and generates each of its submodules, $_{B}A$ is a weak generator and $\mathcal{M}_{^{\ast }%
\mathcal{C}}=\sigma \lbrack A_{^{\ast }\mathcal{C}}];$
6. $A_{^{\ast }\mathcal{C}}$ is a quasiprogenerator and $\mathcal{M}%
_{^{\ast }\mathcal{C}}=\sigma \lbrack A_{^{\ast }\mathcal{C}}];$
7. $_{B}A$ is a weak generator, $\Psi _{M}^{\prime }$ is an isomorphism for every $M\in \mathrm{Gen}(A_{^{\ast }\mathcal{C}})$ and $\mathcal{M}%
_{^{\ast }\mathcal{C}}=\sigma \lbrack A_{^{\ast }\mathcal{C}}];$
8. $A_{^{\ast }\mathcal{C}}$ is a faithful quasiprogenerator and $_{B}A$ is finitely generated;
9. $_{B}A$ is a progenerator and $A_{^{\ast }\mathcal{C}}$ is faithfully balanced;
10. $\mathrm{Hom}_{-^{\ast }\mathcal{C}}(A,-):\mathcal{M}_{^{\ast }%
\mathcal{C}}\longrightarrow \mathcal{M}_{B}$ is an equivalence with inverse $%
\mathrm{Hom}_{-B}(Q,-);$
11. $-\otimes _{^{\ast }\mathcal{C}}Q:\mathcal{M}_{^{\ast }\mathcal{C}%
}\longrightarrow \mathcal{M}_{B}$ is an equivalence with inverse $-\otimes
_{B}A;$
12. $A_{^{\ast }\mathcal{C}}$ and $_{B}A$ are generators;
13. $A_{^{\ast }\mathcal{C}}$ and $_{B}A$ are f.g. and projective;
14. $A_{^{\ast }\mathcal{C}}$ is a progenerator.
Cleft $C$-Galois Extensions
===========================
In what follows $R$ is a *commutative* ring with $1_{R}\neq 0_{R}$ and $%
\mathcal{M}_{R}$ is the category of $R$-(bi)modules. For an $R$-coalgebra $%
(C,\Delta _{C},\varepsilon _{C})$ and an $R$-algebra $(A,\mu _{A},\eta _{A})$ we consider $(\mathrm{Hom}_{R}(C,A),\star ):=\mathrm{Hom}_{R}(C,A)$ as an $R$-algebra with the *usual convolution product* $(f\star g)(c):=\sum
f(c_{1})g(c_{2})$ and unity $\eta _{A}\circ \varepsilon _{C}.$ The unadorned $-\otimes -$ means $-\otimes _{R}-.$
**Entwined modules.**\[ent-str\] A right-right **entwining structure** $(A,C,\psi )$ over $R$ consists of an $R$-algebra $(A,\mu
_{A},\eta _{A}),$ an $R$-coalgebra $(C,\Delta _{C},\varepsilon _{C})$ and an $R$-linear map $$\psi :C\otimes _{R}A\longrightarrow A\otimes _{R}C,\text{ }c\otimes a\mapsto
\sum a_{\psi }\otimes c^{\psi },$$ such that $$\begin{tabular}{llllll}
$\sum (a\widetilde{a})_{\psi }\otimes c^{\psi }$ & $=$ & $\sum a_{\psi }%
\widetilde{a}_{\Psi }\otimes c^{\psi \Psi },$ & $\sum (1_{A})_{\psi }\otimes
c^{\psi }$ & $=$ & $1_{A}\otimes c,$ \\
$\sum a_{\psi }\otimes \Delta _{C}(c^{\psi })$ & $=$ & $\sum a_{\psi \Psi
}\otimes c_{1}^{\Psi }\otimes c_{2}^{\psi },$ & $\sum a_{\psi }\varepsilon
_{C}(c^{\psi })$ & $=$ & $\varepsilon _{C}(c)a.$%
\end{tabular}$$
Let $(A,C,\psi )$ be a right-right entwining structure. An **entwined module** corresponding to $(A,C,\psi )$ is a right $A$-module, which is also a right $C$-comodule through $\varrho _{M},$ such that $$\varrho _{M}(ma)=\sum m_{<0>}a_{\psi }\otimes m_{<1>}^{\psi }\text{ for all }%
m\in M\text{ and }a\in A.$$ The category of right-right entwined modules and $A$-linear $C$-colinear morphisms is denoted by $\mathcal{M}_{A}^{C}(\psi ).$ For $M,N\in \mathcal{M}%
_{A}^{C}(\psi )$ we denote by $\mathrm{Hom}_{A}^{C}(M,N)$ the set of $A$-linear $C$-colinear morphisms from $M$ to $N.$ With $\#_{\psi }^{op}(C,A):=%
\mathrm{Hom}_{R}(C,A),$ we denote the $A$-ring with $(af)(c)=\sum a_{\psi
}f(c^{\psi }),$ $(fa)(c)=f(c)a,$ multiplication $(f\cdot g)(c)=\sum
f(c_{2})_{\psi }g(c_{1}^{\psi })$ and unity $\eta _{A}\circ \varepsilon _{C}$ (see [@Abu03 Lemma 3.3]).
Entwined modules were introduced by T. Brzeziński and S. Majid [@BM98] as a generalization of the Doi-Koppinen modules presented in [@Doi92] and [@Kop95]. By a remark of M. Takeuchi (e.g. [@Brz02 Proposition 2.2]), we have an $A$-coring structure on $\mathcal{C}:=A\otimes
_{R}C,$ where $\mathcal{C}$ is an $A$-bimodule through $a(\widetilde{a}%
\otimes c):=a\widetilde{a}\otimes c,$ $(\widetilde{a}\otimes c)a:=\sum
\widetilde{a}a_{\psi }\otimes c^{\psi }$ and has comultiplication $$\Delta _{\mathcal{C}}:A\otimes _{R}C\longrightarrow (A\otimes _{R}C)\otimes
_{A}(A\otimes _{R}C),\text{ }a\otimes c\mapsto \sum (a\otimes c_{1})\otimes
_{A}(1_{A}\otimes c_{2})$$ and counity $\varepsilon _{\mathcal{C}}:=id_{A}\otimes \varepsilon _{C}.$ Moreover $\mathcal{M}_{A}^{C}(\psi )\simeq \mathcal{M}^{\mathcal{C}},$ $%
\#_{\psi }^{op}(C,A)\simeq $ $^{\ast }\mathcal{C}$ as $A$-rings and $_{A}%
\mathcal{C}$ is flat (resp. f.g., projective), if $_{R}C$ is so (e.g. [@Abu03]).
Inspired by [@Doi94 3.1] we make the following definition:
\[s-rat\]Let $(A,C,\psi )$ be a right-right entwining structure over $R$ and consider the corresponding $A$-coring $\mathcal{C}:=A\otimes _{R}C.$ We say that $(A,C,\psi )$ satisfies the **left** $\alpha $**-condition**, if for every right $A$-module $M$ the following map is injective $$\alpha _{M}^{\psi }:M\otimes _{R}C\longrightarrow \mathrm{Hom}_{R}(\#_{\psi
}^{op}(C,A),M),\text{ }m\otimes c\mapsto \lbrack f\mapsto mf(c)]$$ (equivalently, if $_{A}\mathcal{C}$ is locally projective).
Let $M$ be a right $\#_{\psi }^{op}(C,A)$-module $M$ and consider the canonical map $\rho _{M}:M\longrightarrow \mathrm{Hom}_{R}(\#_{\psi
}^{op}(C,A),M).$ Set $\mathrm{Rat}^{C}(M_{\#_{\psi }^{op}(C,A)}):=(\rho
_{M}^{\psi })^{-1}(M\otimes _{R}C).$ We call $M$ $\#$**-rational**, if $%
\mathrm{Rat}^{C}(M_{\#_{\psi }^{op}(C,A)})=M$ and set $\varrho _{M}:=(\alpha
_{M}^{\psi })^{-1}\circ \rho _{M}.$ The category of $\#$-rational right $%
\#_{\psi }^{op}(C,A)$-modules will be denoted by $\mathrm{Rat}^{C}(\mathcal{M%
}_{\#_{\psi }^{op}(C,A)}).$
\[ent-sg\]*([@Abu03 Theorem 3.10])* Let $(A,C,\psi )$ be a right-right entwining structure and consider the corresponding $A$-coring $%
\mathcal{C}:=A\otimes _{R}C.$
1. If $_{R}C$ is flat, then $\mathcal{M}_{A}^{C}(\psi )$ is a Grothendieck category with enough injective objects.
2. If $_{R}C$ is locally projective *(*resp. f.g. and projective*)*, then $$\mathcal{M}_{A}^{C}(\psi )\simeq \mathrm{Rat}^{C}(\mathcal{M}_{\#_{\psi
}^{op}(C,A)})\simeq \sigma \lbrack (A\otimes _{R}C)_{\#_{\psi }^{op}(C,A)}]\;%
\text{\emph{(}resp. }\mathcal{M}_{A}^{C}(\psi )\simeq \mathcal{M}_{\#_{\psi
}^{op}(C,A)}\text{\emph{)}.} \label{iso-sm}$$
In what follows we fix a right-right entwining structure $(A,C,\psi )$ with $%
\mathcal{C}:=A\otimes _{R}C$ the corresponding $A$-coring and assume that $%
A\in \mathcal{M}_{A}^{C}(\psi )\simeq \mathcal{M}^{\mathcal{C}}$ with $$\varrho _{A}:A\longrightarrow A\otimes _{R}C,\text{ }a\mapsto \sum
a_{<0>}\otimes a_{<1>}=\sum 1_{<0>}a_{\psi }\otimes 1_{<1>}^{\psi }.$$ Then $\sum 1_{<0>}\otimes 1_{<1>}\in \mathcal{C}$ is a group-like element and $$Q\simeq \{q\in \mathrm{Hom}_{R}(C,A)\mid \sum q(c_{2})_{\psi }\otimes
c_{1}^{\psi }=\sum q(c)1_{<0>}\otimes 1_{<1>}\text{ for all }c\in C\}.$$ For every $M\in \mathcal{M}_{A}^{C}(\psi ),$ we set $$M^{co\mathcal{C}}:=\{m\in M\mid \sum m_{<0>}\otimes m_{<1>}=\sum
m1_{<0>}\otimes 1_{<1>}\}.$$ Moreover we set $B:=A^{co\mathcal{C}}.$
Let $x\in C$ be a group-like element. For every right $C$-comodule $M$ we put $M^{coC}:=\{m\in M\mid \varrho _{M}(m)=m\otimes x\}.$ If $\varrho
_{A}(1_{A})=1_{A}\otimes x,$ then we have $M^{coC}=M^{co\mathcal{C}}$ for every $M\in \mathcal{M}_{A}^{C}(\psi ).$
By [@Brz99 Corollaries 3.4, 3.7] $-\otimes _{R}^{c}A:\mathcal{M}%
^{C}\longrightarrow \mathcal{M}_{A}^{C}(\psi )$ is a functor, which is left adjoint to the forgetful functor. Here, for every $N\in \mathcal{M}^{C},$ we consider the canonical right $A$-module $N\otimes _{R}^{c}A:=N\otimes _{R}A$ with the $C$-coaction $n\otimes a\mapsto \sum n_{<0>}\otimes a_{\psi
}\otimes n_{<1>}^{\psi }.$
\[sp\]Let $R$ be a QF ring and assume $C$ be right semiperfect. Let $%
_{R}C$ to be locally projective (projective) and put $C^{\Box }:=\mathrm{Rat}%
(_{^{\ast }C}C^{\ast }).$
1. The following are equivalent:
1. $A$ is a generator in $\mathcal{M}_{A}^{C}(\psi );$
2. $A$ generates $C^{\Box }\otimes _{R}^{c}A$ in $\mathcal{M}%
_{A}^{C}(\psi );$
3. the map $\Psi _{C^{\Box }\otimes _{R}^{c}A}^{\prime }:\mathrm{Hom}%
_{A}^{C}(A,C^{\Box }\otimes _{R}^{c}A)\otimes _{B}A\longrightarrow C^{\Box
}\otimes _{R}^{c}A$ is surjective *(*bijective*)*.
2. The following are equivalent:
1. $A$ is a progenerator in $\mathcal{M}_{A}^{C}(\psi );$
2. $\Psi _{C^{\Box }\otimes _{R}^{c}A}^{\prime }$ is surjective *(*bijective*)* and $_{B}A$ is a weak generator.
By [@MTW01 2.6] $C^{\Box }$ is a generator in $\mathcal{M}^{C},$ hence $%
C^{\Box }\otimes _{R}^{c}A$ is a generator in $\mathcal{M}_{A}^{C}(\psi )$ by the functorial isomorphism $\mathrm{Hom}_{A}^{C}(C^{\Box }\otimes
_{R}^{c}A,M)\simeq \mathrm{Hom}^{C}(C^{\Box },M)$ for every $M\in \mathcal{M}%
_{A}^{C}(\psi ).$
1. The assertions follow form the note above and Theorem \[gen\].
2. \(a) $\Rightarrow $ (b) follows by Theorem \[prog\].
(b) $\Rightarrow $ (a). By the note above $C^{\Box }\otimes _{R}^{c}A$ is a generator in $\mathcal{M}_{A}^{C}(\psi ),$ and the surjectivity of $\Psi
_{C^{\Box }\otimes _{R}^{c}A}^{\prime }$ makes $A$ a generator in $\mathcal{M%
}_{A}^{C}(\psi ).$ So $_{B}A$ is flat by Theorem \[gen\]. The weak generator property makes $_{B}A$ faithfully flat and we are done by Theorem \[prog\].$\blacksquare $
A **(total) integral** for $C$ is a $C$-colinear morphism $\lambda
:C\longrightarrow A$ (with $\sum 1_{<0>}\lambda (1_{<1>})=1_{A}$). We call the ring extension $A/B$ **cleft,** if there exists a $\star $**-invertible integral**. We say $A$ has the **right normal basis property**, if there exists a left $B$-linear right $C$-colinear isomorphism $A\simeq
B\otimes _{R}C.$
\[co-Q\]Let $\lambda \in \mathrm{Hom}_{R}(C,A)$ be $\star $-invertible with inverse $\overline{\lambda }.$ Then:
1. $\lambda \in \mathrm{Hom}^{C}(C,A)$ iff $\overline{\lambda }\in Q.$
2. If $\varrho (a)=\sum a_{\psi }\otimes x^{\psi }$ for some group-like element $x\in C,$ then there exists $\widehat{\lambda }\in Q,$ such that $%
\sum 1_{<0>}\widehat{\lambda }(1_{<1>})=\widehat{\lambda }(x)=1_{A}$ *(*in this case $C$ admits a total integral, namely the $\star $-inverse of $%
\widehat{\lambda }$*)*.
Let $\lambda \in \mathrm{Hom}_{R}(C,A)$ be $\star $-invertible with inverse $%
\overline{\lambda }.$
1. If $\overline{\lambda }\in Q,$ then we have for all $c\in C:$$$\begin{tabular}{lllll}
$\sum \lambda (c_{1})\otimes c_{2}$ & $=$ & $\sum \lambda (c_{1})1_{\psi
}\otimes c_{2}^{\psi }$ & & \\
& $=$ & $\sum \lambda (c_{1})\varepsilon (c_{3})1_{\psi }\otimes c_{2}^{\psi
}$ & & \\
& $=$ & $\sum \lambda (c_{1})(\overline{\lambda }(c_{3})\lambda
(c_{4}))_{\psi }\otimes c_{2}^{\psi }$ & & \\
& $=$ & $\sum \lambda (c_{1})\overline{\lambda }(c_{3})_{\psi }\lambda
(c_{4})_{\Psi }\otimes c_{2}^{\psi \Psi }$ & & \\
& $=$ & $\sum \lambda (c_{1})\overline{\lambda }(c_{22})_{\psi }\lambda
(c_{3})_{\Psi }\otimes c_{21}^{\psi \Psi }$ & & \\
& $=$ & $\sum \lambda (c_{1})\overline{\lambda }(c_{2})1_{<0>}\lambda
(c_{3})_{\Psi }\otimes 1_{<1>}^{\Psi }$ & & \\
& $=$ & $\sum 1_{<0>}\lambda (c)_{\Psi }\otimes 1_{<1>}^{\Psi }$ & & \\
& $=$ & $\sum \lambda (c)_{<0>}\otimes \lambda (c)_{<1>}$ & &
\end{tabular}$$ i.e. $\lambda \in \mathrm{Hom}^{C}(C,A).$ On the other hand, if $\lambda \in
\mathrm{Hom}^{C}(C,A),$ then we have for all $c\in C:$$$\begin{tabular}{lllll}
$\sum \overline{\lambda }(c_{2})_{\psi }\otimes c_{1}^{\psi }$ & $=$ & $\sum
\overline{\lambda }(c_{1})\lambda (c_{2})\overline{\lambda }(c_{4})_{\psi
}\otimes c_{3}^{\psi }$ & & \\
& $=$ & $\sum \overline{\lambda }(c_{1})\lambda (c_{2})_{<0>}\overline{%
\lambda }(c_{3})_{\psi }\otimes \lambda (c_{2})_{<1>}^{\psi }$ & & \\
& $=$ & $\sum \overline{\lambda }(c_{1})1_{<0>}\lambda (c_{2})_{\psi }%
\overline{\lambda }(c_{3})_{\Psi }\otimes 1_{<1>}^{\psi \Psi }$ & & \\
& $=$ & $\sum \overline{\lambda }(c_{1})1_{<0>}(\lambda (c_{2})\overline{%
\lambda }(c_{3}))_{\psi }\otimes 1_{<1>}^{\psi }$ & & \\
& $=$ & $\sum \overline{\lambda }(c)1_{<0>}1_{\psi }\otimes 1_{<1>}^{\psi }$
& & \\
& $=$ & $\sum \overline{\lambda }(c)1_{<0>}\otimes 1_{<1>},$ & &
\end{tabular}$$ i.e. $\overline{\lambda }\in Q.$
2. Assume $\varrho (a)=\sum a_{\psi }\otimes x^{\psi }$ for some group-like element $x\in C.$ Let $\lambda \in \mathrm{Hom}^{C}(C,A)$ with $%
\overline{\lambda }\in Q$ (see (1)). Then $\widehat{\lambda }:=\overline{%
\lambda }\lambda (x)\in Q,$ since $\lambda (x)\in B,$ and moreover $\sum
1_{<0>}\widehat{\lambda }(1_{<1>})=\widehat{\lambda }(x)=\overline{\lambda }%
(x)\lambda (x)=(\overline{\lambda }\star \lambda )(x)=\varepsilon
_{C}(x)1_{A}=1_{A}.\blacksquare $
\[cleft\]Assume $A/B$ to be cleft.
1. $\mathcal{M}_{A}^{C}(\psi )$ satisfies the weak structure theorem $%
\emph{(}$in particular $A/B$ is $\mathcal{C}$-Galois*)*.
2. For every $M\in \mathcal{M}_{A}^{C}(\psi ),$ the $C$-colinear morphism $$\gamma _{M}:M\longrightarrow M^{co\mathcal{C}}\otimes _{R}C,\text{ }m\mapsto
\sum m_{<0>}\overline{\lambda }\otimes m_{<1>}$$ is an isomorphism.
3. $A$ has the right normal basis property.
4. If $_{R}C$ is faithfully flat, then $\mathcal{M}_{A}^{C}(\psi )$ satisfies the strong structure theorem.
Assume there exists a $\star $-invertible $\lambda \in \mathrm{Hom}^{C}(C,A)$ with inverse $\overline{\lambda }\in Q$ (see Lemma \[co-Q\] (1)).
1. Let $M\in \mathcal{M}_{A}^{C}(\psi )$ and consider $$\widetilde{\Psi }_{M}:M\longrightarrow M^{co\mathcal{C}}\otimes _{B}A,\text{
}m\mapsto \sum m_{<0>}\overline{\lambda }\otimes \lambda (m_{<1>}).$$ Then we have for all $n\in M^{co\mathcal{C}},$ $m\in M$ and $a\in A:$$$\begin{tabular}{lllll}
$(\widetilde{\Psi }_{M}\circ \Psi _{M})(n\otimes a)$ & $=$ & $\widetilde{%
\Psi }_{M}(na)$ & & \\
& $=$ & $\sum (na_{<0>})\overline{\lambda }\otimes _{B}\lambda (a_{<1>})$ &
& \\
& $=$ & $\sum na_{<0><0>}\overline{\lambda }(a_{<0><1>})\otimes _{B}\lambda
(a_{<1>})$ & & \\
& $=$ & $\sum n\otimes _{B}a_{<0><0>}\overline{\lambda }(a_{<0><1>})\lambda
(a_{<1>})$ & & \\
& $=$ & $\sum n\otimes _{B}a_{<0>}\overline{\lambda }(a_{<1>1})\lambda
(a_{<1>2})$ & & \\
& $=$ & $n\otimes _{B}a$ & &
\end{tabular}$$ and $$\begin{tabular}{lllll}
$(\Psi _{M}\circ \widetilde{\Psi }_{M})(m)$ & $=$ & $\sum (m_{<0>}\overline{%
\lambda })\lambda (m_{<1>})$ & & \\
& $=$ & $\sum m_{<0><0>}\overline{\lambda }(m_{<0><1>})\lambda (m_{<1>})$ &
& \\
& $=$ & $\sum m_{<0>}\overline{\lambda }(m_{<1>1})\lambda (m_{<1>2})$ & &
\\
& $=$ & $\sum m_{<0>}\varepsilon _{C}(m_{<1>})1_{A}$ & & \\
& $=$ & $m.$ & &
\end{tabular}$$
2. For every $M\in \mathcal{M}_{A}^{C}(\psi ),$ $\gamma _{M}$ is bijective with inverse $$\widetilde{\gamma }_{M}:M^{co\mathcal{C}}\otimes _{R}C\longrightarrow M,%
\text{ }n\otimes c\mapsto n\lambda (c).$$ In fact we have for all $m\in M,$ $n\in M^{co\mathcal{C}}$ and $c\in C:$$$\begin{tabular}{lllll}
$(\widetilde{\gamma }_{M}\circ \gamma _{M})(m)$ & $=$ & $\sum (m_{<0>}%
\overline{\lambda })\lambda (m_{<1>})$ & & \\
& $=$ & $\sum m_{<0><0>}\overline{\lambda }(m_{<0><1>})\lambda (m_{<1>})$ &
& \\
& $=$ & $\sum m_{<0>}\overline{\lambda }(m_{<1>1})\lambda (m_{<1>2})$ & &
\\
& $=$ & $\sum m_{<0>}\varepsilon _{C}(m_{<1>})$ & & \\
& $=$ & $m$ & &
\end{tabular}$$ and $$\begin{tabular}{lllll}
$(\gamma _{M}\circ \widetilde{\gamma }_{M})(n\otimes _{B}c)$ & $=$ & $\sum
(n\lambda (c))_{<0>}\overline{\lambda }\otimes (n\lambda (c))_{<0>}$ & &
\\
& $=$ & $\sum (n\lambda (c)_{<0>})\overline{\lambda }\otimes \lambda
(c)_{<1>}$ & & \\
& $=$ & $\sum (n\lambda (c_{1}))\overline{\lambda }\otimes c_{2}$ & & \\
& $=$ & $\sum n\lambda (c_{1})_{<0>}\overline{\lambda }(\lambda
(c_{1})_{<1>})\otimes c_{2}$ & & \\
& $=$ & $\sum n\lambda (c_{11})\overline{\lambda }(c_{12})\otimes c_{2}$ &
& \\
& $=$ & $n\otimes c.$ & &
\end{tabular}$$
3. By (2) the left $B$-linear right $C$-colinear map $$\gamma _{A}:A\longrightarrow B\otimes _{R}C,a\mapsto \sum
a_{<0>}\leftharpoonup \overline{\lambda }\otimes a_{<1>}$$ is an isomorphism with inverse $b\otimes c\mapsto b\lambda (c).$
4. Assume $_{R}C$ to be faithfully flat. By (3) $A\simeq B\otimes _{R}C$ as left $B$-modules, hence $_{B}A$ is faithfully flat. By (1) $A/B$ is $%
\mathcal{C}$-Glaois and we are done by Theorem \[prog\].$\blacksquare $
\[main\]The following statements are equivalent:
1. $A/B$ is cleft;
2. $\mathcal{M}_{A}^{C}(\psi )$ satisfies the weak structure theorem and $A$ has the right normal basis property;
3. $A/B$ is $\mathcal{C}$-Galois and $A$ has the right normal basis property;
4. $\Lambda :\#_{\psi }^{op}(C,A)\simeq \mathrm{End}(_{B}A)^{op},$ $%
g\mapsto \lbrack a\mapsto a\leftharpoonup $ $g]$ is a ring isomorphism and $%
A $ has the right normal basis property.
If moreover $_{R}C$ is faithfully flat, then *(1)-(4)* are equivalent to
5. $\mathcal{M}_{A}^{C}(\psi )$ satisfies the strong structure theorem and $A$ has the right normal basis property.
(1) $\Rightarrow $ (2). This follows by Proposition \[cleft\].
\(2) $\Rightarrow $ (3). By assumption $\beta :=\Psi _{A\otimes _{R}C}$ is an isomorphism.
\(3) $\Rightarrow $ (4). By assumption $A\otimes _{B}A\simeq A\otimes _{R}C$ as left $A$-modules, hence we have the canonical isomorphisms $$\begin{tabular}{lllll}
$\#_{\psi }^{op}(C,A)$ & $\simeq $ & $\mathrm{Hom}_{A-}(A\otimes _{R}C,A)$ &
$\simeq $ & $\mathrm{Hom}_{A-}(A\otimes _{B}A,A)$ \\
& $\simeq $ & $\mathrm{Hom}_{B-}(A,\mathrm{End}(_{A}A))$ & $\simeq $ & $%
\mathrm{End}(_{B}A).$%
\end{tabular}$$
(4) $\Rightarrow $ (1). Assume $\theta :B\otimes _{R}C\longrightarrow A$ to be a left $B$-linear right $C$-colinear isomorphism and consider the right $%
C $-colinear morphism $\lambda :C\longrightarrow A,$ $c\mapsto \theta
(1_{A}\otimes c)$ and the left $B$-linear morphism $\delta :=(id\otimes
\varepsilon _{C})\circ \theta ^{-1}:A\longrightarrow B.$ Define $\overline{%
\lambda }:=\Lambda ^{-1}(\delta )\in \#_{\psi }^{op}(C,A).$ Then we have for all $c\in C:$$$\begin{tabular}{lllll}
$\sum \lambda (c_{1})\overline{\lambda }(c_{2})$ & $=$ & $\sum \lambda
(c)_{<0>}\overline{\lambda }(\lambda (c)_{<1>})$ & $=$ & $\lambda
(c)\leftharpoonup \overline{\lambda }$ \\
& $=$ & $\delta (\lambda (c))$ & $=$ & $((id\otimes \varepsilon _{C})\circ
\theta ^{-1})(\lambda (c))$ \\
& $=$ & $((id\otimes \varepsilon _{C})\circ \theta ^{-1})(\theta
(1_{A}\otimes c))$ & $=$ & $\varepsilon _{C}(c)1_{A}.$%
\end{tabular}$$ On the other hand we have for all $a\in A:$$$\begin{tabular}{lllll}
$\Lambda (\overline{\lambda }\star \lambda )(a)$ & $=$ & $a\leftharpoonup (%
\overline{\lambda }\star \lambda )$ & $=$ & $\sum a_{<0>}(\overline{\lambda }%
\star \lambda )(a_{<1>})$ \\
& $=$ & $\sum a_{<0>}\overline{\lambda }(a_{<1>1})\lambda (a_{<1>2})$ & $=$
& $\sum a_{<0><0>}\overline{\lambda }(a_{<0><1>})\lambda (a_{<1>})$ \\
& $=$ & $\sum (a_{<0>}\leftharpoonup \overline{\lambda })\lambda (a_{<1>})$
& $=$ & $\sum (a_{<0>}\leftharpoonup \Lambda ^{-1}(\delta ))\lambda
(a_{<1>}) $ \\
& $=$ & $\sum \delta (a_{<0>})\lambda (a_{<1>})$ & $=$ & $\sum \delta
(a_{<0>})\theta (1_{A}\otimes a_{<1>})$ \\
& $=$ & $\sum \theta (\delta (a_{<0>})\otimes a_{<1>})$ & $=$ & $\theta
(\theta ^{-1}(a))=a,$%
\end{tabular}$$ hence $\overline{\lambda }\star \lambda =\eta _{A}\circ \varepsilon _{C}.$
Now assume $_{R}C$ to be faithfully flat. Then (1) $\Rightarrow $ (5) follows by Proposition \[cleft\] (4) and we are done.$\blacksquare $
The following result deals with the special case $\varrho (a)=\sum
a_{\psi }\otimes x^{\psi },$ for some group-like element $x\in C.$ In this case we obtain the equivalent statements (1)-(5) in Theorem \[main\] without any assumptions on $C.$
\[x-case\]Assume that $\varrho (a)=\sum a_{\psi }\otimes x^{\psi }$ for some group-like element $x\in C.$ The following statements are equivalent:
1. $A/B$ is cleft;
2. $\mathcal{M}_{A}^{C}(\psi )$ satisfies the strong structure theorem and $A$ has the right normal basis property;
3. $\mathcal{M}_{A}^{C}(\psi )$ satisfies the weak structure theorem and $A$ has the right normal basis property;
4. $A/B$ is $\mathcal{C}$-Galois and $A$ has the right normal basis property;
5. $\Lambda :\#_{\psi }^{op}(C,A)\simeq \mathrm{End}(_{B}A)^{op},$ $%
g\mapsto \lbrack a\mapsto a\leftharpoonup $ $g]$ is a ring isomorphism and $%
A $ has the right normal basis property.
By Theorem \[main\] it remains to prove that $\Phi _{N}$ is an isomorphism for every $N\in \mathcal{M}_{B},$ if $A/B$ is cleft. But in our special case there exists by Lemma \[co-Q\] some $\widehat{\lambda }\in Q$ with $\sum
1_{<0>}\widehat{\lambda }(1_{<1>})=1_{A}$ and we are done by Corollary \[co=x\] (2).$\blacksquare $
Let $(H,A,C)$ be a right-right resp. a left-right Doi-Koppinen structure. Then $(A,C,\psi )$ is a right-right entwining structure with $$\psi :C\otimes _{R}A\longrightarrow A\otimes _{R}C,\text{ }c\otimes a\mapsto
\sum a_{<0>}\otimes ca_{<1>}$$ resp. a left-right entwining structure with $$\psi :A\otimes _{R}C\longrightarrow A\otimes _{R}C,\text{ }a\otimes c\mapsto
\sum a_{<0>}\otimes a_{<1>}c.$$ If $x$ is a group-like element of $C,$ then $A\in \mathcal{M}(H)_{A}^{C}$ with $\varrho (a):=\sum a_{<0>}\otimes xa_{<1>}$ (resp. $\varrho (a)=\sum
a_{<0>}\otimes a_{<1>}x$) and we get [@DM92 Theorem 1.5] (resp. [@Doi94 Theorem 2.5]) as special cases of Theorem \[x-case\].
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[^1]: Current Address: Department of Mathematical Sciences, Box \# 5046, KFUPM, 31261 Dhahran (Saudi Arabia), **email:** abuhlail@kfupm.edu.sa
[^2]: MSC (2000): 16D90, 16S40, 16W30, 13B02 Keywords: Morita Contexts, Hopf Algebras, (Cleft) Galois Extensions, Entwining Structures, Entwined Modules.
|
---
abstract: 'The observable sector of the “minimal heterotic standard model” has precisely the matter spectrum of the MSSM: three families of quarks and leptons, each with a right-handed neutrino, and one Higgs–Higgs conjugate pair. In this paper, it is explicitly proven that the $SU(4)$ holomorphic vector bundle leading to the MSSM spectrum in the observable sector is slope-stable.'
bibliography:
- 'main.bib'
---
hep-th/0602073\
UPR 1147-T
Stability of the Minimal Heterotic\
Standard Model Bundle
Volker Braun$^{1,2}$, Yang-Hui He$^{3,4}$ and Burt A. Ovrut$^{1}$
${}^1$ Department of Physics, ${}^2$ Department of Mathematics\
University of Pennsylvania, Philadelphia, PA 19104–6395, USA
${}^3$ Merton College, Oxford University, Oxford OX1 4JD, U.K.
${}^4$ Mathematical Institute, Oxford, 24-29 St. Giles’, OX1 3LB, U.K.
\
Email: `vbraun, ovrut@physics.upenn.edu`, `yang-hui.he@merton.ox.ac.uk`.
Introduction {#sec:intro}
============
The $E_8\times E_8$ heterotic string [@Gross:1985fr; @Gross:1985rr; @Horava:1995qa] is, perhaps, the simplest context in which to construct string compactifications giving rise to a realistic matter spectrum; that is, three families of quarks/leptons and one (perhaps several) Higgs–Higgs conjugate pairs without any exotic representations or any other vector-like pairs. Within the last year, there has been significant progress in building such models [@HetSM1; @HetSM2; @HetSM3; @MinimalHetSM; @Buchmuller:2005jr; @Bouchard:2005ag]. In the vacua presented in [@HetSM1; @HetSM2; @HetSM3], called heterotic standard models, the observable sector has the MSSM matter spectrum with the addition of one extra pair of Higgs fields. In [@MinimalHetSM] the number of Higgs pairs was reduced to one, yielding the exact MSSM matter spectrum in the observable sector. Hence, the vacua in [@MinimalHetSM] are called “minimal” heterotic standard models. The MSSM matter spectrum has been obtained, in different contexts, in [@Buchmuller:2005jr; @Bouchard:2005ag].
In this paper, we will confine our discussion to the heterotic standard model vacua presented in [@HetSM1; @HetSM2; @HetSM3; @MinimalHetSM]. Their basic construction is as follows. As is well known, the whole matter content of the standard model, including the right-handed neutrino [@Fukuda:1998mi; @Langacker:2004xy; @Giedt:2005vx] fits into the ${\ensuremath{\mathbf{16}}}$ and ${\ensuremath{\mathbf{10}}}$ representations of $Spin(10)$. To embed this unification of quarks/leptons into the $E_8\times E_8$ heterotic string, one has to break the observable sector $E_8$ gauge group appropriately. This can be done by choosing a suitable gauge instanton [@Donagi:1998xe; @Donagi:1999gc; @Andreas:1999ty; @SM-bundle1; @DonagiPrincipal; @Curio:2004pf; @Andreas:2003zb; @Diaconescu:1998kg] as the vacuum field configuration on a Calabi-Yau threefold. In particular, an $SU(4)$ instanton leaves a $Spin(10)$ gauge group unbroken [@HetSM1]. The corresponding rank $4$ vector bundle is constructed via the method of bundle extensions [@MR1807601; @SU5-z2-1; @SU5-z2-2]. Of course, the $Spin(10)$ gauge group must be further broken to a group containing the standard model gauge group as a factor. The obvious mechanism is to add Wilson lines [@Witten:1985xc; @Sen:1985eb; @Breit:1985ud; @Ibanez:1986tp], thus breaking $Spin(10)$ directly at the compactification scale. In particular, we use a ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ Wilson line to break down to $SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{B-L}$. In order to do so, the Calabi-Yau threefold must have a large enough fundamental group [@SM-bundle1; @Donagi:2003tb; @dP9torusfib; @Ovrut:2003zj], that is, it must contain a ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$. A Calabi-Yau threefold whose fundamental group is exactly ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ was constructed in [@dP9Z3Z3], and is used in [@HetSM1; @HetSM2; @HetSM3; @MinimalHetSM]. The low energy particle spectrum can then be computed using methods of algebraic geometry as discussed in [@Green:1987mn; @Donagi:2004qk; @Donagi:2004ia].
An important phenomenological aspect of heterotic standard model vacua is the $U(1)_{B-L}$ factor occurring in the low energy gauge group. Usual nucleon decay is suppressed in [@HetSM1; @HetSM2; @HetSM3; @MinimalHetSM] by a large compactification mass of $O\big(10^{16}\big)~\text{GeV}$. In addition, these theories exhibit natural doublet-triplet splitting, thus suppressing proton decay via dimension five operators. The role of the gauged $U(1)_{B-L}$ symmetry is to disallow any $\Delta L=1$ and $\Delta B=1$ dimension four terms that would lead to the disastrous decay of nucleons [@Nath:2006ut]. Of course, this symmetry must be spontaneously broken at the order of the electroweak scale. This will be discussed elsewhere [@future:proton]. Hence, only the usual Yukawa couplings and a possible Higgs $\mu$-term can occur in the superpotential at the renormalizable level. Geometrically, these couplings are cubic products of cohomology groups and restricted by classical geometry. The effect of the elliptic fibration of the Calabi-Yau threefold on the Yukawa texture was analyzed in [@Braun:2006me], and leads to one naturally light quark/lepton family.
An essential requirement of these vacua is that the holomorphic vector bundle used in the observable sector be slope-stable. This guarantees [@MR86h:58038; @MR88i:58154] that the associated gauge connection satisfies the hermitian Yang-Mills equations and, hence, preserves $\mathcal{N}=1$ supersymmetry. The vector bundles in the observable sector of [@HetSM1; @HetSM2; @HetSM3] were shown to be slope-stable in [@Gomez:2005ii]. In this paper, we present an analogous proof that the $SU(4)$ vector bundle in the minimal heterotic standard model [@MinimalHetSM] is, indeed, slope-stable as well. Thus, the observable sector containing exactly the matter spectrum of the MSSM is $\mathcal{N}=1$ supersymmetric.
The structure of the hidden sector is less clear. There is a maximal dimensional subcone (codimension zero) of the Kähler cone where the observable sector bundle is slope-stable and the hidden sector satisfies the Bogomolov bound. Hence, there is no obstruction to constructing anomaly free vacua whose hidden sector bundle is slope-stable. However, we have not explicitly constructed such a hidden sector bundle. Nor is it entirely clear that this is desirable. As discussed in [@Kachru:2003aw; @Buchbinder:2003pi; @Buchbinder:2004im], the necessity to stabilize all moduli at a point with a small positive cosmological constant [@Riess:1998cb] might require that the vacuum, in the heterotic case, contain anti-five-branes. If the moduli can be stabilized for such a configuration then, for example, a trivial hidden sector bundle (which is trivially slope-stable) can be chosen. This issue will be discussed in detail elsewhere. We note that the slope-stability of both the observable and hidden sector bundles was proven for the vacuum in [@Bouchard:2005ag].
The Calabi-Yau Manifold {#sec:CY}
=======================
Double Fibration {#sec:fib}
----------------
Let us start by describing the underlying Calabi-Yau threefold. We begin with an elliptic fibration over a rational elliptic (${\ensuremath{dP_{9}}}$) surface. Such an elliptic fibration is automatically a fiber product $${{\ensuremath{\widetilde{X}}}}{ \mathrel{\lower.1mm
\hbox{$\stackrel{\lower.424ex\hbox{\scriptsize def}}{=}$}}
}B_1 \times_{{{\mathbb{P}^{1}}}} B_2$$ of two ${\ensuremath{dP_{9}}}$ surfaces $B_1$ and $B_2$. In the following, we always choose surfaces with suitable ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ automorphisms [@dP9Z3Z3] yielding a free ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ group action on ${{\ensuremath{\widetilde{X}}}}$. There is a commutative square of projections $$\label{eq:CanonicalBundles}
\vcenter{\xymatrix@!0@C=24mm@R=26mm{
\dim_{\mathbb{C}}=3: && &
\Big({{\ensuremath{\widetilde{X}}}},\, K_{{{\ensuremath{\widetilde{X}}}}}= {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}\Big)
\ar[dr]_{\pi_2}^
{\displaystyle K_{{{\ensuremath{\widetilde{X}}}}|{\ensuremath{B_{2}}}}=\chi_1^2{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(\phi)}
\ar[dl]^{\pi_1}_
{\displaystyle K_{{{\ensuremath{\widetilde{X}}}}|{\ensuremath{B_{1}}}}={\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(\phi)}
\\
\dim_{\mathbb{C}}=2: &&
\Big({\ensuremath{B_{1}}},\, K_{{\ensuremath{B_{1}}}}= {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(-f) \Big)
\ar[dr]^{\beta_1}_
{\displaystyle K_{B_1|{{\mathbb{P}^{1}}}}=\chi_1^2{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(f)} & &
\Big({\ensuremath{B_{2}}},\, K_{{\ensuremath{B_{2}}}}=\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(-f) \Big)
\ar[dl]_{\beta_2}^
{\displaystyle K_{{\ensuremath{B_{2}}}|{{\mathbb{P}^{1}}}}={\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(f)} \\
\dim_{\mathbb{C}}=1: && &
\Big({{\mathbb{P}^{1}}},\, K_{{{\mathbb{P}^{1}}}} = \chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-2) \Big)
\,,
}}$$ where $\chi_1$, $\chi_2$ are characters [@HetSM3] of ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ encoding the equivariant action on bundles.
The quotient $$X { \mathrel{\lower.1mm
\hbox{$\stackrel{\lower.424ex\hbox{\scriptsize def}}{=}$}}
}{{\ensuremath{\widetilde{X}}}}\Big/ \big({{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}\big)$$ is a torus-fibered Calabi-Yau threefold with fundamental group $\pi_1\big(X\big)={{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$, which we take to be the base manifold of our string compactification. However, in practice we work with equivariant constructions on the universal cover ${{\ensuremath{\widetilde{X}}}}$. For a free group action these descriptions are equivalent.
Topology {#sec:topology}
--------
It is important to understand the even cohomology groups $H^{\ensuremath{\text{ev}}}\big(X,{\mathbb{Z}}\big)$, because that is where the Chern classes live. Rationally, it is clear that $$H^{\ensuremath{\text{ev}}}\Big({{\ensuremath{\widetilde{X}}}},{\mathbb{Q}}\Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}=
H^{\ensuremath{\text{ev}}}\Big(X,{\mathbb{Q}}\Big)
\,.$$ The degree $2$ invariant integral cohomology of ${{\ensuremath{\widetilde{X}}}}$ is $$H^2\Big({{\ensuremath{\widetilde{X}}}},{\mathbb{Z}}\Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}=
\operatorname{span}_{\mathbb{Z}}\big\{ \tau_1,\tau_2,\phi \big\}
\,.$$ We can compare it with the cohomology of $X$ using the quotient map $$q:~
{{\ensuremath{\widetilde{X}}}}\to X
\qquad \Rightarrow \quad
q^\ast:~
H^\ast\big(X,{\mathbb{Z}}\big) \to H^\ast\big({{\ensuremath{\widetilde{X}}}},{\mathbb{Z}}\big)
\,.$$ In degree $2$, the image is an index $3$ sub-lattice of $H^2\big({{\ensuremath{\widetilde{X}}}},{\mathbb{Z}}\big)\simeq {\mathbb{Z}}^3$ generated by $\tau_1-\tau_2$, $3\tau_1$, $\phi$. In other words, the equivariant line bundles on ${{\ensuremath{\widetilde{X}}}}$ are of the form $${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(x_1 \tau_1 + x_2 \tau_2 + x_3 \phi)
\qquad
x_1,x_2,x_3 \in {\mathbb{Z}}\,,~
x_1+x_2\equiv 0 \mod 3
\,.$$ The products of the degree $2$ generators can easily be determined, and one finds relations $$H^{\ensuremath{\text{ev}}}\Big( {{\ensuremath{\widetilde{X}}}}, {\mathbb{Q}}\Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}=
{\mathbb{Q}}\big[ \tau_1,\tau_2,\phi \big]
\Big/
\left<
\phi^2
,~
\tau_i\phi=3\tau_i^2
\right>
\,.$$ Hence, every even degree cohomology class can be written as a polynomial in $\tau_1$, $\tau_2$, and $\phi$ subject to the relations $\phi^2=0$ and $\tau_i\phi=3\tau_i^2$.
Visible Bundle {#sec:visible}
==============
Construction of the Bundle {#sec:bundledef}
--------------------------
Having presented the Calabi-Yau manifold, we proceed to define a holomorphic rank $4$ vector bundle on it. First, define equivariant rank $2$ vector bundles
$$\begin{aligned}
\label{eq:V1def}
{\ensuremath{\mathcal{V}_{1}}} =&~
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}\big( -\tau_1+\tau_2 \big) \otimes
\pi_1^\ast \big({{\ensuremath{\mathcal{W}_{1}}}}\big)
\\
\label{eq:V2def}
{\ensuremath{\mathcal{V}_{2}}} =&~
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}\big( +\tau_1-\tau_2 \big) \otimes
\pi_2^\ast \big({{\ensuremath{\mathcal{W}_{2}}}}\big)
\,,\end{aligned}$$
where ${{\ensuremath{\mathcal{W}_{1}}}}$ and ${{\ensuremath{\mathcal{W}_{2}}}}$ are rank $2$ vector bundles on ${\ensuremath{B_{1}}}$ and ${\ensuremath{B_{2}}}$ which we will define in detail in Section \[sec:serre\], eqns. and . Using these, we define the desired rank $4$ vector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ as an extension $$\label{eq:Vdef}
0
\longrightarrow
{\ensuremath{\mathcal{V}_{1}}}
\longrightarrow
{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\longrightarrow
{\ensuremath{\mathcal{V}_{2}}}
\longrightarrow
0
\,.$$ Using the fact that the first Chern class of ${{\ensuremath{\mathcal{W}_{i}}}}$ is trivial, $\wedge^2{{\ensuremath{\mathcal{W}_{i}}}}={\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{i}}}}}}$, we first remark that $$c_1\big( {{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\big) = 0
~\in H^2\Big({{\ensuremath{\widetilde{X}}}},{\mathbb{Z}}\Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}\simeq {\mathbb{Z}}^3
\,.$$ But we really want an $SU(4)$ bundle on the quotient $X={{\ensuremath{\widetilde{X}}}}\big/({{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}})$, that is $$c_1\Big( {{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\Big/\big({{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}\big) \Big) = 0
~\in H^2\Big(X,{\mathbb{Z}}\Big) \simeq {\mathbb{Z}}^3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3
\,.$$ The vanishing of the first Chern class including the torsion part follows from $\wedge^4{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}={\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}$, where ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}$ stands for the trivial line bundle with the trivial ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ equivariant group action.
Non-Trivial Extensions {#sec:ext}
----------------------
We defined the rank $4$ bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ as a generic extension of the form eq. . Clearly, we have to make sure that a non-trivial extension exists, since the *trivial extension* ${\ensuremath{\mathcal{V}_{1}}}\oplus{\ensuremath{\mathcal{V}_{2}}}$ cannot give rise to an irreducible $SU(4)$ instanton. The space of extensions is $$\begin{gathered}
\operatorname{Ext}^1\Big( {\ensuremath{\mathcal{V}_{2}}}, {\ensuremath{\mathcal{V}_{1}}} \Big) =
H^1\Big( {{\ensuremath{\widetilde{X}}}}, {\ensuremath{\mathcal{V}_{1}}}\otimes {\ensuremath{\mathcal{V}_{2}}}^{\ensuremath{\vee}}\Big)
= \\ =
H^1\Big( {{\ensuremath{\widetilde{X}}}}, {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2\tau_1+2\tau_2) \otimes
\pi_1^\ast({{\ensuremath{\mathcal{W}_{1}}}}) \otimes \pi_2^\ast({{\ensuremath{\mathcal{W}_{2}}}}^{\ensuremath{\vee}}) \Big)
= \\ =
H^1\Big( {{\ensuremath{\widetilde{X}}}},
\pi_1^\ast\big({{\ensuremath{\mathcal{W}_{1}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(-2t)\big) \otimes
\pi_2^\ast\big({{\ensuremath{\mathcal{W}_{2}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(2t)\big) \Big)
\,.\end{gathered}$$ This cohomology group can directly be computed using the Leray spectral sequence and the push-down eqns. and . One obtains $$H^i\Big( {{\ensuremath{\widetilde{X}}}},~ {\ensuremath{\mathcal{V}_{1}}}\otimes{\ensuremath{\mathcal{V}_{2}}}^{\ensuremath{\vee}}\Big) =
\begin{cases}
0
& i=3 , \\
8 R[{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}]
& i=2 , \\
4 R[{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}]
& i=1 , \\
0
& i=0 .
\end{cases}$$ where $R[{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}]$ stands for the regular representation, that is, the sum of all $9$ irreducible representations of ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$. Of course, only invariant extensions give rise to equivariant vector bundles ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$. The invariant subspace is $$\operatorname{Ext}^1\Big( {\ensuremath{\mathcal{V}_{2}}}, {\ensuremath{\mathcal{V}_{1}}} \Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}=
H^1\Big( {{\ensuremath{\widetilde{X}}}},~ {\ensuremath{\mathcal{V}_{1}}}\otimes{\ensuremath{\mathcal{V}_{2}}}^{\ensuremath{\vee}}\Big)^{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}=
4$$ is indeed non-zero, so suitable extensions do exist.
Low-Energy Spectrum
-------------------
The low energy particle spectrum is determined through the cohomology of ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ and $\wedge^2{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ according to the decomposition $${\ensuremath{\mathbf{248}}} =
\big( {\ensuremath{\mathbf{1}}},{\ensuremath{\mathbf{45}}} \big) \oplus
\big( {\ensuremath{\mathbf{4}}},{\ensuremath{\mathbf{16}}} \big) \oplus
\big( {\ensuremath{\overline{{\ensuremath{\mathbf{4}}}}}}, {\ensuremath{\overline{{\ensuremath{\mathbf{16}}}}}} \big) \oplus
\big( {\ensuremath{\mathbf{6}}},{\ensuremath{\mathbf{10}}} \big) \oplus
\big( {\ensuremath{\mathbf{15}}},{\ensuremath{\mathbf{1}}} \big)$$ under $E_8\supset SU(4)\times Spin(10)$. It is easy to show that $H^i\big({{\ensuremath{\widetilde{X}}}},{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\big)=0$ for $i=0,2,3$. Hence a simple index computation yields $$\label{eq:Vcoh}
H^i\Big( {{\ensuremath{\widetilde{X}}}},~ {{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\Big) =
\begin{cases}
0
& i=3 , \\
0
& i=2 , \\
3 R[{{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}]
& i=1 , \\
0
& i=0 .
\end{cases}$$ Furthermore, interrelated long exact sequences [@HetSM3] together with $$H^\ast\Big( {{\ensuremath{\widetilde{X}}}},~ \wedge^2{\ensuremath{\mathcal{V}_{1}}} \Big) =
H^\ast\Big( {{\ensuremath{\widetilde{X}}}},~ \wedge^2{\ensuremath{\mathcal{V}_{2}}} \Big) =
0$$ yield $$H^i\Big( {{\ensuremath{\widetilde{X}}}},~ \wedge^2{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\Big) =
H^i\Big( {{\ensuremath{\widetilde{X}}}},~ {\ensuremath{\mathcal{V}_{1}}}\otimes{\ensuremath{\mathcal{V}_{2}}} \Big) =
H^i\Big( {{\ensuremath{\widetilde{X}}}},~
\pi_1^\ast\big({{\ensuremath{\mathcal{W}_{1}}}}\big) \otimes \pi_1^\ast\big({{\ensuremath{\mathcal{W}_{1}}}}\big) \Big)
\,.$$ The latter is easily computed using the push-down formula eqns. and and the Leray spectral sequence. The result is that $$\label{eq:wedge2Vcoh}
H^i\Big( {{\ensuremath{\widetilde{X}}}},~ \wedge^2{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\Big) =
H^i\Big( {{\ensuremath{\widetilde{X}}}},~ {\ensuremath{\mathcal{V}_{1}}}\otimes{\ensuremath{\mathcal{V}_{2}}} \Big) =
\begin{cases}
0
& i=3 , \\
\chi_2 \oplus \chi_2^2 \oplus \chi_1\chi_2^2 \oplus \chi_1^2\chi_2
& i=2 , \\
\chi_2 \oplus \chi_2^2 \oplus \chi_1\chi_2^2 \oplus \chi_1^2\chi_2
& i=1 , \\
0
& i=0 .
\end{cases}$$ Finally, the ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ group action on the cohomology is tensored with the Wilson line, and every state that is not invariant under the combined action is projected out. The regular representations in eq. yield $3$ full generations of quarks and leptons, each with a right-handed neutrino. More interesting is the Wilson line action on the ${\ensuremath{\mathbf{10}}}$ of $Spin(10)$, which potentially could lead to exotic color triplets (“triplet Higgs”). We chose the Wilson line such that $${\ensuremath{\mathbf{10}}}=
\Big[
\chi_2^2\big({\ensuremath{\mathbf{1}}},{\ensuremath{\mathbf{2}}},3,0\big) \oplus
\chi_1^2\chi_2^2\big({\ensuremath{\mathbf{3}}},{\ensuremath{\mathbf{1}}},-2,-2\big)
\Big]
\oplus
\Big[
\chi_2\big({\ensuremath{\mathbf{1}}},{\ensuremath{\overline{{\ensuremath{\mathbf{2}}}}}},-3,0\big) \oplus
\chi_1\chi_2\big({\ensuremath{\overline{{\ensuremath{\mathbf{3}}}}}},{\ensuremath{\mathbf{1}}},2,2\big)
\Big]
\label{eq:10Wilson}$$ under the decomposition $$\label{eq:Spin10break}
Spin(10) \supset
SU(3)_C\times SU(2)_L \times U(1)_Y \times U(1)_{B-L} \times {{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}\,.$$ Combining eqns. and , we see that one vector-like pair of Higgs survives the ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ quotient while all color triplets are projected out.
Slope-Stability {#sec:stability}
===============
Conditions for Stability
------------------------
We now proceed and show that the Kähler class $\omega \in
H^2\big({{\ensuremath{\widetilde{X}}}},{\mathbb{R}}\big)$ can be chosen such that the visible sector vector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, eq. , is equivariantly[^1] slope-stable. That means that for all reflexive sub-sheaves ${\ensuremath{\mathcal{F}}}\hookrightarrow {{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, the slope $$\label{eq:slope}
\mu({\ensuremath{\mathcal{F}}}) { \mathrel{\lower.1mm
\hbox{$\stackrel{\lower.424ex\hbox{\scriptsize def}}{=}$}}
}\frac{1}{\operatorname{rank}{\ensuremath{\mathcal{F}}}}
\int_{{\ensuremath{\widetilde{X}}}}c_1({\ensuremath{\mathcal{F}}}) \wedge \omega^2$$ is negative, $$\mu\big({\ensuremath{\mathcal{F}}}\big) < \mu\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\big) = 0$$ The easiest way to prove this is to derive a set of sufficient inequalities for the Kähler class $\omega$, and then to find a common solution [@Gomez:2005ii]. We note that they are not always necessary, that is, the inequalities are not sharp.
For example, consider only ${\ensuremath{\mathcal{V}_{1}}}$ defined by eqns. , . Let ${\ensuremath{\mathcal{L}}}$ be any sub-line bundle, that is $$\label{eq:V1sesL}
\vcenter{\xymatrix{
0 \ar[r] &
\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2-\phi) \ar[r]^-{u} &
{\ensuremath{\mathcal{V}_{1}}}
\ar[r]^-{v} &
\chi_1^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2+\phi) \otimes \pi_1^\ast I_3
\ar[r] &
0
\\
& & {\ensuremath{\mathcal{L}}}\ar[u]_{i} \ar[ur]_{v \circ i} \ar@{-->}[ul]^{w}
}}$$ The composition $v\circ i$ either vanishes or not. We distinguish the two cases:
There exists a non-zero map $$w:~{\ensuremath{\mathcal{L}}}\to\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2-\phi)$$ such that $i=u\circ w$.
There exists a non-zero map $$v\circ i:~{\ensuremath{\mathcal{L}}}\to\chi_1^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2+\phi)$$ whose image vanishes at the codimension two locus where $\pi_1^\ast
I_3$ vanishes.
The existence of these maps restricts the line bundle ${\ensuremath{\mathcal{L}}}$. Now if ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ is stable, then all these line bundles ${\ensuremath{\mathcal{L}}}$ must be of negative slope, $\mu({\ensuremath{\mathcal{L}}})<0$. We only have to check this inequality for the ${\ensuremath{\mathcal{L}}}$ of largest slope, and these form a finite set (see Appendix \[sec:sublinebundle\]):
$${\ensuremath{\mathcal{L}}}={\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2-\phi)
\,.$$
The composition $v\circ i$ cannot be an isomorphism, since that would split the short exact sequence eq. . Hence, ${\ensuremath{\mathcal{L}}}$ can only be a proper sub-line bundle, and those of largest slope are $$\begin{gathered}
\Big\{
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-4\tau_1+\tau_2+2\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-3\tau_1+\phi)
,\,
\\
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2\tau_1-\tau_2+\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1-2\tau_2+2\phi)
\Big\}
\,.
\end{gathered}$$ The first line bundle ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2)$ actually has the same fiber degrees (coefficients of $\tau_1$ and $\tau_2$) as the range of $v\circ i$. Because of the push-down formula eq. , the largest such sub-line bundle whose image vanishes at $\pi_1^\ast
I_3$ is actually $${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2+\phi) \,\otimes\,
\pi_1^\ast\circ\beta_1^\ast\Big( {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-3) \Big)
=
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2-2\phi)
\,.$$ Therefore, the possible line bundles ${\ensuremath{\mathcal{L}}}$ of largest slope are $$\begin{gathered}
{\ensuremath{\mathcal{L}}}\in
\Big\{
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1+\tau_2-2\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-4\tau_1+\tau_2+2\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-3\tau_1+\phi)
,\,
\\
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2\tau_1-\tau_2+\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1-2\tau_2+2\phi)
\Big\}
\,.
\end{gathered}$$
Similarly, one obtains a finite set of potentially destabilizing sub-line bundles of ${\ensuremath{\mathcal{V}_{2}}}$.
Now to prove [@Gomez:2005ii] stability of ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, it suffices to show that
- Sub-line bundles of ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ have negative slope.
- Rank $2$ sub-bundles have negative slope. A sufficient criterion is that $\wedge^2{\ensuremath{\mathcal{V}_{1}}}$ has negative slope and that proper sub-line bundles of $\wedge^2{\ensuremath{\mathcal{V}_{2}}}$ are of negative slope.
- Rank $3$ sub-bundles (reflexive sheaves) have negative slope $\Leftrightarrow$ sub-line bundles of ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}^{\ensuremath{\vee}}$ have negative slope.
This gives a finite set of line bundles which have to have negative slope. One obtains
\[prop:stable\] If all line bundles ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(a_1\tau_1+a_2\tau_2+b\phi)$ with $$\begin{gathered}
(a_1,a_2,b) \in \Big\{
(-1, -2, 2),\,
( 2, -2, -1),\,
( 2, -5, 1),\,
(-4, 1, 2),\,
(-1, 1, -1),\,
\\
(-2, 2, 0),\,
(-2, -1, 2),\,
( 1, -4, 2),\,
( 1, -1, -1)
\Big\}
\end{gathered}$$ have negative slope, then the vector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, eq. , is equivariantly stable.
Kähler Cone Substructure {#sec:kahlersub}
------------------------
The Kähler cone, that is the set of possible Kähler classes, is [@Gomez:2005ii] $$\label{eq:Kcone}
{\ensuremath{\mathcal{K}}}{ \mathrel{\lower.1mm
\hbox{$\stackrel{\lower.424ex\hbox{\scriptsize def}}{=}$}}
}\Big\{ x_1\tau_1 + x_2\tau_2 + y\phi
\,\Big|\, x_1, x_2, y >0
\Big\}
~\subset
H^2\big({{\ensuremath{\widetilde{X}}}},{\mathbb{R}}\big)=\left<\tau_1,\tau_2,\phi\right>_{\mathbb{R}}\,.$$ The slope eq. of a line bundle obviously depends quadratically on the Kähler parameters $x_1$, $x_2$, $y$, and can be computed [@Gomez:2005ii] to be $$\mu\Big( {\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(a_1 \tau_1 + a_2 \tau_2 + b \phi) \Big)
=
3 (x_1 x_2 + 6 y) (a_1 x_2 + a_2 x_1) +
x_1 x_2 (3 a_1+ 3 a_2 + 18 b)
\,.$$ Therefore, according to Proposition \[prop:stable\] the vector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ is stable if the inequalities $$\label{eq:inequalities}
\begin{array}{l@{\,=\,}c@{\,<\,0}c}
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1 -2\tau_2 +2\phi)\big) &
18 x_1 x_2-6 x_1^2-3 x_2^2-18y x_2-36y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}( 2\tau_1 -2\tau_2 -\phi)\big) &
-6 x_1^2+6 x_2^2+36y x_2-36y x_1-18 x_1 x_2
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}( 2\tau_1 -5\tau_2 +\phi)\big) &
-15 x_1^2+6 x_2^2+36y x_2-90y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-4\tau_1 +\tau_2 +2\phi)\big) &
18 x_1 x_2+3 x_1^2-12 x_2^2-72y x_2+18y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1 +\tau_2 -\phi)\big) &
3 x_1^2-3 x_2^2-18y x_2+18y x_1-18 x_1 x_2
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2\tau_1 +2\tau_2 )\big) &
6 x_1^2-6 x_2^2-36y x_2+36y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2\tau_1 -\tau_2 +2\phi)\big) &
18 x_1 x_2-3 x_1^2-6 x_2^2-36y x_2-18y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}( \tau_1 -4\tau_2 +2\phi)\big) &
18 x_1 x_2-12 x_1^2+3 x_2^2+18y x_2-72y x_1
& \\
\mu\big({\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}( \tau_1 -\tau_2 -\phi)\big) &
-3 x_1^2+3 x_2^2+18y x_2-18y x_1-18 x_1 x_2
& \\
\end{array}$$ are simultaneously satisfied.
It is easy to see that there are many solutions. For example, the Kähler class $$\label{eq:omega}
\omega = 3\Big( 2 \tau_1 + 3 \tau_2 + \phi \Big)
~\in H^2\Big({{\ensuremath{\widetilde{X}}}},{\mathbb{R}}\Big)$$ satisfies all the inequalities eq. , the slopes being $-621$, $-378$, $-702$, $-1512$, $-1269$, $-594$, $-918$, $-27$, and $-675$, respectively. The overall factor of $3$ in eq. is not essential, but included to make it a ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$-equivariant integral cohomology class. In other words, the class is actually primitive in the integral cohomology of the quotient $X={{\ensuremath{\widetilde{X}}}}/({{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}})$. Of course, in string theory the Kähler form is not quantized. As usual, the radial part of the Kähler class, that is, the overall volume, does not matter for the stability of vector bundles. We conclude from eq. that the set $${\ensuremath{\mathcal{K}}}^s\subset {\ensuremath{\mathcal{K}}}\subset H^2\big({{\ensuremath{\widetilde{X}}}},{\mathbb{R}}\big)$$ of Kähler classes that make all slopes of the line bundles in Proposition \[prop:stable\] negative is not empty. Therefore, the solution set ${\ensuremath{\mathcal{K}}}^s$ of the strict inequalities eq. must be a maximal-dimensional subcone of the Kähler cone ${\ensuremath{\mathcal{K}}}$.
Note that all cones have their tip at the origin $0\in
H^2\big({{\ensuremath{\widetilde{X}}}},{\mathbb{R}}\big)\simeq {\mathbb{R}}^3$. Hence, we can draw a $2$-dimensional “star map” of these cones as they are seen by an observer at the origin. This is depicted in Figure \[fig:stable\]. One observes that the boundary of the set ${\ensuremath{\mathcal{K}}}^s$ is roughly triangular. On the right hand side in Figure \[fig:stable\], it is bounded by two curved but smooth faces. Those bounds are an artifact of our proof, and are merely sufficient but not necessary conditions. Although it is in general difficult to determine the precise subcone of the Kähler cone where ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ is stable, one expects it to extend even further to the right. On the other hand, the flat face of ${\ensuremath{\mathcal{K}}}^s$ at the left in Figure \[fig:stable\] is a boundary saturating a necessary *and* sufficient inequality. It is precisely the locus where the slope of ${\ensuremath{\mathcal{V}_{1}}}$ changes sign, and if one crosses this line then $\mu\big({\ensuremath{\mathcal{V}_{1}}}\big)>0$ becomes a destabilizing sub-bundle of ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, see eq. . The interpretation is analogous to the picture of D-branes as complexes; this boundary of ${\ensuremath{\mathcal{K}}}^s$ is a line of marginal stability. To its right, the bound state ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ of ${\ensuremath{\mathcal{V}_{1}}}$ and ${\ensuremath{\mathcal{V}_{2}}}$ is stable. To its left, the reversed bound state $$\label{eq:Vrevdef}
0
\longrightarrow
{\ensuremath{\mathcal{V}_{2}}}
\longrightarrow
{{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}_\text{rev}
\longrightarrow
{\ensuremath{\mathcal{V}_{1}}}
\longrightarrow
0
\,.$$ is stable. Using the same methods as above, it is easy to see that ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}_\text{rev}$ is indeed stable in a subcone of ${\ensuremath{\mathcal{K}}}$ extending to the left of ${\ensuremath{\mathcal{K}}}^s$. Although reversing the short exact sequence potentially alters the cohomology groups, it turns out that ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ and ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}_\text{rev}$ give rise to the same low energy spectrum.
To summarize, the observable sector vector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ is slope-stable with respect to any Kähler class $\omega$ in a $3$-dimensional subcone ${\ensuremath{\mathcal{K}}}^s$ of the $3$-dimensional Kähler cone ${\ensuremath{\mathcal{K}}}$. The region ${\ensuremath{\mathcal{K}}}^s$ is show explicitly in Figure \[fig:stable\]. By working harder to strengthen Proposition \[prop:stable\] or by making small changes to the vector bundle it will be possible to enlarge that fraction of the Kähler cone.
Hidden Sector {#sec:hidden}
=============
Although not the main topic of this paper, in this section we will briefly discuss the hidden sector. Denote by ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'$ the holomorphic vector bundle of the hidden sector. For simplicity, we will assume that $c_1\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big)=0$, that is, the hidden sector contains an $SU(n)$ gauge instanton. Given the tangent bundle $T{{\ensuremath{\widetilde{X}}}}$ of the Calabi-Yau threefold and the observable sector bundle ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$, anomaly cancellation imposes the constraint $$c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big)
=
c_2\big(T{{\ensuremath{\widetilde{X}}}}\big) -
c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\big) -
[C_5]
\,.$$ Here, $[C_5]$ is the curve class on which five-branes are wrapped. For simplicity, let us assume that $[C_5]=0$ (both weakly and strongly coupled heterotic string). Then, using $$c_2\big(T{{\ensuremath{\widetilde{X}}}}\big)
=
12\big( \tau_1^2+\tau_2^2 \big)
\,,\qquad
c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}\big)
=
\tau_1^2 + 4 \tau_2^2 + 4\tau_1\tau_2
\,,$$ it follows that $$c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big)
=
11 \tau_1^2 + 8 \tau_2^2 - 4 \tau_1 \tau_2
=
\Big( 3\tau_1^2\Big) + 4 \Big( \tau_1^2+\tau_2^2 \Big)
- 4 \Big( \tau_1\tau_2-\tau_1^2-\tau_2^2 \Big)
\,.$$ Note that $c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big)$ is neither effective nor antieffective, the terms in brackets being pull-backs of effective curves on $X$. If ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'$ is a slope-stable vector bundle with respect to a Kähler class $\omega$, then it must satisfy the Bogomolov inequality [@MR522939] $$\label{eq:bogomolov}
\int_{{\ensuremath{\widetilde{X}}}}c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big) \wedge \omega > 0
\,.$$ Using the parametrization of $\omega$ in eq. , we see that $$\begin{split}
\int_{{\ensuremath{\widetilde{X}}}}c_2\big({{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'\big) \wedge \omega
=&~
\int_{{\ensuremath{\widetilde{X}}}}\Big( 11 \tau_1^2 + 8 \tau_2^2 - 4 \tau_1 \tau_2 \Big)
\wedge
\Big( x_1\tau_1+x_2\tau_2+y\phi \Big)
= \\ =&~
\int_{{\ensuremath{\widetilde{X}}}}\Big( 4 x_1 + 7 x_2 - 12 y \Big) \tau_1^2\tau_2
=
3 \Big( 4 x_1 + 7 x_2 - 12 y \Big)
\,.
\end{split}$$ Therefore, the Bogomolov inequality is satisfied for any Kähler class for which $$\label{eq:bogoineq}
4 x_1 + 7 x_2 - 12 y > 0
\,.$$ This defines a $3$-dimensional cone in the Kähler moduli space which we denote by ${\ensuremath{\mathcal{K}}}^B$. The subcone ${\ensuremath{\mathcal{K}}}^B$ is shown as the white region in Figure \[fig:stable\]. Its complement, where eq. is violated, is drawn in pink. Note that the Kähler class eq. for which the observable sector vector bundle was proven to be stable also satisfies eq. . Hence, $${\ensuremath{\mathcal{K}}}^s \cap {\ensuremath{\mathcal{K}}}^B \not= \emptyset
\,.$$ Since both ${\ensuremath{\mathcal{K}}}^s$ and ${\ensuremath{\mathcal{K}}}^B$ are open (solutions of strict inequalities), their non-empty intersection is automatically a maximal-dimensional subcone of the Kähler cone. It follows that both ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ and ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}'$ can, in principle, be slope-stable with respect to a Kähler class in ${\ensuremath{\mathcal{K}}}^s\cap{\ensuremath{\mathcal{K}}}^B$. Often, the Bogomolov inequality is the only obstruction to finding stable bundles. However, we have not explicitly constructed such a hidden sector bundle.
Serre Construction {#sec:serre}
==================
General Construction {#sec:general}
--------------------
In this section, we are going to construct two $SU(2)$ vector bundles ${{\ensuremath{\mathcal{W}_{1}}}}$ and ${{\ensuremath{\mathcal{W}_{2}}}}$ on the ${\ensuremath{dP_{9}}}$ surfaces ${\ensuremath{B_{1}}}$ and ${\ensuremath{B_{2}}}$, respectively. They are defined as extensions of the form
$$\begin{gathered}
\label{eq:W1def}
0
\longrightarrow
\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(-f)
\longrightarrow
{{\ensuremath{\mathcal{W}_{1}}}}
\longrightarrow
\chi_1^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(f) \otimes I_3
\longrightarrow
0
\\
\label{eq:W2def}
0
\longrightarrow
\chi_2^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(-f)
\longrightarrow
{{\ensuremath{\mathcal{W}_{2}}}}
\longrightarrow
\chi_2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(f) \otimes I_6
\longrightarrow
0\end{gathered}$$
with the ideal sheaves $I_3$ and $I_6$ defined in Subsection \[sec:ideal\]. If they satisfy the Cayley-Bacharach property, then ${{\ensuremath{\mathcal{W}_{1}}}}$ and ${{\ensuremath{\mathcal{W}_{2}}}}$ are rank $2$ vector bundles for generic extensions. We check this in Subsection \[sec:CB\].
Note that the determinant line bundles are trivial by construction, that is $$\wedge^2 {{\ensuremath{\mathcal{W}_{1}}}} = {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}
\,,\qquad
\wedge^2 {{\ensuremath{\mathcal{W}_{2}}}} = {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}
\,.$$ Therefore, the bundles are self-dual, $$\big({{\ensuremath{\mathcal{W}_{1}}}}\big)^{\ensuremath{\vee}}= {{\ensuremath{\mathcal{W}_{1}}}}
\,,\qquad
\big({{\ensuremath{\mathcal{W}_{2}}}}\big)^{\ensuremath{\vee}}= {{\ensuremath{\mathcal{W}_{2}}}}
\,.$$
Ideal Sheaves {#sec:ideal}
-------------
Let $p_1$, $p_2$, $p_3$ be the singular points of the $3I_1$ Kodaira fibers in ${\ensuremath{B_{1}}}\to{{\mathbb{P}^{1}}}$. Similarly, let $q_1$, $q_2$, $q_3$ be the singular points of the $3I_1$ Kodaira fibers in ${\ensuremath{B_{2}}}\to{{\mathbb{P}^{1}}}$. Recall that ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ is generated by $g_1$ and $g_2$, where $g_1$ acts on the base ${{\mathbb{P}^{1}}}$ and $g_2$ does not (it is a translation along the elliptic fiber). The ${{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}$ characters are defined via $$\begin{aligned}
\chi_1(g_1) &= \omega
& \qquad
\chi_1(g_2) &= 1
\\
\chi_2(g_1) &= 1
&
\chi_2(g_2) &= \omega
\,,
\end{aligned}$$ Note that the points $p_i$ and $q_j$ are $g_2$-fixed points, and that $g_2$ acts as $\chi_2\oplus\chi_2^2$ on the tangent spaces $T_{p_i}{\ensuremath{B_{1}}}$ and $T_{q_j}{\ensuremath{B_{2}}}$. First, we define the ideal sheaf $I_3$ as $$\label{eq:I3def}
0
\longrightarrow
I_3
\longrightarrow
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}
\longrightarrow
\bigoplus_{i=1,2,3} {\ensuremath{\mathcal{O}}}_{p_i}
\longrightarrow
0
\,.$$ Furthermore, define for any $G_2\simeq {\mathbb{Z}}_3$ fixed point $p$ the subscheme $Z(p)$ as the point $p$ and its first derivative in $\chi_2^2$-direction. In local coordinates $(x,y)\in {\mathbb{C}}^2$, this ${\mathbb{Z}}_3$ group acts as $$g_2(x,y) = \Big( \chi_2(g_2)\, x, \chi_2^2(g_2)\, y \Big)
=
\big( \omega x, \omega^2 y \big)
\,, \quad
\omega { \mathrel{\lower.1mm
\hbox{$\stackrel{\lower.424ex\hbox{\scriptsize def}}{=}$}}
}e^{\frac{2\pi i}{3}}$$ and the scheme $Z(p)$ is $$Z(p) = \operatorname{spec}\Big( {\mathbb{C}}[x,y]\big/ \left<x, y^2\right> \Big)
\,.$$ Define the ideal sheaf $I_6$ as the sheaf of functions vanishing at ${\mathbb{Z}}(q_1)$, $Z(q_2)$, and $Z(q_3)$. That is, $$\label{eq:I6def}
0
\longrightarrow
I_6
\longrightarrow
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}
\longrightarrow
\bigoplus_{i=1,2,3} {\ensuremath{\mathcal{O}}}_{Z(q_i)}
\longrightarrow
0
\,.$$ In other words, $I_6$ are the functions vanishing at $q_i$ and whose first derivative in the $\chi_2^2$ direction vanishes.
Cayley-Bacharach Property {#sec:CB}
-------------------------
Recall the Cayley-Bacharach property for an extension $$\label{eq:CB}
0
\longrightarrow
{\ensuremath{\mathcal{L}}}\longrightarrow
{\ensuremath{\mathcal{W}}}\longrightarrow
{\ensuremath{\mathcal{M}}}\otimes I_n
\longrightarrow
0$$ of line bundles ${\ensuremath{\mathcal{L}}}$, ${\ensuremath{\mathcal{M}}}$ and ideal sheaf $I_n$ of $n$ points on a surface $B$. It has the Cayley-Bacharach property if the sections $$\label{eq:CBsec}
s
~\in
H^0\Big(B,~ {\ensuremath{\mathcal{L}}}^{\ensuremath{\vee}}\otimes {\ensuremath{\mathcal{M}}}\otimes K_B \Big)$$ vanishing at $n-1$ points of the ideal sheaf automatically vanish at the $n$-th point. The Cayley-Bacharach property implies that ${\ensuremath{\mathcal{W}}}$ is generically a rank $2$ vector bundle.
First, let us check that ${{\ensuremath{\mathcal{W}_{1}}}}$, eq. , has the Cayley-Bacharach property. The sections in question are $$\begin{split}
s_1
~\in&~
H^0 \Big( B_1,~
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(-f)^{\ensuremath{\vee}}\otimes {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(f) \otimes K_{{\ensuremath{B_{1}}}} \Big) =
\\ &~=
H^0 \Big( B_1,~ {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(f) \Big)
=
H^0 \Big( {{\mathbb{P}^{1}}},~ {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(1) \Big)
\,.
\end{split}$$ Furthermore, the ideal sheaf $I_3$ vanishes at $3$ points in $3$ different fibers. But a section of ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(f)$ can only vanish at one fiber, or it is identically zero. Hence, a section $s_1$ vanishing at $2$ of the $3$ points vanishes automatically at the $3$-rd, and the Cayley-Bacharach property holds. The extension ${{\ensuremath{\mathcal{W}_{2}}}}$, eq. , satisfies Cayley-Bacharach analogously. Therefore, ${{\ensuremath{\mathcal{W}_{1}}}}$ and ${{\ensuremath{\mathcal{W}_{2}}}}$ are rank $2$ vector bundles.
Push-Down Formulae {#sec:Wpushdown}
------------------
To compute the cohomology groups of vector bundles, we always utilize the Leray spectral sequence. For that, we need to know the push-down of all bundles involved.
First, consider the ideal sheaves. A standard application of the long exact sequence for the push-down to eq. immediately yields $$\label{eq:I3pushdown}
\beta_{1\ast} \big(I_3\big) = {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-3)
\,, \qquad
R^1 \beta_{1\ast} \big(I_3\big) = \chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1)
\,.$$ For the push-down of $I_6$ defined in eq. , first note that according to the definition of $Z(q_i)$ the push-down of the skyscraper sheaves are $$\beta_{2\ast} {\ensuremath{\mathcal{O}}}_{Z(q_i)} =
{\ensuremath{\mathcal{O}}}_{\beta_{2\ast}(q_i)} \oplus
\chi_2^2 {\ensuremath{\mathcal{O}}}_{\beta_{2\ast}(q_i)}
\,.$$ The long exact sequence for the push-down contains a non-zero coboundary map which can be computed as in [@HetSM3]. One finds that $$\label{eq:I6pushdown}
\beta_{2\ast} \big(I_6\big) = {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-3)
\,, \qquad
R^1 \beta_{2\ast} \big(I_6\big) = {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1) \oplus
\left[ \bigoplus_{i=1}^3 \chi_2^2 {\ensuremath{\mathcal{O}}}_{\beta_2(q_i)} \right]
\,.$$
Using the push-down of the ideal sheaves, we find the long exact sequence $$\label{eq:W1les}
\vcenter{\xymatrix@R=10pt@M=4pt@H+=22pt{
0 \ar[r] &
\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1)
\ar[r] &
\beta_{1\ast} \big( {{\ensuremath{\mathcal{W}_{1}}}} \big)
\ar[r] &
\chi_1^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-2)
\ar`[rd]^<>(0.5){\delta}`[l]`[dlll]`[d][dll] &
\\
&
\chi_1^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-2)
\ar[r] &
R^1\beta_{1\ast} \big( {{\ensuremath{\mathcal{W}_{1}}}} \big)
\ar[r] &
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\ar[r] &
0
\,.
}}$$ From the discussion is Subsection \[sec:CB\] we know that ${{\ensuremath{\mathcal{W}_{1}}}}={{\ensuremath{\mathcal{W}_{1}}}}^{\ensuremath{\vee}}$ is a vector bundle, that is, it satisfies the relative duality for vector bundles $$R^1\beta_{1\ast} \big({{\ensuremath{\mathcal{W}_{1}}}}\big) = \Big( \beta_{1\ast} \big({{\ensuremath{\mathcal{W}_{1}}}}\big)
\otimes K_{{\ensuremath{B_{1}}}|{{\mathbb{P}^{1}}}} \Big)^{\ensuremath{\vee}}\,.$$ This uniquely fixes the coboundary map $\delta$ to be an isomorphism, and one obtains $$\label{eq:W1pushdown}
\begin{split}
\beta_{1\ast} {{\ensuremath{\mathcal{W}_{1}}}}
=&~
\chi_1 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1)
\,,
\\
R^1 \beta_{1\ast} {{\ensuremath{\mathcal{W}_{1}}}}
=&~
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\,.
\end{split}$$ The coboundary map in the analogous push-down of ${{\ensuremath{\mathcal{W}_{2}}}}$ is zero for trivial reasons. We find that $$\label{eq:W2pushdown}
\begin{split}
\beta_{2\ast} {{\ensuremath{\mathcal{W}_{2}}}}
=&~
\chi_2^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1) \oplus \chi_2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-2)
\,,
\\
R^1 \beta_{2\ast} {{\ensuremath{\mathcal{W}_{2}}}}
=&~
\chi_2^2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(1) \oplus \chi_2 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\,.
\end{split}$$
Finally, we need the push-down of ${{\ensuremath{\mathcal{W}_{i}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{i}}}}}}(2t)$. These are simpler to compute since the fiber degrees are large, so $R^1\beta_{i\ast}$ vanishes. First, the push-down of the ideal sheaves twisted by ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{i}}}}}}(2t)$ is
$$\begin{aligned}
\beta_{1\ast} \Big( I_3\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(2t) \Big) =&~
3 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\oplus 3 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1)
\,, &
R^1\beta_{1\ast} \Big( I_3\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(2t) \Big) =&~ 0
\,, \\
\beta_{2\ast} \Big( I_6\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(2t) \Big) =&~
6 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1)
\,, &
R^1\beta_{2\ast} \Big( I_6\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(2t) \Big) =&~ 0
\,.\end{aligned}$$
The push-down long exact sequence for ${{\ensuremath{\mathcal{W}_{1}}}}$, ${{\ensuremath{\mathcal{W}_{2}}}}$ splits [@HetSM3], and we obtain $$\begin{split}
\label{eq:W12tpushdown}
\beta_{1\ast} \Big( {{\ensuremath{\mathcal{W}_{1}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(2t) \Big) =&~
6 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1) \oplus 3 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\oplus 3 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(1)
\,, \\
R^1\beta_{1\ast} \Big( {{\ensuremath{\mathcal{W}_{1}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{1}}}}}}(2t) \Big) =&~ 0
\,, \\
\end{split}$$ and $$\begin{split}
\label{eq:W22tpushdown}
\beta_{2\ast} \Big( {{\ensuremath{\mathcal{W}_{2}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(2t) \Big) =&~
6 {\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}(-1) \oplus 6{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\mathbb{P}^{1}}}}}\,, \\
R^1\beta_{2\ast} \Big( {{\ensuremath{\mathcal{W}_{2}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{2}}}}}}(2t) \Big) =&~ 0
\,.
\end{split}$$ The push-down for ${{\ensuremath{\mathcal{W}_{i}}}}\otimes{\ensuremath{{\ensuremath{\mathcal{O}}}_{{\ensuremath{B_{i}}}}}}(-2t)$ can be obtained by relative duality.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to E. Buchbinder, R. Donagi, P. Langacker, B. Nelson and D. Waldram for enlightening discussions. This research was supported in part by cooperative research agreement DE-FG02-95ER40893 with the U. S. Department of Energy and an NSF Focused Research Grant DMS0139799 for “The Geometry of Superstrings”. Yang Hui-He is supported in part by the FitzJames Fellowship at Merton College, Oxford.
Line Bundles {#sec:sublinebundle}
============
By elementary computation of $\operatorname{Hom}({\ensuremath{\mathcal{L}}},{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}})$, one can easily see that every equivariant sub-line bundle ${\ensuremath{\mathcal{L}}}$ of ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}$ is $$\label{eq:oXtsublb}
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-3 \tau_1 + \phi)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-2 \tau_1 -\tau_2)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-\tau_1-2 \tau_2)
,\,
{\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}(-3 \tau_2 + \phi)$$ or a sub-line bundle thereof. Since a sub-line bundle of a line bundle always has smaller slope, the equivariant sub-line bundles of ${\ensuremath{{\ensuremath{\mathcal{O}}}_{{{\ensuremath{\widetilde{X}}}}}}}$ of largest slope are those listed in eq. .
[^1]: ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}$ being equivariantly stable is the same as ${{\ensuremath{\widetilde{{\ensuremath{\mathcal{V}_{}}}}}}}/\big({{\ensuremath{{\mathbb{Z}}_3\times{\mathbb{Z}}_3}}}\big)$ being stable. For the remainder of this section, everything is equivariant.
|
---
abstract: 'This paper proposes an affinity fusion graph framework to effectively connect different graphs with highly discriminating power and nonlinearity for natural image segmentation. The proposed framework combines adjacency-graphs and kernel spectral clustering based graphs (KSC-graphs) according to a new definition named affinity nodes of multi-scale superpixels. These affinity nodes are selected based on a better affiliation of superpixels, namely subspace-preserving representation which is generated by sparse subspace clustering based on subspace pursuit. Then a KSC-graph is built via a novel kernel spectral clustering to explore the nonlinear relationships among these affinity nodes. Moreover, an adjacency-graph at each scale is constructed, which is further used to update the proposed KSC-graph at affinity nodes. The fusion graph is built across different scales, and it is partitioned to obtain final segmentation result. Experimental results on the Berkeley segmentation dataset and Microsoft Research Cambridge dataset show the superiority of our framework in comparison with the state-of-the-art methods. The code is available at <https://github.com/Yangzhangcst/AF-graph>.'
author:
- 'Yang Zhang, Moyun Liu, Jingwu He, Fei Pan, and Yanwen Guo [^1][^2]'
bibliography:
- 'References.bib'
title: 'Affinity Fusion Graph-based Framework for Natural Image Segmentation'
---
[Zhang : Affinity Fusion Graph-based Framework for Natural Image Segmentation]{}
Natural image segmentation, affinity fusion graph, kernel spectral clustering, sparse subspace clustering, subspace pursuit
[^1]: Y. Zhang, J. He, F. Pan, and Y. Guo are with the National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210023, China (e-mail: yzhangcst@smail.nju.edu.cn; hejw005@gmail.com; felix.panf@outlook.com; ywguo@nju.edu.cn).
[^2]: M. Liu is with the School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: lmomoy8@gmail.com).
|
---
abstract: 'The use of deep learning for medical imaging has seen tremendous growth in the research community. One reason for the slow uptake of these systems in the clinical setting is that they are complex, opaque and tend to fail silently. Outside of the medical imaging domain, the machine learning community has recently proposed several techniques for quantifying model uncertainty (i.e. a model knowing when it has failed). This is important in practical settings, as we can refer such cases to manual inspection or correction by humans. In this paper, we aim to bring these recent results on estimating uncertainty to bear on two important outputs in deep learning-based segmentation. The first is producing spatial uncertainty maps, from which a clinician can observe where and why a system thinks it is failing. The second is quantifying an image-level prediction of failure, which is useful for isolating specific cases and removing them from automated pipelines. We also show that reasoning about spatial uncertainty, the first output, is a useful intermediate representation for generating segmentation quality predictions, the second output. We propose a two-stage architecture for producing these measures of uncertainty, which can accommodate any deep learning-based medical segmentation pipeline.'
author:
- |
Terrance DeVries\
University of Guelph and Vector Institute\
[terrance@uoguelph.ca]{}
- |
Graham W. Taylor\
University of Guelph and Vector Institute\
Canadian Institute for Advanced Research\
[gwtaylor@uoguelph.ca]{}
bibliography:
- 'egbib.bib'
title: Leveraging Uncertainty Estimates for Predicting Segmentation Quality
---
Introduction
============
In recent years, the use of deep learning for medical imaging tasks has increased in prevalence, with these powerful algorithms being applied to a wide variety of medical imaging applications, from metastasis detection for breast cancer [@Liu2017-xi], to improving reconstruction for medical resonance imaging [@Mardani2017-ft]. In some cases, deep learning has even matched or exceeded human performance, such as on the tasks of skin lesion classification [@esteva2017dermatologist] and identifying diabetic retinopathy [@gulshan2016development].
Unfortunately, despite the recent successes reported in the literature, we have yet to see the widespread adoption of deep learning in clinical settings. One possible reason for this delay could be the lack of suitable uncertainty estimates [@litjens2017survey]. Current neural network-based models are often incapable of indicating when their predictions may be faulty, and as a result they fail silently, without any indication that a mistake has been made. This behaviour is worrying for applications that rely on accurate uncertainty estimates for decision making, such as those that a medical professional might encounter when diagnosing a patient based on the results of a predictive model. Proper uncertainty estimates would allow those reviewing the predictions to act accordingly in order to prevent undesirable outcomes [@amodei2016concrete].
One area where this is a problem is the task of segmenting medical images. If the result of an automated segmentation is poor, we would like to refer the case to a qualified human for follow-up. Furthermore, given the spatial nature of the task, it would be useful to provide the human with a map of the model’s uncertainty so that they can better understand where and why the model failed, and perhaps take the uncertainty estimates into consideration when manually correcting the segmentation.
In this work, we consider the specific task of segmenting skin lesions. However, we propose a framework that is general enough to support a variety of medical segmentation tasks. We propose learning spatial uncertainty maps for each segmentation, which can then be used to improve our prediction of the quality of the segmentation, and we demonstrate that this yields an improved performance over alternative techniques that use deep learning to predict segmentation quality. Based on our finding that per-pixel uncertainty is a useful intermediate representation for predicting image-level segmentation quality we compare several contemporary uncertainty estimation methods to assess their relative merits.
The key contribution of this work is unifying two pursuits that have to-date remained disparate: uncertainty estimation in deep neural networks and predicting image-level segmentation quality. Our method is easy to deploy in practice, as it is modular and agnostic to the specific deep-learning architecture or uncertainty estimation technique. Secondary contributions are the extension of Learning Confidence Estimates [@devries2018learning] to pixel-level, rather than scalar output and the empirical finding that at least three recently-proposed methods for quantifying uncertainty can aid almost equally well in predicting segmentation quality.
Related Work
============
Our work attempts to unite two research areas that are of interest to the machine learning and computer vision community: uncertainty estimation and predicting segmentation quality. Here we provide a brief overview of relevant recent work in the respective areas.
Uncertainty Estimation
----------------------
Uncertainty estimates are useful in the context of deployed machine learning systems as they have been shown to be capable of detecting when a neural network is likely to make an incorrect prediction, or when an input may be out-of-distribution.
Traditionally, much of the work done on uncertainty estimation techniques is inspired by Bayesian statistics. A classic example is the Bayesian Neural Network (BNN) [@neal2012bayesian], which attempts to learn a distribution over each of the network’s weight parameters. Such a network would be able to produce a distribution over the output for any given input, thereby naturally producing uncertainty estimates. Unfortunately, Bayesian inference is computationally intractable for these models in practice, so much effort has been put into developing approximations of Bayesian neural networks that are easier to train.
Recent efforts in this area include Monte-Carlo Dropout [@gal2016dropout], Multiplicative Normalizing Flows [@louizos2017multiplicative], and Stochastic Batch Normalization [@atanov2018uncertainty]. These methods have been shown to be capable of producing uncertainty estimates, although with varying degrees of success. The main disadvantage with these BNN approximations is that they require sampling in order to generate the output distributions. As such, uncertainty estimates are often time-consuming or resource-intensive to produce, often requiring 10 to 100 forward passes through a neural network in order to produce useful uncertainty estimates at inference time.
An alternative to BNNs is Deep Ensembles [@lakshminarayanan2017simple], which proposes a frequentist approach to the problem of uncertainty estimation by training many models and observing the variance in their predictions. However, this technique is still quite resource intensive, as it requires inference from multiple models in order to produce the uncertainty estimate.
A promising alternative to sampling-based methods is to instead have the neural network *learn* what its uncertainty should be for any given input, as demonstrated in [@kendall2017uncertainties] and [@devries2018learning]. These methods are more computationally efficient compared to other techniques, and thus better suited when computational resources are limited or when real-time inference is required.
Segmentation Quality Prediction
-------------------------------
When applying uncertainty estimates to the task of semantic segmentation, a number of works have proposed ways to produce spatial *uncertainty maps*, which visualize a model’s confidence in its predictions for each pixel in the image. In most cases, uncertain regions are likely to be misclassified, and the uncertainty maps allow one to see which parts of the image are likely to be problematic [@kampffmeyer2016semantic; @kendall2015bayesian; @kendall2017uncertainties]. This feature gives the model some amount of interpretability, and provides the end user with more information with which they can decide whether the final segmentation is to be trusted or how it should be modified (e.g. in a human-in-the-loop setting).
However, if the semantic segmentation model is part of a larger automated pipeline, pixel-level uncertainty estimates are not as useful, as perfectly acceptable segmentations can still contain some uncertainty. In this case, it is more useful to create a model that can predict the quality of the segmentation at the whole-image level. Previous efforts have attempted to learn the quality of segmentations from hand-crafted image or segmentation features [@kohlberger2012evaluating; @zhang2006meta], but these approaches are limited by the expressiveness of their respective hand-crafted features. They are also limited in their transferability across different medical imaging modalities.
Contemporary approaches have exploited the powerful feature learning capabilities of deep learning. Recently, adversarial training has been used to improve the performance of convolutional segmentation networks by having an auxiliary discriminator network predict the quality of the segmentation (i.e. whether or not the predicted segmentation is discernible from a ground truth segmentation) [@luc2016semantic]. The segmentation network then uses this information to improve its predictions and produce more realistic looking segmentations. This technique has previously been demonstrated to improve segmentation quality in medical imaging tasks such as prostate cancer or brain MRI segmentation [@kohl2017adversarial; @moeskops2017adversarial]. Adversarial training works well to improve segmentation quality, but the quality estimation network has limited utility as the outputs don’t have any human interpretable meaning associated with them beyond whether the segmentation looks realistic or not. As a solution to the interpretability issue, methods have been proposed which attempt to predict segmentation quality in terms of metrics that are more meaningful to humans, such as Jaccard index or Dice coefficient. An example of this is QualityNet [@huang2016qualitynet], which learns a direct mapping between a masked input image and its corresponding segmentation quality via a convolutional neural network (CNN). Another interesting approach is Reverse Classification Accuracy (RCA) [@valindria2017reverse], which evaluates segmentation quality by training a reverse classifier on a predicted segmentation from a new image, and then evaluating on a set of reference images that have ground truths available. While these techniques work well in their respective settings, none of them exploit uncertainty information, which could be used to improve the accuracy of the segmentation quality prediction.
Estimating Segmentation Quality with Uncertainty Information
============================================================
In order to leverage uncertainty information in our predictions of segmentation quality, we first train a neural network-based semantic segmentation model $f$ (as shown in Figure \[fig:system\_diagram\]). The segmentation network takes in some image $x$, and produces two outputs: class prediction logits $\rho$ and corresponding uncertainty (or confidence) estimates $z$: $$\rho, z = f(x) .\$$ As uncertainty is estimated per pixel, we refer to $z$ as an uncertainty map. In this formulation, $z$ may either be calculated using the original outputs of $f$, or the network can produce the $z$ directly. To obtain the final segmentation prediction $\hat{y}$ we take the argmax of the prediction logits or the class prediction probabilities: $$\hat{y} = \text{argmax}(\rho) .\$$ A second network $g$ is then trained to predict the quality of the segmentation $\hat{v}$, given the original input image $x$, as well as the predicted segmentation mask $\hat{y}$ and uncertainty map $z$ from $f$: $$\begin{aligned}
\hat{v} &= g(x, \hat{y}, z) .\\end{aligned}$$ Under our framework, the segmentation quality measurement can be any segmentation-based evaluation metric, or even multiple metrics predicted simultaneously. To obtain the true segmentation quality labels $v$ to train $g$, we evaluate the segmentation predictions from $f$ using the ground truths from the training set. The training set for $g$ can be the same one as used to train $f$, or a separate holdout set, or a combination of the two. In the case that $f$ performs very well on the training set, a holdout set may be necessary, as the lack of examples of poor segmentations will bias $g$ towards always predicting that the segmentation is good.
There have been many methods proposed for attaining uncertainty or confidence estimates from neural networks, but for our experiments we consider four methods: maximum softmax probability, Monte-Carlo Dropout, heteroscedastic classifier neural networks, and learned confidence estimates. We selected these based on their simplicity to implement as well as their diversity.
Maximum Softmax Probability
---------------------------
The first method we evaluate is the maximum softmax probability, which was demonstrated by [@hendrycks2016baseline] to be surprisingly effective at the tasks of misclassification and out-of-distribution detection. The softmax probability can be obtained from any classification neural network for free, making it an appealing choice for confidence estimates. To calculate the maximum softmax probability we simply calculate the maximum across the class dimension of the softmax output from the network $f$: $$\begin{aligned}
z = \text{max}(\text{Softmax}(\rho)) .\\end{aligned}$$ For segmentation, this is done per output pixel in order to obtain an uncertainty map that is of the same resolution as the input image.
Monte-Carlo Dropout
-------------------
The second uncertainty estimation method we consider is Monte-Carlo dropout (MC-dropout) [@gal2016dropout], which has previously seen success in the field of medical imaging [@leibig2017leveraging; @yang2016fast]. MC-dropout approximates a BNN by sampling from a neural network trained with dropout [@srivastava2014dropout] at inference time in order to produce a distribution over the outputs. This approach is very simple to implement in practice, and as many modern neural network architectures already leverage dropout for regularization purposes, uncertainty estimates can often be attained without any changes to the architecture or training paradigm. MC-dropout models epistemic uncertainty, which is the uncertainty associated with the model parameters, such that increasing the amount of training data tends to decrease the epistemic uncertainty associated with the model.
Following the approach used for Bayesian SegNet [@kendall2015bayesian; @kendall2017uncertainties], we apply dropout with $p=0.5$ after each central convolutional block of our U-Net architecture. During test time we sample from the segmentation network $T$ times (we use $T = 20$) and then calculate the average softmax probability over all of the samples in order to approximate Monte Carlo integration: $$p = \frac{1}{T} \sum_{t=1}^{T} \text{Softmax}(\rho_{t}) .\$$ Model uncertainty $z$ is estimated by calculating the entropy of the averaged probability vector across the class dimension: $$z= -\sum_{c=1}^{C} p_{c} \text{log} p_{c}.\$$
Heteroscedastic Classifier Neural Network
-----------------------------------------
The third uncertainty estimation technique we evaluate is one which attempts to model aleatoric uncertainty, which is the uncertainty present in the data itself, such as from noisy labels or measurements. To model aleatoric uncertainty, [@kendall2017uncertainties] introduce the heteroscedastic classifier neural network, which we will refer to as HCNN. In this method, uncertainty estimates are learned by the network, rather than being calculated post-hoc as with MC-dropout. The HCNN produces two outputs via two separate output branches: class prediction logits, and a variance estimate which represents model uncertainty. Again, in the case of segmentation, these two quantities are computed per output pixel. During training, Gaussian noise with magnitude equal to the variance estimate is sampled and added to the probability logits, which are used to calculate the training loss as usual: $$p=\frac{1}{T} \sum_{t=1}^{T}\text{Softmax}(\rho+z \epsilon_{t}), \quad \epsilon_{t} \sim \mathcal{N}(0, 1) .\$$ In our experiments we set T = 100. We also apply a softplus function to the output of the variance estimation branch in order to ensure that it is non-negative.
Learned Confidence Estimates
----------------------------
The final technique we evaluate is Learned Confidence Estimates (LCE), which was introduced by [@devries2018learning]. This method is similar to HCNN in that the network produces two separate outputs: prediction probabilities and a confidence estimate. Confidence estimates are motivated by interpolating between the predicted probability distribution and the target distribution during training, where the degree of interpolation is proportional to the confidence estimate: $$p = z \cdot \text{Softmax}(\rho) + (1-z) \cdot \text{Onehot}(y) .\$$ In this formulation, low confidence estimates are pushed towards the correct answer, while high confidence estimates remain unchanged. To prevent the model from always producing low confidence estimates, a log penalty on the confidence estimate is added to the loss function. As a result, the network can reduce its overall training loss if it correctly infers which samples it is likely to predict incorrectly.
{width="76.00000%"}
**Method**
----------------- ----------------------- -------------------- -------------------- -------------------- --------------------
RCA 0.438 $\pm$ 0.007 43.8 $\pm$ 1.0 53.7 $\pm$ 1.4 74.4 $\pm$ 1.4 30.7 $\pm$ 0.9
QualityNet 0.213 $\pm$ 0.009 25.7 $\pm$ 2.7 80.9 $\pm$ 3.1 89.0 $\pm$ 2.3 69.1 $\pm$ 4.7
No Uncertainty 0.198 $\pm$ 0.011 27.3 $\pm$ 3.3 79.8 $\pm$ 3.8 88.5 $\pm$ 1.9 66.4 $\pm$ 7.9
Max Probability **0.168** $\pm$ 0.014 **18.4** $\pm$ 3.0 **88.4** $\pm$ 2.2 **93.2** $\pm$ 1.6 **80.5** $\pm$ 3.2
MC-dropout **0.163** $\pm$ 0.010 **18.8** $\pm$ 1.4 **88.1** $\pm$ 0.8 **93.5** $\pm$ 1.3 **78.1** $\pm$ 3.0
HCNN 0.196 $\pm$ 0.023 21.3 $\pm$ 1.8 85.5 $\pm$ 1.5 91.6 $\pm$ 1.4 76.2 $\pm$ 4.5
LCE **0.167** $\pm$ 0.019 **19.3** $\pm$ 1.1 **88.3** $\pm$ 1.4 **93.6** $\pm$ 1.5 **79.1** $\pm$ 3.9
Experiments
===========
To evaluate our method we apply it to the problem of skin lesion segmentation, as this application has received a fair amount of attention from the deep learning community [@esteva2017dermatologist]. Specifically, we work with the ISIC 2017 dataset [@codella2017skin], which consists of 2,750 dermoscopic images in three official dataset splits: 2,000 training images, 150 validation images, and 600 test images. Each image depicts a skin lesion from one of three different classes: melanoma, seborrheic keratosis, and benign nevi. Additionally, each image has an accompanying expert-labeled binary segmentation mask. For our experiments, we resize all images and ground truth masks to $224 \times 224$ pixels.
For our semantic segmentation network $f$, we adopt a U-Net style model architecture [@ronneberger2015u]. To facilitate MC-dropout we apply dropout with $p=0.5$ to the central layers of the encoder and decoder, as in Bayesian SegNet [@kendall2015bayesian]. Each model is trained for 120 epochs using batches of 16 images, and the Adam optimizer [@kingma2014adam] with a learning rate of 0.001. Images are randomly flipped and rotated at 90 degree intervals for data augmentation. For each uncertainty estimation method we train five models with random parameter initializations so that we can observe variance in performance. We find that all segmentation networks score within the range of $0.73 \pm 0.02$ Jaccard index, which is competitive with single-model performance for this task.
Our segmentation quality prediction network $g$ is a VGG-style CNN, which is trained to predict the Jaccard index of any given segmentation prediction given the original image, predicted segmentation mask, and confidence. We train our quality prediction network for 30 epochs using batches of 16 images, and the Adam optimizer with a learning rate of 0.001. As with our segmentation network, we apply flipping and rotating transforms for data augmentation.
For comparison, we also train models using two other neural network-based segmentation quality prediction methods: Reverse Classification Accuracy (RCA) [@valindria2017reverse] and QualityNet [@huang2016qualitynet]. As each of these approaches have their own architectural and optimization-based hyper-parameters, we have kept these the same as our technique, where applicable.
Uncertainty Maps
----------------
To compare the quality of the uncertainty maps from each of the different uncertainty (and confidence) estimation techniques, we visualize them in Figure \[fig:uncertainty\_maps\]. In cases where the predicted segmentation is very close to the ground truth segmentation, we find that all techniques act similarly, outputting a tight ring along the segmentation borders. This is what we would expect in such a situation. The more interesting observation is how the uncertainty maps react when the predicted segmentation is very poor. We find that in general, maximum softmax probabilty, MC-dropout, and LCE all display high uncertainty in regions that are segmented incorrectly. Conversely, HCNN usually outputs a small band of low uncertainty around its prediction, but does not highlight other areas that may be incorrect. This output is less useful for identifying failed segmentations, which agrees with our findings in §\[section:segmentation\_quality\_prediction\].
Segmentation Quality Prediction {#section:segmentation_quality_prediction}
-------------------------------
We use a variety of metrics to evaluate how well our models can predict the quality of segmentations: RMSE, detection error, AUROC, and AUPR; each of which is defined below.
**RMSE**: Measures Root Mean Squared Error, which is the difference between the predicted Jaccard index and the true Jaccard index. Predictions that are further from the true value are penalized more heavily in this metric. RMSE is calculated as $\sqrt{\frac{\sum_{t=1}^{n}({\hat{v}}_{t} - v_{t})^{2}}{n}}$ where $t$ indexes the test examples, and $n$ is the total number of test examples.
For practical applications (e.g. human-in-the-loop) we may also want to measure how well our model can detect *failed* segmentations. For the ISIC 2017 dataset, a Jaccard index of below 0.7 is considered to be a failed segmentation [@codella2017skin]. To evaluate how well our model can detect these failures, we threshold the true Jaccard index labels at 0.7 to obtain binary labels, which we can use to calculate detection error, AUROC, and AUPR.
**Detection Error**: Measures the minimum possible misclassification probability over all possible thresholds $\delta$ when detecting segmentation failures, as defined by $\min_{\delta}\left\{0.5 \, P_{\text{pass}}(f(x)\leq \delta) + 0.5 \, P_{\text{fail}}(f(x)> \delta) \right\}$. Here, we equally weight $P_{\text{pass}}$ and $P_{\text{fail}}$ as if they have the same probability of appearing in the test set.
**AUROC**: Measures the Area Under the Receiver Operating Characteristic curve. The Receiver Operating Characteristic (ROC) curve plots the relationship between true positive rate and false positive rate. The area under the ROC curve can be interpreted as the probability that a correctly segmented image will have a higher quality estimate than a failed segmentation.
**AUPR**: Measures the Area Under the Precision-Recall (AUPR) curve, which is calculated by plotting precision versus recall. In our tests, AUPR-Pass indicates that acceptable segmentations are used as the positive class, and AUPR-Fail indicates that failed segmentations are used as the positive class. We evaluate both metrics so that we can see if our model is biased towards either class.
{width="\textwidth"} \[fig:annotated\_jaccard\_scatter\]
\
In Table \[table:seg\_quality\_prediction\], we present the results of our quantitative evaluation of different segmentation quality estimation methods. These are organized by two groupings: 1) recent baselines RCA and QualityNet, which leverage CNNs to predict segmentation quality and 2) variants of our two-stage CNN approach. The variants of our approach apply different quantitative measures of uncertainty, described above. We also include a method which does not explicitly generate uncertainty as an intermediate representation, i.e. $g$ receives only $(x,\hat{y})$ as input rather than $(x,\hat{y},z)$. This is slightly different than QualityNet, in which $g$ receives $(x \odot{\hat{y}})$, where $\odot{}$ is an element-wise matrix multiplication.
We find that treating uncertainty explicitly improves performance significantly compared to the case with no uncertainty information, reducing RMSE by up to 0.03 points, and detection error by up to 8 percentage points. However, surprisingly the particular method by which uncertainty is captured does not have a large effect, at least in this setting. Maximum softmax probability, MC-dropout, and LCE all produce similar improvements in performance. It was expected that MC-dropout or LCE would outperform maximum softmax probability since they have been shown to surpass softmax probability in tasks such as out-of-distribution detection [@devries2018learning], but this was not the case for this particular dataset. Of the uncertainty estimation techniques, HCNN improved performance the least; only 5 points of detection error and no improvement on RMSE. This agrees with [@kendall2017uncertainties], which indicates that aleatoric uncertainty which HCNN aims to capture, is a poor choice for detecting model failures since it mainly models noise in the data itself.
We find that our implementation of QualityNet performs roughly equal to our *no uncertainty* baseline, which is expected given how similar the implementations are. Unfortunately, RCA performs very poorly; only slightly better than random. This is likely because the algorithm was designed to work on datasets of registered images with very little variation between them, such as MRI scans of internal organs. In these datasets each of the objects to be segmented are extremely similar in shape and location. In contrast, the skin lesions from the ISIC 2017 dataset appear with a wide variety of colours, textures, shapes, sizes, and locations, making them very difficult for modern image registration techniques to succeed. In Figure \[fig:all\_jaccard\_scatter\] we plot the true versus the predicted Jaccard index for each of the different uncertainty estimates we tested. We observe that max probability, MC-dropout, and LCE are all better at identifying poor quality segmentations (lower left corner) compared to HCNN or the no uncertainty baseline. Additionally, we note that the majority of false positives (poor quality segmentations that are rated highly) are caused by either corrupted labels or lazy annotations, as shown in Figure \[fig:annotated\_jaccard\_scatter\].
Interestingly, segmentation quality estimates rarely fall below 0.2 for any method. This is likely caused by the rarity of poor quality segmentations in the dataset used to train the segmentation quality estimation network, since it was trained on the same dataset as the original segmentation network. While it is probable that using a separate held-out dataset would result in a greater number of poor quality segmentation examples, and therefore better performance from the segmentation quality estimation network, we do not explore this option due to the small size of the ISIC 2017 dataset.
Conclusion
==========
In this work, we investigated techniques which aid a human operator, such as a clinician, interact with a deep learning-based automated segmentation pipeline. We showed how uncertainty could be derived at the pixel- and image-level within a single end-to-end framework. We demonstrated our method qualitatively and quantitatively on the task of skin lesion segmentation. Though a neural network trained to predict segmentation quality has the capacity to measure uncertainty internally, we showed that making spatial uncertainty explicit aided in predicting a measure of segmentation quality, the Jaccard index. Moreover, we demonstrated that several recent methods for quantifying uncertainty worked well in this setting. In the future, we plan on extending our analysis to other medical segmentation problems, and even tasks outside segmentation that could benefit from a human-in-the-loop. We also used simple, standard losses for our segmentation model. Recent techniques that aim to optimize application-specific metrics like the Jaccard index would likely improve overall performance.
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---
author:
- 'C. Stanghellini[^1]'
- 'D. Dallacasa'
- 'M. Orienti'
title: 'High Frequency Peakers: The Faint Sample'
---
Introduction
============
In the framework of powerful radio-galaxy evolution, GPS and then CSS radio sources are nowadays considered the early stages, as the radio emitting region grows and expands within the interstellar matter of host galaxy, before plunging into the intergalactic medium to originate the extended radio source population (Orienti and Dallacasa 2008,2009).
Most of the samples of powerful CSS and GPS radio sources consist of sources with turnover frequencies ranging from about 100 MHz to about 5 GHz.
Objects with turnover frequencies above 5 GHz would represent [*smaller*]{} and therefore [*younger*]{} radio sources. We call these sources “High Frequency Peakers” (HFPs).
Candidate High Frequency Peakers {#structure}
================================
From the cross correlation of the 87GB catalogue at 4.9 GHz and the NVSS catalogue at 1.4 GHz we selected the sources with inverted spectra with a slope steeper than 0.5 ($S\propto\nu^{-\alpha}$). We defined [**two**]{} samples of candidates based on the flux density in the 87GB catalogue: the “[**bright**]{}” sample (S$_{5GHz}>300$mJy) covering $0^\circ <$ DEC $<+75^\circ$ but excluding objects projected on the galactic plane ($\mid{b_{II}\mid}>10^\circ$) and the “faint” sample (50mJy $<$ S$_{5GHz} <300 $mJy) restricted to an area around the northern galactic cap ($07^h:20^m < $RA$< 17^h:10^m$, $22.5^\circ <$DEC$< 57.5^\circ$) covered by the FIRST survey as well (Becker et al. 1995).
Here we present the “faint” sample. A description of the “bright” sample and the methods used in the selection can be found in Dallacasa et al. (2000).
------------- ------- ------ ------
Date Conf. Obs. code
Time
07 Nov 1998 BnC 150 a
14 Nov 1998 BnC 150 b
19 Dec 1998 C 240 c
14 Jun 1999 AnD 420 d
21 Jun 1999 AnD 180 e
25 Jun 1999 A 120 f
25 Sep 1999 A 240 g
15 Oct 1999 BnA 240 h
25 Feb 2000 BnC 240 i
------------- ------- ------ ------
: VLA observations and Configurations. The total observing time (column 3) is inclusive of the scans on the “bright” HFP candidates.
---------------- --------- -------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --------- -------
$name$ $ID$ z ra-dec $S_{1.4}$ $S_{1.7}$ $S_{4.5}$ $S_{5.0}$ $S_{8.1}$ $S_{8.5}$ $S_{15}$ $S_{22}$ $\nu_m$ $S_m$
$_{J2000}$ $_{GHz}$ $_{mJy}$
J0736+4744$_e$ q20.40r 0111 231 42.9 53.0 61.8 59.2 48.5 47.2 30.0 31.5 3.0 64
J0754+3033$_e$ q17.38r 0.796 4872 554 47.8 54.1 170 182 231 225 201 142 9.3 233
J0804+5431$_i$ g18.33r (0.29) 5925 579 38.9 43.4 83.2 81.0 67.5 67.2 49.3 38.4 5.4 84
J0819+3823$_b$ g21.72r 0081 597 15.8 20.6 104 108 93.2 90.5 44.6 24.2 5.9 115
J0821+3107$_e$ q17.04r 2.624 0761 520 110 123 134 131 112 109 70.5 48r 3.4 140
J0905+3742$_h$ g22.41r 2745 532 53.6 74.1 104 101 71.3 67.8 39.0 24 3.3 113
J0943+5113$_h$ 5179 217 73.6 91.6 161 148 69.6 64.1 25.8 19 3.7 209
J0951+3451$_b$ g19.42r (0.30) 1139 315 19.2 27.8 61.1 62.0 56.7 55.2 38.1 27.6 5.1 62
J0955+3335$_f$ q17.37r 2.477 3797 045 38.6 49.4 101 106 107 105 70.2 38.4 6.6 109
J1002+5701$_h$ g22.42r 4165 116 19.8 32.6 135 133 77.2 70.9 20.1 12.9 4.4 139
J1004+4328$_f$ g21.87r (0.44) 3132 332 12.3 16.9 38.0 38.1 31.9 29.8 32.8 27.7 8.1 36
J1008+2533$_c$ 2073 395 39.0 57.6 111 110 98.2 96.2 74.4 48.1 5.9 109
J1020+2910$_c$ 0865 297 19.3 21.7 24.6 23.6 19.1 19.2 11.4 7.9 3.3 26
J1020+4320$_f$ q18.94r 1.962 2718 564 116 163 239 227 183 175 135 84.5 1.4 245
J1025+2541$_c$ 2379 577 21.7 23.1 50.9 49.3 34.2 32.3 12.6 5.7 4.1 51
J1035+4230$_h$ q19.21r 2.441 3259 200 23.1 29.2 104 109 110 108 74.7 54 5.8 119
J1037+3646$_f$ g22.57r ($>$1) 5729 557 66.8 96.3 147 140 95.9 90.5 56.3 32.5 1.6 157
J1044+2959$_h$ q18.99 2.983 0632 027 38.3 53.8 176 182 169 167 125 92 4.0 183
J1046+2600$_c$ 5727 462 12.5 15.4 36.0 35.0 27.2 25.9 13.0 5.6 6.2 35
J1047+3945$_h$ q20.11r ($>$1) 0324 457 37.8 47.5 55.9 54.4 42.7 42.1 29.8 24 2.1 62
J1052+3355$_f$ q16.99 1.407 5009 041 10.4 15.5 46.3 44.9 29.9 27.6 18.8 25.8 3.0 50
J1053+4610$_i$ g22.40r (0.51) 5347 589 10.4 15.1 36.0 38.1 35.1 34.8 38.1 36.5 11 40
J1054+5058$_i$ 1396 166 11.8 12.3 22.8 27.0 30.6 31.1 39.4 38.8 20 39
J1058+3353$_f$ g18.72r 0.265 0884 040 21.7 30.4 66.9 65.2 52.9 51.3 44.7 36.6 3.0 77
J1107+3421$_f$ g21.42r (0.10) 3431 182 24.6 38.3 77.4 76.1 53.9 50.7 29.3 28.5 2.7 101
J1109+3831$_i$ g21.09r (0.12) 3918 215 9.6 17.6 54.3 58.5 77.0 77.1 53.3 37.7 8.5 73
J1135+3624$_h$ s23.50r 5229 225 29.3 38.5 57.6 57.5 43.0 40.8 22.4 15 4.0 60
J1137+3441$_h$ q23.35r 0.835 0909 562 23.6 28.2 70.1 71.4 76.6 75.6 79.1 91 14.5 87
J1203+4803$_i$ q16.21r 0.817 2992 136 156 191 436 451 596 617 862 909 34 955
J1218+2828$_c$ 0632 250 18.3 24.3 59.9 79.0 94.7 94.1 60.8 32.3 9.5 95
J1239+3705$_a$ 3638 081 8.7? 8.9 86.3 95.2 115 115 89.4 82.9 5.8 123
J1240+2425$_c$ q16.6 0.829 0906 297 36.4 43.4 66.8 62.9 38.7 37.1 25.3 11.5 2.7 67
J1240+2323$_c$ 1776 499 14.5 19.9 54.6 56.2 60.6 60.1 53.0 35.1 7.9 62
J1241+3844$_a$ 4300 037 19.0 22.1 24.2 23.2 20.3 20.9 17.0 21.5 4.1 23
J1251+4317$_f$ q18.81r 1.440 4618 275 13.5 24.0 52.3 54.8 50.0 49.2 43.7 36.0 1.8 53
J1258+2820$_g$ 0202 025 30.4 34.0 39.9 40.5 39.6 38.4 32.4 24.1 4.7 41
J1300+4352$_f$ 2018 248 144 174 235 234 220 219 205 153 5.9 234
J1309+4047$_f$ q19.01r 2.910 4149 549 34.7 53.4 130 128 102 95.2 59.0 35.9 2.0 129
J1319+4851$_g$ q19.18r 1.170 3031 039 20.0 29.2 48.2 45.9 38.2 36.8 26.7 23.2 2.4 53
J1321+4406$_g$ g21.45r (0.32) 2547 337 33.1 39.9 72.3 74.8 74.1 72.6 64.7 54.8 7.6 76
J1322+3912$_g$ q17.61r 2.985 5565 080 108 123 212 215 192 189 128 90.0 5.8 216
J1330+5202$_g$ g21.01r (0.57) 4259 153 83.3 97.2 151 154 153 152 133 110 6.8 156
J1336+4735$_g$ s19.91r 1169 205 28.8 36.8 60.0 58.9 47.3 45.8 31.2 18.6 1.8 60
J1352+3603$_d$ g18.15r (0.33) 0100 515 58.1 65.4 115 117 111 109 72.1 44.9 7.9 118
J1420+2704$_g$ s20.37r 5125 252 8.0 14.3 55.5 55.6 52.3 51.4 37.4 25.9 2.3 57
J1421+4645$_g$ q18.08r 1.668 2302 475 105 125 181 180 164 161 131 110 1.3 180
J1436+4820$_g$ g21.39r ($>$1) 1889 392 17.0 26.7 74.1 75.2 65.2 62.5 40.5 22.8 2.1 75
J1459+3337$_d$ q17.71r 0.645 5849 008 13.2 19.1 162 189 374 394 612 569 17 611
J1512+2219$_g$ g21.08r (0.40) 2823 387 21.7 30.4 34.1 31.7 16.4 15.7 14.3 8.1 2.3 47
J1528+3816$_d$ g20.95r ($>$1) 3702 064 20.1 23.5 47.5 50.3 66.0 65.1 76.6 67.2 14 75
J1530+2705$_d$ g14.31r 0.032 1622 512 10.8 11.1 39.7 44.3 67.3 68.7 67.1 41.1 12 76
J1530+5137$_g$ 1975 303 42.6 53.8 68.3 68.5 70.1 71.9 76.1 76.6 - -
J1547+3518$_d$ s21.37r (0.96) 5413 425 10.4 12.6 43.7 48.7 65.8 66.2 83.9 78.8 17 82
J1602+2646$_d$ g18.58r 0.372 3963 054 33.3 39.8 125 140 226 231 256 213 13 269
J1613+4223$_d$ s20.01r (0.17) 0480 187 35.5 66.1 215 208 131 122 43.9 23.7 2.8 226
---------------- --------- -------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --------- -------
---------------- --------- -------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --------- -------
$name$ $ID$ z ra-dec $S_{1.4}$ $S_{1.7}$ $S_{4.5}$ $S_{5.0}$ $S_{8.1}$ $S_{8.5}$ $S_{15}$ $S_{22}$ $\nu_m$ $S_m$
$_{J2000}$ $_{GHz}$ $_{mJy}$
J1616+4632$_d$ B? 0375 251 65.7 73.0 99.3 101 129 131 157 163 19? 169
J1617+3801$_d$ q19.20r 1.607 4850 407 46.0 21.3 27.5 73.7 78.9 98.7 99.0 100 28 105
J1624+2748$_g$ g21.61r 3565 580 17.1 19.6 97.2 110 173 177.4 185 149 12 198
J1651+3417$_d$ g23.22r (0.69) 4234 006 $\sim$7.3 $\sim$9.4 47.8 52.3 63.0 63.2 49.4 35.7 6.2 64
J1702+2643$_d$ s17.40r (0.21) 0965 148 31.8 34.9 66.0 67.4 76.6 77.3 86.2 88.7
J1719+4804$_d$ q15.3 1.084 3831 121 54.5 68.6 131 141 205 206 188 149 11 229
---------------- --------- -------- ------------ ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --------- -------
We inspected the NVSS and FIRST images to make sure that the component in the catalogue accounted for the whole flux density. The extended objects (typically FRII and a few FRI or complex radio sources, point-like in the 87GB, but resolved in the FIRST) were removed.
The simultaneous radio spectra {#spectra}
==============================
Simultaneous multifrequency observations are necessary to remove flat spectrum variable sources from the sample. Our selection criteria select also variable sources that happened to be in a “high” activity state at the time of the 4.9 GHz observation (e.g. Tinti et al. 2005).
Hence, we have observed at the VLA the whole “faint” sample during several observing runs between November 1998 and February 2000 (Tab.1 ) measuring nearly simultaneous flux densities in L band (with the two IFs at 1.365 and 1.665 GHz), C band (4.535 and 4.985 GHz), X band (8.085 and 8.485 GHz), U band (14.935 and 14.985 GHz) and K band (22.435 and 22.485 GHz).
Each source was observed typically for 40 seconds at each frequency in a single snapshot, cycling through frequencies.
For each observing run we spent one or two scans on the primary flux density calibrators 3C286 or 3C147 or 3C48. Secondary calibrators were observed for 1 minute at each frequency about every 25 minutes; they were chosen aiming to minimize the telescope slewing time and therefore we could not derive accurate positions for the radio sources we observed.
The data reduction has been carried out following the standard procedures for the VLA implemented in the NRAO AIPS software. Separate images for each IF were obtained at L, C and X bands in order to improve the spectral coverage of our data.
In general one iteration of phase-only self-calibration have been performed before the final imaging. On the final image we perfomed a Gaussian fit to measure the flux density of the radio source, and checked the total flux density to find evidence of resolved radio emission. Generally all the HFP candidates were unresolved by the present observations.
The r.m.s. noise levels in the image plane is relevant only for measured flux densities of a few mJy, the major contribution comes from the amplitude calibration error.
The overall amplitude error is dominated by the calibration error, and we estimate it is (1$\sigma$) 3% at L,C and X bands, 5% at U band and finally 10% at K band.
The “Faint” HFP Sample
======================
We derived the spectral indices between any pair of adjacent frequencies. We considered genuine HFPs the radio sources showing an inverted spectral index steeper than 0.5 between 1.37 and 1.67 GHz [**or**]{} between 1.67 and 4.54 GHz. The sources not fullfilling this requirement are discarded as genuine HFPs. The final sample consists of the 61 HFPs listed in Table 2 with the results of our observations.
The optical ID and redshift are from the NED database, when available or from our optical ID on the Sloan Digital Sky Survey (SDSS) when no other optical information is available.
We fitted the radio data with a purely analytic function, with no physics behind, given it is used to determine only “analytical” quantities, namely the peak and the frequency at which it occurs. We used the function of an hyperbola:
$Log~S~=~aLog(\nu)~+~b~-~\sqrt{{c^2\over d^2}\times (Log(\nu)-e)^2+c^2}$
When the data did not sampled the region above the peak we simplified the function omitting $aLog(\nu)$, thus resulting in an hyperbola symmetric with respect to the vertical axis.
From this fitting curve we derived the spectral peak ($S_m$ and $\nu_m$, last two columns of Table 2), representing the actual maximum, regardless the point where the optical depth is unity, or any other physical measure. Given the assumptions the statistical error of the fit does not represent the real uncertainty on the estimate of $S_m$ and $\nu_m$, and a conservative value is about $10\%$.
The analysis of the properties of the faint sample and a comparison with the bright sample will be presented in a following paper.
Becker, R.H., White, R.L., Helfand, D.J.: 1995, ApJ 450, 559 Dallacasa, D., Stanghellini, C., Centonza, M., Fanti, R.: 2000, A&A 363, 887 Orienti, M., Dallacasa, D.: 2008, A&A 487, 885 Orienti, M., Dallacasa, D.: 2009, in preparation Tinti, S., de Zotti, G.: 2005, A&A 445, 889
[^1]: Corresponding author:
|
---
abstract: '[ The cross sections of the nuclear reactions induced by neutrons at $E_n$= 14.6 MeV on the isotopes of Dy, Er, Yb with emission of neutrons, proton and alpha-particle are studied by the use of new experimental data and different theoretical approaches. New and improved experimental data are measured by the neutron-activation technique. The experimental and evaluated data from EXFOR, TENDL, ENDF libraries are compared with different systematics and calculations by codes of EMPIRE 3.0 and TALYS 1.2. Contribution of pre-equilibrium decay is discussed. Different systematics for estimations of the cross-sections of considered nuclear reactions are tested. ]{}'
author:
- 'A.O. Kadenko'
- 'I.M. Kadenko'
- 'V.A. Plujko'
- 'O.M. Gorbachenko'
- 'G.I. Primenko'
title: |
\
\
\
\
\
\
Cross Sections of Neutron Reactions $(n,~p)$, $(n,~\alpha )$, $(n,~2n)$ on Isotopes of Dysprosium, Erbium and Ytterbium at $\sim$ 14 MeV Neutron Energy
---
INTRODUCTION
=============
Studies of the nuclear reaction cross sections induced by neutrons provide information on the properties of excited states of atomic nuclei and nuclear reaction mechanisms [@09Capo]. Data on nuclear reaction cross section are also needed in applications, specifically, such as the design of fusion reactor protection and modernization of existing nuclear power plants [@06Forr; @09Koni]. Despite of a large amount of information [@12EXFO] on observed characteristics of neutron interactions with nuclei, there are disagreements between existing experimental data and evaluated data both within different systematics and calculations by the different codes.
In this contribution, experimental and theoretical cross sections of reactions (n, p), (n, $\alpha $), (n, 2n) on Dy, Er and Yb isotopes at neutron energies near 14.6 MeV are determined and compared. Neutron generator (NG-300), installed in Nuclear Physics Department of Taras Shevchenko National University of Kyiv, was used as a source of neutrons with $E_n$= 14.6 $\pm$ 0.2 MeV. The neutron activation method was applied for measurements of the cross-sections(see [@12Gorb; @12Kade] for details). Theoretical calculations were performed by the EMPIRE 3.0 and TALYS 1.2 codes[@07Herm; @08Koni]. The experimental cross-sections were also compared with data from the latest versions of evaluated nuclear data libraries: ENDF/B-VII, TENDL-2010, JENDL-4.0. The reliability of the different systematics [@71LuWe]-[@09Kono] for estimation of the nuclear reaction cross sections on isotopes of Dy, Er and Yb was analyzed.
RESULTS OF MEASUREMENTS AND CALCULATIONS
=========================================
![Cross sections of the (n, 2n), (n, p) and (n, $\alpha $) reactions on the isotopes of Dysprosium.[]{data-label="fig1"}](fig1f.eps){width="0.95\columnwidth"}
Figure \[fig1\] shows cross sections of the (n, 2n), (n, p), (n, $\alpha$) reactions on the isotopes of Dy as a function of the number of neutrons in comparison with theoretical calculations and evaluated data. For reactions with transitions on ground (g) and metastable (m) states, the residual nuclei are additionally indicated on this and next figures.
Calculations of the nuclear reaction cross sections by the EMPIRE 3.0 code were performed with and without including pre-equilibrium processes (PCROSS = 1.5 and 0). For nuclear level density, the generalized superfluid model was taken. The global optical potential of Koning-Delaroche [@03Koni] was used. In calculations by the TALYS 1.2 code default parameters were set. It can be seen, that the measured (n,2n) cross section on the $^{158}$Dy (N=92) with allowance for uncertainties coincides with the available experimental data. The measured (n,2n) cross section on the $^{156}$Dy (N=90) is less than previous ones, but it coincides with evaluated value from the TENDL-2010 library. The (n, 2n) cross sections on Dy isotopes increase with the neutron number increasing.
The allowance for pre-equilibrium processes leads to strong increasing the (n, p) cross sections (approximately in five times). A behavior of the (n, 2n) cross sections is opposite, and pre-equilibrium emission reduce the values of these cross sections mainly due to increase cross-sections of competing binary reactions with emission of charge particles.
![Cross sections of the (n, 2n), (n, p) and (n, $\alpha $) reactions on the isotopes of Erbium.[]{data-label="fig2"}](fig2f.eps){width="0.95\columnwidth"}
On the whole, the results of calculations by EMPIRE 3.0 code with allowance for pre-equilibrium processes are in best agreement with experimental data. For calculations using different systematics, the results according to [@99Kono] are more suited for description of the experimental data.
Cross sections of the (n, 2n), (n, p), (n, $\alpha$) reactions on the Er isotopes are given on figure \[fig2\]. Cross sections of (n, 2n) reaction on the Er isotopes have the similar peculiarities as on Dy. There is a rather good agreement between presented measurements and the results of other authors.
Figures \[fig3\]-\[fig5\] show cross section of the (n, 2n), (n, p), (n, $\alpha$) reactions the isotopes of Yb. Rather good agreement between experimental data for the $^{168}$Yb and $^{170}$Yb isotopes (N = 98 and 100) is observed for (n,2n) reaction, but presented cross section on $^{176}$Yb (N = 106) is placed higher. The cross section of the reaction $^{172}$Yb(n, p)$^{172}$Tm (N=102) calculated by the EMPIRE 3.0 with pre-equilibrium processes is agree better with measured value. The results of calculation using systematics from [@06Broe; @05Belg] is also closer to measured one. The cross sections of the (n,$\alpha$) reaction were measured with higher precision and they agree better with calculation by systematics from [@08Kade; @09Kono].
![Cross sections of the (n, 2n) reaction on the isotopes of Ytterbium.[]{data-label="fig3"}](fig3f.eps){width="0.95\columnwidth"}
![Cross sections of the (n, p) reaction on the isotopes of Ytterbium.[]{data-label="fig4"}](fig4f.eps){width="0.95\columnwidth"}
![Cross sections of the (n, $\alpha $) reaction on the isotopes of Ytterbium.[]{data-label="fig5"}](fig5f.eps){width="0.95\columnwidth"}
CONCLUSIONS
===========
The results of measurements of the cross sections of the nuclear reactions (n, p), (n, $\alpha $), (n, 2n) on isotopes of Dy, Er and Yb at the neutron energy 14.6 $\pm$ 0.2 MeV are presented. They were compared with available experimental data, evaluated nuclear data and the theoretical calculations by the EMPIRE and TALYS codes. In the most cases, the presented data correlate well with available experimental data. On the whole, the cross sections calculated by the EMPIRE 3.0 code with pre-equilibrium processes agree better with experimental data than the results obtained by TALYS 1.2 code with default set of parameters. Amongst the systematics, the cross section values calculated by expressions from [@06Broe; @05Belg] are more consistent with measured cross-sections.
[9]{}
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R.A. Forrest, [Fusion Eng. and Design]{} [**81**]{}, 2143 (2006).
A.J. Koning [*et al.*]{}, [JRC Scientific and Tech. Rep]{} EUR23977EN (2009).
; http://www.nndc.bnl.gov/exfor/exfor00.htm .
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A.O. Kadenko [*et al.*]{}, [Book of abstract of the 20th Int. Sem. on Inter. of Neutrons with Nuclei: ISINN-20]{}, Dubna, 52 (2012); [Proc. of the 20th Int. Sem. on Inter. of Neutrons with Nuclei: ISINN-20]{}, Dubna (in press) (2013).
M. Herman [*et al.*]{}, [Nuclear Data Sheets]{} [**108**]{}, 2655 (2007); http://www.nndc.bnl.gov/empire/ .
A.J. Koning [*et al.*]{}, [Proc. of the Int. Conf. on Nuclear Data for Science and Technology - ND2007]{}, 211 (2008); http://www.talys.eu/home/ .
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|
---
bibliography:
- 'biblio.bib'
title: The Design and Performance of IceCube DeepCore
---
Introduction {#sec:Introduction}
============
DeepCore Design and Schedule {#sec:Design}
============================
Ice Properties {#subsec:IceProperties}
--------------
Photomultiplier Tubes {#subsec:New_Photocathode_PMTs}
---------------------
Geometry {#subsec:Geometry}
--------
Simulation and Selection of DeepCore Events {#sec:Simulation}
===========================================
Simulation Tools {#subsec:Simulation_Tools}
----------------
Trigger {#subsec:Triggering}
-------
Online Filter {#subsec:Filtering}
-------------
Conclusions {#sec:Conclusions}
===========
Acknowledgements
================
|
---
abstract: 'Despite of their success, the results of first-principles quantum mechanical calculations contain inherent numerical errors caused by various approximations. We propose here a neural-network algorithm to greatly reduce these inherent errors. As a demonstration, this combined quantum mechanical calculation and neural-network correction approach is applied to the evaluation of standard heat of formation ${\Delta\!_{\rm f}\!H^{\minuso}}$ and standard Gibbs energy of formation ${\Delta\!_{\rm f}\!G^{\minuso}}$ for 180 organic molecules at 298 K. A dramatic reduction of numerical errors is clearly shown with systematic deviations being eliminated. For examples, the root–mean–square deviation of the calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ (${\Delta\!_{\rm f}\!G^{\minuso}}$) for the 180 molecules is reduced from 21.4 (22.3) kcal$\cdotp$mol$^{-1}$ to 3.1 (3.3) kcal$\cdotp$mol$^{-1}$ for B3LYP/6-311+G([*d,p*]{}) and from 12.0 (12.9) kcal$\cdotp$mol$^{-1}$ to 3.3 (3.4) kcal$\cdotp$mol$^{-1}$ for B3LYP/6-311+G(3[*df*]{},2[*p*]{}) before and after the neural-network correction.'
author:
- 'LiHong Hu, XiuJun Wang, LaiHo Wong and GuanHua Chen'
date: '; submitted to Phys. Rev. Lett.'
title: 'Combined first–principles calculation and neural–network correction approach as a powerful tool in computational physics and chemistry '
---
One of the Holy Grails of computational science is to quantitatively predict properties of matters prior to experiments. Despite the facts that the first-principles quantum mechanical calculation [@wtyang; @schaefer] has become an indispensable research tool and experimentalists have been increasingly relying on computational results to interpret their experimental findings, the practically used numerical methods by far are often not accurate enough, in particular, for complex systems. This limitation is caused by the inherent approximations adopted in the first-principles methods. Because of computational cost, electron correlation has always been a difficult obstacle for first-principles calculations. Finite basis sets chosen in practical computations are not able to cover entire physical space and this inadequacy introduces further inherent computational errors. Effective core potential is frequently used to approximate the relativistic effects, resulting inevitably in errors for systems that contain heavy atoms. The accuracy of a density-functional theory (DFT) calculation is mainly determined by the exchange-correlation (XC) functional being employed [@wtyang], whose exact form is however unknown. Nevertheless, the results of first-principles quantum mechanical calculation can capture the essence of physics. For instance, the calculated results, despite that their absolute values may poorly agree with measurements, are usually of the same tendency among different molecules as their experimental counterpart. The quantitative discrepancy between the calculated and experimental results depends predominantly on the property of primary interest and, to a less extent, also on other related properties, of the material. There exists thus a sort of quantitative relation between the calculated and experimental results, as the aforementioned approximations, to a large extent, contribute to the systematic errors of specified first-principles methods. Can we develop general ways to eliminate the systematic computational errors and further to quantify the accuracies of numerical methods used? It has been proven an extremely difficult task to determine the calculation errors from the first-principles. Alternatives must be sought.
We propose here a neural–network algorithm to determine the quantitative relationship between the experimental data and the first-principles calculation results. The determined relation will subsequently be used to eliminate the systematic deviations of the calculated results, and thus, reduce the numerical uncertainties. Since its beginning in the late fifties, Neural Networks has been applied to various engineering problems, such as robotics, pattern recognition, speech, and etc. [@PRNN; @nature533] As the first application of Neural Networks to quantum mechanical calculations of molecules, we choose the standard heat of formation ${\Delta\!_{\rm f}\!H^{\minuso}}$ and standard Gibbs energy of formation ${\Delta\!_{\rm f}\!G^{\minuso}}$ at 298.15 K as the properties of interest.
A total of 180 small- or medium-sized organic molecules, whose ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ values are well documented in Refs., are selected to test our proposed approach. The tabulated values of ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ in the three references differ less than 1.0 kcal$\cdotp$mol$^{-1}$ for same molecule. The uncertainties of all ${\Delta\!_{\rm f}\!H^{\minuso}}$ values are less than 1.0 kcal$\cdotp$mol$^{-1}$, while those of ${\Delta\!_{\rm f}\!G^{\minuso}}$s are not reported in Refs.. These selected molecules contain elements such as H, C, N, O, F, Si, S, Cl and Br. The heaviest molecule contains 14 heavy atoms, and the largest has 32 atoms. We divide these molecules randomly into the training set (150 molecules) and the testing set (30 molecules). The geometries of 180 molecules are optimized via B3LYP/6-311+G([*d,p*]{}) [@g98] calculations and the zero point energies (ZPEs) are calculated at the same level. The enthalpy and Gibbs energy of each molecule are calculated at both B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}). [@g98] B3LYP/6-311+G(3[*df*]{},2[*p*]{}) employs a larger basis set than B3LYP/6-311+G([*d,p*]{}). The unscaled B3LYP/6-311+G([*d,p*]{}) ZPE is employed in the ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ calculations. The strategies in reference are adopted to calculate ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$. The calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ for B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}) are compared to their experimental counterparts in Figs. \[fig:wide1\] and \[fig:wide2\], respectively. The horizontal coordinates are the raw calculated data, and the vertical coordinates are the experimental values. The dashed lines are where the vertical and horizontal coordinates are equal, [*i.e.*]{}, where the B3LYP calculations and experiments would have the perfect match. The raw calculation values are mostly below the dashed line, [*i.e.*]{}, most raw ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ are larger than the experimental data. In another word, there are systematic deviations for both B3LYP ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$. Compared to the experimental measurements, the root–mean–square (RMS) deviations for ${\Delta\!_{\rm f}\!H^{\minuso}}$ (${\Delta\!_{\rm f}\!G^{\minuso}}$) are 21.4 (22.3) and 12.0 (12.9) kcal$\cdotp$mol$^{-1}$ for B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculations, respectively. In Table \[tab:table1\] we compare the B3LYP and experimental ${\Delta\!_{\rm f}\!H^{\minuso}}$s for 10 of 180 molecules. Overall, B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculations yield better agreements with the experiments than B3LYP/6-311+G([*d,p*]{}). In particular, for small molecules with few heavy elements B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculations result in very small deviations from the experiments. For instance, the ${\Delta\!_{\rm f}\!H^{\minuso}}$ deviations for CH$_4$ and CS$_2$ are only -0.5 and 0.6 kcal$\cdot$mol$^{-1}$, respectively. Our B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculation results are also in good agreements with those of reference which employed a similar calculation strategy except that their ZPEs were scaled by a factor of 0.98 or 0.96 and their geometries were optimized at B3LYP/6-31+G([*d*]{}). For large molecules, both B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculations yield quite large deviations from their experimental counterparts.
Our neural network adopts a three-layer architecture which has an input layer consisted of input from the physical descriptors and a bias, a hidden layer containing a number of hidden neurons, and an output layer that outputs the corrected values for ${\Delta\!_{\rm f}\!H^{\minuso}}$ or ${\Delta\!_{\rm f}\!G^{\minuso}}$ (see Fig. \[fig:nnst\]). The number of hidden neurons is to be determined. The most important issue is to select the proper physical descriptors of our molecules, which are to be used as the input for our neural network. The calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ contain the essence of exact ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$, respectively, and are thus obvious choices of the primary descriptor for correcting ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$, respectively. We observe that the size of a molecule affects the accuracies of calculations. The more atoms a molecule has, the worse the calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ are. This is consistent with the general observations in the field. [@jcp97g2] The total number of atoms $N_t$ in a molecule is thus chosen as the second descriptor for the molecule. ZPE is an important parameter in calculating ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$. Its calculated value is often scaled in evaluating ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$, [@jcp97g2] and it is thus taken as the third physical descriptor. Finally, the number of double bonds, $N_{db}$, is selected as the fourth and last descriptor to reflect the chemical structure of the molecule.
![\[fig:nnst\] Structure of our neural network.](nnst)
To ensure the quality of our neural network, a cross-validation procedure is employed to determine our neural network. [@rbfnn] We divide further randomly 150 training molecules into five subsets of equal size. Four of them are used to train the neural network, and the fifth to validate its predictions. This procedure is repeated 5 times in rotation. The number of neurons in the hidden layer is varied from 2 to 10 to decide the optimal structure of our neural network. We find that the hidden layer containing two neurons yields best overall results. Therefore, the 5-2-1 structure is adopted for our neural network as depicted in Fig. \[fig:nnst\]. The input values at the input layer, $x_1$, $x_2$, $x_3$, $x_4$ and $x_5$, are scaled ${\Delta\!_{\rm f}\!H^{\minuso}}$ (or ${\Delta\!_{\rm f}\!G^{\minuso}}$), $N_t$, ZPE, $N_{db}$ and bias, respectively. The bias $x_5$ is set to 1. The weights $\{ Wx_{ij}\}$s connect the input layer $\{x_i\}$ and the hidden neurons $y_1$ and $y_2$, and $\{ Wy_{j} \}$s connect the hidden neurons and the output Z which is the scaled ${\Delta\!_{\rm f}\!H^{\minuso}}$ or ${\Delta\!_{\rm f}\!G^{\minuso}}$ upon neural-network correction. The output Z is related to the input $\{x_i\}$ as $$\begin{aligned}
Z = \sum_{j=1,2}Wy_{j}~ Sig(\sum_{i=1,5} Wx_{ij}~ x_i),\end{aligned}$$ where $Sig(v) = {1\over 1+exp(-\alpha v)}$ and $\alpha$ is a parameter that controls the switch steepness of Sigmoidal function $Sig(v)$. An error back-propagation learning procedure [@nature533] is used to optimize the values of $Wx_{ij}$ and $Wy_{j}$($i=1,2,3,4,5$ and $j=1,2$). In Figs. \[fig:wide1\]c, \[fig:wide1\]d, \[fig:wide2\]c and \[fig:wide2\]d, the triangles belong to the training set and the crosses to the testing set. Compared to the raw calculated results, the neural-network corrected values are much closer to the experimental values for both training and testing sets. More importantly, the systematic deviations for ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ in Figs. \[fig:wide1\]a, \[fig:wide1\]b, \[fig:wide2\]a and \[fig:wide2\]b are eliminated, and the resulting numerical deviations are reduced substantially. This can be further demonstrated by the error analysis performed for the raw and neural-network corrected ${\Delta\!_{\rm f}\!H^{\minuso}}$s and ${\Delta\!_{\rm f}\!G^{\minuso}}$s of all 180 molecules. In the inserts of Figs. \[fig:wide1\] and \[fig:wide2\], we plot the histograms for the deviations (from the experiments) of the raw B3LYP ${\Delta\!_{\rm f}\!H^{\minuso}}$s and ${\Delta\!_{\rm f}\!G^{\minuso}}$s and their neural–network corrected values. Obviously, the raw calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$s and ${\Delta\!_{\rm f}\!G^{\minuso}}$s have large systematic deviations while the neural–network corrected ${\Delta\!_{\rm f}\!H^{\minuso}}$s and ${\Delta\!_{\rm f}\!G^{\minuso}}$s have virtually no systematic deviations. Moreover, the remaining numerical deviations are much smaller. Upon the neural-network corrections, the RMS deviations of ${\Delta\!_{\rm f}\!H^{\minuso}}$s (${\Delta\!_{\rm f}\!G^{\minuso}}$s) are reduced from 21.4 (22.3) kcal$\cdot$mol$^{-1}$ to 3.1 (3.3) kcal$\cdot$mol$^{-1}$ and 12.0 (12.9) kcal$\cdot$mol$^{-1}$ to 3.3 (3.4) kcal$\cdot$mol$^{-1}$ for B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}), respectively. Note that the error distributions after the neural–network correction are of approximate Gaussian distributions (see Figs. \[fig:wide2\]c and \[fig:wide2\]d). Although the raw B3LYP/6-311+G([*d,p*]{}) results have much larger deviations than those of B3LYP/6-311+G(3[*df*]{}, 2[*p*]{}), the neural–network corrected values of both calculations have deviations of the same magnitude. This implies that it is sufficient to employ the smaller basis set 6-311+G([*d,p*]{}) in our combined DFT calculation and neural–network correction (or DFT-NEURON) approach. The neural–network algorithm can correct easily the deficiency of a small basis set. Therefore, the DFT-NEURON approach can potentially be applied to much larger systems. In Table \[tab:table1\] we also list the neural–network corrected ${\Delta\!_{\rm f}\!H^{\minuso}}$s of the 10 molecules. The deviations of large molecules are of the same magnitude as those of small molecules. Unlike other quantum mechanical calculations that usually yield worse results for larger molecules than for small ones, the DFT-NEURON approach does not discriminate against the large molecules.
Analysis of our neural network reveals that the weights connecting the input for ${\Delta\!_{\rm f}\!H^{\minuso}}$ or ${\Delta\!_{\rm f}\!G^{\minuso}}$ have the dominant contribution in all cases. This confirms our fundamental assumption that the calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ (${\Delta\!_{\rm f}\!G^{\minuso}}$) captures the essential values of exact ${\Delta\!_{\rm f}\!H^{\minuso}}$ (${\Delta\!_{\rm f}\!G^{\minuso}}$). The input for the second physical descriptor, $N_{t}$, has quite large weights in all cases. In particular, when the smaller basis set 6-311+G([*d,p*]{}) is adopted in the B3LYP calculations, $N_t$ has the second largest weights. It is found that the raw ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$ deviations are roughly proportional to $N_{t}$, which confirms the importance of $N_{t}$ as a significant descriptor of our neural network. The bias contributes to the correction of systematic deviations in the raw calculated data, and has thus significant weights. When the larger basis set 6-311+G(3[*df*]{},2[*p*]{}) is used, the bias has the second largest weights for all cases. ZPE has been often scaled to account for the discrepancies of ${\Delta\!_{\rm f}\!H^{\minuso}}$s or ${\Delta\!_{\rm f}\!G^{\minuso}}$s between calculations and experiments, [@jcp97g2] and it is thus expected to have large weights. This is indeed the case, especially when the smaller basis set 6-311+G([*d,p*]{}) is adopted in calculations. In all cases the number of double bonds, $N_{db}$, has the smallest but non-negligible weights. In Table \[tab:table2\] we list the values of $\{Wx_{ij}\}$ and $\{Wy_{j}\}$ of the two neural networks for correcting ${\Delta\!_{\rm f}\!G^{\minuso}}$s of B3LYP/6-311+G([*d,p*]{}) and B3LYP/6-311+G(3[*df*]{},2[*p*]{}) calculations.
------------ -- -- -- ------- -- -- -- ------- -- -- -- -- ------- -- -- -- -------
Weights y$_1$ y$_2$ y$_1$ y$_2$
Wx$_ {1j}$ 0.78 -0.72 0.83 -0.73
Wx$_{2j}$ -0.60 0.02 -0.30 0.02
Wx$_{3j}$ 0.44 0.02 0.18 0.02
Wx$_{4j}$ 0.07 0.24 0.05 0.17
Wx$_{5j}$ -0.42 -0.04 -0.46 0.01
Wy$_{j}$ 1.48 -0.57 1.44 -0.47
------------ -- -- -- ------- -- -- -- ------- -- -- -- -- ------- -- -- -- -------
: \[tab:table2\]Weights of DFT-Neural Networks for ${\Delta\!_{\rm f}\!G^{\minuso}}$
Our DFT-NEURON approach has a RMS deviation of $\sim$3 kcal$\cdot$mol$^{-1}$ for the 180 small- to medium-sized organic molecules. This is slightly larger than their experimental uncertainties. [@McGraw; @crchandbook; @thermodata] The physical descriptors adopted in our neural network, the raw calculated ${\Delta\!_{\rm f}\!H^{\minuso}}$ or ${\Delta\!_{\rm f}\!G^{\minuso}}$, the number of atoms $N_t$, the number of double bonds $N_{db}$ and the ZPE are quite general, and are not limited to special properties of organic molecules. The DFT-NEURON approach developed here is expected to yield a RMS deviation of $\sim$3 kcal$\cdot$mol$^{-1}$ for ${\Delta\!_{\rm f}\!H^{\minuso}}$s and ${\Delta\!_{\rm f}\!G^{\minuso}}$s of any small- to medium-sized organic molecules. G2 method [@jcp97g2] results are more accurate for small molecules. However, our approach is much more efficient and can be applied to much larger systems. To improve the accuracy of the DFT-NEURON approach, we need more and better experimental data, and possibly, more and better physical descriptors for the molecules. Besides ${\Delta\!_{\rm f}\!H^{\minuso}}$ and ${\Delta\!_{\rm f}\!G^{\minuso}}$, the DFT-NEURON approach can be generalized to calculate other properties such as ionization energy, dissociation energy, absorption frequency, band gap and etc. The raw first-principles calculation property of interest contains its essential value, and is thus always the primary descriptor. Since the raw calculation error accumulates as the molecular size increases, the number of atoms $N_t$ should thus be selected as a descriptor for any DFT-NEURON calculations. Additional physical descriptors should be chosen according to their relations to the property of interest and to the physical and chemical structures of the compounds. Others have used Neural Networks to determine the quantitative relationship between the experimental thermodynamic properties and the structure parameters of the molecules. [@rbfnn] We distinct our work from others by utilizing specifically the first-principles methods and with the objective to improve quantum mechanical results. Since the first-principles calculations capture readily the essences of the properties of interest, our approach is more reliable and covers much a wider range of molecules or compounds.
To summarize, we have developed a promising new approach to improve the results of first-principles quantum mechanical calculations and to calibrate their uncertainties. The accuracy of DFT-NEURON approach can be systematically improved as more and better experimental data are available. As the systematic deviations caused by small basis sets and less sophisticated methods adopted in the calculations can be easily corrected by Neural Networks, the requirements on first-principles calculations are modest. Our approach is thus highly efficient compared to much more sophisticated first-principles methods of similar accuracy, and more importantly, is expected to be applied to much larger systems. The combined first-principles calculation and neural-network correction approach developed in this work is potentially a powerful tool in computational physics and chemistry, and may open the possibility for first-principles methods to be employed practically as predictive tools in materials research and design.
We thank Prof. YiJing Yan for extensive discussion on the subject and generous help in manuscript preparation. Support from the Hong Kong Research Grant Council (RGC) and the Committee for Research and Conference Grants (CRCG) of the University of Hong Kong is gratefully acknowledged.
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---
abstract: 'We develop a Gaussian-process mixture model for heterogeneous treatment effect estimation that leverages the use of transformed outcomes. The approach we will present attempts to improve point estimation and uncertainty quantification relative to past work that has used transformed variable related methods as well as traditional outcome modeling. Earlier work on modeling treatment effect heterogeneity using transformed outcomes has relied on tree based methods such as single regression trees and random forests. Under the umbrella of non-parametric models, outcome modeling has been performed using Bayesian additive regression trees and various flavors of weighted single trees. These approaches work well when large samples are available, but suffer in smaller samples where results are more sensitive to model misspecification – our method attempts to garner improvements in inference quality via a correctly specified model rooted in Bayesian non-parametrics. Furthermore, while we begin with a model that assumes that the treatment assignment mechanism is known, an extension where it is learnt from the data is presented for applications to observational studies. Our approach is applied to simulated and real data to demonstrate our theorized improvements in inference with respect to two causal estimands: the conditional average treatment effect and the average treatment effect. By leveraging our correctly specified model, we are able to more accurately estimate the treatment effects while reducing their variance.'
author:
- Abbas Zaidi
- Sayan Mukherjee
bibliography:
- 'refs.bib'
title: '**Gaussian Process Mixtures for Estimating Heterogeneous Treatment Effects**'
---
Gaussian Process Mixture Models, Treatment Effect Estimation, Bayesian Machine Learning.
Introduction
============
The estimation of treatment effects is one of the core problems in causal inference. A treatment effect is a measure used to compare interventions in randomized experiments, policy analysis, and medical trials. The treatment effect measures the difference in outcomes between units assigned to the treatment versus those assigned to the control. There have been a variety of related approaches for estimating treatment effects including those based on graphical models [@pearl2009causal] and the potential outcomes framework [@rubin1978bayesian]. In this paper, we develop methodology that builds on the potential outcomes framework as defined in [@rubin2005causal] to estimate treatment effects.
In the potential outcomes framework we compare the observed outcome to the outcome under the counterfactual, that is, what the outcome would be under a different set of treatment conditions. If the counterfactual outcome were known then the treatment effect on an individual unit is the difference between the outcome under the observed and counterfactual interventions. The *fundamental problem of causal inference* is that in general for any unit one can only observe the outcome under a single treatment condition. As a consequence unit level causal effects are not identifiable. However, population level causal effects can be identified under some standard assumptions (see Section \[former\]). An estimator of population level effects is the average treatment effect (ATE) which is a measure of the difference in the mean outcomes between units assigned to the treatment and units assigned to the control. If treatment effects are homogenous across individuals then estimators such as the ATE that consider causal effects at an aggregate level are reasonable, however such estimators will overlook subgroup or covariate-level specific heterogeneity in treatment effects. There is evidence that heterogeneity in treatment effects is more the rule than the exception [@heckman2006understanding; @green2012modeling; @xie2012estimating].
A quantity in addressing heterogeneous treatment effects is the conditional average treatment effect (CATE) which is the average treatment effect conditional on the covariate level of a unit of observation. One can consider the CATE as a difference of two regression functions – the average response given treatment at a set of covariate levels minus the the average response assuming the control condition and the same set of covariate levels. One can estimate the ATE by marginalizing the CATE over the joint distribution of the covariates. There are a number of approaches for estimating the two aforementioned causal estimands. The main approach for modeling heterogeneous treatment effects based on the CATE is conditional mean regression. Under this approach, we model the CATE as a difference between the conditional mean outcome given the treatment for particular covariate levels minus the mean outcome given the control at the same covariate levels [@ding2017causal]. The implementation of these models can be approached both parametrically and non-parametrically.
The most popular parametric methods for estimating the difference between the conditional mean outcomes include linear and polynomial regression [@pearl2009causal], along with penalized regression approaches such as least absolute subset selection operator and ridge regression [@tibshirani1996regression]. At the other end of the spectrum are non-parametric regression models to estimate the difference between the conditional means. Examples include boosting [@powers2017some], Bayesian additive regression trees (BART) [@hill2011bayesian; @hahn2017bayesian; @chipman2010bart] as well as classical regression trees [@athey2015machine; @breiman1984classification] and random forests [@wager2017estimation; @foster2011subgroup; @breiman2001random]. These methods have some limitations to their use and we provide a brief discussion of these.
The use of random forests for CATE estimation as defined in [@wager2017estimation] provides some interesting theoretical results that allow for probabilistically valid statistical inference. These methods are theorized to outperform classical methods particularly in the presence of irrelevant covariates. This technique however, has been demonstrated to be outperformed in application [@hahn2017bayesian]. In addition, without a procedure for imposing a degree of regularization, random forests are difficult to actually deploy for heterogeneous treatment effect estimation [@wendling2018comparing]. BART and its variants [@hahn2017bayesian; @hill2011bayesian] present a persuasive argument for their use in application, but there is limited work on their formal inferential properties [@wager2017estimation] for learning heterogeneous treatment effects. Specifically for BART, formal statistical analysis is hurdled by the lack of theory arguing posterior concentration around the true conditional mean function – the key quantity of interest in heterogeneous treatment effect estimation via conditional mean regression.
An alternative to modeling the difference in conditional mean outcomes is the use of transformed responses or outcome variables (TRV) [@dudik2011doubly; @beygelzimer2009offset] that is ideologically similar to concepts of *inverse probability weighting* (IPW) [@hirano2003efficient]. The TRV approach introduces a transformation for the outcome and the treatment indicator variable for which the conditional expectation given a covariate level is equivalent to the CATE. This allows it to be used with *off-the-shelf* machine learning techniques and has been applied to optimal treatment policy estimation in the same vein as ideas of *double-robustness* as reviewed in [@ding2017causal] that combine regression adjustment with weighting. More recent work on the TRV has attempted to model it as a function of the observed covariates via regression trees [@athey2015machine] and boosting [@powers2017some]. This has raised questions of estimation quality of the approach given the high variance of the procedure. We assert that this is a consequence of the properties of the TRV that have not been explicitly accounted for in the model since past work has relied on using it as a benchmark for other methods [@athey2016recursive].
In this paper we introduce a novel non-parametric Bayesian model based on Gaussian process regression [@singh2016gaussian; @rasmussen2006Gaussian] for inference of the TRV that allows us to infer a posterior distribution on the CATE. The model we propose is a finite mixture of Gaussian-processes [@rasmussen2000infinite] that leverages the distribution implied by the transformation. This formulation is aimed at improving the overall quality of inference on the treatment effects with a correctly specified model.
This approach has benefits over both conditional mean regression and other TRV based techniques. In practice, we never estimate either the treatment nor the control function perfectly and different covariate distributions for the treatment and control groups can lead to biases in the treatment effect estimation [@powers2017some]. The TRV allows for the joint modeling of information from both the treated and control groups which can help circumvent the aforementioned estimation challenge which for instance has been discussed as a specific limitation of conditional mean regression with random forests [@wager2017estimation]. This joint modeling is also an improvement over Bayesian techniques that place individual vague priors on the treatment and control outcome models since the prior on the treatment effect as the difference of the two is possibly *doubly* vague [@hahn2017bayesian]. This can make inference a challenge since it is difficult to control the degree of heterogeneity that the model adapts to. Furthermore the TRV generates unbiased estimates for the CATE [@powers2017some].
In addition to its benefits over conditional mean regression methods, the model we introduce offers four advantages over other TRV modeling approaches. First, we significantly improve the accuracy of point estimation by explicitly modeling the distribution of the transformed outcome. Second, by modeling the distribution of the transformed outcome specifically we are able to greatly reduce the variance of causal estimands i.e. the average treatment effect and the conditional average treatment effect. Reducing the variance of the estimators is crucial since this has been the main criticism of the TRV approach [@athey2015machine; @powers2017some]. This provides tighter uncertainty intervals relative to the approaches discussed in [@athey2015machine] and [@wager2017estimation]. Third, our approach is well suited for instances when the treated and control groups share information since our proposed mechanism jointly models the behavior of both via the transformation.
The methodology we introduce makes a number of significant contributions to the estimation of heterogeneous treatment effects. Our main contribution is that we improve the overall quality of inference by improving the point estimation with a correctly specified model. In addition, the proposed framework is flexible in that we do not assume a functional form for how heterogeneity of treatment effects are driven by the levels of the observed covariates. Finally, our proposed framework is easily adapted to studies where the mechanism by which individuals receive the treatment is unknown. For this problem, past work has relied on a two-stage procedure for learning this treatment assignment mechanism first and then utilizing this in the model. We instead propose an approach whereby the treatment assignment mechanism and the treatment effects are jointly learnt in a unified framework. By working under this paradigm we have a twofold gain. First, the uncertainty quantification from our proposed model reflects uncertainty from all stages of inference including the learning of the assignment mechanism, and the treatment effects. Second as a by-product of the *feedback* in between the two estimation stages, the assignment mechanism makes more complete use of the data, which can improve estimation of causal effects.
The remainder of this paper is organized as follows: in Section \[former\], we introduce the TRV, the relevant notation and the assumptions inherent to the TRV approach. We state our new model in Section \[model\]. Our approach is benchmarked against to TRV regression trees and random forests, along with non-TRV weighted tree methods as discussed in [@athey2016recursive], as well as Bayesian tree models in [@hahn2017bayesian; @hill2011bayesian] on both simulated and real data in Section \[data\]. We close with a summary of our findings and possible areas of future work.
Transformed Response Variables Framework {#former}
========================================
In order to formulate the approach of transformed outcomes, we first define some notation that we will use throughout this paper. The observed data $\mathcal{D}$ consists of a sample of size $n$ where for each unit of observation we are given a response variable $Y_{i} \in \mathbb{R}$ and a covariate vector$X_{i} \in \mathbb{R}^{p}$. In addition to the observed data, we denote as $W_{i} \in \{0, 1\}$ the treatment assignment. The corresponding treatment assignment probability is denoted as $e_{i} = \mathbb{P}(W_{i} = 1)$. Finally, the potential outcome is denoted as $Y_{i}(W_{i} = w)$.
Under the potential outcomes framework, in order to estimate treatment effects from observational data certain assumptions about the treatment assignment mechanism need to be satisfied. Briefly, these assumptions are that the treatment assignment is *individualistic* (A1), *probabilistic* (A2) and *ignorable*(A3). Details of these assumptions are left to the reader in [@imbens2015causal]. A1 and A2 are implied under the assumption that the units of observation are a simple random sample from the target population that are independent and identically distributed.
Assumptions A2 and A3 are together known as the *strong ignorability* assumption and grants the indentifying equivalence between the potential outcome and the causal conditioning, $Y(W = w) \stackrel{P}{=} Y \mid W = w$. All three of the assumptions summarized here are always satisfied in randomized trials; In observational studies the assumptions may hold to varying degrees.
For instance, A2, which is also sometimes referred to as the overlap condition can be directly assessed. However, by comparison A3 is untestable and therefore indirect techniques are needed to determine the degree to which it is satisfied most commonly via sensitivity analyses [@rosenbaum1982assessing]. These assumptions are necessary for the formal results in the transformed response variable framework to hold.
Beyond these, we make one additional assumption that allows us to simplify the statistical the model we specify in this paper: Stable Unit Treatment Value Assumption (SUTVA) — This condition assumes no interference between observations, and that there are no multiple versions of the treatment (A4). In its absence, we would need to define a different potential outcome for the unit of observation not just for each treatment received by that unit but for each combination of treatments received by every other observation in the experiment. Relaxing these assumptions will be discussed in Section \[futurework\] as an avenue that our future work will aim to explore.
The causal estimands considered are the conditional average treatment effect (CATE), that we denote as $\tau^{CATE}$ and the average treatment effect (ATE) that we denote as $\tau^{ATE}$. $\tau^{CATE}$ is the primary estimate of interest in modeling heterogeneous treatment effects and is defined as, $$\label{eq:cate}
\tau^{CATE} = \mathbb{E}_{Y}[Y(1) - Y(0) \mid X = x],$$ the ATE can be derived by integrating over the the joint distribution of the covariates $$\label{eq:ate}
\tau^{ATE} = \mathbb{E}_{X}\Big[ \mathbb{E}_{Y}[Y(1) -Y(0) \mid X =x ] \Big] =\mathbb{E}_{X}[\tau^{CATE}].$$
The idea behind the transformed response variable apporach is to define a variable $Y_i^{*}$ for which the conditional expectation with respect to the response recovers the CATE under A3 (see Appendix \[app:A\] for a proof of this result). A transformation that satisfies the above condition is,
$$\label{eq:trv}
Y^{*}_{i} = f(W_{i}, Y_{i}, e_{i})= \frac{W_{i} - e_{i}}{e_{i}(1-e_{i})} Y_{i}.$$
The transformation requires knowledge of the probability of receiving the treatment. We assume that the treatment assignment probability depends on the observed covariate levels, or $e_{i} = e_{i}(X = x_{i})$ is a propensity score. A trivial example is when the propensity score is a fixed covariate independent value, $e_{i} = e$. This is not an example commonly seen in real observational causal inference problems and is as such not considered as a part of the model presented here, albeit [@athey2015machine; @Athey:2015:MLC:2783258.2785466; @athey2016recursive] consider it as a means of model validation.
Strengths and Weaknesses of Past Work in TRV Modeling {#LMS}
-----------------------------------------------------
TRV modeling offers three main advantages when used for estimating treatment effects as demonstrated in prior studies. Foremost amongst these is that the TRV can easily be modeled with any supervised learning method. For instance, regression trees and random forests have been used [@athey2015machine; @Athey:2015:MLC:2783258.2785466; @wager2017estimation] as has boosting [@powers2017some]. This is not an exhaustive list, and there are a myriad of other methods that can be used in conjunction with the TRV to estimate heterogeneous treatment effects. Furthermore, relative to conditional mean regression, this method does not ignore the propensity score which explicitly enters the estimation via the transformation. Finally, based on the modeling approach used, we can address treatment effect heterogeneity flexibly and therefore avoid issues arising from model misspecification since it is likely that there are complex relationships between the covariates and heterogeneity of the treatment effects. Despite their usefulness, the TRVs have some key weaknesses.
First, as mentioned in [@athey2015machine] and [@powers2017some] using TRVs as CATE estimators results in high variance estimates of the causal estimands. By construction the treatment assignment probability and the assignment itself only enter the model implicitly via the transformation and are therefore only accounted for indirectly. In addition, the treatment assignment probability only appears in the denominator, and if this is close to zero or one, the variance can spike. Similar difficulties have been seen in IPW [@hirano2003efficient] estimators, that like this transformation grant more weight to tail (read: unlikely) observations. Combining supervised learning techniques with inverse-probability weighting, gives rise to double-robust estimators, which in spirit is also similar to our modeling of the transformed outcome. [@ding2017causal] summarize that the instability of the estimator due to extreme treatment assignment probabilities is even worse in this case than in inverse-probability weighting, since there are potentially two sources of model misspecification. While we can address concerns of model misspecification using flexible machine-learning models, this flexibility is a double-edged sword. When the model generates predictions that are inherently high variance such as those of regression trees, this means that the method suffers in terms of efficiency and the quality of inference is degraded.
Second, uncertainty quantification using methods built atop inverse-probability weighting in general and transformed outcomes in specific is difficult. As discussed at length earlier, there are theoretical concerns due to the the impact of extreme weights which is a limitation of the transformation. There are also practical concerns with uncertainty quantification under specific models for the TRV as it relates to generating intervals. For single regression trees as well as the other ensemble learning methods which have been used for TRV modeling, intervals have been generated using the bootstrap. Prior work [@wager2014confidence] has suggested that in certain applications the Monte Carlo error can dominate the uncertainty quantification produced. In conjunction with the high variance inherent to the aforementioned approaches, we might be unable to gather useful insights. If treatment effects are small (near zero), the conflation of the Monte Carlo noise with the underlying sampling noise may lead us to overstate the variance and therefore lower the power of our analysis. In addition note that when the sample size is small, [@powers2017some] demonstrate that the variance of the TRV is small as well – it increases with increasing sample size. Hence, in situations where bootstrapping is likely to do well for the uncertainty in the model i.e. in large samples, the high variance of the TRV is even more so an issue.
Based on these limitations, we propose the Gaussian process mixture model in Section \[model\]. Our proposed model attempts to overcome the aforementioned limitations by leveraging the mixture distribution implied by the transformation. In addition, we still aim to model the TRV flexibly and capture the complexity of treatment effect heterogeneity. We achieve gains in the quality of inference by constructing a likelihood that reflects the error structure imposed by the TRV under some basic assumptions that earlier work with this technique has ignored. The details of these findings will be discussed in greater depth in Section \[data\] where these approaches are applied to real and simulated data.
The Gaussian Process Mixture Model {#model}
==================================
We specify a non-parametric Bayesian model based on a mixture of Gaussian processes to model heterogeneous treatment effects. Our model is based on the transformed response variable framework. It is motivated by three objectives: (1) to explicitly model the distribution implied by the transformed outcome with the goal of reducing the variance of the TRV generated estimates that have hitherto been produced using non-probabilistic models, (2) model the two treatment groups jointly so we can borrow strength and therefore improve inference even relative to non-TRV based methods for estimating treatment effects, and (3) making more complete use of the data by jointly modeling the transformed response as well as the treatment assignment probabilities in a one step model. The *feedback* between the two stages in joint modeling can improve the point estimation of treatment effects and the propensity scores [@zigler2013model]. Throughout this section we assume A1-A4 are satisfied.
Model Specification {#understanding}
-------------------
A natural starting point is to consider two non-parametric regression functions for the response under treatment and control, respectively $$\begin{aligned}
Y_{i}(1) &=& f_{1}(x_{i}) + \varepsilon_{i}(1), \quad \epsilon_{i}(1) \stackrel{\mathrm{iid}}{\sim} \mathrm{N}(0, \sigma^{2}), \\
Y_{i}(0) &=& f_{0}(x_{i}) + \varepsilon_{i}(0), \quad \epsilon_{i}(0) \stackrel{\mathrm{iid}}{\sim} \mathrm{N}(0, \sigma^{2}).\end{aligned}$$
In expectation, the difference of these two non-parametric functions is the conditional average treatment effect. Substituting these non-parametric regression functions under the treatment and control cases in the definition of the TRV in yields the following mixture model,
$$\label{eq:newmodel}
Y_{i}^{*} = g(x_{i}) + \varepsilon_{i}^{*},$$
$$\varepsilon_{i}^{*} \sim e_{i} \mathrm{N}\bigg((1-e_{i}) h(x_{i}), \frac{1}{e_{i}^{2}}\sigma^{2}\bigg) + (1-e_{i})\mathrm{N}\bigg(-e_{i} h(x_{i}), \frac{1}{(1-e_{i})^{2}}\sigma^{2}\bigg).$$
where $g(\cdot)$ is interpreted as the conditional average treatment effect,
$$g(x_{i}) = f_{1}(x_{i}) - f_{0}(x_{i}).$$
while the function $h(\cdot)$, helps expresses the multi-modal nature of the error distribution that is implied by the transformation,
$$h(x_{i}) = \frac{f_{1}(x_{i})}{e_{i}} + \frac{f_{0}(x_{i})}{1-e_{i}}.$$
A detailed derivation of this model is given in Appendix \[app:B\].
The argument for specifying the TRV mixture model rather than individual models for the treatment and control is that the conditionals $Y_i \mid X_i, W_i =1$ and $Y_i \mid X_i, W_i =0$ may not be perfectly estimable. Past work has indicated that ignoring shared information between the treated and untreated groups is a potential source of bias in the treatment effect estimation [@powers2017some]. Under the Bayesian paradigm, methods that place individual vague priors on the aforementioned conditionals make it challenging to control the degree of heterogeneity the model adapts to since the implied priors on their differences is potentially extremely vague [@hahn2017bayesian].
Our model formulation can be considered under two specifications – when the treatment assignment probabilities are known and when they need to be inferred from the data. The details of each specification are given in Sections \[simspec\] and \[compspec1\] for the two cases respectively.
### Model specification with known assignment probabilities {#simspec}
We will place Gaussian process priors on both $g$ and $h$ and will specify an inverse gamma prior on $\sigma^2$ to leverage conjugacy. Therefore, for the case where the treatment probabilities are known we specify the following model,
$$\label{basicmodel}
\begin{aligned}
Y_{i}^{*} &= g(x_{i}) + \varepsilon_{i}^{*},\\
\varepsilon_{i}^{*} \sim e_{i} \mathrm{N}\bigg((1-e_{i}) h(x_{i}), \frac{1}{e_{i}^{2}}\sigma^{2}\bigg) &+ (1-e_{i})\mathrm{N}\bigg(-e_{i} h(x_{i}), \frac{1}{(1-e_{i})^{2}}\sigma^{2}\bigg),\\
g & \sim\mathrm{GP}(0, \kappa_{g}), \\
h & \sim\mathrm{GP}(0, \kappa_{h}), \\
\sigma^2 & \sim \mathrm{IG}(a,b).
\end{aligned}$$
Here $\mathrm{IG}(a, b)$ is the inverse gamma distribution with hyper-parameters $a$ and $b$ and $\mathrm{GP}(\textbf{0}, \boldsymbol{\kappa})$ denotes the Gaussian process priors on the function $g$ and $h$. Both priors are zero mean and have covariance kernels specified (1) a non-stationary linear kernel $\kappa_{g}(u, v) = s^{2}_{0} + \sum_{i=1}^{p}s^{2}_{i}(u_{i}-c_{i})(v_{i}-c_{i})$, with hyper-parameters $s^{2}_{0}, \ldots s^{2}_{p}$ on $g$ and (2) a square exponential, $\kappa_{h}(u, v) = s_{h}^{2}\exp\{- \tau^2 \| u-v\|^2\} $ with hyper-parameters $\tau, s^{2}$ on $h$. These kernels rely on the notion of similarity between data points – if the inputs are closer together than the target values of the response, in this case the TRV are also likely to be close together. Under the Gaussian process prior, the kernel functions described above formally define what is near or similar.
The hyper-parameters $s^{2}_{0}, \ldots s^{2}_{p}$ can be interpreted in the context of linear regression with $\{\mathrm{Normal}\sim(0, s^{2}_{j})\}_{j=0}^{p}$ priors on the $p+1$ regression coefficients including the intercept. The offset $\{c_{i}\}_{i=1}^{p}$ determines the $x$ coordinate of the point that all the lines in the posterior is meant to go through. This provides some insight into how these can be set for applied modeling problems. In cases where there is a large number of covariates, many of which are thought to share information, the prior variance for those dimensions can be made small, with a higher degree of mass concentrated near zero to induce more shrinkage. In contrast, where there is a small number of important covariates the prior variance can be set to make the prior more diffuse. The offset can be set to the average of each covariates observed value. This is a general overview of the strategy that we have employed.
### Model specification with unknown assignment probabilities {#compspec1}
Computing the TRV requires knowledge of the treatment assignment probabilities $\{e_i\}_{i=1}^{n}$. In the case where these are unknown we consider them as latent variables and add extra levels to the hierarchical model specified in to model the treatment assignment probabilities. We model the assignment probabilities individually so for notational ease, later in this paper we use $\pmb{e} = \{e_i\}_{i=1}^{n}$. Our specification, *apriori*, assumes that the assignment mechanism and the outcome model are independent.
#### Modeling the Propensity Score
In order to learn the treatment assignment probabilities, we specify a probit regression model that is layered onto the model defined in . $$\label{indassign}
\begin{aligned}
W_{i} &\sim\mathrm{Ber}(e_{i}),\\
e_{i} &= \Phi(X_{i} \boldsymbol{\beta}), \\
\boldsymbol{\beta} & \sim\mathrm{N}_{p+1}(0, \Psi_{p+1 \times p + 1}).
\end{aligned}$$ Where $\Phi$ denotes the standard Normal cumulative distribution function. In this paper we will only consider the above Gaussian prior on $\boldsymbol \beta$ with prior covariance $\Psi$. However, additional complexity can be added by allowing the coefficient vector $\boldsymbol \beta$ to vary via a hierarchical prior structure as may be motivated by more complex multi-stage clustered data.
Posterior Sampling with Known Assignment Probabilities {#inference}
------------------------------------------------------
Inference for the model specified in Section \[simspec\] involves sampling from a posterior distribution via straightforward Gibbs-sampling.
We define $\mathbf{g} = (g(x_{1}), \ldots, g(x_{n}))$ and $\mathbf{h} = (h(x_{1}), \ldots, h(x_{n}))$ as the values of the two regression functions on the training data. We denote the TRV as $\textbf{Y}^{*} = (Y_{1}^{*}, \ldots, Y_{n}^{*})$ . In this case the target joint posterior distribution is $$\label{eq:jd1}
\pi(\mathbf{g}, \mathbf{h}, \sigma^{2} \mid \mathcal{D}).$$
Due to prior conjugacy the conditional distributions: $\pi(\mathbf{g} \mid \mathbf{h}, \sigma^{2}, \mathcal{D})$, $\pi(\mathbf{h} \mid \mathbf{g}, \sigma^{2}, \mathcal{D})$ and $\pi(\sigma^{2} \mid \mathbf{h}, \mathbf{g}, \mathcal{D})$ all have simple forms that we can easily sample from. We first state some matrices and vectors that will enter our calculations: $\mathbf{D}$ is an $n\times n$ diagonal matrix with entries $\mathbf{D}_{ii} = \bigg( \frac{W_i }{e_{i}^{2}} \sigma^2 + \frac{1-W_i}{(1-e_{i})^{2}}\sigma^{2}\bigg)$, $\boldsymbol{\Lambda}$ is also an $n\times n$ diagonal matrix with entries $\boldsymbol{\Lambda} _{ii} = \bigg(W_{i}(1-e_{i}) + (1-W_{i})(-e_{i})\bigg)$, $\mathbf{K}$ is also an $n\times n$ diagonal matrix with entries $\mathbf{K}_{ii} = \sigma^{2}\mathbf{D}_{ii}$, and $\mathbf{m} = \Lambda \textbf{H}$. We also denote the covariance matrix $\pmb{\kappa}_{g}$ with the $ij$-th entry as taking the value $\kappa_g(x_i,x_j)$ and similarly $\pmb{\kappa}_{h}$ is a matrix with the $ij$-th entry taking the value $\kappa_h(x_i,x_j)$. We now state the conditional distributions that will enter our Gibbs sampler, $$\label{eq:fcg}
\begin{aligned}
\pi(\mathbf{g} \mid \mathbf{h}, \sigma^{2}, \mathcal{D}) &\sim \mathrm{N}\big((\pmb{\kappa}_{g}^{-1}+\mathbf{D}^{-1})^{-1}(\mathbf{D}^{-1}\textbf{Y}^{*} - \mathbf{m}\}, \{\pmb{\kappa}_{g}^{-1}+\mathbf{D}^{-1})^{-1}\big), \\
\pi(\mathbf{h} \mid \mathbf{g}, \sigma^{2}, \mathcal{D}) &\sim \mathrm{N}\bigg((\pmb{\kappa}_{h}^{-1} + \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} \boldsymbol{\Lambda})^{-1} \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} (\textbf{Y}^{*}-\mathbf{g}), (\pmb{\kappa}_{h}^{-1} + \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} \boldsymbol{\Lambda})^{-1} \bigg), \\
\pi(\sigma^{2} \mid \mathbf{h}, \mathbf{g}, \mathcal{D}) & \sim \mathrm{IG} \bigg(a+\frac{n}{2}, b + \frac{(\textbf{Y}^{*}-\mathbf{g} - \mathbf{m})^{T}\mathbf{K}^{-1}(\textbf{Y}^{*}-\mathbf{g}- \mathbf{m})}{2}\bigg).
\end{aligned}$$ The Gibbs steps that would be used to sample from these full conditional distributions are given appendix \[app:D\].
Posterior Sampling with Unknown Assignment Probabilities {#complexinference}
--------------------------------------------------------
There are two additional problems with respect to inference when the assignment probabilities are unknown: one needs to estimate the assignment probabilities $\pmb{e}$ and use these to compute the TRV $\textbf{Y}^{*}$. The following target posterior distribution corresponds to the model when the treatment probabilities are modeled as specified by the probit augmentation to the model in .
$$\begin{aligned}
&\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*}, \sigma^{2},\boldsymbol{e}, \boldsymbol{\beta} \mid \mathcal{D}). \label{post3}\end{aligned}$$
In this setting the joint posterior is more complicated than equation and is harder to sample from since it cannot be completely decomposed into Gibbs steps. Generating samples requires incorporating the full conditional distributions from the previous section, along with additional steps to sample the treatment assignment probability by using a Metropolis-within-Gibbs step and constructing the transformed outcome.
The Metropolis-Hastings step consists of specifying a proposal distribution $q(\boldsymbol{\beta})$, and given a candidate value $\pmb{\beta}^* \sim q(\pmb{\beta})$ is accepted with acceptance probability, $$\label{eq:mhratio}
\alpha = \min\left(1, \frac{\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*},\sigma^{2}, \boldsymbol{\beta}^{*}, \pmb{e} \mid \mathcal{D}) \, q(\pmb{\beta})}{\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*}, \sigma^{2}, \boldsymbol{\beta}, \pmb{e} \mid \mathcal{D}) \, q(\pmb{\beta}^{*})}\right).$$ where the posterior for evaluation is given in . We have used a symmetric random walk proposals[^1] in order to reduce the overall computational burden. Once we have sampled the coefficients for the probit model, we can deterministically compute the treatment assignment probability and the TRV. The complete algorithm for this sampling scheme is detailed in appendix \[app:D\].
#### Joint Bayesian modeling and the feedback problem:
The joint Bayesian model specified in this paper for learning the assignment mechanism $\pmb{e}$ and the transformed outcome $\textbf{Y}^{*}$ leads to a feedback problem of the type described in [@zigler2013model]. The treatment assignment probability $\pmb{e}$ appears in the joint posterior distribution both as a part of the transformed outcome model through as well as its own model in . Therefore its posterior samples involve information from both. In the specific context of the assignment model, this means that the posterior samples of parameters in learning $\pmb{e}$ are informed by information from the outcome stage.
Under the classical method of using $\pmb{e}$ as a dimension reduced covariate representation in the outcome stage model (an analog to our transformed outcome), [@zigler2013model] demonstrate that the estimation of causal effects is poor. There is a possibility of considerable bias due to the distortion of the causal effects. Furthermore, the usefulness of the propensity score adjustment as a replacement for the covariates is also compromised.
However, this is not the concern in the modeling scheme proposed in this paper. [@zigler2013model] show that the nature of the feedback between the two stages is altered when the outcome stage model is augmented with adjustment for the individual covariates and that this method can recover causal effects akin to when a classical two stage procedure is used. Our approach via the kernels of the Gaussian processes provides individual covariate adjustment therefore alleviating concerns created by the feedback. Therefore we reap the benefits of the joint estimation, but by means of suitably elicited priors, and individually controlled covariates, we bypass the concerns of feedback. In fact, by making more complete use of the data, we are arguably able to improve the overall quality of estimation.
Results on Simulated and Real Data {#data}
==================================
In this section we validate our Gaussian process based TRV model on simulated and real data. We use the simulations to show that our approach outperforms other techniques (both TRV as well as conditional mean regression type methods). This holds true both when the treatment assignment probabilities are known or need to be inferred from the data. We also observe on the simulated data that our model does in fact recover the causal effects in the TRV framework in the presence of feedback as theorized earlier. Our assertion is based on comparisons of mean squared error, bias and point-wise coverage of the uncertainty intervals generated by the model.
The real data analyzed here comes from a study of the causal effects of debit card ownership on household spending in Italy [@mercatanti2014debit] – we will refer to these data as the SHIW data. In the analysis of the SHIW data we jointly infer treatment effects as well as the treatment assignment probability for each individual, as these are not observed.
The most interesting aspect of our analysis of the SHIW data is that we are able to identify heterogeneity in the treatment effects. We find that the impact of debit card usage on aggregate household spending is found to vary based on income and this variability is highest at the lowest levels of income – a notion that is validated under behavioral economic theory which further lends credibility to our proposed model.
Estimands Used and Modelling Approaches Compared
------------------------------------------------
In this section we state the estimands that we will use for comparing our method to other non-parametric methods. We will also state in detail how we compute the relevant estimand for both our method and the other techniques considered. The analysis is focused on the estimation of the CATE.
#### Gaussian process mixture model:
We first specify the procedure we use to estimate the CATE for our model. The model is trained on data $(x_1,...,x_n)$ and the values of the two functions are $$\begin{aligned}
{\mathbf{g}} = (g(x_{1}), \ldots, g(x_{n})), \\
{\mathbf{h}} = (h(x_{1}), \ldots, h(x_{n})).\end{aligned}$$ We will use the function values to evaluate the accuracy of our estimators.
Depending on whether the treatment assignment probabilities are observed or not we obtain posterior samples $\left(\mathbf{g}^{(j)},\mathbf{h}^{(j)}\right)_{j=1}^K$ or $\left(\mathbf{g}^{(j)},\mathbf{h}^{(j)}, \boldsymbol{e}^{(j)}\right)_{j=1}^K$, respectively, using which we can compute posterior samples for the conditional average treatment effect at each location $x_i$, $i=1,...,n$ as $${\tau_{i}^{CATE}}^{(j)}(x_i) = g^{(j)}(x_i).$$ Given the posterior samples we can compute a posterior mean as a point estimate, $\widehat{\tau_{i}^{CATE}}$ along with its corresponding credible intervals. Where applicable, marginalizing over the values $x_{i}$ allows us to compute posterior estimates of the average treatment effect $\widehat{\tau_{i}^{ATE}}$.
Based on the quantities that we have specified above, the mean squared error, bias and coverage used for model validation are specified as follows,
$$\mathrm{Mean \>\>\> Squared \>\>\> Error} = \frac{1}{n}\sum_{i=1}^{n}(\tau_{i}^{CATE}-\widehat{\tau_{i}^{CATE}})^{2},$$
$$\mathrm{Bias} = \frac{1}{n}\sum_{i=1}^{n}(\tau_{i}^{CATE}-\widehat{\tau_{i}^{CATE}}),$$
$$\mathrm{Coverage} = \frac{1}{n}\sum_{i=1}^{n} \textbf{1}(\tau_{i}^{CATE}\in[\tau_{i}^{CATE, \>\>\> lwr}, \tau_{i}^{CATE, \>\>\> upr}]).$$
#### Summary of alternative methods used:
We will compare our proposed Gaussian process mixture model approach to other regression based methods for estimating treatment effects. We have considered random forests and single regression trees for treatment effect estimation via TRV modeling as well as *fit based trees*, *causal trees* [@athey2016recursive], and BART[@hahn2017bayesian; @hill2011bayesian] as non-TRV alternatives [^2].
None of the aforementioned methods have an obvious framework for learning the treatment assignment probabilities internally. This a crucial step in computing the CATE and ATE both via TRV and non-TRV based estimation techniques. In the case of the regression trees and random forests for TRV modeling, the TRV needs to be computed from the learnt propensity score first before any modeling can commence. The BART model uses the propensity score as an additional covariate, while causal and fit based trees use the propensity score as a weighting mechanism.
Therefore, we will use a two-step procedure where we first use the data to infer the treatment assignment probabilities and then given these estimates, apply the aforementioned regression methods to estimate the treatment effect. The treatment assignment probability vector $\pmb{e}$ is estimated via logistic regression [@rubin1996matching], a standard approach for estimating propensity scores in the causal inference literature.
Results on Simulated Data
-------------------------
The objective of the simulation studies presented in this section is to compare the performance of the Gaussian process mixture model to, BART, causal trees, fit based trees, the random forest and single regression tree models. We consider two criteria in our comparison. The first criteria is a comparison of the accuracy of the CATE, in terms of mean squared error and bias. The second criterion involves assessing how well the methods quantify uncertainty by considering the coverage of the intervals produced by all the models.
### Simulated Data Model {#simstudy}
In order to evaluate the proposed model as well as the other aforementioned approaches, we consider two simulation settings – one high dimensional case (with 40 covariates) and one low dimensional case (with 5 covariates) each with its own covariate level heterogeneity and a sample size of $n = 250$. For the remainder of this analysis, the high dimensional case is referred to as Case A, and the low dimensional case is referred to as Case B. By design neither of these simulation cases has a meaningful average treatment effect. We start with a detailed description of Case A.
In this framework, covariates $X_{1}, \ldots X_{30}$ are independent covariates, $X_{31}, \ldots X_{35}$ depend on pairs of covariates, while $X_{36}, \ldots, X_{40}$ depend on groups of three as follows,
$$\begin{aligned}
X_{k} &\sim \mathrm{Normal}(0, 1); \>\>\> k = 1, \ldots, 15,\\
X_{k} &\sim \mathrm{Uniform}(0, 1); \>\>\> k = 16, \ldots, 30,\\
X_{k} &\sim \mathrm{Bernoulli}(q_{k}); \>\>\> q_{k} = \mathrm{logit}^{-1}(X_{k-30} - X_{k-15}); \>\>\> k = 31,\ldots, 35,\\
X_{k} &\sim \mathrm{Poisson}(\lambda_{k}); \>\>\> \lambda_{k} = 5 + 0.75 X_{k-35}(X_{k-20} + X_{k-5}); \>\>\> k = 36,\ldots, 40.\\\end{aligned}$$
Next, we simulate the propensity score and the corresponding treatment assignments. This has been done as a simple linear transformation since the focus of the paper is not propensity score modeling but rather CATE modeling. The propensity scores and the treatment effects of interest for Case A are given in figure \[fig:simLarge\].
$$\begin{aligned}
p_{i} &= \mathrm{logit}^{-1}(0.3 \sum_{k = 1}^{5}X_{k} -0.5 \sum_{k = 21}^{25}X_{k} -0.0001 \sum_{k = 26}^{35}X_{k} + 0.055 \sum_{k = 36}^{40}X_{k} ),\\
W &\sim \mathrm{Bernoulli}(p_{i}).\end{aligned}$$
Finally we generate the potential outcomes and the observed outcomes. $$\begin{aligned}
f(\mathbf{X}) &= \frac{\sum_{k = 16}^{19} X_{k}\exp(X_{k+14})}{1+\sum_{k = 16}^{19}X_{k} \exp(X_{k+14})},\\
Y(0) &= 0.15 \sum_{k=1}^{5}X_{k} + 1.5 \exp(1+1.5 f(\mathbf{X}))+ \epsilon_{i},\\
Y(1) &= \sum_{k = 1}^{5}\{ 2.15 X_{k} + 2.75 X_{k}^{2} + 10 X_{k}^{3}\} + 1.25 \sqrt{0.5 + 1.5\sum_{k = 36}^{40}X_{k}} + \epsilon_{i},\\
Y &= WY(1) + (1-W)Y(0);\>\>\> \epsilon_{i} \stackrel{\mathrm{IID}}{\sim} \mathrm{Normal}(0, 0.0001).\end{aligned}$$
\
The lower dimensional case, which we have adapted from the simulation study in [@hahn2017bayesian] is presented similarly. We start by simulating the following 5 covariates, $$\begin{aligned}
X_{k} &\sim \mathrm{Normal}(0, 1); \>\>\> k = 1, \ldots, 3,\\
X_{4} &\sim \mathrm{Bernoulli}(p = 0.25), \\
X_{5} &\sim \mathrm{Binomial}(n = 2, p = 0.5).\end{aligned}$$
In this scheme, unlike Case A, all the covariates are independent. The propensity score model analogous to the previous case is a linear transformation of the covariates.
$$\begin{aligned}
p_{i} &= \mathrm{logit}^{-1}(0.1X_{1}-0.001X_{2}+.275X_{3}-0.03X_{4}),\\
W &\sim \mathrm{Bernoulli}(p_{i}).\end{aligned}$$
Finally we generate the potential outcomes and the observed outcomes. The results of this simulation are presented in figure \[fig:simSmall\].
$$\begin{aligned}
f(\mathbf{X}) &= -6 + h(X_{5}) + |X_{3}-1|,\\
h(0) &= 2, \>\>\> h(1) = -1, \>\>\> h(2) = -4,\\
Y(0) &= f(X) - 15 X_{3} + \epsilon_{i},\\
Y(1) &= f(X)+ (1 + 2X_{2}X_{3}) + \epsilon_{i},\\
Y&= WY(1) + (1-W)Y(0); \>\>\> \epsilon_{i} \stackrel{\mathrm{IID}}{\sim} \mathrm{Normal}(0, 0.0001).\end{aligned}$$
\
### Comparison of Methods {#ressim}
The first stage of our analysis compares the CATE estimation in instances when the treatment assignment probability is assumed to be known. We focus on the mean squared error, bias and coverage of the CATE under Case A and Case B along with visual analyses of model adaptability to gauge fit quality. For the proposed model the samplers were run for $K = 6, 000$ steps with $1,000$ initial steps burned off. No thinning of the samples generated was needed. Similarly, for the non-Bayesian methods, $K = 5, 000$ replications of the bootstrap were generated. The comparison of point estimates of the CATE under Case A is presented in figure \[fig:caseAComparisonKnown\] and Case B in figure \[fig:caseBComparisonKnown\] for the sub-case where the treatment assignment mechanism is known; the corresponding diagnostic measures are presented in tables \[tb:caseASummaryKnown\] and \[tb:caseBSummaryKnown\].
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\
\
In Case A, both in terms of point estimation, as well as uncertainty quantification, we can conclude that when the treatment assignment is known, the proposed model is the overall winner. As we can see, it adapts well to the heterogeneity of the treatment effects in the data, and is able to recover the effects to a high degree as observed in figure \[fig:caseAComparisonKnown\](a). It also has the lowest mean squared error of the models presented and the point-wise coverage of its uncertainty intervals, while low relative to tree based methods, is better than BART (see table \[tb:caseASummaryKnown\]). Furthermore, the bias of the model is generally lower than causal trees, fit based trees and transformed outcome trees.
It warrants mention that BART only adapts to heterogeneity minimally. We can attribute this to the complexity of regularization in causal inference problems [@hahn2017bayesian] from the shrinkage prior as well as poor mixing of the MCMC used for BART in high dimensions [@pratola2016efficient]. We see similar behavior from transformed outcome trees, where post-estimation *pruning* can lead to regularization induced bias as well. An elaborate discussion on bias in causal inference applications from regularized models originally designed for prediction is given in [@hahn2017bayesian] and [@hahn2018regularization].
\
\
\
In Case B, the model performs well in terms of recovering the high degree of heterogeneity but it suffers in terms of mean square error and bias. The model still adapts well to the heterogeneity inherent in the data, and is able to recover the effects as observed in figure \[fig:caseBComparisonKnown\](a), albeit with a higher degree of overall noise. This noisiness translates to high mean squared error and bias, where the other alternative models perform better, with one minor caveat. Due to the piece-wise nature of the tree based models, they do not adapt to the heterogeneity as well as the proposed model and BART do. Furthermore, the model also has the highest degree of point-wise uncertainty interval coverage (see table \[tb:caseBSummaryKnown\]).
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian-Process Mixture 4191.665 13.207 0.780
2 Bayesian Additive Regression Tree 5856.135 -5.351 0.596
3 Transformed Outcome Tree 7769.077 14.374 0.876
4 Fit Based Tree 6154.396 15.633 0.928
5 Causal Tree 8390.039 21.923 0.964
6 Transformed Outcome Random Forest 4993.576 0.317 0.932
: Case A - Conditional Average Treatment Effect Summary (Known)[]{data-label="tb:caseASummaryKnown"}
We also compare the CATE estimation for both cases when the treatment assignment probabilities are unknown and need to be inferred from the data. The comparison of the point estimation is given in figures \[fig:caseAComparisonUnknown\] and \[fig:caseBComparisonUnknown\] respectively for the two cases, with the corresponding summary measurements of fit in tables \[tb:caseASummaryUnknown\] and \[tb:caseBSummaryUnknown\].
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian Process Mixture 50.262 3.174 0.988
2 Bayesian Additive Regression Tree 5.498 0.229 0.808
3 Transformed Outcome Tree 16.421 0.202 0.900
4 Fit Based Tree 15.620 0.282 0.952
5 Causal Tree 21.143 0.974 0.972
6 Transformed Outcome Random Forest 118.745 -0.582 0.816
: Case B - Conditional Average Treatment Effect Summary (Known)[]{data-label="tb:caseBSummaryKnown"}
For Case A, the performance of the model is far superior in terms of adapting to the heterogeneity, as indicated in figure \[fig:caseAComparisonUnknown\](a), in particular compared to the performance of the transformed outcome random forest and BART given in figures \[fig:caseAComparisonUnknown\](c) and \[fig:caseAComparisonUnknown\](f). The deterioration in the quality of the estimates from BART is particularly noticeable and can be attributed to the same over-regularization observed before which is even more of a concern since there is additional uncertainty from the learning of the assignment mechanism. Furthermore, while the point-wise coverage of the uncertainty interval is lower relative to the other models, the Gaussian process mixture is the clear winner in terms of the mean square error. The proposed model also outperforms the tree based models (causal and fit based trees as well as transformed outcome trees) in terms of bias (see table \[tb:caseASummaryUnknown\]) and its point-wise interval coverage is stable relative to BART, which speaks to the models overall robustness despite the added layer of complexity from learning the assignment mechanism.
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\
\
We see that for Case B, the results of the analysis are similar to when the treatment assignment was known. The performance of the model is comparable in terms of adapting to the heterogeneity relative to the other models, as indicated in figure \[fig:caseBComparisonUnknown\](a) – albeit again with a similar degree of noisiness as earlier. However, we again out-perform transformed outcome random forests in terms of point estimation with lower mean squared error. The only aspect in which the model out performs all the other methods considered is in terms of point-wise interval coverage.
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Our conclusion is that the model performs well when there are a large number of covariates present, and the degree of heterogeneity in the treatment effects is high. The flexibility of the mixture of Gaussian processes ensures adaptability, where tree based models fail particularly when there is shared information in the covariates (as is true in Case A) since the prior provides some degree of built-in regularization that is not as excessive as that of BART. However, when the number of covariates is small, the flexibility of the model hurts its overall performance since we observe that our estimates are generally noisier. These limitations of the model are discussed as avenues for future work in the last section of this paper.
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- --------- -----------------
1 Gaussian Process Mixture 3916.562 13.207 0.780
2 Bayesian Additive Regression Tree 6754.058 -5.569 0.624
3 Transformed Outcome Tree 6289.891 7.061 0.880
4 Fit Based Tree 6154.396 15.633 0.932
5 Causal Tree 8390.039 21.923 0.968
6 Transformed Outcome Random Forest 12124.426 -21.958 0.960
: Case A - Conditional Average Treatment Effect Summary (Unknown)[]{data-label="tb:caseASummaryUnknown"}
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian Process 31.517 1.898 1.000
2 Bayesian Additive Regression Tree 6.259 0.118 0.776
3 Transformed Outcome Tree 16.421 0.202 0.892
4 Fit Based Tree 15.620 0.282 0.956
5 Causal Tree 19.652 0.876 0.972
6 Transformed Outcome Random Forest 115.329 -0.349 0.820
: Case B - Conditional Average Treatment Effect Summary (Unknown)[]{data-label="tb:caseBSummaryUnknown"}
Results on the Italy Survey on Household Income and Wealth (SHIW) {#real}
-----------------------------------------------------------------
Our application of the GP mixture model to a real data aimed at the estimation the causal effects of debit card ownership on household spending. A causal analysis of this question was developed in [@mercatanti2014debit] using data from the Italy Survey on Household Income and Wealth (SHIW) to estimate the population average treatment effect for the treated (PATT). The SHIW is a biennial, national population representative survey run by Bank of Italy. The subset of the SHIW data we considered consists of $n = 564$ observations with 385 untreated and 179 treated observations. The outcome variable is the monthly average spending of the household on all consumer goods. The treatment condition is whether the household possesses one and only one debit card, and the control condition is that the household does not possess *any* debit cards. The covariates we used include: cash inventory held by the household, household income, average interest rate in the province where the household resides, measurement of wealth, and the number of banks in the province in which the household resides. See [@mercatanti2014debit] for more details about the data. Our analysis of these data will consist of comparing estimates of the ATE and CATE (with respect to household income) of our GP mixture model to the same alternative models as the previous section.
Decile $Mean \quad Income$ $\widehat{\tau^{CATE}}$ $\widehat{\tau^{CATE}_{lwr}}$ $\widehat{\tau^{CATE}_{upr}}$
-------- --------------------- ------------------------- ------------------------------- ------------------------------- --
1 -1.137 0.629 0.404 0.857
2 -0.831 0.567 0.374 0.761
3 -0.638 0.558 0.381 0.734
4 -0.472 0.459 0.298 0.620
5 -0.310 0.425 0.270 0.578
6 -0.114 0.396 0.245 0.546
7 0.103 0.343 0.190 0.490
8 0.397 0.272 0.097 0.441
9 0.848 0.172 -0.050 0.389
10 2.143 -0.125 -0.513 0.251
: Conditional average treatment effect with average income by decile[]{data-label="tb:cateIncomeRealModel"}
We start with a presentation of the CATE under our model against income in \[fig:realComparison\](a).The proposed model estimates an overall downward trend in the effect of owning a debit card, i.e. as the level of income increases, the effect of owning a debit card declines. In order to summarize this effect, we consider the CATE for binned deciles of income for the proposed model in figure \[fig:realComparison\](b) and the alternative models in figure \[fig:realComparison\](c). We find that the proposed model detects a statistically meaningful effect for the first eight deciles of income, and this effect is estimated to decline in size. For the final two deciles, the model concludes that there is no statistically meaningful effect of owning a debit card. These results are summarized in table \[tb:cateIncomeRealModel\]. By comparison, the inference from the alternative approaches is not quite as clear. BART and transformed outcome trees, detect minimal heterogeneity. With BART, this flattening can be attributed to over-regularization due to the prior, as seen in the simulated data case, while for transformed outcome trees, the axis-parallel splits used to estimate the model are not always suitable for partitioning the covariates. By comparison transformed outcome random forests, transformed outcome trees and causal trees demonstrate the most heterogeneity at the highest two deciles of income. These results are summarized in table \[tb:comparisonTable\] in Appendix \[app:C\].
In order to be comprehensive and comparable to past work, we have also produced estimates of the average treatment effect in table \[tb:ateReal\]. The proposed Gaussian process mixture detects a statistically meaningful ATE. This result is consistent with the findings of [@mercatanti2014debit]. Furthermore, we also see that the uncertainty interval for the Gaussian process mixture is the tightest of the methods used here, all of which with the exception of BART generate similar inference. This result is consistent with the findings on simulated data presented in the last section since the BART model does not adapt to heterogeneity well in instances where the number of covariates is high with large contributions to the variation in the treatment effects. Again this argues that the GP mixture model may be outperforming the other methods.
Model Type $\widehat{\tau^{ATE}}$ $\widehat{\tau^{ATE}_{lwr}}$ $\widehat{\tau^{ATE}_{upr}}$
--- ----------------------------------- ------------------------ ------------------------------ ------------------------------
1 Gaussian Process Mixture 0.369 0.220 0.518
2 Transformed Outcome Tree 0.470 0.210 0.555
3 Fit Based Tree 0.378 0.214 0.608
4 Causal Tree 0.475 0.360 0.939
5 Bayesian Additive Regression Tree 0.115 -1.129 1.397
6 Transformed Outcome Random Forest 0.414 0.229 0.604
: Comparison of average treatment effects.[]{data-label="tb:ateReal"}
Based on the economic concepts of *income elasticity of demand*, *consumer choice* and *substitution effects* [@varian2014intermediate], the heterogeneity identified at the lowest levels of standardized income is a more sensible result relative to the implication of the other approaches. At the lowest levels of income, economic agents are more likely to substitute debit card use for cash in an effort to maximize spending. The debit cards act as an inflator of perceived financial resources and this effect is expected to diminish as the overall income grows. Therefore, the GP mixture model makes a more convincing case for capturing the true nature of how holding a debit card influences spending.
\
Discussion and Future Work {#futurework}
==========================
We have proposed a novel non-parametric Bayesian model to estimate heterogeneous treatment effects. Our approach combines the *transformed response variable* framework with a mixture of Gaussian-processes. The motivation for the GP mixture model was to improve the accuracy of our point estimates as well as to better quantify uncertainty relative to other models particularly those from the Bayesian non-parametrics literature. We compared the performance of our technique to a single regression tree and random forest model within the TRV framework as well as two conditional mean regression type weighted tree based methods and BART. We used simulation studies to show instances where our approach is a better estimator with respect to both point estimation and uncertainty quantification. Furthermore, our approach also has the advantage in that we can address the case where treatment assignment probabilities are unknown within our model; other methods require a two-stage process where another model is required to infer the treatment assignment probabilities. This tandem estimation provides better insight into the data generating process and also captures uncertainty from all levels of inference.
In addition, a Bayesian model of treatment effects with a single likelihood for the design and analysis stages creates concerns of feedback since the TRV depends on the assignment mechanism. We demonstrate that our model is robust to this feedback due to both our prior specification as well as individual covariate adjustment via the Gaussian process covariance functions. However, this raises the theoretical question of whether there is a weaker condition that can be satisfied and still lead to effective inference of treatment effects which is the first area that we aim to explore in future work.
There are several ways we can extend our model to be more robust and flexible. In the context of robustness, the GP prior covariance functions specified impose smoothness assumptions on the treatment effects that may not be realistic in a myriad of applied settings. Relaxing the smoothness and using non-parametric models that have been developed to model dose-response curves may result in richer and more reliable inference. Furthermore, as noted earlier inference using the TRV is sensitive to the probability of receiving the treatment and can create biases and instability when the assignment probability are close to their extremes. While we have addressed instability in the estimation of effects using a correctly specified model and indirectly improved propensity score estimation, we have not directly curbed the susceptibility of the method to extreme weights. The variance of the mixture model is still influenced by the reciprocal of the treatment assignment probability (as is the case generally with IPW estimators). Extending our model to be more insensitive to these extreme cases is vital in application.
Under the theme of model flexibility, we are currently fixing the hyper-parameter values within the kernels of the Gaussian-processes since attempting to learn these from the data creates two problems that we need to carefully study. First, learning these parameters is difficult from a sampling perspective since the target distributions are often extremely multi-modal. A promising avenue for addressing this is the use of a combination of sampling and optimization [@levine2001implementations] – this is particularly important since Bayesian non-parametric methods are known to be sensitive to prior calibration. This is crucial in instances where the degree of heterogeneity in treatment effects is small as we have seen via simulation study. Second, the scalability of Gaussian processes is very limited [@johndrow2015approximations] and hence increasing the number of parameters that we are attempting to learn hurts the scalability even more. This broadly summarizes the areas that we will explore in future work.
Proof of Equivalence {#app:A}
====================
We now show that the transformation presented in section \[former\] in expectation recovers the CATE i.e. $$\mathbb{E}_{Y}[Y^{*} \mid X = x] = \tau^{CATE}.$$
First observe that $Y_{i} = Y_{i}(W_{i}) = W_{i}Y_{i}(1) + (1 - W_{i})Y_{i}(0).$
By the definition of the TRV $$\begin{aligned}
A = \mathbb{E}_{Y}[Y^{*} \mid X = x, \mathcal{D}] &=& \mathbb{E}_{Y} \left[\frac{W-e_{i}}{e_{i}(1-e_{i})}Y \mid X = x, \mathcal{D}\right], \\
&=& \frac{1}{e_{i}(1-e_{i})}\left( \mathbb{E}_{Y}[YW \mid X = x, \mathcal{D}] - e_{i} \mathbb{E}_{Y}[Y \mid X=x, \mathcal{D}]\right).\end{aligned}$$
Due to the ignorability of the treatment assignment the following holds $$\begin{aligned}
A & = & \frac{1}{e_{i}(1-e_{i})}(e_{i}\mathbb{E}_{Y}[Y \mid W = 1, X=x, \mathcal{D}] - e_{i} \mathbb{E}_{Y}[Y \mid X = x, \mathcal{D}]) \\
&=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid X = x, W =1,\mathcal{D}] - \frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid X=x, \mathcal{D}].\end{aligned}$$ By iterating expectations the following holds: $$\begin{aligned}
A &=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W=1, X=x, \mathcal{D}] - \frac{1}{1-e_{i}}\mathbb{E}_{W}[\mathbb{E}_{Y}[Y \mid W=1,X=x, \mathcal{D}]], \\
&=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W = 1,X_{i}=x, \mathcal{D}] - \frac{e_{i}}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W= 1,X=x, \mathcal{D}] - \\
& &\mathbb{E}_{Y}[Y \mid W = 0, X=x, \mathcal{D}].\end{aligned}$$ Collecting the first two terms provides the desired result $$A=\mathbb{E}_{Y}[Y \mid W = 1, X= x, \mathcal{D}] - \mathbb{E}_{Y}[Y \mid W =0, X=x, \mathcal{D}].$$
Derivation of Model {#app:B}
===================
The derivation of the model presented in the paper begins with the transformation of interest given as follows, with $Y_{i}$ denoting the observed response, $W_{i}$ the assigned treatment and $e_{i} = P(W_{i} = 1)$ $$Y_{i}^{*} = \frac{W_{i} - e_{i}}{e_{i}(1-e_{i})}Y_{i}.$$ In addition, we define the two regression functions for the outcome, one under the treatment and one under the control, $$\begin{aligned}
(Y_{i}|W_{i} = 0) = f_{0}(X_{i})+\epsilon_{i}(0),\\
(Y_{i}|W_{i} = 1) = f_{1}(X_{i})+\epsilon_{i}(1).\\\end{aligned}$$ Using the transformation, and substituting the regression functions under the two cases i.e. when $W_{i} = 1$ and when $W_{i} = 0$ and assuming further that $\epsilon(1), \epsilon(0) \stackrel{IID}{\sim}\mathrm{N}(0, \sigma^{2})$, we can define with probability $e_{i}$,
$$\begin{aligned}
(Y_{i}^{*}|W_{i} = 1) &= \frac{f_{1}(X_{i}) - e_{i}f_{1}(X_{i})+e_{i}f_{0}(X_{i})}{e_{i}} + f_{1}(X_{i})-f_{0}(X_{i}) + \frac{\epsilon_{i}(1)}{e_{i}},\\
&= f_{1}(X_{i})-f_{0}(X_{i}) + (1-e_{i})\bigg(\frac{ f_{1}(X_{i})}{e_{i}}+\frac{f_{0}(X_{i})}{1-e_{i}} \bigg)+\frac{\epsilon_{i}(1)}{e_{i}},\\
&= g(X_{i})+ (1-e_{i})h(X_{i})+\frac{\epsilon_{i}(1)}{e_{i}}.\end{aligned}$$
and similarly, with probability $1-e_{i}$ that,
$$\begin{aligned}
(Y_{i}^{*}|W_{i} = 0) &= \frac{-(1-e_{i})f_{1}(X_{i}) +(1- e_{i})f_{0}(X_{i})-f_{0}(X_{i})}{e_{i}} + f_{1}(X_{i})-f_{0}(X_{i}) - \frac{\epsilon_{i}(0)}{1-e_{i}},\\
&= f_{1}(X_{i})-f_{0}(X_{i}) + (-e_{i})\bigg(\frac{ f_{1}(X_{i})}{e_{i}}+\frac{f_{0}(X_{i})}{1-e_{i}} \bigg)-\frac{\epsilon_{i}(0)}{1-e_{i}},\\
&= g(X_{i})+ (-e_{i})h(X_{i})+\frac{\epsilon_{i}(0)}{e_{i}-1}.\end{aligned}$$
This yields the mixture model model that we have presented in the paper, $$\begin{aligned}
Y_{i}^{*} &= g(X_{i}) + \varepsilon_{i},\\
\varepsilon_{i}\sim(e_{i})\mathrm{Normal}((1-e_{i})h(X_{i}), \frac{\sigma^{2}}{e_{i}^{2}})&+(1-e_{i})\mathrm{Normal}(-e_{i}h(X_{i}), \frac{\sigma^{2}}{(1-e_{i})^{2}}).\end{aligned}$$
Comparison of SHIW Data {#app:C}
=======================
This section presents comparative analysis using various methods for the CATE estimation for the SHIW data using the Gaussian process mixture in section \[real\]. Point estimates of the CATE along with $95\%$ uncertainty intervals for each decile of income, along with the average value of income in that decile are presented in table \[tb:comparisonTable\].
Sampling Algorithms for Model Specifications {#app:D}
============================================
#### Algorithm for inference with known assignment probabilities:
For the full conditional distributions specified in we can run the following Gibbs sampling procedure to generate a sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{((j)})_{j=1}^K$ as follows,
1. Initialize ${\mathbf h}^{(0)}$, $\sigma^{(0)}$, and ${\mathbf g}^{(0)}$;
2. For $j= 1,...,K$
1. $\mathbf{g}^{(j)} \sim \pi(\mathbf{g} \mid \mathbf{h}^{(j-1)}, \sigma^{(j-1)}, \mathcal{D})$;
2. $\mathbf{h}^{(j)} \sim \pi(\mathbf{h} \mid \mathbf{g}^{(j)}, \sigma^{(j-1)}, \mathcal{D})$ ;
3. $ \sigma^{(j)} \sim \pi(\sigma \mid \mathbf{h}^{(j)}, \mathbf{g}^{(j)}, \mathcal{D}).$
Given the sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{(j)})_{j=1}^K$ we discard an initial $K_0$ of the samples to address burn-in of the chain and we thin the remaining samples by a small factor $\gamma$ to obtain independent samples from the joint posterior distribution in section \[simspec\] and in equation . We will specify the burn-in and thinning settings whenever we discuss applications of the method.
#### Algorithm for inference with unknown assignment probabilities:
For the full posterior stated in equation a standard Gibbs sampling procedure of the type specified above cannot be used for sampling the treatment assignment probabilities. We use a näive approach to sampling the assignment probabilities in addition to the other model parameters with an additional Metropolis-within-Gibbs step. This results in the following procedure:
1. Initialize ${\mathbf h}^{(0)}$, $\sigma^{(0)}$, ${\mathbf g}^{(0)}$, and $\boldsymbol{\beta}^{(0)}$. Use $\boldsymbol{\beta}^{(0)}$to compute $\boldsymbol{e}^{(0)}$;
2. Compute $\mathbf{Y}^*$ from the initial $\boldsymbol{e}^{(0)}$ and data;
3. For $j= 1,...,K$
1. $\mathbf{g}^{(j)} \sim \pi(\mathbf{g} \mid \mathbf{h}^{(j-1)}, \sigma^{(j-1)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$;
2. $\mathbf{h}^{(j)} \sim \pi(\mathbf{h} \mid \mathbf{g}^{(j)}, \sigma^{(j-1)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$ ;
3. $ \sigma^{(j)} \sim \pi(\sigma \mid \mathbf{h}^{(j)}, \mathbf{g}^{(j)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$;
4. Use Metropolis-Hastings step to sample $\boldsymbol{\beta}^{(j)}$;
5. Compute $\boldsymbol{e}^{(j)}$ from $\boldsymbol{\beta}^{(j)}$ and data;
6. Compute $\mathbf{Y}^*$ from $\boldsymbol{e}^{(j)}$ and data.
Therefore using the steps in the algorithm above we simulate a sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{(j)}, \pmb{\beta}^{(j)},$ $\pmb{e}^{(j)},\mathbf{Y}^{*(j)})_{j=1}^K$ akin to earlier with burn-in and thinning considerations that reflects draws from the joint distribution in section \[compspec1\] in .
Decile $Mean \quad Income$ $tot$ $fit$ $ct$ $BART$ $RF$ $lwr_{tot}$ $upr_{tot}$ $lwr_{fit}$ $upr_{fit}$ $lwr_{ct}$ $upr_{ct}$ $lwr_{BART}$ $upr_{BART}$ $lwr_{RF}$ $upr_{RF}$
-------- --------------------- ------- ------- -------- -------- ------- ------------- ------------- ------------- ------------- ------------ ------------ -------------- -------------- ------------ ------------ --
1 -1.137 0.470 0.234 0.637 0.118 0.420 0.089 0.678 0.087 0.741 0.202 1.103 -1.166 1.394 0.082 0.772
2 -0.831 0.470 0.417 0.500 0.117 0.383 0.087 0.672 0.096 0.730 0.191 1.117 -1.211 1.397 0.083 0.770
3 -0.638 0.470 0.538 0.515 0.105 0.461 0.067 0.679 0.083 0.733 0.183 1.094 -1.204 1.398 0.082 0.772
4 -0.472 0.470 0.258 0.430 0.089 0.376 0.076 0.676 0.094 0.725 0.193 1.105 -1.224 1.393 0.085 0.744
5 -0.310 0.470 0.093 0.307 0.095 0.276 0.077 0.670 0.082 0.740 0.198 1.112 -1.191 1.398 0.075 0.766
6 -0.114 0.470 0.598 0.681 0.116 0.391 0.074 0.676 0.097 0.732 0.204 1.103 -1.107 1.397 0.083 0.772
7 0.103 0.470 0.471 0.552 0.096 0.407 0.081 0.660 0.086 0.748 0.212 1.112 -1.136 1.370 0.087 0.760
8 0.397 0.470 0.414 -0.103 0.122 0.269 0.086 0.682 0.092 0.744 0.192 1.104 -1.120 1.385 0.082 0.760
9 0.848 0.470 0.363 0.384 0.134 0.442 0.082 0.667 0.094 0.744 0.178 1.108 -1.119 1.400 0.087 0.776
10 2.143 0.470 0.396 0.835 0.156 0.706 0.075 0.673 0.107 0.735 0.190 1.108 -1.068 1.406 0.081 0.760
Acknowledgements
================
The authors gratefully acknowledge the support of Andrea Mercatanti (Department of Statistics, Bank of Italy) for providing data for this paper and sincerely thank Elizabeth Lorenzi (Duke University) for providing insightful commentary and expertise on the topic of causal inference.
[^1]: We generate proposals as $\pmb{\beta}^{*}\sim\mathrm{N}(\mu = \pmb{\beta}^{j-1}, \sigma^{2})$ i.e. from a Gaussian distribution that is centered at the last accepted value. The variance controls the step size of the proposals and needs to be tuned for the application.
[^2]: We use the implementations of these methods in the `R` packages `causalTree` [@causal2016tree], `rpart`[@rpart2002], `randomForest`[@rf2018] and `BART`[@bart2018]
|
---
abstract: 'In this paper, we present our general results about traversing flows on manifolds with boundary in the context of the flows on surfaces with boundary. We take advantage of the relative simplicity of $2D$-worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary, in which the boundary effects take the central stage.'
address: '5 Bridle Path Circle, Framingham, MA 01701, USA'
author:
- Gabriel Katz
title: 'Flows in Flatland: A Romance of Few Dimensions '
---
Introduction
============
This paper is about the gradient flows on compact surfaces, thus the reference to Abbott’s *Flatland* [@Ab] in the title. The paper is an informal introduction into the philosophy and some key results from [@K] -[@K6], as they manifest themselves in $2D$.
The remarkable convergence of topological, geometrical, and analytical approaches to the study of closed surfaces is widely recognized by the practitioners for more than a century. We will exhibit a similar convergence of different investigative approaches to vector flows on surfaces *with boundary*.
We will take advantage of the relative simplicity of $2D$ flows to illustrate and popularize the main ideas of our recent research of *traversally generic* flows on manifolds with boundary. When the results are specific to the dimension two, their validation will be presented in detail. The multidimensional arguments that resist significant simplifications in $2D$ will be described and explained in general terms.
Throughout the investigation, we focus on the interactions of gradient flows with the boundary, rather than on the critical points of Morse functions. So, in our approach to the Morse Theory, the boundary effects rule.
On Morse Theory on surfaces with boundary and beyond
====================================================
*Morse Theory*, the classical book of John W. Milnor [@Mi], starts with the canonical picture of a Morse function $f: T^2 \to \R$ on a 2-dimensional torus $T^2$ (see Fig. 1). It is portrayed as the height function $f$ on the torus $T^2$ residing in the space $\R^3$. The height $f$ has four *critical points*: $a, b, c$, and $d$ so that $$f(a) > f(b) > f(c) > f(d).$$ A point $z$ is called critical if the differential $df$ of $f$ vanishes at $z$. In the vicinity of each critical point $z$, $T^2$ admits a pair of local coordinate functions, say $x$ and $y$, so that locally the function $f$ acquires the form $$f(x, y) = f(0, 0) \pm x^2 \pm y^2,$$ where the signs may form four possible combinations.
\[fig1.1\]
{height="2in" width="2.8in"}
We call a vector field $v$, tangent to $T^2$, *gradient-like* if $df(v) > 0$ everywhere outside of the set $Cr(f)$ of critical points.
If the torus is “slightly slanted" with respect to the vertical coordinate $f$ in $\R^3$, then the following picture emerges. The majority of downward trajectories of the $f$-gradient flow $\{\Phi_t\}_{t \in \R}$ that emanate from $a$, asymptotically reach $d$. There are two trajectories that asymptotically link $a$ with $b$, and two trajectories that link $a$ with $c$. No (unbroken) trajectory asymptotically connects $b$ to $c$.
Perhaps, a more transparent depiction of the gradient flow $\{\Phi_t\}_{t \in \R}$ is given in Fig. 2, where the torus is shown in terms of its fundamental domain, the square. To form $T^2$, the opposite sides of the square are identified in pairs.
The Morse Theory is concerned with the sets of *constant level* $\{f^{-1}(\a)\}_{\a \in \R}$ and the *below constant level* sets $\{f^{-1}((-\infty, \a))\}_{\a \in \R}$. The main observation is that the topology of these sets is changing in an essential way only when the rising $\a$ crosses the critical values $$Cr(f) = \{f(a), f(b), f(c), f(d)\}.$$ Each such “critical crossing" results in an *elementary surgery* on the set $\{f^{-1}((-\infty, \a))\}_{\a \in \R}$, where $\a$ is just below a critical value $\a_\star \in Cr(f)$. For a small $\e > 0$, an elementary surgery $$f^{-1}((-\infty, \a_\star -\e)) \Rightarrow f^{-1}((-\infty, \a_\star + \e))$$ attaches the *handle* $ f^{-1}((\a_\star -\e, \a_\star + \e))$ to the set $f^{-1}((-\infty, \a - \e))$. Eventually, when $\a$ rises above $f(a)$, the entire topology of torus $T^2$ is captured by a sequence of these elementary surgeries.
From a different angle, the knowledge of how the critical points $a, b, c, d$ interact via the trajectories of the $\Phi_t$-flow is also sufficient for reconstructing the surface $T^2$ as Fig. 2 suggests (see [@C]).
\[fig1.2\]
{height="1.8in" width="3in"}
Note that, in the vicinity of each critical point, the gradient flow exhibits *discontinuity*: small changes in the initial position of a point $z$, residing in the vicinity of a critical point, result in significant differences in the position of $\Phi_t(z)$ for big positive/small negative values of $t$ (see Fig. 3). In fact, this discontinuity of the gradient flow, expressed in terms of the *stable* and *unstable manifolds* of critical points (see [@Mi]), captures the topology of the surface (as the left diagram in Fig. 2 suggests)!
\[fig1.3\]
{height="1.3in" width="2.2in"}
As a result of gradient flow discontinuity, the *space of trajectories* $\mathcal T(v)$ is pathological (non-separable). The space $\mathcal T(v)$ is constructed by declaring equivalent any two points that reside on the same trajectory.
When a compact connected surface $X$ has a nonempty boundary $\d X$, traditionally, the Morse function $f: X \to \R$ is assumed to be constant on $\d X$ and its gradient flow interacts with the boundary in constrained way. Then the *relative topology* of the pair $(X, \d X)$ can be captured in the ways analogous to the previous description of the Morse Theory on torus. In fact, the Morse Theory on manifolds with boundary can be viewed as a very special instance of the Morse Theory on stratified spaces (the two strata $\d X$ and $X$ form the stratification). The latter was developed by Goresky and MacPherson in \[GM\] -\[GM2\].
In this paper, we propose a different philosophy for the Morse Theory on compact surfaces/manifolds $X$ *with boundary*. To formulate it, let us revisit our favorite closed surface, the torus. By deleting from $T^2$ small disks, centered on the points of the critical set $Cr(f)$, we manufacture a surface $X$ whose boundary is a disjoint union of four circles. Evidently, $f: X \to \R$ has no critical points at all. Still it has a nontrivial topology! Can this topology be reconstructed from some data, provided by the critical point-free $f$ and its gradient-like field $v \neq 0$? An experienced reader would notice that the restriction $f|: \d X \to \R$ has critical points (maxima and minima), some of which interact *along* the boundary (with the help of a gradient-like field $v^\d$, tangent to $\d X$). However, it is quite clear that these interactions are not sufficient for a reconstruction of the topology of $X$! In fact, a reconstruction of the surface $X$ becomes possible if one introduces additional interactions between the points of $Cr(f|_{\d X})$ that occur “through the bulk $X$" and are defined with the help of *both* vector fields $v$ and $v^\d$. This observation has been explored by a number of authors, but it is not the world view that we are promoting here...
To dramatize further the situation we are facing, let us place four small disks, centered on the critical points of $f: T^2 \to \R$, into a single open disk $D^2$ and form $X = T^2 \setminus D^2$ (see Fig 2, the right diagram). Again, $f|: X \to \R$ has no critical points, the gradient field $v|_X \neq 0$, but its topology of $X$ is nontrivial. This time, the boundary $\d X$ of the punctured torus $X$ is just a single circle! Let us keep this challenge in mind.
Can one propose a “Morse Theory" that is not centered on critical points? The answer is affirmative. It relies on the following observation. Typically, in the vicinity of $\d X$, the $v$-trajectories are interacting with the boundary in a number of very particular and stable ways: they are either transversal to $\d X$, or are tangent to it in a *concave* or *convex* fashion[^1] (see Fig. 5). So the boundary $X$ may be “wiggly" with respect to the flow. We claim that this geometry of the $v$-flow in connection to the boundary $\d X$ is the crucial ingredient for reconstructions of $X$ in terms of the flow (see Section 8, especially Theorem \[th7.1\]).
In the vicinity of a *concave* tangency point, the $v$-flow is *discontinuous* in the same sense as the gradient flow is discontinuous in the vicinity of its critical point: in time, close initial points become distant. In this context, the divergence of initially close points occurs due to very different *travel times* available to them; unlike the infinite travel time for the gradient flows of the Morse theory on *closed* surfaces, in the case of the non-singular gradient flows on surfaces with boundary, every point exits the surface in finite time. In particular, the surface is *not* flow-invariant. And again, these discontinuities of the flow reflect the topology of the surface. Let us clarify this point.
Fig. 4 shows a gradient flow $v$ on a surface $X \subset \R^2$, the disk with $4$ holes. The nonsingular function $f: X \to \R$ is the vertical coordinate in $\R^2$. Each $v$-trajectory is either a closed segment, or a singleton. By collapsing each trajectory to a point, we create a quotient space $\mathcal T(v)$ of trajectories. Since the flow trajectories are closed segments or singletons, this time, the trajectory space $\mathcal T(v)$ is “decent", a finite graph with verticies of valency $1$ or $3$ only. The verticies of valency $3$ correspond to the points on $\d X$ where the boundary is concave with respect to the flow, and the univalent verticies to the points on $\d X$ where the flow is convex.
The obvious map $\Gamma: X \to \mathcal T(v)$ cellular. Moreover, because the fibers of $\Gamma$ are contractable, $\Gamma$ is a *homotopy equivalence*. In particular, the fundamental groups $\pi_1(X)$ and $\pi_1(\mathcal T(v))$ are isomorphic with the help of $\Gamma$. So the trajectory spaces of generic non-vanishing vector fields $v$ of the gradient type on connected surfaces $X$ with boundary deliver $1$-dimensional *homotopy theoretical models* of $X$.
\[fig1.4\]
{height="2.3in" width="3in"}
Vector felds and Morse stratifications on surfaces
==================================================
Following [@Mo], for any vector field $v$ on a compact surface $X$ with boundary such that $v|_{\d X} \neq 0$, we consider the closed locus $\d_1^+X(v)$, where the field is pointing inside $X$ and the closed locus $\d_1^-X(v)$, where it points outside. The intersection $$\d_2X(v) =_{\mathsf{def}} \d_1^+X(v)\cap \d_1^+X(v)$$ is the locus where $v$ is *tangent* to the boundary $\d X$. Points $z \in \d_2X(v)$ come in two flavors: by definition, $z \in \d^+_2X(v)$ when $v(z)$ points inside of the locus $\d_1^+X(v)$; otherwise $z \in \d^-_2X(v)$. To achieve some uniformity of notations, put $\d_0^+X =_{\mathsf{def}} X$ and $\d_1X =_{\mathsf{def}} \d X$.
\[fig1.5\]
{height="1.2in" width="4in"}
We say that a vector field $v$ on a compact surface $X$ is *boundary generic* if:
- $v|_{\d X}$, viewed as a section of the normal $1$-dimensional (quotient) bundle $$n_1 =_{\mathsf{def}} T(X)|_{\d X} \big/ T(\d X),$$ is transversal to its zero section,
- $v|_{\d_2 X(v)}$, viewed as a section of the normal $1$-dimensional bundle $n_2 =_{\mathsf{def}} T(\d X)|_{\d X}$, is transversal to its zero section. $\diamondsuit$
In particular, for a boundary generic $v$, the loci $\d_1^\pm X(v)$ are finite unions of closed intervals and circles, residing in $\d X$; and the loci $\d_2^\pm X(v)$ are finite unions of points, residing in $\d X$ (see Fig. 4).
We denote by $\mathcal V^\dagger(X)$ the space (in the $C^\infty$-topology) of all boundary generic fields on a compact surface $X$.
Let $\chi(Z)$ denote the Euler number of a space $Z$. Recall that $\chi(Z)$ is the alternating sum of dimensions of the homology spaces $\{H_i(Z; \R)\}_i$.
Since for a connected surface $X$ with boundary $H_2(X; \R) = 0$, we get $$\chi(X) = 1 - \dim_\R(H_1(X; \R)).$$ For a closed connected surface, $$\chi(X) = 2 - \dim_\R(H_1(X; \R)).$$
Given a vector field $v$ with isolated zeros, we can associate an integer $\mathsf{ind}_x(v)$ with each zero $x$ of $v$. This integer is the *degree* of the map which, crudely speaking, takes each point $z$ on a small circle $C_x$ with its center at $x$ to the unit vector $v(z)/\|v(z)\|$. Then we define $\mathsf{Ind}(v)$, the (global) index of $v$, as the sum $\sum_{\{x \in \text{ zeros of } v\}} \mathsf{ind}_x(v)$.
The Morse formula [@Mo], in the center of our investigation, computes the index $\mathsf{Ind}(v)$ of a given boundary generic vector field $v$ on a surface $X$ as the alternating sum of the Euler numbers of the Morse strata $\{\d_j^+X(v)\}_{0 \leq j \leq 2}$: $$\begin{aligned}
\label{eq1.1}
\mathsf{Ind}(v) = \chi(X) - \chi(\d_1^+X(v)) + \chi(\d_2^+X(v)).\end{aligned}$$ In the case of a connected surface $X$ with boundary, $\chi(X) = 1 - \dim_\R(H_1(X; \R))$, and this formula reduces to $$\mathsf{Ind}(v) = 1 - \dim_\R(H_1(X; \R)) - \#\{ \mathsf{arcs}\; in \; \d_1^+X(v)\} + \#\{\d_2^+X(v)\}$$ $$= 1 - \dim_\R(H_1(X; \R)) + \frac{1}{2}\big(\#\{\d_2^+X(v)\}- \#\{\d_2^-X(v)\}\big).$$ In particular, if $v \neq 0$, then $\mathsf{Ind}(v) = 0$, and we get $$\begin{aligned}
\label{eq1.2}
\frac{1}{2}\big(\#\{\d_2^+X(v)\}- \#\{\d_2^-X(v)\}\big) = \dim_\R(H_1(X; \R)) - 1,\end{aligned}$$ where the RHS of the equation is the topological invariant $|\chi(X)| = -\chi(X)$ of $X$. In contrast, the cardinality $\#\{\d_2^+X(v)\}$ depends on $v$.
\[lem3.1\] Let a surface $X$ be formed by removing $k$ open disks from a closed surface $Y$, the sphere with $g$ handles. Then, for any boundary generic field $v \neq 0$ on $X$, $$\#\{\d_2^+X(v)\}\, \geq \, 4g - 4 + 2k.$$ Moreover, $\#\{\d_2^+X(v)\} = 4g - 4 + 2k$ only when $\#\{\d_2^-X(v)\} = 0$.
The Euler number is additive under gluing surfaces along their boundary components. Therefore, if $k$ disks are removed from $Y$, the sphere with $g$ handles, then $\chi(X) = 2-2g - k$. Thus the Morse formulas (\[eq1.1\]) and (\[eq1.2\]) imply $$\#\{\d_2^+X(v)\} \geq 4g - 4 + 2k$$ for any $v \neq 0$. Moreover, $\#\{\d_2^+X(v)\} = 4g - 4 + 2k$ if and only if $\#\{\d_2^-X(v)\} = 0$, the main feature of the *boundary concave* fields (see Definition \[def1.2\]).
In particular, for any non-vanishing boundary generic field $v$ on a torus with a single hole, $\#\{\d_2^+X(v)\} \geq 2$ (cf. Fig. 2).
Recall that an *immersion* is a smooth map of manifolds, whose differential has the trivial kernel.
Consider a smooth map $\a: X \to \R^2$, which is an immersion in the vicinity of $\d X$. Any such $\a$ gives rise to the Gauss map $G: \d X \to S^1$, defined by the formula $G(x) = \a_\ast(\tau_x)/\| \a_\ast(\tau_x)\|$, where $\tau_x$ is the tangent vector to $\d X$ at $x$. The direction of $\tau_x$ is consistent with the preferred orientation of $\d X$, induced by the preferred orientation of $X$ .
Let $\hat v \neq 0$ be a constant field on $\R^2$. Since the kernel of the differential of $D\a: TX \to T\R^2$ is trivial along $\d X$, the field $\hat v$ defines a vector field $\tilde v = \a^\ast(\hat v)$ on $X$ in the vicinity of $\d X$. The pull-back field $\tilde v$ extends to a vector field $v$ on $X$, possibly with zeros (see \[G\] for engaging discussions of vector field transfers and the Gauss-Bonnet Theorem).
Then the degree of the Gauss map is given by a classical Hopf formula ([@H]) $$\deg(G) = \chi(X) - \mathsf{Ind}(v).$$
When $\a: X \to \R^2$ is an immersion everywhere, the pull-back field $v = \a^\ast(\hat v) \neq 0$ everywhere. Thus $\mathsf{Ind}(v) = 0$, and, for a connected $X$ with $\d X \neq \emptyset$, we get $$\deg(G) = \chi(X) =_{\mathsf{def}} 1 - \dim(H_1(X; \R)).$$ So, for an immersions $\a$, we get a new interpretation of formula (\[eq1.2\]): $$\begin{aligned}
\label{eq1.3}
\deg(G) = \chi(X) = \frac{1}{2} \Big(\#\{\d_2^-X(v)\} - \#\{\d_2^+X(v)\}\Big).\end{aligned}$$ This global-to-local formula has another classical geometrical interpretation. Let $\mathsf g = \a^\ast(\mathsf g_E)$ be the Riemannian metric on $X$, the pull-back of the Euclidean metric on $\R^2$. Let $K_\nu$ denote the normal curvature of $\d X$ with respect to $\mathsf g$. Then $$\deg(G) = \frac{1}{2\pi}\int_{\d X} K_\nu\, d\mathsf g,$$ which leads to another pleasing global-to-local connection: $$\frac{1}{\pi}\int_{\d X} K_\nu\, d\mathsf g = \#\{\d_2^-X(v)\} - \#\{\d_2^+X(v)\}.$$
In particular, for a connected orientable surface $X$ of genus $g$ with a *single* boundary component, $$\begin{aligned}
\label{eq1.4}
\chi(X) = 1 - 2g = \frac{1}{2}\big(\#\{\d_2^-X(v)\} - \#\{\d_2^+X(v)\}\big).\end{aligned}$$
So the number of $v$-trajectories $\g$ in $X$ that are tangent to $\d X$, but are not singletons (they correspond to points of $\d_2^+X(v)$), as a function of genus $g$, grows at least as fast as $4g - 2$!
On the other hand, when $\d X$ is connected, by the Whitney index formula [@W], the degree of the Gauss map $G: \d X \to S^1$ can be also calculated as $\mu + N^+ - N^-$, where $N^\pm$ denotes the number of positive/negative self-intersections of the curve $\a(\d X) \subset \R^2$, and $\mu = \pm 1$. Here is a brief description of the rule by which the self-intersections acquire polarities. Let $p \in \a(\d X)$ be a point where the coordinate function $y: \R^2 \to \R$ attends its minimum on the curve $\a(\d X)$. If the tangent vector $\tau_p$ at $p$, which defines the orientation of $\a(\d X)$, is $\d_x$, then we put $\mu = +1$; if $\tau_p = - \d_x$, then $\mu = -1$. Starting at $p$ and moving in the direction of $\tau_p$, we visit each self-intersection $a$ twice and in a particular order. The first visitation defines a tangent vector $\tau_1(a)$, the second visitation defines a tangent vector $\tau_2(a)$. When the ordered pair $(\tau_1(a), \tau_2(a))$ defines the clockwise orientation of the $xy$-plane, then we attach “$-$" to $a$. Otherwise, the polarity of $a$ is “$+$".
Therefore we get a somewhat mysterious connection between the self-intersections of $\d X$ under immersions $\a: X \to \R^2$ and the tangency patterns of the flows in $X$ that are the $\a$-pull-backs of non-vanishing flows in the plane.
\[th1.2\] Let $\hat v \neq 0$ be a vector field in the plane $\R^2$. Let $X$ be a connected orientable surface with a connected boundary. Consider an immersion $\a: X \to \R^2$ such that the loop $\a(\d X)$ has transversal self-intersections only. Assume that the pull-back $v = \a^\ast(\hat v)$ is a boundary generic field on $X$. Then $$\frac{1}{2} \Big(\#\{\d_2^+X(v)\} - \#\{\d_2^-X(v)\}\Big) = N^+ - N^- \pm 1 = 2g -1,$$ $$\frac{1}{2} \Big(\#\{\d_2^+X(v)\} - \#\{\d_2^-X(v)\}\Big) + 2\, \leq N^+ + N^-,$$ the latter inequality being sharp by an appropriate choice of $\a$.
The first formula is the result of combining the Whitney formula for $\deg(G)$ with formulas (\[eq1.3\]), (\[eq1.4\]).
By a theorem of Guth [@Gu], for any immersion $\a: X \to \R^2$, the total number of self-intersections of the loop $\a(\d X)$ admits an estimate $$N^+ + N^- \geq \, 2g + 2.$$ Moreover, this lower bound is realized by an immersion $\a: X \to \R^2$! Therefore, by formula (\[eq1.4\]), the Guth inequality is transformed into $$N^+ + N^- \geq %3 - \deg(G) =
2+ \frac{1}{2}\big(\#\{\d_2^+X(v)\} - \#\{\d_2^-X(v)\}\big).$$ Moreover, for some optimal immersion $\a$, $$N^+ + N^- = 2+ \frac{1}{2}\big(\#\{\d_2^+X(v)\} - \#\{\d_2^-X(v)\}\big) = 2 - \frac{1}{2\pi}\int_{\d X} K_\nu\, d\mathsf g.$$
When a surface $X$ is oriented and a field $v$ is boundary generic, then the points from $\d_2^+X(v)$ come in two *new flavors*: “$\oplus, \ominus$". By definition, a point $a \in \d_2^+X(v)$ has the polarity “$\oplus$" if the orientation of $T_aX$ determined by the pair $(\nu_a, v(a))$, where $\nu_a$ is the inner normal to $\d X$, agrees with the preferred orientation of $X$. Otherwise, the polarity of $a$ is defined to be “$\ominus$".
Thus, for each choice of orientation of $X$ (and hence of $\d X$) we get a partition $$\d_2^+X(v) = \d_2^{+, \oplus}X(v) \coprod \d_2^{+, \ominus}X(v).$$ Switching the orientation of $X$ switches the second polarities in the partition.
Convexity, concavity, and complexity of flows in 2D
===================================================
\[def1.2\] We say that a boundary generic vector field $v$ is *boundary convex* if $\d_2^+X(v) = \emptyset$. We say that a boundary generic $v$ is *boundary concave* if $\d_2^-X(v) = \emptyset$ (see Fig 5). $\diamondsuit$
The existence of a boundary convex field puts severe restrictions on the topology of the surface.
\[lem1.2\] If a compact connected surface $X$ with boundary $\d X \neq \emptyset$ admits a boundary convex gradient-like vector field $v \neq 0$, then $X$ is either a disk $D^2$, or an annulus $A^2$.
The convexity of the field $v$ implies that $X$ admits a $(-v)$-directed continuous retraction on the locus $\d_1^+X(v)$. Since $X$ is connected, it follows that $\d_1^+X(v)$ is connected as well. Thus, $\d_1^+X(v)$ is either a circle, or a segment. In the first case, $X$ is diffeomorphic to an annulus $S^1 \times [0, 1]$; in the second case, $X$ is diffeomorphic to a disk $D^2$.
The same phenomenon occurs in any dimension: if a compact connected smooth $(n+1)$-manifold $X$ with a *connected* boundary admits a *boundary convex* gradient-like vector field $v \neq 0$, then $H_n(X; \Z) = 0$ ([@K1]). In other words, $H_n(X; \Z) \neq 0$ is a topological obstruction to the existence of a boundary convex non-vanishing gradient field on $X$.
In contrast, the boundary concave non-vanishing gradient fields are plentiful. For example, consider a radial vector field $v$ on an annulus $A^2$. Delete from $A^2$ any number of convex disks and restrict $v$ to the resulting $2$-disk with holes. The convexity of the disks that we have removed implies that any disk with holes admits a boundary concave gradient-like vector $v \neq 0$.
Many other surfaces admit such concave fields as well. For example, consider a Morse function $f: Y \to \R$ on a closed surface $Y$ and its gradient field $v$. Then removing small convex (in the local Morse coordinates) balls, centered on the critical points, from $Y$, produces a *boundary concave* non-vanishing gradient field on $X$. In particular if $Y$ is a sphere with $g$ handles, then one can find a Morse function with $2g + 2$ critical points (see Fig. 1). So the surface $X$, obtained from $Y$ by removing $2g + 2$ balls, admits a concave gradient-like field $v \neq 0$.
In fact, by Theorem \[th6.2\], any connected orientable surface with boundary, but the disk, admits a boundary concave non-vanishing gradient field!
We view the integer $c^+(v) =_{\mathsf{def}} \#(\d_2^+X(v))$ as a *measure of complexity* of the $v$-flow, subject to the condition $\mathsf{Ind}(v) = 0$ or, alternatively, subject to the condition $v \neq 0$ .
We define the *complexity* of a compact connected surface $X$ with boundary as the minimum $$c^+(X) = \min_{v \neq 0} \{c^+_2(v)\},$$ where $v$ runs over all non-vanishing boundary generic fields on $X$.
By varying $v$ within different spaces of fields, one may consider a variety of such minima; non-vanishing fields and non-vanishing gradient-like fields are the two most important cases. So we introduce the *gradient complexity* $$gc^+(X) =_{\mathsf{def}} \min_{v \neq 0\; \text{of the gradient type}} \{c^+_2(v)\},$$ where $v$ runs over all non-vanishing gradient-like fields on $X$.
Evidently $gc^+(X) \geq c^+(X)$. Let $M^\circ$ denote the Möbius band. In Section 6, we will show that $gc^+(M^\circ) = 1$, while $c^+(M^\circ) = 0$, so the two notions of complexity are different.
In terms of this complexity, we can restate the Lemma 3.1 as follows.
\[cor 4.1\] Let $X$ be a connected compact surface with boundary. Let $v$ be a boundary generic vector field on $X$, subject to the condition $\mathsf{Ind}(v) = 0$.
Then the complexity of $v$ satisfies the inequality $$c^+(v) \geq 2\cdot \dim_\R H_1(X; \R) - 2 = -2\cdot \chi(X).$$ When $\chi(X) \leq 0$, this inequality turns into the equality $c^+(v) = -2\cdot \chi(X)$ if and only if $v$ is boundary concave.
As a result, for any natural $N$, there are finitely many connected compact surfaces of bounded complexity $c^+(X) \leq N$. In fact, the number of such surfaces (counted up to a homeomorphism) grows as a quadratic function in $N$. $\diamondsuit$
[**Example 4.1.**]{} For any non-vanishing boundary concave field $v$ on the torus with a single hole, $\#\{\d_2^+X(v)\} = 2$. In fact, the constant field $v$, being restricted to the complement to a convex disk in $T^2$, is boundary concave and has the property $\#\{\d_2^+X(v)\} = 2$. Thus, by Corollary 4.1, $c^+(X) = 2$. $\diamondsuit$
Lemma \[lem3.1\] leads immediately to
\[cor4.2\] Let $X$ be a sphere with $g$ handles and $k$ holes, where $g, k \geq 1$. If $X$ admits a non-vanishing boundary concave field $v$, then $\#\{\d_2^+X(v)\} = 4g - 4 + 2k$. $\diamondsuit$
Given a compact surface $X$ with boundary, we form its *double* $DX =_{\mathsf{def}} X \cup_{\d X} X$ by attaching two copies of $X$ along their boundaries. Note that $\chi(DX) = 2\cdot \chi(X)$. Therefore, $\chi(X) < 0$ if and only if $\chi(DX) < 0$.
Recall that any closed orientable surface with a negative Euler number admits a metric of constant negative curvature $-1$. So if $\chi(X) < 0$, then $DX$ admits such hyperbolic metric.
Let $vol(DX)$ denote the hyperbolic volume of $DX$, and let $vol(\D^2)$ denote the volume of an ideal hyperbolic triangle $\D^2$ in the hyperbolic plane $\mathbf H^2$.
In $2D$, a remarkable convergence of topology and geometry takes place. In the spirit of this convergence, since $\chi(DX) = -vol(DX)/ vol(\D^2)$, Corollary 4.1 admits a more geometric reformulation:
\[th1.3\] Let $v \neq 0$ be a boundary generic vector field on a compact connected and orientable surface $X$ with boundary. Assume that $\chi(X) < 0$[^2]. Then the complexity of the $v$-flow satisfies the inequality: $$c^+(v)\; \geq \; vol(DX)/ vol(\D^2).$$ Moreover, $c^+(v) = vol(DX)/ vol(\D^2)$ if and only if $v$ is boundary concave. $\diamondsuit$
Theorem \[th1.3\] admits far reaching multidimensional generalizations (see [@AK], [@K5]). They are valid for so called *traversally generic* vector fields (see Definitions 5.1 and 5.2 and [@K2]) on arbitrary smooth compact $(n+1)$-dimensional manifolds $X$ with boundary. Such fields $v$ naturally generate stratifications of trajectory spaces $\mathcal T(v)$, whose strata are labeled by the combinatorial patterns of tangency from the universal partially ordered set $\Omega^\bullet_{'\langle n]}$ (see the end of Section 6 and [@K3]). In high dimensions, we use the simplicial semi-norms $\|\sim\|_\D$ of Gromov [@Gr] on the homology $H_\ast(X; \R)$ and $H_\ast(DX; \R)$ (as a substitute of the hyperbolic volume) to provide lower bounds on the number of connected components of the $\Omega^\bullet_{'\langle n]}$-strata of any given dimension.
On spaces of vector fields
==========================
We say that a vector field $v \neq 0$ on a compact surface $X$ is *traversing* if all its trajectories are closed segments or singletons[^3]. $\diamondsuit$
Each trajectory $\g$ of a traversing field $v$ must reach the boundary both in positive and negative times: otherwise $\g$ is not homeomorphic to a closed interval.
We denote by $\mathcal V_{\mathsf{trav}}(X)$ the space (in the $C^\infty$-topology) of all traversing fields on $X$.
We denote by $\mathcal V_{\mathsf{grad}}(X)$ the space (in the $C^\infty$-topology) of all gradient-like fields on a given compact surface $X$ and by $\mathcal V_{\neq 0}(X)$ the space of all non-vanishing fields on $X$.
The next lemma says that $v$ is traversal if and only if it is non-vanishing and of a gradient type (see [@K1] for the proof).
\[5.1\] For any compact connected surface $X$ with boundary, $$\mathcal V_{\mathsf{trav}}(X) = \mathcal V_{\mathsf{grad}}(X) \cap \mathcal V_{\neq 0}(X).$$ $\diamondsuit$
The surfaces $X$ and vector fields $v$ we consider are all smooth. We can add an external collar to $X$ to form a diffeomorphic surface $\hat X \supset X$ and to extend $v$ to a smooth field $\hat v$ on $\hat X$. Let $\hat \g$ be a $\hat v$-trajectory (or rather its germ) through a point $x$ of $\d X$. We can talk about *order of tangency* of two smooth curves, $\hat\g$ and $\d X$, at $x \in \hat\g \cap\d X$ in $\hat X$ (see Definition \[def1.5\]). We say that the tangency of $\hat\g$ to $\d X$ is *simple* if its degree is 2. When the two curves are transversal at $x$ we say that the order of tangency is $1$. In fact, this notions depend only on $(X, v)$ and not on the extension $(\hat X, \hat v)$.
\[def1.4\] A traversing vector field $v$ on a compact surface $X$ is called *traversally generic*, if two properties are valid: [**(1)**]{} if a trajectory $\g$ is tangent to the boundary $\d X$, then the tangency is simple, and [**(2)**]{} no $v$-trajectory $\g$ contains more then one simple point of tangency to $\d X$.[^4] $\diamondsuit$
We denote by $\mathcal V^\ddagger(X)$ the space of all traversally generic vector fields on a compact surface $X$. In fact, the notion of traversally generic field is available in any dimension (see [@K2]).
As the name suggests, the traversally generic fields are typical among all traversing fields; furthermore, a perturbation of any traversally generic field is traversally generic. This is the content of the next theorem. Its validation requires an involved argument, which even in $2D$ resists a significant simplification [@K2].
\[th1.4\] For any compact connected surface $X$ with boundary, the space $\mathcal V^\ddagger(X)$ traversally generic fields is open and dense in the space $\mathcal V_{\mathsf{trav}}(X) = \mathcal V_{\mathsf{grad}}(X) \cap \mathcal V_{\neq 0}(X)$. $\diamondsuit$
Graph-theoretical approach to the concavity of traversing fields in 2D
======================================================================
We start with a couple of very natural questions.
[**Question 6.1.**]{} Which compact connected surfaces with boundary admit boundary concave gradient-like vector fields $v \neq 0$? $\diamondsuit$
Recall that $c^+(X) \leq gc^+(X)$.
[**Question 6.2.**]{} Are there compact connected surfaces $X$ with boundary for which $c^+(X) < gc^+(X)$? $\diamondsuit$
On many occasions we took advantage of the fact that, for traversally generic vector fields $v$, the trajectory spaces $\mathcal T(v)$ are finite graph whose verticies have valency $1$ and $3$ only (see Fig. 4). Moreover, for a traversally generic boundary concave field $v$, all the verticies of $\mathcal T(v)$ have valency $3$. Now we will take a closer look at the graph-theoretical models of the boundary concave and traversally generic fields in 2D.
Let $G$ be a finite connected trivalent graph with $a$ verticies. We denote by $\b G$ its barycentric subdivision: each edge $e$ of $G$ is divided by a new vertex $v_e$, its center. We consider the finite set $\mathsf{Tri}(G)$ of all *colorings* of the edges of $\b G$ with *tree* colors so that, at each vertex of $G$, exactly three distinct colors are applied. Thus, $\#\mathsf{Tri}(G) = 6^a$.
\[th1.5\] Let $G$ be a finite connected trivalent graph. Each coloring $\a \in \mathsf{Tri}(G)$ produces (in a canonical way) a compact connected surface $X(G, \a)$ with boundary. The surface $X(G, \a)$ admits a traversally generic *concave* vector field $v(G, \a)$. The cardinality of the locus $\d_2^+X(G, \a)\big(v(G, \a)\big)$ is the number of verticies in $G$.
Moreover, every connected surface with boundary, which admits a traversally generic concave vector field, can be produced in this way.
\[fig1.6\]
{height="3.5in" width="4in"}
Let $\mathsf A, \mathsf B, \mathsf C$ denote the three distinct colors, and $\mathcal P = \{\mathsf A, \mathsf B, \mathsf C\}$ the entire pallet.
Consider a $2$-dimensional space $Z = G \times (0, 4)$. It has singularities in the form of binders of the-page open books (see Fig.1.6 ). The binders correspond to the verticies of $G$.
First, employing a given coloring $\a$, we will construct a piecewise linear surface $\hat X(G, \a) \subset Z$. The vector field on $\hat X(G, \a)$ will be induced by the product structure in $Z$.
For each edge $e \subset G$ and its barycenter $v_e \in \b G$, we place the interval $v_e \times [2, 3]\subset Z$ over $v_e$. Let $\hat e$ be half of the interval $e \subset G$, bounded two vericies $v_e \in \b G$ and $w \in G$. Over $\hat e$, we place a strip $E \subset Z$; its construction depends on the color attached to the interval $[v_e, w]$ as follows:
- if the color of $[v_e w]$ is $\mathsf A$, then we link the vertex $v_e \times 2$ with the vertex $w \times 1$ by a line in the rectangle $R = [v_e, w] \times (0, 4)$, and the vertex $v_e \times 3$ with the vertex $w \times 2$ by another line in $R$;
- if the color of $[v_e, w]$ is $\mathsf B$, then we link by a line in $R$ the vertex $ v_e \times 2$ with the vertex $w \times 2$, and the vertex $v_e \times 3$ with the vertex $w \times 3$ by another line;
- if the color of $[v_e, w]$ is $\mathsf C$, then we link by a line in $R$ the vertex $v_e \times 2$ with the vertex $w \times 1$, and the vertex $v_e \times 3$ with the vertex $3 \times w$ by another line.
By definition, $E(e, w)$ is the strip in $[v_e, w] \times (0, 4)$, bounded by the two lines whose construction is has been described above. Thanks to the monotonicity of the bijections $A: \{2, 3\} \to \{1, 2, 3\}$, $B: \{2, 3\} \to \{1, 2, 3\}$, and $C: \{2, 3\} \to \{1, 2, 3\}$ that correspond to the colors $\mathsf A, \mathsf B, \mathsf C$, the lines that bound the strips $E(e, w)$ do not intersect. We denote by $\hat X(G, \a)$ the union of all such strips.
The local model of each binder implies that indeed $\hat X(G, \a)$ is piecewise linear surface, imbedded in the singular space $Z$. Inside $Z$, one can smoothen the sharp edges of the boundary $\d \hat X(G, \a)$ in order to get a smooth surface $X(G, \a)$ (can you visualize this smoothing in the vicinity of point $w \times 2$ from Fig. 6?). The restriction of the product structure in $Z$ to its subspace $X(G, \a)$ produces a smooth non-vanishing vector field $v(G, \a)$ on $X(G, \a)$. Its trajectories (the vertical lines in $Z$) will be simply tangent to $\d X(G, \a)$ exactly at the points of the type $2 \times w$, where $w$ runs over the set of verticies of $G$. By Theorem \[th1.3\], this field $v(G, \a)$ is of the gradient type.
Conversely, any traversally generic and concave vector field $v$ on a connected compact surface $X$ with boundary, produces a map $\Gamma: X \to \mathcal T(v)$, where the space of trajectories is a finite trivalent graph. Its verticies are in 1-to-1 correspondence with the points of the locus $\d_2^+X(v)$.
As a point $v_e$ in the open edge $e$ of the graph $\mathcal T(v)$ approaches a vertex $w$, the intersection of the $v$-trajectory $\g = \Gamma^{-1}(v_e)$ with the boundary $\d X$ defines a bijection of the $v$-ordered set $\g \cap \d X$ of cardinality $2$ to a $v$-ordered set $\Gamma^{-1}(w) \cap \d X$ of cardinality $3$, the orders being respected by the bijections. This determines one of three colors we attach to the half-edge $[v_e, w]$. Therefore the geometry of the flow determines a tricoloring of the graph $\b \mathcal T(v)$.
The next theorem answers Questions 6.1 and 6.2.
\[th6.2\]
- For any orientable connected surface $X$ with boundary, the two complexities are equal: $gc^+(X) = c^+(X)$. Moreover, any such $X$, but the disk, admits a boundary concave traversally generic vector field. As a result, for any orientable connected $X$ with boundary, but the disk, the two complexities are equal to $-2\chi(X)$.
- For any non-orientable connected surface $X$ with boundary, which is a boundary connected sum of several punctured Klein bottles and annuli, $gc^+(X) = c^+(X) = -2\chi(X)$ as well. Again, any such $X$ admits a boundary concave traversally generic vector field.
- In contrast, the Möbius band $M^\circ$ does not admit a boundary concave traversally generic vector field. In fact, $c^+(M^\circ) = 0$ and $gc^+(M^\circ) = 1$.
- Moreover, $c^+(M^\circ \#_\d X) \leq c^+(X) + 2$ and $gc^+(M^\circ \#_\d X) \leq gc^+(X) + 3$ for any $X$ as in the first two bullets[^5].
Consider the boundary connected sum $X_1 \#_\d X_2$ of two compact surfaces $X_1$ and $X_2$. The Euler number of the sum satisfies the rule $$\chi(X_1 \#_\d X_2) = \chi(X_1) + \chi(X_2) - 1.$$ On the other hand, given two boundary generic fields $v_1$ and $v_2$, there exists a traversally generic field $w$ on $X_1 \#_\d X_2$ such that $$|\d_2^+(X_1 \#_\d X_2)(w)| = |\d_2^+(X_1)(v_1)| + |\d_2^+(X_2)(v_2)| + 2.$$ Indeed, we may attach a $1$-handle $H$ to $\d_1^-X_1(v_1) \coprod \d_1^+X_2(v_2)$ so that an $H$ has a neck with respect to the extension $w$. Such field $w$ contributes two points to $\d_2^+(X_1 \#_\d X_2)(w)$. Of course, this construction fails when $\d_1^-X_1(v_1) \coprod \d_1^+X_2(v_2) = \emptyset$; however, for traversing fields $v$, both loci $\d_1^\pm X(v) \neq \emptyset$. By Corollary 4.1, if $X$ admits a boundary concave field, then $c^+(X) = -2 \chi(X)$, provided $\chi(X) \leq 0$. In particular, if $X$ with a non-positive Euler number admits a *boundary concave* traversally generic $v$, then $$gc^+(X) = c^+(X) = -2 \chi(X).$$
Let $v_1$ and $v_2$ be some boundary generic/ traversally generic fields which deliver the two gradient complexities. The previous arguments about extending $v_1$ and $v_2$ across the handle $H$ imply that if $gc^+(X_1) = -2\chi(X_1)$ and $gc^+(X_2) = -2\chi(X_2)$ (say both surfaces admit boundary concave and traversally generic fields), then $$gc^+(X_1 \#_\d X_2) \leq gc^+(X_1) + gc^+(X_2) +2 = -2\chi(X_1 \#_\d X_2),$$ provided that $\chi(X_1 \#_\d X_2) \leq 0$. Since the reverse inequality holds by Corollary 4.1, we get $$gc^+(X_1 \#_\d X_2) = -2 \chi(X_1 \#_\d X_2)$$ when $gc^+(X_1) = -2\chi(X_1)$ and $gc^+(X_2) = -2\chi(X_2)$.
Recall the topological classification of closed connected surfaces. Any such surface is either a sphere, or a connected sum of several tori (the orientable case), or a connected sum of several projective spaces (the non-orientable case). Therefore any connected surface with boundary is obtained from the surfaces in this list by deleting at least one disk.
Let $T^\circ$ denote the complement to an open disk in a $2$-torus, and $M^\circ$ denote the complement to an open disk in a projective plane—the Möbius band—, and let $A$ denote the annulus. Thus any connected surface with boundary is either a disk $D$, or a boundary connected sum of several copies of punctured tori $T^\circ$ and annuli $A$ (the orientable case), or a boundary connected sum of several copies of Möbius bands $M^\circ$ and annuli $A$ (the non-orientable case).
Let us now compute the complexities of the basic blocks in this decomposition. Note that $c^+(D) = 0 = gc^+(D)$, since $D$ admits a convex traversing flow. Also $c^+(A) = 0 = gc^+(A)$, the latter equality being delivered by the radial gradient field.
We claim that $c^+(T^\circ) = 2 = gc^+(T^\circ)$. Indeed, since $\chi(T^\circ) = -1$, by Corollary 4.1, we get $c^+(T^\circ) \geq 2$. On the other hand, there exists a trivalent graph $G_T$ with an appropriate tricoloring and exactly *two* verticies such that, applying the construction from Theorem \[th1.5\], we produce a traversally generic field $v(G_T, \a)$ on the surface $X(G_T, \a) = T^\circ$ with the cardinality $2$ locus $\d_2^+X(G_T, \a)(v(G_T, \a))$. As a result, both complexities of $T^\circ$ equal to $2$.
Similar considerations apply to the punctured Klein bottle $K^\circ = M^\circ \#_\d M^\circ$ and a different trivalent graph $G_K$ with two verticies and an appropriate tricoloring. Since $\chi(K^\circ) = -1$, we conclude that $c^+(K^\circ) = 2 = gc^+(K^\circ)$.
The third trivalent graph $G_A$ with two verticies and an appropriate tricoloring delivers a traversally generic boundary concave flow on a punctured annulus $A^\circ$, the disk with two holes. Thus, $c^+(A^\circ) = 2 = gc^+(A^\circ)$.
In fact, Theorem \[th1.5\] implies that $T^\circ, K^\circ, A^\circ$ are the only connected surfaces of the gradient complexity $2$ that admit concave traversally generic fields. Indeed, just start with the tree “$>\bullet-\bullet<$" with two trivalent verticies and consider the ways one can identify its four leaves in pairs. Then consider all admissible tricologings of the resulting graphs $G$. This cases will deliver the three model tricolored graphs $G_T, G_K, G_A$.
Now the “quasi-additivity" of Euler numbers and gradient complexities under the connected sum operations imply that the gradient complexity of boundary connected sums $$X = (T^\circ \#_\d \dots \#_\d T^\circ) \#_\d (A \#_\d \dots \#_\d A) \#_\d(K^\circ \#_\d \dots \#_\d K^\circ)$$ of several copies of the model surfaces $T^\circ, K^\circ, A$ is equal to $2|\chi(X)|$. Indeed, these properties imply that $gc^+(X) \leq 2 \cdot |\chi(X)|$, while in general $gc^+(X) \geq 2 \cdot |\chi(X)|$. Moreover, every such surface $X$ admits a *boundary concave* traversally generic field (by the $1$-handle-with-a-neck argument), since the basic blocks $T^\circ, K^\circ$ and $A$ do.
The Möbius band $M^\circ$ is different. We notice that $M^\circ$ admits a non-vanishing vector field $v$ with a single closed trajectory—the core of the Möbius band—and transversal to the boundary $\d M^\circ$. Thus, $c^+(M^\circ) = 0$. Now consider a trivalent graph $G_M$ with a single vertex of valency $3$ and a single vertex of valency $1$ (this $G_M$ is a circle to which a radius is attached). The construction from Theorem \[th1.5\] applies to produce a remarkable embedding of the Möbius band in the product $G_M \times [0, 4]$. So we conclude that $M^\circ$ admits a traversally generic field $v$ (not concave!) with $\d_2^+X(v)$ being a singleton ($\d_2^-X(v)$ is a singleton as well). As a result, $gc^+(M^\circ) \leq 1$. On the other hand, any traversally generic field $v$ on $M^\circ$ must produce the graph $\mathcal T(v)$ which is homotopy equivalent to a circle, the homotopy type of $M^\circ$. If $gc^+(v) = 0$, this graph $\mathcal T(v)$ has no trivalent verticies, in which case, $\mathcal T(v)$ is homeomorphic to a circle. So $M^\circ \to \mathcal T(v)$ must be a fibration whose fibers (the $v$-trajectories) are segments. Moreover, thanks to the field $v$, this fibration is orientable, a contradiction with the non-orientability of $M^\circ$. Therefore, we conclude that $gc^+(M^\circ) = 1$, while $c^+(M^\circ) = 0$.
Finally, for any $X$ which is a boundary connected sum of $T^\circ$’s, $K^\circ$’s, and $A$’s, by the same arguments, the inequalities $c^+(M^\circ \#_\d X) \leq -2\chi(X) + 2$ and $gc^+(M^\circ \#_\d X) \leq -2\chi(X) + 3$ hold. This validates the claim in the last bullet.
Combinatorics of tangency for traversing flows in 2D
====================================================
Pick an extension $\hat X$ of a given compact surface $X$ by adding an external collar to $X$. Let $\hat v$ be an extension of a given field $v$ into $\hat X$. Pick a smooth auxiliary function $z: \hat X \to \R$ such that:
- $0$ is a regular value of $z$,
- $z^{-1}(0) = \d X$,
- $z^{-1}((-\infty, 0]) = X$,
$$\begin{aligned}
\label{eq1.6}\end{aligned}$$
\[def1.5\] Let $\hat\g$ be a $\hat v$-trajectory through a point $x \in \d X$. We say that $\hat\g$ has the *order/multiplicity of tangency* $k$ to $\d X$ at $x$, if $\mathcal L_{\hat v}^{\{j\}} (z) = 0$ for all $j < k$, and $\mathcal L_{\hat v}^{\{k\}} (z) \neq 0$ at $x$ [^6]. Here $\mathcal L_{\hat v}^{\{j\}} (z)$ denotes the $j^{th}$ iterated $\hat v$-directional derivative of the function $z$. $\diamondsuit$
Given a traversally generic vector field $v$ on a compact connected surface $X$, we will attach the *combinatorial pattern* $(1,1)$ to a typical $v$-trajectory $\g \subset X$ that corresponds to the edges of the graph $\mathcal T(v)$, the pattern $(121)$ to the trajectories that correspond to the trivalent verticies of $\mathcal T(v)$, and the pattern $(2)$ to the univalent verticies (see Fig. 4). In fact, the numbers 1 and 2 in these patterns reflect the order of tangency of the curves $\hat\g$ and $\d X$ at the points of $\g \cap \d X$ (see Definition \[def1.5\]). On a given compact surface $X$, for *traversally generic* fields $v$ no other patterns (say, like $(1221)$ or $(13)$) occur. In $2D$, this conclusion follows from Definition \[def1.5\].
The lemma below is another way to state this fact. Its proof, relying on the Malgrange Preparation Theorem [@Mal], can be found in [@K2].
Let $v$ be a traversally generic field on $X$. Extend $(X, v)$ to a pair $(\hat X, \hat v)$. In the vicinity of each $v$-trajectory $\g$, there exist special local coordinates $(u, x)$ in $\hat X$ and a real polynomial $P(u, x)$ of degree $2$ or $4$ such that:
- each $\hat v$-trajectory is given by the equation $\{x = const\}$,
- the boundary $\d X$ is given by the polynomial equation $\{P(u, x) = 0\}$,
- $X$ is given by the polynomial inequality $\{P(u, x) \leq 0\}$.
The polynomial $P(u, x)$ takes three canonical forms:
1. $u(u -1)$, which corresponds to the combinatorial pattern $\mathbf{(11)}$,
2. $u^2 - x,$ which corresponds to the combinatorial pattern $\mathbf{(2)}$,
3. $u\big((u -1)^2 + x\big)(u -2),$ which corresponds to the pattern $\mathbf{(121)}$. $\diamondsuit$
To summarize, at $\d_2X(v)$ the order of tangency is $2$; the trajectories through $\d_2^+X(v)$ have the combinatorial tangency pattern $(121)$, and through $\d_2^-X(v)$ the combinatorial tangency pattern $(2)$. The rest of trajectories have the pattern $(11)$.
We denote by $\Omega^\bullet_{'\langle 1]}$ the partially ordered set whose elements are $(11), (2), (121)$ and the order is defined by $(11) \succ (2)$ and $(11) \succ (121)$. This combinatorics does not look impressive. However, in higher dimensions, traversally generic fields on $(n+1)$-manifolds with boundary generate a rich and interesting partially ordered finite list $\Omega^\bullet_{'\langle n]}$ of combinatorial tangency patters. The poset $\Omega^\bullet_{'\langle n]}$ is *universal* in each dimension $n+1$. They are discussed in [@K3].
Holography of traversing flows on surfaces
==========================================
Let $v$ be a traversing and boundary generic vector field on a compact connected surface $X$ with boundary. For any point $z \in \d_1^+X(v)$, consider the closest point $w(z) \in \d_1^-X(v)$ that can be reached by moving along the trajectory $\g_z$ through $z$ in the direction of $v$ (see Fig. 7). Note that $w(z) = z$ if and only if $z \in \d_2^-X(v)$.
The correspondence $z \to w(z)$ defines a map $$C_v: \d_1^+X(v) \to \d_1^-X(v)$$ which we call the *causality map*. It is a distant relative of the classical Poincaré Return Map.
Alternatively, one can think of $C_v$ as determining a *partial order* “$z \prec w(z)$" among the points of the boundary $\d X$.
The word “causality" in the name of $C_v$ is motivated by the following pivotal special case.
\[fig1.7\]
{height="2.7in" width="4in"}
[**Example 8.1.** ]{} Let $w = w(\theta, t)$ be a smooth time-dependent vector field on the circle $S^1$ (equipped with the angular coordinate $\theta$). It gives rise to a vector field $v = (w, 1)$ on the cylinder $S^1 \times \R$. We think about the factor $S^1$ as *space* and about the factor $\R$ as *time* $t$. So we call $S^1 \times \R$ *the space of events*. Note that $v \neq 0$ is a gradient-like field with respect to the time function $T: S^1 \times \R \to \R$.
Pick any smooth compact and connected surface $X \subset S^1 \times \R$. Such a surface has a boundary $\d X$. We call $X$ the *event domain*, and its boundary $\d X$ the *event horizon*.
Since the field $v \neq 0$ is traversing in $X$, the map $C_v$ is well-defined. Then the map $C_v: \d_1^+X(v) \to \d_1^-X(v)$ indeed gives rise the *causality relation* on the event horizon: the correspondence $C_v$ reflects the *evolution* of an event $z$ into the event $C_v(z)$. $\diamondsuit$
Let $\mathcal C(\d_2^+X(v))$ denotes the union of $v$-trajectories through the points of the concavity locus $\d_2^+X(v)$.
The causality map is *discontinuous* at the points of the intersection $\mathcal C(\d_2^+X(v)) \cap \d_1^+X(v)$ (see Fig. 7). On the positive side, the discontinuities of the causality map $C_v$ are not too bad: in a sense, the map has “left" and “right" limits.
Given a pair $(X, v)$, the $v$-trajectories, viewed as *un*parametrized $v$-oriented curves, produce an oriented $1$-dimensional foliation $\mathcal F(v)$ on $X$.
\[th7.1\][**(The Causal Holography Principle in 2D).**]{}
Let $(X_1, v_1)$ and $(X_2, v_2)$ be two compact connected surfaces with boundaries, carrying traversally generic vector fields $v_1$ and $v_2$, respectively. Assume that there is a diffeomorphism $\Phi^\d : \d X_1 \to \d X_2$ which conjugates the two causality maps: $$C_{v_2} \circ \Phi^\d = \Phi^\d \circ C_{v_1}.$$
Then $\Phi^\d$ extends to a diffeomorphism $\Phi: X_1 \to X_2$ which maps the oriented foliation $\mathcal F(v_1)$ to the oriented foliation $\mathcal F(v_2)$.
We will only sketch the argument. A fully developed proof of the multidimensional analogue of this theorem is contained in [@K4].
First, we notice that since $C_{v_1}$ and $C_{v_2}$ are $\Phi^\d$-conjugate, the diffeomorphism $\Phi^\d$ induces a well-defined continuous map $\Phi_{\mathcal T}: \mathcal T(v_1) \to \mathcal T(v_2)$ of the trajectory spaces. Moreover, $\Phi_{\mathcal T}$ preserves the stratifications of the two trajectory spaces/graphs by the combinatorial type of trajectories. That is, the trivalent verticies of $\mathcal T(v_1)$ are mapped to the trivalent verticies of $\mathcal T(v_2)$, the univalent verticies are mapped to univalent verticies, and the interior of the edges to the interior of the edges.
Then we pick a smooth function $f_2: X_2 \to \R$ such that $df_2(v_2) > 0$. With the help of $\Phi^\d$, we pull-back $f_2|_{\d X_2}$ to get a smooth function $f_1^\d: \d X_1 \to \R$ such that $$f_1^\d(z) < f_1^\d(C_{v_1}(z))$$ for all $z \in \d_1^+X_1(v_1)$.
Then we argue that $f_1^\d$ extends to a smooth function $f_1$ such that $df_1(v_1) > 0$.
We use $f_1$ to embed $X_1$ in the product $\mathcal T(v_1) \times \R$ by the formula $$\a_{(v_1, f_1)}(z) = (\g_z, f_1(z)),$$ where $\g_z$, the $v_1$-trajectory through $z$, is viewed as the point $\Gamma_1(z)$ of the graph $\mathcal T(v_1)$. Similarly, we use $f_2$ to embed the surface $X_2$ in the product $\mathcal T(v_2) \times \R$ with the help of the map $\a_{(v_2, f_2)}$.
Finally, we employ $\Phi_{\mathcal T}$, $f_1$ and $f_2$ to construct a map $$\hat \Phi: \mathcal T(v_1) \times \R \to \mathcal T(v_2) \times \R$$ by the formula $$\hat\Phi(\g, t) = \big(\Phi_{\mathcal T}(\g),\, f_1(f_2^{-1}(t))\big),$$ where $t$ belongs to the $f_2$-image of the trajectory $\Gamma_2^{-1}\big(\Phi_{\mathcal T}(\g)\big)$.
Crudely, the restriction of $\hat\Phi$ to $\a_{v_1, f_1}(X_1) \subset \mathcal T(v_1) \times \R$ is the desired diffeomorphism $\Phi: X_1 \to X_2$.
Note that, in general, the pull-back $\Phi^\ast(f_2)$ is not $f_1$; so the parametrizations of the trajectories are not respected by the diffeomorphism $\Phi$, but the $1$-foliations $\mathcal F(v_1)$ and $\mathcal F(v_2)$ are.
\[cor7.1\] Let $X$ be a compact connected surface with boundary, and $v$ a smooth traversally generic vector field on it.
Then the knowledge of the causality map $C_v: \d_1^+X(v) \to \d_1^-X(v)$ is sufficient for a reconstruction of the pair $(X, \mathcal F(v))$, up to a diffeomorphism that is constant on $\d X$. $\diamondsuit$
The world “holography" is present in the name of Theorem \[th7.1\] since the surface $X$ and the $2D$-dynamics of the $v$-flow in it are recorded on two $1$-dimensional screens, $\d_1^+X(v)$ and $\d_1^-X(v)$.
Theorem \[th7.1\] and Corollary \[cor7.1\] are valid in any dimension ([@K4]).
[**Example 8.2.**]{} Let $v$ be a traversally generic field on a connected surface $X$ whose boundary $\d X$ is a *single* loop. Then the boundary $\d X$ is divided into $q$ disjoint arcs $a_1, \dots , a_q$ that form $\d_1^+X(v)$ and $q$ complementary arcs $b_1, \dots , b_q$ that form $\d_1^-X(v)$. The causality map $$C_v: \coprod_{i=1}^q a_i \to \coprod_{i=1}^q b_i$$ can be represented by its graph $G(C_v) \subset \prod_{i, j} a_i \times b_j$.
The map $C_v$ (the curve $G(C_v)$) is discontinuous at exactly $c^+(v)$ points in $\coprod_{i=1}^q a_i $ that correspond to the points of the intersection $\mathcal C(\d_2^+X(v)) \cap \d_1^+X(v)$. There the map $C_v$ has distinct left and right limits.
According to the Corollary \[cor7.1\], the curve $G(C_v) \subset \prod_{i, j} a_i \times b_j$ determines $X$ and the *un*-parametrized dynamic of the $v$-flow, up to a diffeomorphism $\Phi: X \to X$ that is the identity on $\d X$. Note that the number $q$ alone is not sufficient even to determine the genus of the surface $X$. $\diamondsuit$
Revisiting Example 8.2, we get the following interpretation of Corollary \[cor7.1\]:
\[cor7.2\] For any smooth time-dependent vector field $w$ on the circle $S^1$, the causality relation on the event horizon $\d X$ is sufficient for a reconstruction of the event domain $X$ and the un-parametrized dynamics of the $(w, 1)$-flow, up to a diffeomorphism of $X$ that is the identity on $\d X$. $\diamondsuit$
The theory of billiards on Riemmanian surfaces $X$ with boundary benefits from applying the $3D$-version of the Causal Holography to the geodesic flow on the $3$-fold $SX$, the space of unit tangent vectors on $X$. See [@K4] for some of these applications. In addition to geodesic billiards, they include the classic inverse geodesic scattering problems.
Convex quasi-envelops and characteristic classes of traversing flows on orientable surfaces
===========================================================================================
Traversing flows have interesting *characteristic classes*—elements of certain cohomology—associated with them. In dimension two, they are quite primitive, but for high-dimensional flows, surprisingly rich (see [@K6]).
We have seen that the traversally generic flows exhibit a very particular combinatorial patterns of tangency to the boundary $\d X$. In particular, for generic $2D$-flows, no tangencies of orders $\geq 3$ occur.
There is a nice link between this behavior and the *spaces of smooth functions* $f: \R \to \R$ or even polynomials that have no zeros of multiplicities $\geq 3$. To explain the connection, we will need the following definition/construction.
Let $\mathcal F$ denote the space (in the $C^\infty$-topology) of smooths functions $f: \R \to \R$ which are identically $1$ outside of a compact set. Let $\mathcal F_{\leq 2}$ be its subspace, formed by functions that have zeros only of multiplicity $\leq 2$.
Such spaces of functions with “*moderate singularities*" have been studied in depth by V. I. Arnold [@Ar] and V. A. Vassiliev [@V]. In $2D$, we employ just a tiny portion of their results. The main theorem of Arnold-Vassiliev describes the weak homotopy/homology types of the spaces $\mathcal F_{\leq k}$ for all $k \geq 2$. In particular, the homology of the space $\mathcal F_{\leq 2}$ is isomorphic to the homology of $\Omega S^2$, the space of loops on a $2$-sphere ([@V])! Arnold proved also that the fundamental group $\pi_1(\mathcal F_{\leq 2}) \approx \Z$ [@Ar].
For an even non-negative integer $d$, we will also explore the subspaces $\mathcal F^d_{\leq 2} \subset \mathcal F_{\leq 2}$, formed by functions whose *degree*—the sum of multiplicities of all its zeros—is even and does not exceed $d$.
Let $\hat v$ be a boundary convex traversing vector field on an annulus $A$. With the help of $\hat v$, we can introduce a product structure $A \approx S^1 \times [0, 1]$ so that the fibers of the projection $A \to S^1$ are the $\hat v$-trajectories.
\[def1.6\] Consider a collection $L$ of several smooth immersed loops in the annulus $A$ which intersect and self-intersect transversally and do not have triple intersections.
We say that a boundary convex traversing vector field $\hat v$ is *generic with respect to* $L$, if no $\hat v$-trajectory $\g$ contains more than one point of self-intersection from $L$ and no more than one point of simple tangency to $L$, but not both. $\diamondsuit$
For a given $L$, by standard techniques of the singularity theory, we can find a perturbation of $\hat v$ within the space $\mathcal V_{\mathsf{trav}}(A)$ so that the perturbed field is generic with respect to $L$.
Since an *immersion* is a smooth map of manifolds, whose differential has the trivial kernel, the immersions allow for a *transfer* of a given vector field on the target manifold to a vector field in the source manifold. The transfer of a non-vanishing field is a non-vanishing field.
All surfaces in this section are orientable. Note that any orientable surface $X$ admits an *immersion* $\a: X \to A$ (or even in the plane $\R^2$) (see Fig. 8). We will use this fact to pull-back non-vanishing fields on the target space $A$ to $X$.
\[def1.7\] Consider an *immersion* $\a: X \to A$ of a given compact orientable surface $X$ into an annulus $A$, equipped with a traversal boundary convex (“radial") field $\hat v$. We call such $\a$ *generic relative to* $\hat v$, if $\hat v$ is generic with respect to the curves $\a(\d X)$ in the sense of Definition 9.1.
Given a transversally generic field $v$ on a connected compact surface $X$, we call a map $\a: (X, v) \to (A, \hat v)$ a *convex quasi-envelop of* $(X, v)$ if there exists an immersion $\a: X \to A$ which is generic relative to the radial field $\hat v$ on $A$, and $v = \a^\ast(\hat v)$, the pull-back of $\hat v$. $\diamondsuit$
\[fig1.8\]
{height="2.8in" width="3.4in"}
Given a boundary generic relative to $\hat v$ immersion $\a: X \to A$, the $\a$-pullback (transfer) of the field $\hat v$ defines a vector field $v \neq 0$ on $X$. Since $\a$ is an immersion, evidently the pull-back $v$ is traversing on $X$. Moreover, $v$ is taversally generic in the sense of Definition \[def1.4\], since no $v$-trajectory $\g$ has more than one point of simple tangency to $\d X$.
\[def1.8\] Let $\a: X \to A$ be a *regular embedding* of a given compact surface $X$ into an annulus $A$, carrying a traversal boundary convex field $\hat v$. We denote by $v$ the pull-back of $\hat v$ under $\a$. If $\a$ is traversally generic relative to $\hat v$, then we say that the pair $(A, \hat v)$ is a *convex envelop* of $(X, v)$.
$\diamondsuit$
The existence of a convex envelop puts significant restrictions of the topology of $X$: such orientable surfaces $X$ do not have $1$-handles. In other words, they are disks with holes.
If a compact connected surface $X$ with boundary has a pair of loops whose transversal intersection is a singleton, then no traversal flow on $X$ admits a convex envelop. In other words, if a connected surface $X$ with boundary has a handle, then no traversal flow on $X$ can be convexly enveloped.
By Lemma 1.2, the space $\hat X$ of a convex envelop is either a disk or an annulus, both surfaces residing in the plane. No two loops in the plane intersect transversally at a singleton. Thus, for surfaces with a handle, no convex envelops exist.
So the existence of a convex envelop severally restricts the topology of surface $X$. To incorporate surfaces with handles into our constructions, we have introduced the notion of a convex quasi-envelop (Definition \[def1.7\]).
Now we are in position to explore a connection between immersions $\a: (X, v) \subset (A, \hat v)$ of a given surface $X$ in the annulus $A$, such that $v = \a^\ast(\hat v)$ and $\hat v$ is generic with respect to $\a(\d X)$ on one hand, and loops in the functional spaces $\mathcal F_{\leq 2}$ on the other.
Let $\a(\d X)^\times$ denote the set of self-intersections of the curves forming the image $\a(\d X)$. Let $\a(\d X)^\circ$ denote the set $\a(\d X) \setminus \a(\d X)^\times$.
With the pattern $\a(\d X)$ we associate an auxiliary smooth function $z_\a: A \to \R$, subject to the following properties:
- $z_\a^{-1}(0) = \a(\d X)$,
- $0$ is the regular value of $z_\a$ at the points of $\a(\d X)^\circ$,
- in the vicinity of each point $a \in \a(\d X)^\times$, consider local coordinates $(x_1, x_2)$ such that $\{x_1 =0\}$ and $\{x_2 =0\}$ define the two intersecting branches of $\a(\d X)$; then locally $z_\a = c\cdot x_1x_2$, where the constant $c \neq 0$.
- $z_\a =1$ in the vicinity of $\d A$,
- the sign of $z_\a$ changes to opposite as a path crosses an arc from $\a(\d X)^\circ$ transversally[^7].
$$\begin{aligned}
\label{eq1.7}\end{aligned}$$
Here we denote by $A^\circ$ the interior of the annulus $A$. Let $\phi: A^\circ \to \R$ be a smooth function so that $d\phi(\hat v) > 0$ in $A^\circ$ and $\phi(\hat\g \cap A^\circ) = \R$ for all $\hat v$–trajectories $\hat\g$ in $A$. Then, with the help of $z_\a$ and $\phi$, we get a map $J_{z_\a}: \mathcal T(\hat v) \to \mathcal F_{\leq 2}$ whose target is the space of smooth functions $f: \R \to \R$ with no zeros of multiplicity $\geq 3$ and that are identically $1$ outside of a compact set in $\R$. We define the map $J_{z_\a}$ by the formula $$\begin{aligned}
\label{eq1.8}
J_{z_\a}(\hat \g) = (z_\a|_{\hat \g}) \circ (\phi |_{\hat\g})^{-1},\end{aligned}$$ where, abusing notations, $\hat \g$ stands for both a $\hat v$-trajectory in $A$ and for the corresponding point in the trajectory space $\mathcal T(\hat v) \approx S^1$.
For a fixed $\a$, it is easy to check that the homotopy class $[J_{z_\a}]$ of $J_{z_\a}$ does not depend on the choice of the auxiliary function $z_\a$, subject to the five properties in (\[eq1.7\]) (the space of such $z_\a$’s is convex and thus contractible).
We pick a generator $\kappa \in \pi_1(\mathcal F_{\leq 2}) \approx \Z$ (see [@Ar]) and define the integer $J^\a$ by the formula $J^\a \cdot \kappa = [J_{z_\a}]$. As a result, any immersion $\a: X \to A$, which is generic with respect to $\hat v$, produces a homotopy class $[J_{z_\a}] \in \pi_1(\mathcal F_{\leq 2})$ and an integer $J^\a$.
The isomorphism $\pi_1(\mathcal F_{\leq 2}) \approx \Z$ follows from the work of V. I. Arnold [@Ar] by a slight modification of his arguments, which we will describe next (see Theorem \[th1.7\]). The main difference between our constructions and the ones from [@Ar] is that Arnold uses the critical loci of functions from $\mathcal F_{\leq 2}$, while we are using the zero loci.
Generic loops in $\b: S^1 \to \mathcal F_{\leq 2}$ have an interpretation in terms of finite collections $C$ of smooth closed curves in the annulus $A$ with *no inflection points with respect to their tangent lines of the form* $\{\theta = const\}$ in the $(u, \theta)$-coordinates. We call such tangent lines $\theta$-*vertical*. Furthermore, the generic homotopy between such loops $\b$ correspond to some *cobordism* relation between the corresponding plane curves, the cobordism also avoids the $\theta$-vertical inflections.
First, let us spell out the genericity requirements on the collections $C$ of closed curves in the annulus $A$:
1. $C \subset A$ is a finite collectionof closed smooth *immersed* curves $\{C_j\}_j$,
2. the projections $\{\theta: C_j \to S^1\}_j$ have Morse type singularities only[^8],
3. the self-intersections and mutual intersections of the curves $\{C_j\}_j$ are transversal and no triple intersections are permited,
4. at each double intersection, the two banches of $C$ are not parallel to the $u$-coordinate,
5. the $\theta$-images of the intersections and of the critical values of $\{\theta: C_j \to S^1\}_j$ are all distinct in $S^1$,
6. the cardinality of each fiber of $\theta: C \to S^1$ does not exceed a given natural number $d$.
$$\begin{aligned}
\label{eq1.9}\end{aligned}$$
\[def1.9\] Given two collections $C_0$ and $C_1$ of immersed closed curves as in (\[eq1.9\]), we say that they are *cobordant with no $\theta$-vertical inflections*, if there is a smooth function $F: A \times [0, 1] \to \R$ such that:
- $0$ is a regular value of $F$,
- the restriction of the projection $T: A \times [0, 1] \to [0, 1]$ to the zero set $W =_{\mathsf{def}} F^{-1}(0)$ is a Morse function,
- $C_0 = W \cap (A \times \{0\})$ and $C_1 = W \cap (A \times \{1\})$,
- for each $t \in [0, 1]$, the section $C_t =_{\mathsf{def}} W \cap (A \times \{t\})$ is such that $C_t$ has no $\theta$-horizontal inflections[^9]
- for each $t \in [0, 1]$ the cardinality of the fibers of $\theta: C_t \to S^1$ does not exceed a given natural number $d$. $\diamondsuit$
It is possible to verify that the cobordism with no $\theta$-vertical inflections is an *equivalence relation* among collections of curves as in (\[eq1.9\]). Indeed, if $C$ is cobordant to $C'$ with the help of $F$, and $C'$ to $C''$ with the help of $F'$, then there exists a piecewise smooth function $F \cup F': A \times [0, 2] \to \R$ whose restriction to $A \times [0, 1]$ is $F$ and to $A \times [1, 2]$ is a $(+1)$-shift of $F'$. Smoothing $F \cup F'$ along $A \times \{1\}$ in the normal direction and scaling down the interval $[0, 2]$ to $[0,1]$, produces the desired function-cobordism $F \ast F': A \times [0, 1] \to \R$. So we can talk about *the set of bordisms* $\mathbf B_{\mathsf{no\, \theta-inflect.}}$, based on collections of closed curves in the annulus with no $\theta$-vertical inflections. This set is a *group*: the operation $C, C' \Rightarrow C \ast C'$ is defined by the union $\tilde C \cup \tilde C' \subset A$, where $\tilde C \subset S^1 \times (0, 0.5)$ and $\tilde C' \subset S^1 \times (0.5, 1)$ are the images of $C$ and $C'$, scaled down in the $u$-direction by the factor $0.5$ and placed in sub-annuli of $A = S^1 \times [0, 1]$. The role of $-C$ is played by the mirror image of $C$ with respect to a vertical (equivalently, horizontal) line, a fiber of $\theta: A \to S^1$.
Note that this operation $\ast$ may affect the maximal cardinalities $d$ and $d'$ of the fibers $\theta: C \to S^1$ and $\theta: C' \to S^1$ in a somewhat unpredictable way. In any case, the fiber cardinality of $\theta: C\ast C' \to S^1$ has the upper boundary $d + d'$.
The previous constructions deliver the following proposition, a slight modification of Theorem from [@Ar].
\[th1.7\] The fundamental group $\pi_1(\mathcal F_{\leq 2})$ is isomorphic to the bordism group $\mathbf B_{\mathsf{no\, \theta-inflect.}}$, based on finite collections of immersed loops with no $\theta$-vertical inflections in the annulus $A$ and subject to the constraints (\[eq1.9\]). The isomorphism is induced by the correspondence $$K: \{\b: S^1 \to \mathcal F_{\leq 2}\} \Rightarrow \{\b(\theta)^{-1}(0)\}_{\theta \in [0, 2\pi]} \subset A.$$ $\diamondsuit$
This theorem is a foundation of a *graphic calculus* that converts homotopies of loops in the functional space $\mathcal F_{\leq 2}$ into cobordisms of closed loop patterns in the annulus $A$ with no $\theta$-vertical inflections.
Figures 10 - 14 show an application of this calculus. They explain why any loop in $\mathcal F_{\leq 2}$ is homotopic to an integral multiple of a generator $\kappa \in \pi_1(\mathcal F_{\leq 2})$, represented by a model loop pattern $K \subset A$ as in Fig. 9, diagram (a) or (b).
We orient the annulus $A = S^1 \times [0, 1]$ so that the the $\theta$-coordinate, corresponding to $S^1$, is the first, and the $u$-coordinate, corresponding to $[0, 1]$, is the second.
We fix an orientation of $X$, thus picking orientations for each component of $\d X$. Given an orientation-preserving immersion $\a: (X, v) \subset (A, \hat v)$ such that $\a(\d X)$ has the properties as in (\[eq1.9\]), we notice that the polarity of $a \in \d_2^+X(v)$ is $\oplus$ if and only if $\a_\ast(\nu_a)$, where $\nu_a$ is the inner normal to $\d X$ at $a$, points in the direction of $\theta$. Otherwise, the polarity of $a$ is $\ominus$ (see Fig. 9).
\[th1.8\] Any orientation-preserving immersion $\a: (X, v) \subset (A, \hat v)$ such that $\hat v$ is generic with respect to $\a(\d X)$[^10] produces a map $J_{z_\a} : S^1 \to \mathcal F_{\leq 2}$ (see (\[eq1.8\])). Its homotopy class $[J_{z_\a}] = J^\a \cdot \kappa$, where $\kappa$ denots a generator of $\pi_1( \mathcal F_{\leq 2}) \approx \Z$.
The integer $J^\a$ can be computed by the formula: $$J^\a = \#\{\d_2^{+, \oplus}X(v)\} - \#\{\d_2^{+, \ominus}X(v)\}$$ and thus does not depend on $\a$ (as long as the transfer $\a^\ast(\hat v) = v$).
Moreover, $|J^\a| \leq c_2^+(v)$, the complexity of the $v$-flow.
Let $d =_{\mathsf{def}} \max_{\hat\g} \# \{\hat\g \cap \a(\d X)\}$ be the maximal cardinality of the intersections of the $\hat v$-trajectories $\hat\g$ with the loops’ pattern $\a(\d X)$. Since $X$ bounds $\d X$, $d$ is even.
For any $\hat v$-generic immersion $\a: X \subset A$, we pick an auxiliary function $z_\a : A \to \R$, adjusted to $\a$ as in (\[eq1.9\]). By the previous arguments, this choice produces the loop $J_{z_\a}: S^1 \to \mathcal F^d_{\leq 2}$. Although the loop $J_{z_\a}$ is generated by an immersion $\a: X \to A$, in the process of deforming $J_{z_\a}$ by a cobordism $F: A \times [0, 1] \to \R$ with no $\theta$-vertical inflections as in Definition 9.4, we may destroy this connection with the original $\a$: the new curve patterns $\{C_t\}_{t \in [0, 1]}$ in $A$ may not be produced by immersions $\{\a_t: X \to A\}_{t \in [0, 1]}$.
Let us describe an algorithm (see Figures 10 - 14) that reduces a given pattern $C_0 = J_{z_\a}^{-1}(0) \subset A$ to a pattern from *the canonical set of patterns* $\{n\cdot K\}_{n \in \Z}$ (as in Fig. 9) by a cobordism $F: A \times [0, 1] \to \R$. We will perform a sequence of elementary surgeries on the set $C_0$, executed inside of the cylindrical shell $A \times [0, 1]$. It is sufficient to construct a smooth surface $W \subset A \times [0, 1]$ as in Definition 9.4, for which $W \cap A \times \{0\} = C$ and $W \cap A \times \{1\} = J^\a \cdot K$; then one can define a function $F: A \times [0, 1] \to \R$, appropriately adjusted to $W$, so that $0$ is a regular value of $F$ and $F^{-1}(0) = W$.
\[fig1.9\]
{height="2.6in" width="4.5in"}
As we modify the $t$-section $C_t \subset A \times \{t\}$, we keep track of the checker board polarities $+, -$, attached to the regions of $A \setminus C_t$; through the process, the polarity of the region adjacent to $\d A$ remains “$+$". Let us denote by $A_t^-$ the region of the negative polarity that is “bounded" by the curve pattern $C_t \subset A \times \{t\}$. $A_t^+$ denotes the complementary set. Informally, the regions of polarity $+$ are the regions where the function $F$ from Definition \[def1.9\] is non-negative.
With the help of this polarization $\{A_t^+, A_t^-\}$ of the annulus $A$, the points $a \in \d_2(C_t, \hat v)$, where $\hat v$-flow is tangent to $C_t$, acquire the polarization “$+$" or “$-$": if the germ of the trajectory $\hat\g_a$ is contained in $A_t^-$, then the polarity of $a$ is defined to be “$+$", otherwise it is “$-$". Moreover, if the inner normal $\nu_a$ to the region $A^-_t$ at $a$ has the same direction as the coordinate $\theta$ on $A$, then the second polarity of $a$ is defined to be “$\oplus$", otherwise it is “$\ominus$". As a result, we can talk about the four sets: $\d_2^{+, \oplus}(A_t^-, \hat v)$, $\d_2^{+, \ominus}(A_t^-, \hat v)$, $\d_2^{-, \oplus}(A_t^-, \hat v)$, $\d_2^{-, \ominus}(A_t^-, \hat v)$. We simplify the notations for these loci as: $\d_2^{+, \oplus}C_t, \; \d_2^{+, \ominus}C_t, \; \d_2^{-, \oplus}C_t, \; \d_2^{-, \ominus}C_t.$
\[fig1.10\]
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\[fig1.11\]
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\[fig1.12\]
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\[fig1.13\]
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\[fig1.14\]
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Let us describe an algorithm that constructs a cobordism with no $\theta$-vertical inflextions between a given loop pattern $\a(\d X)$ and a few copies of the canonical pattern as in Fig. 9.
[**(1)**]{} At any stage of this construction, we can resolve each crossing $a \in(C_t)^\times$ in a preferred way. The two branches of the preferred resolution will be transversal to the $\theta$-fiber through $a$. When the vector $\hat v(a)$ points inside $A_t^-$, the resolution will add a 1-handle to $A_t^+$, when the vector $\hat v(a)$ points inside $A_t^+$, the resolution will add a 1-handle to $A_t^-$. In any case, the sets $\d_2^{+, \oplus}C_t$, $\d_2^{+, \ominus}C_t$ are not affected. As a result of these resolutions, the new pattern $C'_t \subset A$ is a disjoint union of *simple* smooth curves with no $\theta$-vertical inflections. Moreover, it shares with $C_0$ the same sets $\d_2^{+, \oplus}\sim$, $\d_2^{+, \ominus}\sim$ [^11] (see Fig. 10).
[**(2)**]{} Next we will pick and fix one regular value $\theta_\star \in S^1$ of the map $\theta: C_t \to S^1$. The intersection of $\theta^{-1} \cap A_t^+$ consists of several intervals. By performing $1$-surgery of $C_t$ along each of these intervals we get a new loop pattern $C'_t$ which has an empty intersection with the $\theta$-fiber over the point $\theta_\star$. Moreover no $\theta$-vertical inflections were introduced in the process. The original loci $\d_2^{+, \oplus}\sim$, $\d_2^{+, \ominus}\sim$ are preserved (while the loci $\d_2^{-, \oplus}\sim$, $\d_2^{-, \ominus}\sim$ are changed). Therefore we may assume that $C_t = C'_t$ is contained in a rectangle $R \subset A$ and $C_t$ shares the numbers of points from the loci $\d_2^{+, \oplus}\sim$, $\d_2^{+, \ominus}\sim$ with the original $C_0$ (see Fig. 11, diagram 3).
[**(3)**]{} Consider the set $\Theta^+_t \subset S^1$ of critical values of $\theta: C_t \to S^1$ for the critical points from $\d_2^+C_t$. We pick a regular value $\theta^\sharp_i$ in-between each pair of adjacent critical values $\theta_i, \theta_{i+1} \in \Theta^+_t $. Then we apply $1$-surgery on $A_t^+$ as in (2) to empty the region $A_t^-$ in the vicinity of the fiber $\theta^{-1}(\theta_i^\sharp)$ (see Fig. 11, diagram 4).
[**(4)**]{} As a result of these surgeries, $A_t^-$ turns into a disjoint union of connected regions, each of which contains a single point of the set $\d_2^+(C_t)$ at most (see Fig. 12, diagram 5).
[**(5)**]{} By Lemma \[lem1.2\], any connected region of $A_t^- \subset R \subset A$ with no points from $\d_2^+\sim$ is a disk. It can be eliminated by a $0$-surgery (see Fig. 12, diagram 6).
[**(6)**]{} Pairs of points $a \in \d_2^{+, \oplus} \sim$ and $b \in \d_2^{+, \ominus}\sim$ can be cancelled via a surgery on their regions $D_a, D_b \subset A_t^-$ (as shown in Fig. 13, diagrams 7 and 8, and Fig. 14, diagrams 9 and 10). This cancellation of pairs will be executed gradually and with some care.
Any strip $S_i \subset A$, bounded by the vertical lines $\{\theta = \theta^\sharp_i \}$ and $\{\theta = \theta^\sharp_{i+1} \}$, contains a single region $D_a$ with $a \in \d_2^+C_t$. If the points of opposite second polarity ($\oplus, \ominus$) exist, then there are two *adjacent* vertical strips $S_i, S_{i+1}$ such that $a_i \in S_i \cap \d_2^{+, \oplus} C_t$ and $b_i \in S_{i+1} \cap \d_2^{+, \oplus} C_t$ have opposite second polarities. We attach to $D_{a_i} \coprod D_{b_i}$ two $1$-handles to form an annulus $A_{a_ib_i}$ as in Figures 12 and 13. To complete the cancellation of opposite pairs, we perform $2$-surgery on the inner circles of the annuli $\{A_{a_ib_i}\}$. This converts the annuli into disks, residing in $A_t^-$. They can be eliminated by $0$-surgery along the outer circles of $\{A_{a_ib_i}\}$’s (see Fig. 14, diagram 10).
It may happen that the model domain $D_{a_i}$ as in Fig. 9, (b), and “its mirror image" $D_{b_i}$ with respect to a vertical line $\{\theta = \theta_i^\sharp\}$ are positioned so that their horns are pointing in opposite directions. In such a case, they can be cancelled by a slightly different sequence of elementary surgeries (see [@Ar]). Alternatively, taking a trip around the annulus $A$, we will find a pair of adjacent strips such that their domains $D_a$ and $D_b$ of opposite second polarity can “lock horns". For them, the previous recipe will apply.
This cancellation procedure can be repeated by considering the remaining adjacent pairs of regions with the opposite second polarity untill no regions with the opposite second polarity are left. [**(7)**]{} As a result of all these steps, $A^-_t$ is either empty, or a disjoint union of disks (as in Fig. 9), each of which contains a single point from $\d_2^+\sim$ (and tree points from $\d_2^-\sim$); *the second polarities of such points are the same for all disks*. Thus we got an integral multiple of the basic pattern as in Fig. 9 and proved that $\pi_1(\mathcal F_{\leq 2})\approx \Z$.
Note that the original difference $\#\{\d_2^{+, \oplus}X(v)\} - \#\{\d_2^{+, \ominus}X(v)\}$ between the numbers of $\hat v$-trajectories with polarities $\oplus$ and $\ominus$ and of the combinatorial types $(\dots 121 \dots)$ is preserved under the modifications in (1)-(7).
The original maximal cardinality $d$ of the $\theta$-fibers evidently does not increase under the steps (1)-(7). Finally, we notice that $$c^+(v) =_{\mathsf{def}} \#\{\d_2^{+, \oplus}(v)\} + \#\{\d_2^{+, \ominus}(v)\}$$ $$\geq \quad | \#\{\d_2^{+, \oplus}(v)\} - \#\{\d_2^{+, \ominus}(v)\}|.$$
[**Remark 9.1.**]{} It is interesting and somewhat surprising to notice that the invariant $J^\a = \#\{\d_2^{+, \oplus}(v)\} - \#\{\d_2^{+, \ominus}(v)\}$ reflects more the topology of the field $v= \a^\ast(\hat v)$ than the topology of the surface $X$: in fact, any integral value of $J^\a$ can be realized by a traversally generic field $v$ on a *disk* $D$ which even admits a convex envelop! A portion of the boundary $\d D$ looks like a snake with respect to the field $\hat v$ of the envelop. For any $X$, the effect of deforming a portion of $\d X$ into a snake is equivalent to adding several times a spike (an edge and a pair of univalent and trivalent verticies) to the graph $\mathcal T(v)$. Evidently, these operations do not affect $H_1(\mathcal T(v); \Z) \approx H_1(X; \Z)$.
In contrast, $\#\{\d_2^{+, \oplus}(v)\} + \#\{\d_2^{+, \ominus}(v)\} \geq 2|\chi(X)|$ has a topological significance for $X$.
For example, for $\a$ as in Fig. 8, $J^\a =0$. If we subject $\a$ to an isotopy that introduces a snake-like pattern of Fig. 9, (a), then for the new immersion $\a'$, the invariant $J^{\a'} = 1$. $\diamondsuit$
[**Remark 9.2.**]{} Consider a connected oriented surface $X$ with a connected boundary. It is a boundary connected sum of a few copies of $T^\circ$, the torus with a hole. A punctured torus admits an immersion $\a: T^\circ \to A$ in the annulus so that the cardinality of the fibers of $\theta: \a(\d T^\circ) \to S^1$ does not exceed $6$ (see Fig. 8). Therefore, any connected oriented surface $X$ with boundary admits an immersion $\a: X \to A$ with the property $\#\{ \theta^{-1}(\theta_\star) \cap \a(\d X)\} \leq 6$ for all $\theta_\star \in S^1$. $\diamondsuit$
Let us glance at the implications of Theorems \[th1.7\] and \[th1.8\] and give them a new, perhaps, more natural spin.
The finite-dimensional space $\mathcal P^d_{\leq 2}$ of real monic polynomials of an even degree $d$ and with no real roots of multiplicity $\geq 3$ is a natural “approximation" of the functional space $\mathcal F^d_{\leq 2}$. Of course, a polynomial from $\mathcal P^d_{\leq 2}$ is not a function from $\mathcal F_{\leq 2}$: it is not identically $1$ outside of a compact set. However, there is an embedding $\mathcal I_d: \mathcal P^d_{\leq 2} \to \mathcal F_{\leq 2}$ that, in the vicinity of $\pm \infty$, “levels down to $1$" any real polynomial $P$ of an even degree $d$. Its image belongs to the subspace $\mathcal F^d_{\leq 2}$. This embedding is described by an analytic formula (see [@V]) as follows. Fix an auxiliary smooth function $\chi: \R \to [0, 1]$ such that $\chi(u) = 0$ for $|u| \leq 1$, $\chi(u) = 1$ for $|u| \geq 2$, and $\d\chi/\d u \neq 0$ for $1 < |u| < 2$. Let $\mu(P)$ denote the sum of absolute values of the coefficients of the monic $P$. Then $$\mathcal I_d(P)(u) =_{\mathsf{def}} P(u) + (1 - P(u))\cdot \chi(u/\mu(p)).$$
In fact, the zeros of any polynomial $P$ are in 1-to-1 correspondence with the zeros of the function $\mathcal I_d(P)$ and their multiplicities are preserved.
Consider the “forbidden set" $\mathcal F^d_{\geq 3} \subset \mathcal F^d$ of functions $f$ that have at least one zero of multiplicity $\geq 3$. Among them, the functions $f$ that have exactly one zero of multiplicity $3$ form an open and dense subset $(\mathcal F^d_{\geq 3})^\circ$.
For each $f \in (\mathcal F^d_{\geq 3})^\circ$, let $u^\star_f$ be the unique zero of multiplicity $3$.
The set $\mathcal F^d_{\geq 3}$ has codimension $2$ in $\mathcal F^d$; so loops in $\mathcal F^d_{\leq 2}$ may be *linked* with the locus $\mathcal F^d_{\geq 3}$ in $\mathcal F^d$. Here is a model example of such a link (see Fig. 15).
\[fig1.16\]
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For any $u^\ast \in (-2, 2)$, consider a $\theta$-family of quartic $u$-polynomials $$\big\{P_{u^\star}(u, \theta) =_{\mathsf{def}} (u-2)\big[(u -u^\star)^3 + \cos(\theta) (u -u^\star) + \sin(\theta)\big]\big\}_{\theta \in [0, 2\pi]}.$$ Each $P_{u^\star}(u, \theta)$ belongs to the space $\mathcal P^4_{\leq 2}$. The $\mathcal I_4$-image of this $\theta$-family forms a loop $\{h_{u^\star}(u, \theta)\}_{\theta \in [0, 2\pi]}$ in $\mathcal F^4_{\leq 2}$. The loop bounds a $2$-disk $$D^2_{u^\star} =_{\mathsf{def}} \big\{\mathcal I_4\big((u-2)[(u -u^\star)^3 + x_1(u -u^\star) + x_0]\big)\big\}_{\{x_0^2 + x_1^2 \leq 1\}}$$ in $\mathcal F^4$, which hits the subspace $\mathcal F^d_{\geq 3}$ at the singleton $\mathcal I_4\big((u-2)(u -u^\star)^3\big)$.
Similarly, for any $f \in (\mathcal F^d_{\geq 3})^\circ$ and $d \geq 4$, the loop $$L_f =_{\mathsf{def}}\big\{f(u, \theta) = h_{u^\star_f}(u, \theta) \cdot \big[f(u)/(u- u^\star_f)^3\big]\big\}_{\theta \in [0, 2\pi]}$$ resides in $\mathcal F^d_{\leq 2}$ and is linked with the component of $(\mathcal F^d_{\geq 3})^\circ$ that contains $f$.
Since the correspondence $f \to u^\star_f$ is continuous for $f \in (\mathcal F^d_{\geq 3})^\circ$, the homotopy class $[L_f] $ of the loop $L_f$ does not depend on the choice of $f$ within each component of $(\mathcal F^d_{\geq 3})^\circ$.
In fact, by Theorem \[th1.8\], $[L_f]$ is a generator $\kappa$ of $\pi_1(\mathcal F^d_{\leq 2})$. In particular, the $\mathcal I_4$-image of the loop $$\big\{P(\theta, u) =_{\mathsf{def}}(u-2)[u^3 + \cos(\theta) u + \sin(\theta)]\big\}_{\theta \in [0, 2\pi]}$$ in $\mathcal P^4_{\leq 2}$ is a generator $\kappa$ of $\pi_1(\mathcal F_{\leq 2}) \approx \Z$. Its zero set in the annulus $A = S^1 \times [-3, 3]$ (equipped with the coordinates $(\theta, u)$) is a union of two loops, similar to the ones shown in Fig. 9, (a).
Theorem 11 from [@V], makes an important for us claim: the $\mathcal I_d$-induced map in homology $$(\mathcal I_d)_\ast: H_j(\mathcal P^d_{\leq 2};\, \Z) \to H_j(\mathcal F_{\leq 2};\, \Z)$$ is an isomorphism for all $j \leq d/3$. In particular, $$(\mathcal I_4)_\ast: H_1(\mathcal P^4_{\leq 2};\, \Z) \to H_1(\mathcal F_{\leq 2};\, \Z) \approx \Z$$ is an isomorphism. Moreover, $$(\mathcal I_4)_\ast: \pi_1(\mathcal P^4_{\leq 2}) \to \pi_1(\mathcal F_{\leq 2}) \approx \Z$$ is an isomorphism as well [@Ar]. As we proceed, let us keep these facts in mind.
Of course it is much easier to visualize events in the $4$-dimensional space $\mathcal P^4_{\leq 2}$ than their analogues in the infinite-dimensional $\mathcal F^4_{\leq 2}$. This will be our next task. It will lead us to explore the beautiful stratified geometry of the Swallow Tail discriminant surface.
\[fig1.15\]
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Consider the subspace $\tilde{\mathcal P}^4_{\leq 2} \subset \mathcal P^4_{\leq 2}$, formed by the monic depressed[^12] polynomials $P(u)$. Since $\tilde{\mathcal P}^4_{\leq 2}$ is a deformation retract of $\mathcal P^4_{\leq 2}$ ([@Ar]), these two are homotopy equivalent. So it is a bit easier to visualize the generator $\tilde\kappa \in \pi_1(\tilde{\mathcal P}^4_{\leq 2})$ (rather than $\kappa \in \pi_1(\mathcal P^4_{\leq 2})$), since $\tilde{\mathcal P}^4_{\leq 2}$ is a domain in $\R_{\mathsf{coef}}^3$, the space of the coefficients $x_0, x_1, x_2$.
The polynomials with real roots of multiplicity $\geq 2$ form a singular surface $H \subset \R_{\mathsf{coef}}^3$, the famous Swallow Tail (see Fig. 16). The point-polynomial $u^4$ (the strongest singularity $O$ of $H$), together with the polynomials that have one root of multiplicity $3$, must be excluded from $\R_{\mathsf{coef}}^3$ to form $\tilde{\mathcal P}^4_{\leq 2}$. These excluded polynomials form two branches of a curve $\mathcal C \subset H$, whose apex $O$ is the origin and whose branches extend to infinity. One branch $OB$ of $\mathcal C$ correspond to polynomials with the smaller simple root followed by the root of multiplicity $3$; the other branch $OA$ of $\mathcal C$ correspond to polynomials with the smaller root of multiplicity $3$, followed by the simple root. The curve $OC$, the self-intersection locus of $H$, represents polynomials with two distinct roots of multiplicity $2$. It belongs to the set $\tilde{\mathcal P}^4_{\leq 2}$ .
Thus $\pi_1(\tilde{\mathcal P}^4_{\leq 2}) = \pi_1(\R_{\mathsf{coef}}^3 \setminus \mathcal C) \approx \Z$. Therefore the generator $\tilde\kappa \in \pi_1(\tilde{\mathcal P}^4_{\leq 2})$ is represented by an oriented loop in $\R_{\mathsf{coef}}^3$ that winds once around the curve $\mathcal C$. Such loop $\tilde\kappa$ must hit once the locus $H_{(121)} \subset H$, formed by polynomials whose real zeros conform to the pattern $(121)$: indeed, the curve $\mathcal C = OA \cup OB \cup O$ is the boundary of the surface $H_{(121)}$.
Traversing $H_{(121)}$ from the chamber of polynomials with $4$ real roots to the chamber of polynomials with $2$ real roots picks a normal to $H_{(121)}$ orientation. Similar rule of orientation can be applied to the stratum $H_{(112)}$ bounded by the curves $OC \cup OB$, the stratum $H_{(211)}$ bounded by the curves $OC \cup OA$, and the stratum $H_{(2)}$ that separates the chamber with two real roots from the chamber with no real roots at all.
For any smooth loop $\b: S^1 \to \tilde{\mathcal P}^4_{\leq 2}$ which is in general position to $H$ and its strata, consider the zero set $$\D(\b) =_{\mathsf{def}} \{(\theta, u)|\; \b(\theta)(u) = 0\}\subset S^1 \times \R.$$ Thanks to the very definition of the target space $\tilde{\mathcal P}^4_{\leq 2}$, the set $\D(\b)$ is a collection of closed curves with transversal self-intersections, no triple intersections, no self-tangencies, and no $\theta$-vertical inflections. The cardinality of the fiber of $\theta: \D(\b) \to S^1$ does not exceed $4$.
Now the linking number $J^\b =_{\mathsf{def}} \mathsf{lk}(\b(S^1), \mathcal C)$ equals the algebraic intersection of the loop $\b(S^1)$ with the surface $H_{(121)}$ that bounds $\mathcal C$. Each time the loop $\b(S^1)$ intersects the surface $H_{(121)}$ transversally in the direction of the positive normal, a point from $\d_2^{+, \oplus}\D(\b)$ is generated, and each time $\b(S^1)$ intersects $H_{(121)}$ is in the direction of the negative normal, a point from $\d_2^{+, \ominus}\D(\b)$ is generated. In particular, the generator $\kappa: S^1 \to \tilde{\mathcal P}^4_{\leq 2}$ of $\pi_1(\tilde{\mathcal P}^4_{\leq 2})$ has the property $\mathsf{lk}(\b(S^1), \mathcal C) = 1$. Therefore we get $$\mathsf{lk}(\b(S^1), \mathcal C) = \b(S^1) \circ H_{(121)} = \#\{\d_2^{+, \oplus}\D(\b)\} - \#\{\d_2^{+, \ominus}\D(\b)\},$$ the baby model of the formula from Theorem \[th1.8\]. This number is an invariant of the homotopy class of the loop $\b$.
Similarly, each time the loop $\b(S^1)$ intersects the surface $H_{(2)}$ transversally in the direction of the positive normal, a point from $\d_2^{-, \oplus}\D(\b)$ is generated, and each time $\b(S^1)$ intersects $H_{(2)}$ is in the direction of the negative normal, a point from $\d_2^{-, \ominus}\D(\b)$ is generated.
Note that perhaps not any set $\D(\b)$ is the image $\a(\d X)$ of an immersion $\a: X \to S^1 \times \R$ for some orientable surface $X$. But the generator $\kappa$ in Fig. 9, (a), is. Also, if we insist that the cardinality of the $\theta$-fibers $\leq 4$, we cannot accommodate surfaces $X$ with handles. According to Remark 9.2, to accommodate them, we need to deal with polynomials/functions of degree $6$ at least.
Let us describe briefly how this “degree $6$ polynomial model" works. We will see that the increasingly complex combinatorics of tangency begins to play a significant role.
To simplify the notations, we identify $\mathcal P^6$ with its image $\mathcal I_6(\mathcal P^6) \subset \mathcal F^6$ and the cylinder $S^1 \times \R$ with the interior of the annulus $A$.
The combinatorial patterns $\omega$ of real divisors of monic degree $6$ real polynomials are numerous:
- (111111), (1111), (11), $\emptyset$;
- (21111), (12111), (11211), (11121), (11112), (211), (121), (112), (2);
- (2211), (1221), (1122), (2121), (2112), (1212), (22);
- (3111), (1311), (1131), (1113), (31), (13);
- etc.
We denote by $\mathcal P^6_\omega$ the set of real monic polynomials whose real divisors conform to the combinatorial pattern $\omega = (\omega_1, \omega_2, \dots , \omega_s)$, where $\{\omega_i\}_{1 \leq i \leq s}$ are natural numbers. We denote by $|\omega|$ the $l_1$-norm of the vector $\omega$. Evidently, $|\omega| \leq 6$.
The sets $\{\mathcal P^6_\omega\}_\omega$ form a partition of the space $\mathcal P^6$. In fact, each $\mathcal P^6_\omega$ is homeomorphic to an open ball of dimension $6 - |\omega|'$, where $|\omega|' = |\omega | - s$ ([@K3]). The closure $\bar{\mathcal P}^6_\omega$ of $\mathcal P^6_\omega$ in $\R^6_{\mathsf{coef}}$ is an affine semi-algebraic variety. By resolving $\bar{\mathcal P}^6_\omega$ appropriately, one can show that the partition $\{\mathcal P^6_\omega\}_\omega$ defines a structure of a $CW$-complex on $\mathcal P^6$, or rather, on its one-point compactification ([@K3]). So we may think of $\mathcal P^6_\omega$’s as being “cells" (although $\bar{\mathcal P}^6_\omega$ may not be homeomorphic to an infinite cone over a closed ball).
Let $H \subset \mathcal P^6$ denote the set of monic polynomials with multiple roots, the $5$-dimensional discriminant variety. The first bullet lists the four $6$-dimensional chambers-cells in which $H$ divides $\mathcal P^6$. The second bullet lists all $5$-dimensional strata in which $H$ is divided by the strata of dimension $4$. The first three bullets list the monic polynomials that form the space $\mathcal P^6_{\leq 2}$. The third and the fourth bullets list the $4$-dimensional cells-strata. The forbidden locus $\mathcal P^6_{\geq 3}$ is the union of strata, labeled by the combinatorial types in the fourth bullet and on. Then $\mathcal P^6_{\geq 3}$ is the closure of the set $$\mathcal P^6_{(3111)} \cup \mathcal P^6_{(1311)} \cup \mathcal P^6_{(1131)} \cup \mathcal P^6_{(1113)} \cup \mathcal P^6_{(31)} \cup \mathcal P^6_{(13)}.$$
We can orient each cell $\mathcal P^6_\omega$ so that $$\d \bar{\mathcal P}^6_{(12111)} = \bar{\mathcal P}^6_{(3111)} - \bar{\mathcal P}^6_{(1311)} + \bar{\mathcal P}^6_{(1221)} -\bar{\mathcal P}^6_{(1212)},$$ $$\d \bar{\mathcal P}^6_{(11121)} = \bar{\mathcal P}^6_{(1131)} - \bar{\mathcal P}^6_{(1113)} + \bar{\mathcal P}^6_{(2121)} - \bar{\mathcal P}^6_{(1221)},$$ $$\d \bar{\mathcal P}^6_{(121)} = \bar{\mathcal P}^6_{(31)} -\bar{\mathcal P}^6_{(13)} + \bar{\mathcal P}^6_{(2121)} - \bar{\mathcal P}^6_{(1212)}.$$ The operator $\d$ in these formulas should be understood in the spirit of algebraic topology as the boundary operator on cellular chains (and not as a topological boundary of the appropriate sets) [@K6].
Adding the three formulas above, we get that the forbidden set, viewed as a $4$-chain, is an algebraic boundary of a $5$-chain:
$$\mathcal P^6_{\geq 3} =_{\mathsf{def}}\, \bar{\mathcal P}^6_{(3111)} - \bar{\mathcal P}^6_{(1311)} + \bar{\mathcal P}^6_{(1131)} - \bar{\mathcal P}^6_{(1113)} + \bar{\mathcal P}^6_{(31)} - \bar{\mathcal P}^6_{(13)}$$ $$= \d \big(\bar{\mathcal P}^6_{(12111)} + \bar{\mathcal P}^6_{(11121)} + \bar{\mathcal P}^6_{(121)}\big).$$
Now consider a smooth loop $\b: S^1 \to \mathcal P^6_{\leq 2}$. By a small perturbation we may assume that $\b(S^1)$ is transversal to the hypersurfaces $\mathcal P^6_{(12111)}, \mathcal P^6_{(11121)}, \mathcal P^6_{(121)}$ that bound the cycle $\bar{\mathcal P}^6_{\geq 3}$. Therefore
$$\mathsf{lk}\big(\b(S^1), \mathcal P^6_{\geq 3}\big) = \b(S^1) \circ \big(\mathcal P^6_{(12111)} \cup \mathcal P^6_{(11121)} \cup \mathcal P^6_{(121)}\big).$$
Again, we form the set $$\D(\b) =_{\mathsf{def}} \{(\theta, u)|\; \b(\theta)(u) = 0\}\subset S^1 \times \R.$$
Then $$\b(S^1) \circ \big(\mathcal P^6_{(12111)} \cup \mathcal P^6_{(11121)} \cup \mathcal P^6_{(121)}\big) = \#\{\d_2^{+, \oplus}\D(\b)\} - \#\{\d_2^{+, \ominus}\D(\b)\}.$$[^13] Therefore we get $$\mathsf{lk}\big(\b(S^1), \mathcal P^6_{\geq 3}\big) = \#\{\d_2^{+, \oplus}\D(\b)\} - \#\{\d_2^{+, \ominus}\D(\b)\},$$ a version of the formula from Theorem \[th1.8\], being applied to the loop $\b = J_{z_\a}$. The loop is produced by a generic with respect to $\hat v$ immersion $\a: X \to A$, such that the cardinality of the fibers of $\theta: \a(\d X) \to S^1$ does not exceed $6$, and by an auxiliary function $z_{\a(\d X)}$.
These considerations are not restricted to polynomials/functions of degree $6$: they apply to any even degree $d$. The application requires a deeper dive into the combinatorics of real polynomial divisors and their modifications, but the spirit is captured by the arguments that deal with degree $6$ (see \[K3\]).
Any $\hat v$-generic immersion $\a: X \to A$ also produces a well-defined element $[K_\a]$ in the set of *homotopy classes* $[\mathcal T(v), \mathcal F_{\leq 2}]$ of maps from the trajectory graph $\mathcal T(v)$ to the functional space $\mathcal F_{\leq 2}$. Its construction is similar to the one of $J_{z(\a)}$. Consider the $\hat v$-generated obvious map $Q_\a: \mathcal T(v) \to \mathcal T(\hat v) \approx S^1$ (each $v$-trajectory is contained in the unique $\hat v$-trajectory). Put $K_\a =_{\mathsf{def}} J_{z(\a)} \circ Q_\a$.
[**Remark 9.3.**]{} Note that, for some immersions $\a : X \to A$, the invariant $J^\a$ may be different from $0$, but $[K_\a]$ may be trivial. For example, this is the case when $X$ is a disk with a snake-like boundary $\a(\d X)$ with respect to $\hat v$. However, there exist immersions $\a$ with a nontrivial $[K_\a]$. For example, such is the immersion in Fig. 10, (1). At the same time, for $\a$ in Fig. 11, (3), $[K_\a]$ is trivial. $\diamondsuit$
Since $\pi_1(\mathcal F_{\leq 2}) \approx \Z$, it follows that $H_1(\mathcal F_{\leq 2}; \Z) \approx \Z$. In turn, this implies that the $1$-dimensional cohomology $H^1(\mathcal F_{\leq 2}; \Z) \approx \Z$.
Thus $K_\a$ induces a map $$K_\a^\ast: H^1(\mathcal F_{\leq 2}; \Z) \to H^1(\mathcal T(v); \Z) \approx H^1(X; \Z).$$ In particular, we get an element $K_\a^\ast(\kappa^\ast) \in H^1(X; \Z)$, where $\kappa^\ast$ is a generator of $H^1(\mathcal F_{\leq 2}; \Z) \approx \Z$. This cohomology class $K_\a^\ast(\kappa^\ast)$ is a characteristic class of the given $\hat v$-generic immersion $\a$.
Theorem \[th1.7\] implies that if two $\hat v$-generic immersions $\a, \a_1: X \to A$ are such that the pull-backs $\a^\ast(\hat v) = \a^\ast_1(\hat v) = v$, then $K_\a^\ast(\kappa^\ast) = K_{\a_1}^\ast(\kappa^\ast)$. So the cohomology class $K_\a^\ast(\kappa^\ast)$ is, in fact, a characteristic class of $v$. It is desirable to be able to reach this conclusion without relying on the cobordisms of curves’ patterns in $A$ with no $\theta$-vertical inflections.
Based on the partial evidence, provided by the two polynomial models $\mathcal P^4_{\leq 2}$ and $\mathcal P^6_{\leq 2}$, we may conjecture that the value of $K_\a^\ast(\kappa^\ast)$ on any loop ($1$-cycle) $\delta: S^1 \to X$ equals to the linking number $\mathsf{lk}(K_\a(\delta), \mathcal F_{\geq 3})$. The validation of this conjecture requires to extend our analysis of the stratified geometry of $\mathcal P^6_{\leq 2}$ to $\mathcal P^d_{\leq 2}$ and to show that $\mathcal I_d: \mathcal P^d_{\leq 2} \to \mathcal F^d_{\leq 2}$ is a weak homotopy equivalence for all even $d$. Both steps are realizable with the techniques developed in [@K3] and in [@K5].
In dimensions higher than two, similar considerations apply to produce characteristic classes of traversally generic flows. They are based on computations of homology of spaces of real monic polynomials with *restricted combinatorics* of their real divisors. It turns out that the topology of high-dimensional convex envelops is as intricate as the homotopy groups of spheres [@K6].
Our investigation of vector flows in Flatland reached its conclusion. To find out how things flow in other lands—“the romances of many dimensions"—([@Ab]), the reader could consult with the references below.
[\[GM2\]]{}
Abbott, E., [*Flatland: A Romance of Many Dimensions*]{}, Dover Publications, New York, 1992.
Alpert, H., Katz, G., [*Using Simplicial Volume to Count Multi-tangent Trajectories of Traversing Vector Fields*]{}, Geometriae Dedicata DOI 10.1007/s10711-015-0104-6 (arXiv:1503.02583v1 \[math.DG\] (9 Mar 2015)).
Arnold V.I., [*Spaces of Functions with Moderate Singularities*]{}, Func. Analysis & Its Applications, 23(3), 1-10 (1989) (Russ).
Cohen, R.L., [*Topics in Morse Theory*]{}, Stanford University, 1991.
Goresky, M., MacPherson, R., [*Stratified Morse Theory*]{}, Proceedings of Symposia in Pure Mathematics, Vol. 40 (1983), Part 1, 517-533.
Goresky, M., MacPherson, R., Morse theory for the intersection homology groups, Analyse et Topologie sur les Espaces Singulieres, Astérisque \#101 (1983), 135-192, Société Mathématique de France.
Goresky, M., MacPherson, R., Stratified Morse Theory, Springer Verlag, N. Y. (1989), Ergebnisse vol. 14. Also translated into Russian and published by MIR Press, Moscow, 1991.
Gottlieb, D.H., [*All the Way with Gauss-Bonnet and the Sociology of Mathematics*]{}, Math. Monthly, 103 (1996), 457-469.
Gromov, M., [*Volume and bounded cohomology*]{} Publ. Math. I.H.E.S., tome 56 (1982), 5-99.
Guth, L., [*Minimal number of self-intersections of the boundary of an immersed surface in the plane*]{}, arXiv:0903.3112v1 \[math.DG\] 18 Mar 2009.
Hopf, H., [*Vectorfelder in $n$-dimensionalen Mannigfaltigkeiten*]{}, Math. Annalen 96 (1937), 225-250.
Katz, G., [*Convexity of Morse Stratifications and Spines of 3-Manifolds*]{}, math.GT/0611005 v1(31 Oct. 2006). Katz, G., [*Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary*]{}, Applied Mathematics, 2014, 5, 2823-2848, (also arXiv:1406.6907v1 \[mathGT\] (26 June, 2014)).
Katz, G., [*Traversally Generic & Versal Flows: Semi-algebraic Models of Tangency to the Boundary*]{}, to appear in Asian J. of Math. (arXiv:1407.1345v1 \[mathGT\] (4 July, 2014)).
Katz, G., [*The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows*]{}, arXiv:1407.2984v3 \[math.GT\] (6 Aug 2014).
Katz, G., [*Causal Holography of Traversing Flows*]{}, arXiv:1409.0588v1\[mathGT\] (2 Sep 2014).
Katz, G., [*Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume*]{}, to appear in J. of Topology and Analysis (arXiv:1503.09131v2 \[mathGT\] (24 Apr 2015)).
Katz, G., [*Morse Theory, Gradient Flows, Concavity, and Complexity on Manifolds with Boundary*]{}, monograph, to be published by World Scientific.
Malgrange, B., [*The preparation theorem for differential functions*]{}, Differential Analysis (Papers presented at the Bombay Colloquium, 1964), 203-208.
Milnor, J., [*Morse Theory*]{}, Princeton University Press, Princeton, New Jersey, 1965.
Morse, M. [*Singular points of vector fields under general boundary conditions*]{}, Amer. J. Math. 51 (1929), 165-178.
Vasiliev, V.A., [*Complements of Discriminants of Smooth Maps: Topology and Applications*]{}, Translations of Mathematical Monographs, vol. 98, American Math. Society publication, 1994.
Whitney, H., [*On regular closed curves in the plane*]{}, Comp. Math. 4 (1937) 276-284.
[^1]: It is possible to have a field $v$ for which some trajectories will be cubically tangent to the boundary, but the majority of vector fields $v$ avoid such cubic tangencies.
[^2]: This excludes disk and annulus.
[^3]: It easy to see that the ends of these segments, as well as the singletons, reside in $\d X$.
[^4]: In particular, a traversally generic $v$ is boundary generic.
[^5]: These inequalities, together with the computations of the complexities in the first three bullets, cover the entire variety of compact connected surfaces with boundary.
[^6]: this is equivalent to saying that the $(k-1)$-st jet at $x$ of $z|_\g$ vanishes, but the $k$-th jet does not.
[^7]: Thus the sign of $z_\a$ provides a “checker board" coloring of the domains in $A \setminus \a(\d X)$.
[^8]: This excludes the $\theta$-vertical inflections.
[^9]: Note that the second bullet excludes the triple intersections of $C_t$.
[^10]: for any convex quasi-envelop $\a$ of $(X, v)$
[^11]: Later on, we may be forced to introduce momentarily new crossings for the exceptional sections $C_t$’s, which eventually will be eliminated.
[^12]: that is, with the zero coefficient next to $u^3$
[^13]: Note that the points of $\b(S^1) \circ \mathcal P^6_{(11211)}$ belong to the locus $\d_2^{-, \sim}\D(\b)$.
|
Two-parameter Sturm-Liouville problems
B. Chanane[^1] and A. Boucherif
Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals
Dhahran 31261, Saudi Arabia.
chanane@kfupm.edu.sa
May 4, 2012
**Abstract** This paper deals with the computation of the eigenvalues of two-parameter Sturm-Liouville (SL) problems using the *Regularized Sampling Method*, a method which has been effective in computing the eigenvalues of broad classes of SL problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.
Introduction
============
In an interesting paper published in 1963, F. M. Arscott \[2\] showed that the method of separation of variables used in solving boundary value problems for Laplace’s equation leads to a two-parameter eigenvalue problem for ordinary differential equations with the auxiliary requirement that the solutions satisfy boundary conditions at several points. This has led to an extensive development of multiparameter spectral theory for linear operators (see for instance \[3-4\], \[7-10\], \[12\], \[15\], \[30-33\], \[35\], \[37\], \[39-42\]). In the paper \[12\], the authors give an overview results on two-parameter eigenvalue problems for second order linear differential equations. Several properties of corresponding eigencurves are given.In \[15\] the authors have obtained interesting geometric properties of the eigencurves (for instance transversal intersections is equivalent to simplicity of the eigenvalues in the sense of Chow and Hale). All the above works are concerned with the theoretical aspect of existence of eigenvalues. Also, several authors have dealt with the theoretical numerical analysis of two-parameter eigenvalue problems (see \[5\], \[11\], \[13\], \[34\], \[36\], \[38\] and the references therein). Eigenvalue problems have played a major role in the applied sciences. Consequently, the problem of computing eigenvalues of one-parameter problems has attracted many researchers (see for example \[1, 6, 16,17,19-29\] and the references therein).
Concerning the computations of eigenvalues of one-parameter Sturm-Liouville problems, the authors in \[16\] introduced a new method based on Shannon’s sampling theory. It uses the analytic properties of the boundary function. The method has been generalized to a class of singular problems of Bessel type \[17\], to more general boundary conditions, separable \[19\] and coupled \[21\], to random Sturm-Liouville problems \[23\] and to fourth order regular Sturm-Liouville problems \[28\]. The books by Atkinson \[4\], Chow and Hale \[30\] , Faierman \[33\] , McGhee and Picard \[37\], Sleeman \[39\], Volkmer \[42\] and the long awaited mongraph by Atkinson and Mingarelli [@AM2011] contains several results on eigenvalues of multiparameter Sturm-Liouville problems and the corresponding bifurcation problems. However, no attempt has been made to compute the eigenvalues of two-parameter Sturm-Liouville problems using the approach based on the *Regularized Sampling Method* introduced recently by the first author in [@c2005] to compute the eigenvalues of general Sturm-Liouville problems and extended to the case of Singular [@c2007d], Non-Self-Adjoint [@c2007c], Non-Local [@c2008], Impulsive SLPs [@c2007b],[@c2007a]. We shall consider, in this paper the computation of the eigenpairs of two-parameter Sturm-Liouville problems with three-point boundary conditions using the *Regularized Sampling Method*.
The Characteristic Function
===========================
Consider the two-parameter Sturm-Liouville problem $$\left\{
\begin{array}
[c]{c}-y^{\prime\prime}+qy=(\mu_{1}^{2}w_{1}+\mu_{2}^{2}w_{2})y\text{ , }0<x<1\\
y(0)=0\text{ , }y(c)=0\text{ , }y(1)=0
\end{array}
\right. \label{2.1}$$ where $w_{1}$, $w_{2}$ are positive and in $\mathcal{C}^{2}[0,1]$ and $q\in
L[0,1]$ and $c\in(0,1)$ some given constant.
By an eigenvalue of (\[2.1\]) we mean a value of the couple $(\mu_{1},\mu_{2})$ for which problem (\[2.1\]) has a non trivial solution. Conditions that insure the existence of eigenvalue are given in \[3\], \[9\], \[10\], \[15\], \[30\], \[31\], \[40\] and \[41\]. In fact, under fairly general conditions it has been shown (see \[15\], \[30\]) that there are smooth curves of eigenvalues (actually eigenpairs). Our objective is to effectively localize the eigencurves in the parameter $(\mu_{1},\mu_{2})$-plane.We should point out that we have restricted our attention to Dirichlet boundary conditions in order to eliminate technical details that might obscur the ideas.
We shall associate to (\[2.1\]) the initial value problem $$\left\{
\begin{array}
[c]{c}y^{\prime\prime}+(\mu_{1}^{2}w_{1}+\mu_{2}^{2}w_{2})y=qy\text{ , }0<x<1\\
y(0)=0\text{ , }y^{\prime}(0)=1
\end{array}
\right. \label{2.2}$$ and deal first with the unperturbed case ($q=0$) then with the perturbed case ($q\neq0$).
The unperturbed case ($q=0$)
----------------------------
In this case, (\[2.2\]) reduces to $$\left\{
\begin{array}
[c]{c}\varphi_{1}^{\prime\prime}+w\varphi_{1}=0\text{ , }0<x<1\\
\varphi_{1}(0)=0\text{ , }\varphi_{1}^{\prime}(0)=1
\end{array}
\right. \label{2.3}$$ where $w=\mu_{1}^{2}w_{1}+\mu_{2}^{2}w_{2}$.
The solution $\varphi_{1}$ of (\[2.3\]) is an entire function of $(\mu
_{1},\mu_{2})\in\mathbb{C}^{2}$ for each fixed $x\in(0,1]$ of order $(1,1) $ and type $(\sigma_{1}(x),\sigma_{2}(x))$ and satisfies the estimate,$$\left\vert \varphi_{1}(x)\right\vert \leq K_{1}\exp\left[ \sigma
_{1}(x)\left\vert \mu_{1}\right\vert +\sigma_{2}(x)\left\vert \mu
_{2}\right\vert \right] \text{ , }(\mu_{1},\mu_{2})\in\mathbb{C}^{2}$$ for each fixed $x\in(0,1]$ where, $\sigma_{i}(x)=2\left\{ x\int_{0}^{x}w_{i}(\xi)d\xi\right\} ^{\frac{1}{2}}$for $i=1,2$.
From (\[2.3\]) we get the integral equation $$\varphi_{1}(x)=x-\int_{0}^{x}(x-\xi)\left[ \mu_{1}^{2}w_{1}(\xi)+\mu_{2}^{2}w_{2}(\xi)\right] \varphi_{1}(\xi)d\xi\label{2.9}$$
Let $$\left\{
\begin{array}
[c]{c}\varphi_{1,0}(x)=x\\
\varphi_{1,n+1}(x)=\int_{0}^{x}(x-\xi)\left[ \mu_{1}^{2}w_{1}(\xi)+\mu
_{2}^{2}w_{2}(\xi)\right] \varphi_{1,n}(\xi)d\xi\text{ , }n\geq0
\end{array}
\right. \label{2.10}$$ We shall show, by induction on $n$, that $$\left\vert \varphi_{1,n}(x)\right\vert \leq\frac{x}{n!(n+1)!}\left\{
x\int_{0}^{x}\left[ \left\vert \mu_{1}\right\vert ^{2}w_{1}(\xi)+\left\vert
\mu_{2}\right\vert ^{2}w_{2}(\xi)\right] d\xi\right\} ^{n}\text{ , }n\geq0\label{2.11}$$ It is true for $n=0$. Assume it is true for $n$. We shall show that it is true for $n+1$. Indeed, from (\[2.10\]), we have $$\begin{aligned}
\left\vert \varphi_{1,n+1}(x)\right\vert & \leq\int_{0}^{x}(x-\xi)\left[
\left\vert \mu_{1}\right\vert ^{2}w_{1}(\xi)+\left\vert \mu_{2}\right\vert
^{2}w_{2}(\xi)\right] \times\nonumber\\
& \frac{\xi}{n!(n+1)!}\left\{ \xi\int_{0}^{\xi}\left[ \left\vert \mu
_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert \mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau\right\} ^{n}d\xi\text{ }\label{2.12}$$ Using the fact that the expression $(x-\xi)\xi^{n+1}$ attains its maximum at $\xi=\frac{n+1}{n+2}x$, we get $$\begin{aligned}
\left\vert \varphi_{1,n+1}(x)\right\vert & \leq\frac{x}{n!(n+1)!}\frac
{1}{n+1}\frac{1}{n+2}\left( \frac{n+1}{n+2}x\right) ^{n+1}\left\{ \int
_{0}^{x}\left[ \left\vert \mu_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert
\mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau\right\} ^{n+1}\nonumber\\
& \leq\frac{x}{(n+1)!(n+2)!}\left\{ x\int_{0}^{x}\left[ \left\vert \mu
_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert \mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau\right\} ^{n+1}\label{2.13}$$ that is (\[2.11\]) is true for $n+1$. Hence, it is true for all $n\geq0$.
Now, $\varphi_{1}(x)=\sum_{n\geq0}(-1)^{n}\varphi_{1,n}(x)$ and the series is absolutely and uniformly convergent since $$\begin{aligned}
\left\vert \varphi_{1}(x)\right\vert & =\left\vert \sum_{n\geq0}(-1)^{n}\varphi_{1,n}(x)\right\vert \leq\sum_{n\geq0}\left\vert \varphi
_{1,n}(x)\right\vert \nonumber\\
& \leq x\sum_{n\geq0}\frac{1}{n!(n+1)!}\left\{ x\int_{0}^{x}\left[
\left\vert \mu_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert \mu_{2}\right\vert
^{2}w_{2}(\tau)\right] d\tau\right\} ^{n}\nonumber\\
& =xI_{1}(\left\{ 2\sqrt{x\int_{0}^{x}\left[ \left\vert \mu_{1}\right\vert
^{2}w_{1}(\tau)+\left\vert \mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau
}\right\} )\times\nonumber\\
& \left\{ x\int_{0}^{x}\left[ \left\vert \mu_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert \mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau\right\}
^{-\frac{1}{2}}\label{2.14}$$ where $I_{1}$ is the modified Bessel function of the first kind order 1 $$I_{1}(z)=\sum_{n\geq0}\frac{1}{n!(n+1)!}\left( \frac{z}{2}\right) ^{2n+1}.$$ Using the fact that $I_{1}(z)\sim\frac{e^{z}}{\sqrt{2\pi z}}$ as $z\rightarrow\infty,$ we get $$\begin{aligned}
\left\vert \varphi_{1}(x)\right\vert & \leq K_{1}\exp\left[ 2\left\{
x\int_{0}^{x}\left[ \left\vert \mu_{1}\right\vert ^{2}w_{1}(\tau)+\left\vert
\mu_{2}\right\vert ^{2}w_{2}(\tau)\right] d\tau\right\} ^{\frac{1}{2}}\right] \nonumber\\
& \leq K_{1}\exp\left[ \sigma_{1}(x)\left\vert \mu_{1}\right\vert +\sigma
_{2}(x)\left\vert \mu_{2}\right\vert \right] \label{2.15}$$ Therefore, $\varphi_{1}$ is an entire function of $(\mu_{1},\mu_{2})\in\mathbb{C}^{2}$, as a uniformly convergent series of entire functions, for each fixed $x\in(0,1],$ of order $(1,1)$ and type $(\sigma_{1}(x),\sigma
_{2}(x)).$This concludes the proof.
We shall make use of the Liouville-Green’s transformation $$\left\{
\begin{array}
[c]{c}t(x)=\int_{0}^{x}\sqrt{w(\xi)}d\xi\\
z(t)=\left\{ w(x)\right\} ^{\frac{1}{4}}\varphi_{1}(x)
\end{array}
\right. \label{2.25}$$ to bring (\[2.2\]) to the form $$\left\{
\begin{array}
[c]{c}\frac{d^{2}z}{dt^{2}}+\left\{ 1+r(t)\right\} z=0\text{ , }0<t<\int_{0}^{1}\sqrt{w(\xi)}d\xi\\
z(0)=0\text{ , }\frac{dz}{dt}(0)=\left\{ w(0)\right\} ^{-\frac{1}{4}}\end{array}
\right. \label{2.5}$$ which can be written as an integral equation $$z(t)=\left\{ w(0)\right\} ^{-\frac{1}{4}}\sin t-\int_{0}^{t}\sin
(t-\tau)r(\tau)z(\tau)d\tau\label{2.6}$$ where $r(t)=\left[ \left\{ w(x)\right\} ^{-\frac{3}{4}}\frac{d^{2}}{dx^{2}}\left\{ w(x)\right\} ^{-\frac{1}{4}}\right] _{|x=x(t)}$.
Returning to the original variables, we deduce that $\varphi_{1}$ satisfies the integral equation $$\varphi_{1}(x)=\left\{ w(0)w(x)\right\} ^{-\frac{1}{4}}\sin\left\{ \int
_{0}^{x}\sqrt{w(\xi)}d\xi\right\} -\int_{0}^{x}\sin(\int_{\overline{x}}^{x}\sqrt{w(\xi)}d\xi)\psi(x,\overline{x})\varphi_{1}(\overline{x})d\overline{x}\label{2.7}$$ where $$\psi(x,\overline{x})=\left\{ w(x)\right\} ^{-\frac{1}{4}}\left\{
w(\overline{x})\right\} ^{-\frac{3}{4}}\left[ -\frac{1}{4}w^{\prime\prime
}(\overline{x})\left\{ w(\overline{x})\right\} ^{-\frac{1}{2}}+\frac{5}{16}\left\{ w^{\prime}(\overline{x})\right\} ^{2}\left\{ w(\overline
{x})\right\} ^{-\frac{3}{2}}\right] \label{2.8}$$
We shall present next some estimates whose proofs are immediate and left to the reader.
The function $\psi(x,\overline{x})$ satisfies the estimate $$\left\vert \psi(x,\overline{x})\right\vert \sim\frac{K_{2}}{\left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert }\text{, as }\left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert \rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}\text{ }\label{2.16}$$
The function $\alpha$ defined by $$\alpha(x)=\left( \text{sinc}\left\{ \sigma_{1}(x)\mu_{1}+\sigma_{2}(x)\mu_{2}\right\} \right) ^{m}\text{ }$$ where $sinc(z)=z^{-1}\sin z$ and $m$ is a positive integer, is an entire function of $(\mu_{1},\mu_{2})\in\mathbb{C}^{2}$ for each fixed $x\in(0,1]$ of order $(1,1)$ and type $(\sigma_{1}(x),\sigma_{2}(x))$. Furthermore, $\alpha$ satisfies the estimate $$\left\vert \alpha(1)\right\vert \sim\frac{K_{3}}{\left\vert \mu_{1}\right\vert
+\left\vert \mu_{2}\right\vert }\text{, as }\left\vert \mu_{1}\right\vert
+\left\vert \mu_{2}\right\vert \rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.$$
The function $\varphi_{1}$ satisfies the estimate $$\left\vert \varphi_{1}(1)\right\vert \sim\frac{K_{4}}{\left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert }\text{, as }\left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert \rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}\text{ }\label{2.17}$$
Combining the above results, we obtain the following theorem,
The function $\alpha\varphi_{1}$ is an entire function of $(\mu_{1},\mu
_{2})\in\mathbb{C}^{2}$ for each fixed $x\in(0,1]$ of order $(1,1)$ and type $((m+1)\sigma_{1}(x),(m+1)\sigma_{2}(x))$ and satisfies the estimate $$\left\vert \alpha(x)\varphi_{1}(x)\right\vert \sim\frac{K(x)}{\left(
\left\vert \mu_{1}\right\vert +\left\vert \mu_{2}\right\vert \right) ^{m+1}}\text{, as }\left\vert \mu_{1}\right\vert +\left\vert \mu_{2}\right\vert
\rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.$$
where $K$ depends on $x\in(0,1]$ but is independent of $(\mu_{1},\mu_{2})$ .
Let $PW_{\beta_{1},\beta_{2}}$ denote the Paley-Wiener space,$$PW_{\beta_{1},\beta_{2}}=\left\{
\begin{array}
[c]{c}h(z_{1},z_{2})\text{ entire / }\left\vert h(z_{1},z_{2})\right\vert \leq
C\exp\left\{ \beta_{1}\left\vert z_{1}\right\vert +\beta_{2}\left\vert
z_{2}\right\vert \right\} \text{,}\\
\text{ }\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\text{ }\left\vert
h(z_{1},z_{2})\right\vert ^{2}dz_{1}dz_{2}<\infty
\end{array}
\right\}$$ we have,
$\alpha(x)\varphi_{1}(x)$ , as a function of $(\mu_{1},\mu_{2})$ belongs to the Paley-Wiener space $PW_{\beta_{1},\beta_{2}}$ where $\left( \beta
_{1},\beta_{2}\right) =((m+1)\sigma_{1}(x),(m+1)\sigma_{2}(x))$ for each fixed $x\in(0,1].$
The perturbed case ($q\neq0$)
-----------------------------
Let $\varphi_{1}$ , $\varphi_{2}$ be two linearly independent solutions of $\varphi^{\prime\prime}+w\varphi=0$ satisfying $\varphi
_{1}(0)=\varphi_{2}^{\prime}(0)=0$ , $\varphi_{1}^{\prime}(0)=\varphi
_{2}(0)=1$ then the method of variation of parameters shows that (\[2.2\]) can be written as the integral equation $$y(x)=\varphi_{1}(x)+\int_{0}^{x}\Phi(x,\xi)q(\xi)y(\xi)d\xi\label{2.19}$$ where $\Phi(x,\xi)=\varphi_{1}(\xi)\varphi_{2}(x)-\varphi_{2}(\xi)\varphi
_{1}(x)$.
Here again, it is not hard to show that $y(x)$ is an entire function of $(\mu_{1},\mu_{2})$ for each $x\in(0,1]$, of order $(1,1)$ and type $(\sigma_{1}(x),\sigma_{2}(x))$. Multiplication by $\alpha$ gives a function $\alpha(x)y(x)$ of $(\mu_{1},\mu_{2})$ in a Paley-Wiener space $PW_{\beta
_{1},\beta_{2}}$ for each $x\in(0,1]$. More specifically, we have the following,
The function $\widetilde{y}$ defined by $\widetilde{y}(x)=\alpha(x)y(x)$ , belongs to $PW_{\beta_{1},\beta_{2}}$ where $\left( \beta_{1},\beta
_{2}\right) =((m+1)\sigma_{1}(x),(m+1)\sigma_{2}(x))$ as a function of $(\mu_{1},\mu_{2})\in\mathbb{C}$ for each $x\in(0,1]$, and satisfies the estimate,$$\left\vert \alpha(x)y(x)\right\vert \sim\frac{K(x)}{\left( \left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert \right) ^{m+1}},\text{as
}\left\vert \mu_{1}\right\vert +\left\vert \mu_{2}\right\vert \rightarrow
\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.$$ where $K$ depends on $x\in(0,1]$ but is independent of $(\mu_{1},\mu_{2})$ .
Since $\Phi_{xx}+w\Phi=0$ , $\Phi(t,t)=0$ and $\Phi_{x}(t,t)=1$, we have, $$\begin{aligned}
\Phi(x,t) & =\left\{ w(t)w(x)\right\} ^{-\frac{1}{4}}\sin\left( \int
_{t}^{x}\sqrt{w(\xi)}d\xi\right) \nonumber\\
& -\int_{t}^{x}\sin\left( \int_{\overline{x}}^{x}\sqrt{w(\xi)}d\xi\right)
\psi(x,\overline{x})\Phi(\overline{x},t)d\overline{x}\label{2.20}$$ so that, $$\left\vert \Phi(x,t)\right\vert \sim\frac{K_{4}}{\left\vert \mu_{1}\right\vert
+\left\vert \mu_{2}\right\vert }\leq K_{5}\text{, as }\left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert \rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.\label{2.22}$$ from which we get, after using Gronwall’s lemma on (\[2.19\])and the estimate for $\ \varphi_{1},$ $$\left\vert y(x)\right\vert \leq K_{1}\exp\left[ \sigma_{1}(x)\left\vert
\mu_{1}\right\vert +\sigma_{2}(x)\left\vert \mu_{2}\right\vert \right]
\exp\left\{ K_{5}\int_{0}^{x}|q(t)|dt\right\}$$$$\left\vert y(1)\right\vert \leq K_{7}\exp\left[ \sigma_{1}(1)\left\vert
\mu_{1}\right\vert +\sigma_{2}(1)\left\vert \mu_{2}\right\vert \right]
\text{, as }\left\vert \mu_{1}\right\vert +\left\vert \mu_{2}\right\vert
\rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.\label{2.26}$$ and $$\left\vert \alpha(1)y(1)\right\vert \leq K_{8}\exp\left[ (m+1)\sigma
_{1}(1)\left\vert \mu_{1}\right\vert +(m+1)\sigma_{2}(1)\left\vert \mu
_{2}\right\vert \right] \text{, as }\left\vert \mu_{1}\right\vert +\left\vert
\mu_{2}\right\vert \rightarrow\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.\label{2.28}$$ Furthermore, we have,$$\left\vert \alpha(x)y(x)\right\vert \sim\frac{K(x)}{\left( \left\vert \mu
_{1}\right\vert +\left\vert \mu_{2}\right\vert \right) ^{m+1}},\text{as
}\left\vert \mu_{1}\right\vert +\left\vert \mu_{2}\right\vert \rightarrow
\infty,(\mu_{1},\mu_{2})\in\mathbb{R}^{2}.$$ where $K$ depends on $x\in(0,1]$ but is independent of $(\mu_{1},\mu_{2})$ .
To summarize, in both cases, unperturbed and perturbed, the transform $\widetilde{y}(x;\mu_{1},\mu_{2})$ of the solution $y(x;\mu_{1},\mu_{2})$ of (\[2.2\]) is in a Paley-Wiener space $PW_{\beta_{1},\beta_{2}}$ where $\left( \beta_{1},\beta_{2}\right) =((m+1)\sigma_{1}(x),(m+1)\sigma
_{2}(x)).$ Thus $\widetilde{y}(x;\mu_{1},\mu_{2})$ can be recovered at each $x\in(0,1]$ from its samples at the latice points $(\mu_{1j},\mu_{2k})=(j\frac{\pi}{(m+1)\sigma_{1}(x)},k\frac{\pi}{(m+1)\sigma_{2}(x)})$, $(j,k)\in\mathbb{Z}^{2}$ using the rectangular cardinal series ([@50; @43; @44]),
Let $f\in PW_{\beta_{1},\beta_{2}}$ then $$f(\mu_{1},\mu_{2})=\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}f(\mu_{1j},\mu_{2k})\frac{\sin\beta_{1}(\mu_{1}-\mu_{1j})}{\beta_{1}(\mu
_{1}-\mu_{1j})}\frac{\sin\beta_{2}(\mu_{2}-\mu_{2k})}{\beta_{2}(\mu_{2}-\mu_{2k})}$$ the convergence of the series being uniform and in $L_{d\mu_{1}d\mu_{2}}^{2}(\mathbb{R}^{2})$, and $\mu_{mn}=n\pi/\beta_{m}$ , $m=1,2$, $n\in
\mathbb{Z}$.
Let $\sigma_{11}=\sigma_{1}(1),$ $\sigma_{21}=\sigma_{2}(1),$ $\sigma_{12}=\sigma_{1}(c),$ $\sigma_{22}=\sigma_{2}(c).$
The eigenpairs are therefore $(\mu_{1}^{2},\mu_{2}^{2})$ where $(\mu_{1},\mu_{2})$ solve the nonlinear system$$\left\{
\begin{array}
[c]{c}B_{1}(\mu_{1},\mu_{2})=0\\
B_{2}(\mu_{1},\mu_{2})=0
\end{array}
\right. \label{2.30}$$ where,$$\left\{
\begin{array}
[c]{c}B_{1}(\mu_{1},\mu_{2})\triangleq\frac{1}{\alpha(1)}\sum_{j=-\infty}^{\infty
}\sum_{k=-\infty}^{\infty}\widetilde{y}(1;\mu_{1j},\mu_{2k})\frac
{\sin2(m+1)\sigma_{11}(\mu_{1}-\mu_{1j})}{2(m+1)\sigma_{11}(\mu_{1}-\mu_{1j})}\frac{\sin2(m+1)\sigma_{21}(\mu_{2}-\mu_{2k})}{2(m+1)\sigma_{21}(\mu_{2}-\mu_{2k})}\\
B_{1}(\mu_{1},\mu_{2})\triangleq\frac{1}{\alpha(c)}\sum_{j=-\infty}^{\infty
}\sum_{k=-\infty}^{\infty}\widetilde{y}(c;\mu_{1j},\mu_{2k})\frac
{\sin2(m+1)\sigma_{12}(\mu_{1}-\mu_{1j})}{2(m+1)\sigma_{12}(\mu_{1}-\mu_{1j})}\frac{\sin2(m+1)\sigma_{22}(\mu_{2}-\mu_{2k})}{2(m+1)\sigma_{22}(\mu_{2}-\mu_{2k})}\end{array}
\right. \label{2.27}$$
A numerical example
===================
We shall consider in this section the two-parameter Sturm-Liouville problem with three- point boundary conditions given by $$\left\{
\begin{array}
[c]{c}-y^{\prime\prime}=(\mu_{1}^{2}+\mu_{2}^{2}x)y\text{ , }0<x<1\\
y(0)=y(0.7)=y(1)
\end{array}
\right. \label{3.10}$$
The general solution $y$ of the first differential equation can be expressed in terms of Ai and Bi functions and their first dirivatives as$$y(x;\mu_{1},\mu_{2})=\frac{(-1)^{2/3}\left( \text{Ai}\left( \frac
{\sqrt[3]{-1}\mu_{1}^{2}}{\mu_{2}^{4/3}}\right) \text{Bi}\left(
\frac{\sqrt[3]{-1}\left( \mu_{1}^{2}+\mu_{2}^{2}x\right) }{\mu_{2}^{4/3}}\right) -\text{Ai}\left( \frac{\sqrt[3]{-1}\left( \mu_{1}^{2}+\mu_{2}^{2}x\right) }{\mu_{2}^{4/3}}\right) \text{Bi}\left( \frac{\sqrt[3]{-1}\mu_{1}^{2}}{\mu_{2}^{4/3}}\right) \right) }{\mu_{2}^{2/3}\left(
\text{Ai}^{\prime}\left( \frac{\sqrt[3]{-1}\mu_{1}^{2}}{\mu_{2}^{4/3}}\right) \text{Bi}\left( \frac{\sqrt[3]{-1}\mu_{1}^{2}}{\mu_{2}^{4/3}}\right) -\text{Ai}\left( \frac{\sqrt[3]{-1}\mu_{1}^{2}}{\mu_{2}^{4/3}}\right) \text{Bi}^{\prime}\left( \frac{\sqrt[3]{-1}\mu_{1}^{2}}{\mu
_{2}^{4/3}}\right) \right) }\label{3.11}$$
Thus the eigenpairs $(\mu_{1}^{2},\mu_{2}^{2})$ can be obtained from the solutions $(\mu_{1},\mu_{2})$ of the system $$\left\{
\begin{array}
[c]{c}\widetilde{y}(1;\mu_{1},\mu_{2})=0\\
\widetilde{y}(c;\mu_{1},\mu_{2})=0
\end{array}
\right. \label{3.4}$$
For numerical purposes we have truncated the associated series to $|j|,|k|\leq
N=50$ and took $m=5$ in the function $\alpha$. Thus, the approximate eigenpairs are seen as solutions of the system $$\left\{
\begin{array}
[c]{c}\widetilde{y}^{[N]}(1;\mu_{1},\mu_{2})=0\\
\widetilde{y}^{[N\}}(c;\mu_{1},\mu_{2})=0
\end{array}
\right. \label{3.5}$$ The next table shows the exact eigenpairs together with their approximations using the *Regularized Sampling Method* (RSM).
\[c\][|c|c|c|c|]{}$\mu_{1}$ (exact) & $\mu_{2}$ (exact) & $\mu_{1}$ (RSM) & $\mu_{2}$ (RSM)\
$7.788149097670813$ & $7.5908224365786845$ & $7.78814883048352$ & $7.590823605399021$\
$4.194056047341936$ & $22.273796542861913$ & $4.194057357011064$ & $22.273795699572364$\
$15.597607295800684$ & $15.165889997583099$ & $15.597605904429917$ & $15.1658929384375$\
$13.87615754699624$ & $30.597837626955204$ & $13.876157718920586$ & $30.59783713797321$\
$23.40242244972004$ & $22.744341450749697$ & $23.40242640980619$ & $22.744329536815076$\
$9.51130982713563$ & $44.296491367611544$ & $9.511333341150682$ & $44.29647446214639$\
$39.31833491621106$ & $15.670248767892955$ & $39.31833152483279$ & $15.670257662031721$\
$47.18632817417246$ & $24.620489734469363$ & $47.186345272524804$ & $24.62038504864854$\
$46.81208951678993$ & $45.483251580157955$ & $46.813097550330646$ & $45.480149444567346$\
$30.03437464767369$ & $46.558148101315844$ & $30.034392317076748$ & $46.55811263809656$\
Conclusion
==========
In this paper, we have successfully computed the eigenpairs of two-parameter Sturm-Liouville problems using the regularized sampling method. A method which has been very efficient in computing the eigenvalues of broad classes of Sturm-Liouville problems (Singular, Non-Self-Adjoint, Non-Local, Impulsive,...). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.
**Acknowledgements** The authors are grateful to KFUPM for its usual support.
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[^1]: Corresponding author
|
---
abstract: 'A “fiducial” kinematical region for our calculations of instanton ($I$)-induced processes at HERA within $I$-perturbation theory is extracted from recent lattice simulations of QCD. Moreover, we present the finalized $I$-induced cross-sections exhibiting a strongly reduced residual dependence on the renormalization scale.'
author:
- |
A. Ringwald and F. Schrempp\
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
title: ' **QCD-Instantons at HERA[^1]**'
---
Introduction {#s0}
============
In this contribution, we briefly summarize some recent progress in our ongoing systematic study [@rs; @grs; @dis97-phen; @dis97-theo; @mrs1; @mrs3; @crs] of the discovery potential of DIS events induced by QCD instantons.
Instantons [@bpst] are non-perturbative gauge field fluctuations. They describe [*tunnelling*]{} transitions between degenerate ground states (vacua) of [*different topology*]{} in non-abelian gauge theories like QCD. Correspondingly, (anti-)instantons carry an [*integer topologigal*]{} charge $\mid Q\mid =1$, while the usual perturbation theory resides in the sector $Q=0$. Unlike the latter, instantons violate [*chirality*]{} ($Q_5$) in (massless) QCD and the sum of baryon plus lepton number ($B+L$) in QFD, in accord [@th] with the general ABJ-anomaly relation. An experimental discovery of instanton ($I$)-induced events would clearly be of basic significance.
The deep-inelastic regime is distinguished by the fact that here hard instanton-induced processes may both be [*calculated*]{} [@bb; @mrs1] within instanton-perturbation theory and possibly [*detected experimentally*]{} [@rs; @grs; @dis97-phen; @crs]. As a key feature we have recently shown [@mrs1], that in DIS the generic hard scale ${\cal Q}$ cuts off instantons with [*large size*]{} $\rho>>{\cal Q}^{-1}$, over which we have no control theoretically.
There has been much recent activity in the lattice community to “measure” topological fluctuations in [*lattice simulations*]{} [@lattice; @ukqcd] of QCD. Being independent of perturbation theory, such simulations provide “snapshots” of the QCD vacuum including all possible non-perturbative features like instantons. They also provide crucial support for important prerequisites of our calculations in DIS, like the validity of $I$-perturbation theory and the dilute $I$-gas approximation for [*small*]{} instantons of size $\rho \leq {\cal Q}^{-1}$. As one main point of this paper (Sect. 2), these lattice constraints will be exploited and translated into a “fiducial” kinematical region for our predictions of the instanton-induced DIS cross-section based on $I$-perturbation theory. In Sect. 3 we display the finalized calculations of the various instanton-induced cross-sections [@mrs3]. The essential new aspect here is the strong reduction of the residual dependence on the renormalization scale $\mu_r$ resulting from a recalculation based on improved instanton densities [@morretal], which are renormalization group (RG) invariant at the 2-loop level.
Validity of Instanton Perturbation Theory in DIS\
– Restrictions from Lattice-QCD Simulations {#s1}
=================================================
The leading instanton (I)-induced process in the DIS regime of $e^\pm
P$ scattering is displayed in Fig.1. The non-trivial [*topology*]{} of instantons is reflected in a violation of [*chirality*]{} by $\mid\Delta Q_5\mid =2 n_f$, in accord [@th] with the general ABJ-anomaly relation (while in pQCD always $\Delta Q_5\equiv 0$). The dashed box emphasizes the so-called instanton-[*subprocess*]{} with its own Bjorken variables, $$Q^{\prime\,2}=-q^{\prime\,2}>0;\hspace{0.5cm}
0\le{{x^\prime}}=\frac{Q^{\prime\,2}}{2 p\cdot q^\prime}\le 1.$$
The cross-section of interest may be shown [@rs; @dis97-phen; @mrs3] to exhibit a convolution-type structure, consisting of a smooth, calculable “flux factor” $P^{(I)}({{x^\prime}},\ldots)$ from the the $\gamma^\ast \bar{q}q\prime$ vertex, and the $I$-subprocess total cross-section $\sigma^{(I)}_{q^\prime\,g}(Q^\prime,{{x^\prime}})$, containing the essential instanton dynamics. We have evaluated the latter [@mrs3] by means of the optical theorem and the so-called ${I\overline{I}}$-valley approximation [@valley] for the relevant $q^\prime g\Rightarrow q^\prime g$ forward elastic scattering amplitude in the ${I\overline{I}}$ background. This method resums the exponentiating final state gluons in form of the known valley action $S^{{I\overline{I}}}$ and reproduces standard $I$-perturbation theory at larger ${I\overline{I}}$ separation $\sqrt{R^2}$.
Corresponding to the symmetries of the theory, the instanton calculus introduces at the classical level certain (undetermined) “collective coordinates” like the $I\,(\overline{I})$-size parameters $\rho\,(\overline{\rho})$ and the ${I\overline{I}}$ distance $\sqrt{R^2/\rho\overline{\rho}}$ (in units of the size). Observables like $\sigma^{(I)}_{q^\prime\,g}(Q^\prime,{{x^\prime}})$, must be independent thereof and thus involve integrations over all collective coordinates. Hence, denoting the [*density*]{} of $I\,(\overline{I})$’s by $D(\rho (\overline{\rho}))$ (see Eq.(\[density\])), we have generically, $$\begin{array}{lcl}
\sigma^{(I)}_{q^\prime g}( Q^\prime,x^\prime)
&=&\int\limits_0^\infty {\rm d}\rho
\int\limits_0^\infty {\rm d}\overline{\rho}\underbrace{D(
\rho)D(\overline{\rho})}_
{I,\overline{I}-{\rm densities}
\Leftarrow {\bf \rm Lattice!}}
e^{-({\rho + \overline{\rho}})Q^\prime}\\
&\times&\int\limits^\infty {\rm d}\xi
{\mathcal M}(\xi, x^\prime, Q^\prime,\ldots)
e^{-\frac{4\pi}{\alpha_s}(S^{{I\overline{I}}}(\xi)-1)},\\
\end{array}\label{sigma}$$ where $\xi=R^2/\rho\overline{\rho}+\rho/\overline{\rho}+\overline{\rho}/\rho$ is a convenient conformally invariant variable characterizing the ${I\overline{I}}$ distance. In Eq.(\[sigma\]), the crucial exponential cut-off [@mrs1] $e^{-(\rho+\overline{\rho})\,Q^\prime}$ is responsible for the [*finiteness*]{} of the $\rho,\,\overline{\rho}$ integrations. In addition, it causes the integrals (\[sigma\]) to be dominated by a [*single, calculable*]{} (saddle) point ($\rho^\ast=\overline{\rho}^\ast\sim Q^{\prime\,-1},
\xi^\ast({{x^\prime}},Q^\prime)$), in [*one-to-one*]{} relation to the conjugate momentum variables (${{x^\prime}}, Q^\prime$). This effective one-to-one mapping of the conjugate $I$-variables allows for the following important strategy: We may determine [*quantitatively*]{} the range of validity of $I$-perturbation theory and the dilute $I$-gas approximation in the instanton collective coordinates ($\rho<\rho_{\rm max},
R/\rho>(R/\rho)_{\rm min}$) from recent (non-perturbative) lattice simulations of QCD and translate the resulting constraints via the mentioned one-to-one relations into a “fiducial” kinematical region (${{x^\prime}}>{{x^\prime}}_{\rm min},Q^\prime>
Q^\prime_{\rm min}$) at HERA! Experimentally, these cuts must be implemented via a (${{x^\prime}},\,Q^\prime$) reconstruction from the final state topology [@crs], while theoretically, they are incorporated into our $I$-event generator [@grs] “QCDINS 1.6.0” and the resulting prediction of $\sigma^{(I)}_{\rm HERA}({{x^\prime}}>{{x^\prime}}_{\rm min},Q^\prime>Q^\prime_{\rm
min})$ (see Sect.3).
In lattice simulations 4d-Euclidean space-time is made discrete, e.g. in case of the “data” from the UKQCD collaboration [@ukqcd] which we shall use here,
-------------------- --- -----------------------------------------
lattice spacing: a = 0.055 - 0.1 fm
lattice volume: V = $l_{\rm space}^{\,3}\cdot l_{\rm time}=
[16^3\cdot 48 - 32^3\cdot 64]\,a^4$
-------------------- --- -----------------------------------------
In principle, such a lattice allows to study the properties of an ensemble of (anti-)instantons with sizes $a < \rho < V^{1/4}$. However, in order to make instanton effects visible, a certain “cooling” procedure has to be applied first. It is designed to filter out (dominating) fluctuations of [*short*]{} wavelength ${\cal O}(a)$, while affecting the instanton fluctuations of much longer wavelength $\rho >> a$ comparatively little. For a discussion of lattice-specific caveats, like possible lattice artefacts and the dependence of results on “cooling” etc., see Refs. [@lattice; @ukqcd].
The first important quantity of interest, entering Eq.(\[sigma\]), is the $I$-density, $D(\rho)$ (tunnelling probability!). It has been worked out a long time ago in the framework of $I$-perturbation theory: (renormalization scale $\mu_r$) $$D(\rho)\equiv \frac{{\rm d}n}{{\rm d}^4 x {\rm d}\rho}=
d \left(\frac{2\pi}{\alpha_s(\mu_r)}\right)^6
\exp{(-\frac{2\pi}{\alpha_s(\mu_r)})}\frac{(\rho\, \mu_r)^b}{\rho^{\,5}}.
\label{density}$$ Note the [*power law*]{} in the instanton size $\rho$ with the power $b$ given in Table1,
----------------------------------------------------------------------------------------------------------------------------------------------
$b$ $\rule[-3mm]{0mm}{7mm}\frac{1}{D}\frac{{\rm d}D}{{\rm d}\mu_r}$ Ref.
-------------------------------------------------- ----------------------------------------------------------------- -------------------------
$\beta_0$ ${\mathcal O}(\alpha_s)$ [’t Hooft [@th]]{}
Morris, Ross,
\[-1.5ex\][$\beta_0+\frac{\alpha_s(\mu_r)}{4\pi} \[-1.5ex\][ ${\mathcal O}(\alpha_s^2)$]{} & Sachrajda [@morretal]
(\beta_1-12\beta_0)$]{}
----------------------------------------------------------------------------------------------------------------------------------------------
: The power $b$ in Eq.(\[density\]) from Ref. [@th] and Ref. [@morretal], making the $I$-density $D(\rho)$ RG-group invariant at the 1-loop and 2-loop level, respectively.
in terms of the QCD $\beta$-function coefficients: $\beta_0=11-\frac{2}{3}{n_f};\ \beta_1=102-\frac{38}{3} {n_f}$. This power law $D(\rho)_{\mid n_f=0}\propto \rho^6$ of $I$-perturbation theory is confronted in Fig.2(top) with recent lattice “data”, which strongly suggests $I$-perturbation theory to be valid for $\rho {\mbox{\,\raisebox{.3ex}
{$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}\rho_{\rm max}=0.3$ fm. Next, consider the square of the total topological charge, $Q^2=(n\cdot(+1)+\bar{n}\cdot(-1))^2$ along with the total number of charges $N_{\rm tot}=n+\bar{n}$. For a [*dilute gas*]{}, the number fluctuations are [*poissonian*]{} and correlations among the $n$ and $\bar{n}$ distributions absent, implying $\langle (n-\bar{n})^2\rangle = \langle n+\bar{n}\rangle$, or $\langle \frac{Q^2}{N_{\rm tot}} \rangle =1$. From Fig.2(bottom), it is apparent that this relation characterizing the validity of the dilute $I$-gas approximation, is well satisfied for sufficiently [*small*]{} instantons! Again, we find $\rho_{\rm max}\simeq 0.3$ fm, quite independent of the number of cooling sweeps. For increasing $\rho_{\rm max}{\mbox{\,\raisebox{.3ex}
{$>$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}}\,}0.3$ fm, the ratio $\langle \frac{Q^2}{N_{\rm tot}}\rangle$ rapidly and strongly deviates from one. Crucial information about a second instanton parameter of interest, the average ${I\overline{I}}$ distance $ \langle R \rangle$, may be obtained as well from the lattice [@lattice; @ukqcd]. Actually, the ratio [@ukqcd] $ \frac{\langle R_{{I\overline{I}}}\rangle}{\langle \rho \rangle}\simeq 0.83$ has good stability against “cooling”, from which we shall take $R/\rho{\mbox{\,\raisebox{.3ex}
{$>$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}}\,}1$ as a reasonable lower limit for our $I$-perturbative DIS calculations.
Finally, the “fiducial” kinematical region for our cross-section predictions in DIS is found from lattice constraints and the discussed saddle-point translation as $$\left.\begin{array}{lcl}\rho&{\mbox{\,\raisebox{.3ex}
{$<$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}\,}}& 0.3 {\rm\ fm};\\
\frac{R}{\rho}&{\mbox{\,\raisebox{.3ex}
{$>$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}}\,}&1\\
\end{array}\right\}\Rightarrow
\left\{\begin{array}{lclcl}Q^\prime&{\mbox{\,\raisebox{.3ex}
{$>$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}}\,}&Q^\prime_{\rm min}&\simeq&
8 {\rm\ GeV};\\
x^\prime&{\mbox{\,\raisebox{.3ex}
{$>$}$\!\!\!\!\!$\raisebox{-.9ex}{$\sim$}}\,}&x^\prime_{\rm min}(Q^\prime_{\rm min})&\simeq &0.35.\\
\end{array} \right .
\label{fiducial}$$
$I$-Induced Cross-Sections for HERA {#s2}
===================================
We have achieved great progress in stability by using the [*2-loop RG invariant*]{} form of the $I$-density $D(\rho)$ from Eq.(\[density\]) and Table1 in a recalculation of the $I$-subprocess cross-sections [@mrs3]: The residual dependence on the renormalization scale $\mu_r$ turns out to be [*strongly reduced*]{} (Fig.3). As “best” scheme we use $\mu_r = 0.15\ Q^\prime$ throughout, for which $\partial \sigma^{(I)}_{q^\prime g}/\partial \mu_r \simeq 0$. The quantitative calculations of $\sigma^{(I)}_{\rm q^\prime g}$
(Fig.4) nicely illustrate the qualitative arguments from Sect.2, that the $Q^\prime$ dependence probes the effective instanton [*size $\rho$*]{} (top), while the $x^\prime$ dependence maps the ${I\overline{I}}$ [*distance $R$*]{} in units of the $I$-size $\rho$ (bottom).
Fig.5 displays the finalized $I$-induced cross-section at HERA, as function of the cuts $x^\prime_{\rm min}$ and $Q^\prime_{\rm min}$ in leading semi-classical approximation, as obtained with the new release “QCDINS 1.6.0” of our $I$-event generator.
For the minimal cuts (\[fiducial\]) extracted from lattice simulations, we specifically obtain $$\sigma^{(I)}_{\rm HERA}({{x^\prime}}\ge0.35,Q^\prime\ge 8\, {\rm GeV})
\simeq 126\, {\rm pb};
\ x_{\rm Bj}\ge 10^{-3};\ 0.9\ge y_{\rm Bj}\ge 0.1 .$$ In view of the fact that the cross-section varies strongly as a function of the (${{x^\prime}},Q^\prime$) cuts, the constraints from lattice simulations are extremely valuable for making concrete predictions.
[99]{} A. Ringwald and F. Schrempp, hep-ph/9411217, in: Proc. [*Quarks ‘94*]{}, Vladimir, Russia, May 1994, eds. D. Gigoriev et al., pp. 170-193. M. Gibbs, A. Ringwald and F. Schrempp, DESY 95-119, hep-ph/9506392, in: [*Proc. DIS 95*]{}, Paris, France, April 1995, eds. J.-F. Laporte and Y. Sirois, pp. 341-344. A. Ringwald and F. Schrempp, DESY 97-115, hep-ph/9706399, in: [*Proc. DIS 97*]{}, Chicago, IL, April 1997, eds. J. Repond and D. Krakauer, pp. 781-786. S. Moch, A. Ringwald and F. Schrempp, DESY 97-114, hep-ph/9706400, in: [*Proc. DIS 97*]{}, Chicago, IL, April 1997, eds. J. Repond and D. Krakauer, pp. 1007-1013. S. Moch, A. Ringwald and F. Schrempp, . S. Moch, A. Ringwald and F. Schrempp, to be published. T. Carli, A. Ringwald and F. Schrempp, in preparation. A. Belavin, A. Polyakov, A. Schwarz and Yu. Tyupkin, . G. ‘t Hooft, ; ; (Erratum). I. Balitsky and V. Braun, . For a recent review, see: P. van Baal, INLO-PUB-7/97, hep-lat/9709066, Review at Lattice ‘97, Edinburgh. D. Smith and M. Teper, Edinburgh preprint 98-1, hep-lat/9801008. T. Morris, D. Ross and C. Sachrajda, . A. Yung, .
V.V. Khoze and A. Ringwald, .
[^1]: Talk presented at the 6th International Workshop on Deep-Inelastic Scattering and QCD (DIS98), Brussels, April 1998; to be published in the Proceedings (World Scientific).
|
---
abstract: 'We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.'
address:
- 'Institut Camille Jordan, Université Claude Bernard Lyon 1, France'
- 'Fakultät für Mathematik, Universität Bielefeld, Germany'
author:
- Lie Fu
- Charles Vial
title: A motivic global Torelli theorem for isogenous K3 surfaces
---
[^1]
[^2]
Introduction {#introduction .unnumbered}
============
Torelli theorems for K3 surfaces {#torelli-theorems-for-k3-surfaces .unnumbered}
--------------------------------
A compact complex surface is called a K3 surface if it is simply connected and has trivial canonical bundle. The Hodge structure carried by the second singular cohomology contains essential information of the surface. Indeed, the global Torelli theorem says that the isomorphism class of a K3 surface $S$ is determined by the Hodge structure ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Z}})$ together with the intersection pairing on it [@PSS; @BR].
The following more flexible notion due to Mukai [@Mukai] turns out to be crucial in the study of their derived categories: two complex projective K3 surfaces $S$ and $S'$ are called *isogenous* if there exists a *Hodge isometry* $\varphi\colon {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})$, *i.e.* an isomorphism of rational Hodge structures compatible with the intersection pairing on both sides. Recently, Buskin [@buskin] proved that such an isometry $\varphi$ is induced by an algebraic correspondence, as had previously been conjectured by Shafarevich [@Shaf] as a particularly interesting case of the Hodge conjecture. Let us call any such representative a *Shafarevich cycle* for this isogeny. Shortly afterwards, Huybrechts [@huybrechts-isogenous] gave another proof and showed that in fact $\varphi$ is induced by a chain of exact linear equivalences between derived categories of twisted K3 surfaces; thereby establishing the *twisted derived global Torelli theorem* [@huybrechts-isogenous Corollary 1.4]: $$\label{eq:TwDerTorelli}
\text{$S$ and $S'$ are isogenous} \Longleftrightarrow \text{$S$ and $S'$ are
twisted derived equivalent.}$$ Here, following Huybrechts [@huybrechts-isogenous], we say that two K3 surfaces $S$ and $S'$ are *twisted derived equivalent* if there exist K3 surfaces $S=S_0, S_1,\ldots, S_n=S'$ and Brauer classes $\alpha = \beta_0 \in
\mathrm{Br}(S), \alpha_1 ,\beta_1 \in \mathrm{Br}(S_1),\ldots, \alpha_{n-1},
\beta_{n-1} \in \mathrm{Br}(S_{n-1})$ and $\alpha' = \alpha_n \in
\mathrm{Br}(S')$ and exact linear equivalences between bounded derived categories of twisted coherent sheaves $$\label{eq:tde}
\begin{array}{rlll}
\mathrm{D}^b(S,\alpha)\simeq&\!\!\!\mathrm{D}^b(S_1,\alpha_1),&&\\
&\!\!\!\mathrm{D}^b(S_1,\beta_1)\simeq&\!\!\!\mathrm{D}^b(S_2,\alpha_2) ,&\\
&&\,\,\,\vdots&\\
&&\!\!\!\mathrm{D}^b(S_{n-2},\beta_{n-2})\simeq&\!\!\!\mathrm{D}^b(S_{n-1},\alpha_{n-1}),\\
&&&\!\!\!\mathrm{D}^b(S_{n-1},\beta_{n-1})\simeq\mathrm{D}^b(S',\alpha').
\end{array}$$ Note that by [@cs] any exact linear equivalence between bounded derived categories of twisted coherent sheaves on smooth projective varieties is of Fourier–Mukai type, so that in , each equivalence $\mathrm{D}^b(S_i,\beta_i) \stackrel{\sim}{\longrightarrow}
\mathrm{D}^b(S_{i+1},\alpha_{i+1})$ is induced by a Fourier–Mukai kernel $\mathcal{E}_i \in \mathrm{D}^b(S_i\times S_{i+1},\beta_i^{-1} \boxtimes
\alpha_{i+1})$ (unique up to isomorphism).
Combined with his previous work [@huybrechts-derivedeq] generalized to the twisted case, Huybrechts deduced that isogenous complex projective K3 surfaces have isomorphic Chow motives[^3]. However, the converse does not hold in general: there are K3 surfaces having isomorphic Chow motives (hence isomorphic rational Hodge structures) without being isogenous. Examples of such K3 surfaces were constructed geometrically in [@BSV]; see Remark \[R:BSV\] of Appendix \[sect:Appendix\]. We provide in Theorem \[thm:IsomChowMot\] two further constructions of infinite families of pairwise non-isogenous K3 surfaces with isomorphic Chow motives. The motivation of the paper is to complete the picture by giving a motivic characterization of isogenous K3 surfaces (Corollary \[cor:torelli\]). Our main result is the following:
\[thm:main\] Let $S$ and $S'$ be two twisted derived equivalent K3 surfaces over a field $k$. Then the Chow motives of $S$ and $S'$ are isomorphic as algebra objects, in fact even as Frobenius algebra objects (Definition \[def:FrobAlg\]), in the category of rational Chow motives over $k$.
In concrete terms, there exists a correspondence $\Gamma \in {\ensuremath\mathrm{CH}}^{2}(S\times
S')$ with $\Gamma \circ \delta_S = \delta_{S'} \circ (\Gamma \otimes \Gamma)$ (“algebra homomorphism”) and such that $\Gamma$ is invertible as a morphism between the Chow motives of $S$ and $S'$ with $\Gamma^{-1}={}^{t}\Gamma$ (“orthogonality”), where ${}^t\Gamma$ denotes the transpose of $\Gamma$ and where $\delta_S$ is the small diagonal in $S \times S\times S$ viewed as a correspondence between $S\times S$ and $S$. Equivalently, $\Gamma$ is an isomorphism such that $(\Gamma\otimes\Gamma)_{*}\Delta_{S}=\Delta_{S'}$ and $(\Gamma\otimes\Gamma\otimes\Gamma)_{*}\delta_{S}=\delta_{S'}$. The notion of Frobenius algebra object provides a conceptual way to pack these conditions. Roughly speaking, a Frobenius algebra object in a rigid tensor category is an algebra object together with an isomorphism to its dual object[^4] with some compatibility conditions. The Chow motive of a smooth projective variety carries a natural structure of Frobenius algebra object in the rigid tensor category of Chow motives. We refer to § \[sec:Frob\] for more details on Frobenius algebra objects.
Note that a particular consequence of Theorem \[thm:main\] is that the induced action $\Gamma_* : {\ensuremath\mathrm{CH}}^*(S)\to {\ensuremath\mathrm{CH}}^*(S')$ on the Chow rings is an isomorphism of graded ${\ensuremath\mathds{Q}}$-algebras – this can in fact be deduced from the previous work of Huybrechts [@huybrechts-derivedeq]; see Remark \[rmk:ChowRingIso\]. However, having an isomorphism of Frobenius algebra objects allows us to derive the following much stronger result:
\[cor:Powers\] Let $S$ and $S'$ be two twisted derived equivalent K3 surfaces defined over a field $k$. Then for any positive integers $n_{1}, \ldots, n_{r}$, there is an isomorphism of Frobenius algebra objects $${\ensuremath\mathfrak{h}}\left(\operatorname{Hilb}^{n_{1}}(S)\times
\cdots
\times \operatorname{Hilb}^{n_{r}}(S)\right)\simeq {\ensuremath\mathfrak{h}}\left(\operatorname{Hilb}^{n_{1}}(S')\times \cdots
\times \operatorname{Hilb}^{n_{r}}(S')\right).$$ As a consequence, there is an algebraic correspondence inducing an isomorphism of *graded* ${\ensuremath\mathds{Q}}$-algebras: $${\ensuremath\mathrm{CH}}^{*}\left(\operatorname{Hilb}^{n_{1}}(S)\times \cdots \times
\operatorname{Hilb}^{n_{r}}(S)\right)\simeq {\ensuremath\mathrm{CH}}^{*}\left(\operatorname{Hilb}^{n_{1}}(S')\times \cdots
\times
\operatorname{Hilb}^{n_{r}}(S')\right).$$ Here $\operatorname{Hilb}^{n}(S)$ denotes the Hilbert scheme of length-$n$ subschemes on $S$.
Now let the base field be the field of complex numbers ${\ensuremath\mathds{C}}$. Combining Theorem \[thm:main\] with Huybrechts’ twisted derived global Torelli theorem mentioned above, we can establish:
\[cor:torelli\] Let $S$ and $S'$ be two complex projective K3 surfaces. The following statements are equivalent:
(i) $S$ and $S'$ are isogenous;
(ii) $S$ and $S'$ are twisted derived equivalent;
(iii) ${\ensuremath\mathfrak{h}}(S)$ and ${\ensuremath\mathfrak{h}}(S')$ are isomorphic as Frobenius algebra objects.
In the appendix, we construct in Theorem \[thm:IsomChowMot\] an infinite family of pairwise non-isogenous K3 surfaces whose motives are all isomorphic as algebra objects. This justifies introducing the Frobenius structure. In addition, Proposition \[prop:IsoChowRing3\] gives some evidence that the isogeny class of a K3 surface cannot be determined by its Chow ring.
Finally, with integral coefficients, an algebra isomorphism between the motives of two K3 surfaces must respect the Frobenius structure. Therefore, the classical global Torelli theorem [@PSS] can be upgraded to a *motivic global Torelli theorem*: $$\label{eq:MotGlobTorelli}
\text{$S$ and $S'$ are isomorphic} \Longleftrightarrow \text{their integral
Chow
motives are isomorphic as algebra objects.}$$ We refer to Theorem \[thm:motglobtor\] for a proof.
Orlov conjecture and multiplicative structure {#orlov-conjecture-and-multiplicative-structure .unnumbered}
---------------------------------------------
The proof of Theorem \[thm:main\] relies on the Beauville–Voisin decomposition of the small diagonal of a K3 surface (see Theorem \[thm:bv\]): given an exact linear equivalence $
\operatorname{D}^b(S,\alpha)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S',\alpha')$ between twisted K3 surfaces, we are reduced to exhibiting a correspondence $\Gamma \in
{\ensuremath\mathrm{CH}}^2(S\times S')$ such that $\Gamma \circ {}^t\Gamma = \Delta_{S'}$, ${}^t\Gamma \circ \Gamma = \Delta_{S}$ and $\Gamma_*o_{S} = o_{S'}$, where $o_{S}=\frac{1}{24}c_{2}(S)$ is the Beauville–Voisin 0-cycle [@bv]. The key point then consists in showing that if $v_2(\mathcal{E})$ denotes the dimension-2 component of the Mukai vector of the Fourier–Mukai kernel of an exact linear equivalence $\Phi_{\mathcal{E}} : \operatorname{D}^b(S,\alpha)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S',\alpha')$ between twisted smooth projective surfaces, then $v_2(\mathcal{E})$ induces an isomorphism ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S) \stackrel{\sim}{\longrightarrow}
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$ of the transcendental motives of $S$ and $S'$ with inverse given by $v_2(\mathcal{E}^\vee\otimes p^* \omega_S)$, where $\mathcal{E}^\vee$ denotes the derived dual of $\mathcal{E}$ and $p : S\times S' \to S$ is the natural projection. (In the case of K3 surfaces, we have $v_2(\mathcal{E}^\vee\otimes p^* \omega_S)= {}^tv_2(\mathcal{E})$.) This is achieved by exploiting known cases of Murre’s Conjecture \[conj:murre\], and we thereby give an alternative proof of Huybrechts’ [@huybrechts-derivedeq Theorem 0.1], generalized to all surfaces: *two twisted derived equivalent smooth projective surfaces have isomorphic Chow motives*; see Theorem \[thm:huybrechts\]. This confirms the two-dimensional case of the following conjecture due to Orlov:
\[conj:orlov\] Let $X$ and $Y$ be two derived equivalent smooth projective varieties. Then their Chow motives are isomorphic.
We illustrate also in §\[subsec:3-4\] how the same techniques can be used to establish Orlov’s Conjecture \[conj:orlov\] in some new cases in dimension 3 and 4; see Proposition \[prop:generalization\].
In view of Theorem \[thm:main\], we naturally ask that under what circumstances one could expect a “multiplicative Orlov conjecture”, namely whether two derived equivalent smooth projective varieties have isomorphic Chow motives as algebra objects, or even as Frobenius algebra objects. According to the celebrated theorem of Bondal–Orlov [@bo], this holds true for varieties with ample or anti-ample canonical bundle, since any two such derived equivalent varieties must be isomorphic. The situation gets more intriguing for varieties with trivial canonical bundle and we cannot expect in general that derived equivalent varieties have isomorphic Chow motives as Frobenius algebra objects: there exists counter-examples for Calabi–Yau threefolds and abelian varieties, where even the graded cohomology algebras of the two derived equivalent varieties are not isomorphic as Frobenius algebras (see Example \[ex:BC\] and Proposition \[prop:AV\] $(ii)$). We notice that, on the other hand, two derived equivalent abelian varieties are isogenous and have isomorphic Chow motives as algebra objects (see Proposition \[prop:AV\] $(i)$).
Although we do not provide much evidence beyond the case of K3 surfaces, we are tempted to ask, because of the (expected) similarities of the intersection product on hyper-Kähler varieties with that on abelian varieties (*cf.* Beauville’s seminal [@beauville-splitting], and also [@sv; @fv]), the following
\[conj:main\] Let $X$ and $Y$ be two twisted derived equivalent projective hyper-Kähler varieties. Are their Chow motives isomorphic as algebra objects or even as Frobenius algebra objects? In particular, are their cohomology ${\ensuremath\mathrm{H}}^*(-,{\ensuremath\mathds{Q}})$ isomorphic as graded ${\ensuremath\mathds{Q}}$-algebras or even as Frobenius algebras?
Corollary \[cor:Powers\] gives an example in higher dimensions. See § \[sect:MultOrlov\] for other examples, conjectures and rudiment discussions on this subject.
Canonicity of the Shafarevich cycle {#canonicity-of-the-shafarevich-cycle .unnumbered}
-----------------------------------
In [@huybrechts-isogenous], Huybrechts shows that the restriction to the transcendental cohomology of an isogeny $\varphi\colon
{\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})$ is induced by the cycle $v_2(\mathcal{E}_{n-1}) \circ \cdots \circ
v_2(\mathcal{E}_{0}) \in {\ensuremath\mathrm{CH}}^2(S\times S')$, where $\mathcal{E}_0,\ldots,\mathcal{E}_{n-1}$ are the Fourier–Mukai kernels in . This provides a Shafarevich cycle for the isogeny $\varphi$. In §\[sec:v2\], we give some evidence for the above cycle to be canonical, that is, independent of the choice of a chain of twisted derived equivalence inducing the isogeny. This depends on extending a result of Huybrechts and Voisin (Theorem \[thm:hv\]) to twisted equivalences. We do however prove unconditionally in Theorem \[thm:product-c2\] that the intersection of the second Chern classes of two objects $\mathcal{E}_1$ and $\mathcal{E}_2$ in $\mathrm{D}^b(S\times S')$ inducing an equivalence $\mathrm{D}^b(S) \stackrel{\sim}{\longrightarrow}
\mathrm{D}^b(S')$ is proportional to $c_2(S)\times c_2(S')$ in ${\ensuremath\mathrm{CH}}^2(S\times S')$. This suggests that the Mukai vectors of twisted derived equivalences between K3 surfaces can be added to the Beauville–Voisin ring; see §\[sec:v2\].
Notation and Conventions {#notation-and-conventions .unnumbered}
------------------------
We fix a base field $k$. By a derived equivalence between smooth projective $k$-varieties, we mean a $k$-linear exact equivalence of triangulated categories between their bounded derived categories of coherent sheaves. Chow groups will always be considered with rational coefficients. Concerning the category of Chow motives over $k$, we follow the notation and conventions of [@andre]. This category is a pseudo-abelian rigid tensor category, whose objects consist of triples $(X,p,n)$, where $X$ is a smooth projective variety of dimension $d_X$ over $k$, $p\in
{\ensuremath\mathrm{CH}}^{d_X}(X\times_k X)$ with $p\circ p = p$, and $n\in {\ensuremath\mathds{Z}}$. Morphisms $f:
M=(X,p,n) \to N=(Y,q,m)$ are elements $\gamma \in {\ensuremath\mathrm{CH}}^{d_X+m-n}(X\times_k Y)$ such that $\gamma \circ p = q\circ \gamma = \gamma$. The tensor product of two motives is defined in the obvious way, while the dual of $M=(X, p, n)$ is $M^\vee =
(X,{}^tp,-n+d_X)$, where ${}^tp$ denotes the transpose of $p$. The Chow motive of a smooth projective variety $X$ is defined as ${\ensuremath\mathfrak{h}}(X) := (X,\Delta_X,0)$, where $\Delta_X$ denotes the class of the diagonal inside $X\times X$, and the *unit motive* is denoted $\mathds{1} := {\ensuremath\mathfrak{h}}(\operatorname{Spec}(k))$. In particular, we have ${\ensuremath\mathrm{CH}}^l(X) = \operatorname{Hom}(\mathds{1}(-l),{\ensuremath\mathfrak{h}}(X))$. The *Tate motive* of weight $-2i$ is the motive $\mathds{1}(i) := (\operatorname{Spec}(k),
\Delta_{\operatorname{Spec}(k)},
i)$. A motive is said to be of *Tate type* if it is isomorphic to a direct sum of Tate motives.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank Benjamin Bakker, Chiara Camere and Jean-Yves Welschinger for helpful discussions on the Appendix. This paper was completed at the Institut Camille Jordan in Lyon where the second author’s stay was supported by the CNRS. The first author is supported by ECOVA (ANR-15-CE40-0002), HodgeFun (ANR-16-CE40-0011), LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon and *Projet Inter-Laboratoire* 2019 by Fédération de Recherche en Mathématiques Rhône-Alpes/Auvergne CNRS 3490.
Derived equivalent surfaces
===========================
The aim of this section is to provide an alternative proof to the following result of Huybrechts [@huybrechts-derivedeq; @huybrechts-isogenous]:
\[thm:huybrechts\] Let $S$ and $S'$ be two (twisted) derived equivalent smooth projective surfaces defined over a field $k$. Then $S$ and $S'$ have isomorphic Chow motives.
The reason for including such a proof is threefold: first it provides anyway all the prerequisites and notation for the proof of Theorem \[thm:main\] which will be given in §\[S:mainthm\]; second by avoiding Manin’s identity principle[^5] (as is employed in [@huybrechts-derivedeq]) we obtain an explicit inverse to the isomorphism $v_2(\mathcal{E}) : {\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)
\stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$ which will be essential to the proof of Theorem \[thm:main\]; and third it provides somehow a link between Orlov’s Conjecture \[conj:orlov\] and Murre’s Conjecture \[conj:murre\] which itself is intricately linked to the conjectures of Bloch and Beilinson (see [@jannsen]).
Murre’s conjectures
-------------------
We fix a base field $k$ and a Weil cohomology theory ${\ensuremath\mathrm{H}}^*(-)$ for smooth projective varieties over $k$. Concretely, we think of ${\ensuremath\mathrm{H}}^*(-)$ as Betti cohomology in case $k\subseteq {\ensuremath\mathds{C}}$, or as $\ell$-adic cohomology when $\mathrm{char}(k) \neq
\ell$.
\[conj:murre\] Let $X$ be a smooth projective variety of dimension $d$ over $k$.
(A) \[A\] The Chow motive ${\ensuremath\mathfrak{h}}(X)$ has a Chow–Künneth decomposition (also called *weight decomposition*) ${\ensuremath\mathfrak{h}}(X) =
{\ensuremath\mathfrak{h}}^0(X) \oplus \cdots \oplus {\ensuremath\mathfrak{h}}^{2d}(X)$, meaning that ${\ensuremath\mathrm{H}}^*({\ensuremath\mathfrak{h}}^i(X)) =
{\ensuremath\mathrm{H}}^i(X)$ for all $i$.
(B) \[B\] ${\ensuremath\mathrm{CH}}^l({\ensuremath\mathfrak{h}}^i(X)) := \operatorname{Hom}(\mathds{1}(-l),{\ensuremath\mathfrak{h}}^i(X)) = 0$ for all $i>2l$ and for all $i<l$.
(C) \[C\] The filtration $\mathrm{F}^k{\ensuremath\mathrm{CH}}^l(X) := {\ensuremath\mathrm{CH}}^l\big(
\bigoplus_{i\leq 2l-k} {\ensuremath\mathfrak{h}}^i(X) \big)$ does not depend on the choice of a Chow–Künneth decomposition.
(D) \[D\] $\mathrm{F}^1{\ensuremath\mathrm{CH}}^l(X) = {\ensuremath\mathrm{CH}}^l(X)_{\mathrm{hom}} := \ker
({\ensuremath\mathrm{CH}}^l(X) \xrightarrow{\operatorname{cl}}
{\ensuremath\mathrm{H}}^{2l}(X))$.
The filtration defined in is conjecturally the Bloch–Beilinson filtration [@Beilinson] (see also [@VoisinBook Chapter 11]). In fact, as shown by Jannsen [@jannsen], the conjecture of Murre holds for all smooth projective varieties if and only if the conjectures of Bloch–Beilinson hold. We refer to [@jannsen] for precise statements.
\[prop:formal\] Let $X$ and $Y$ be smooth projective varieties over $k$. Assume that ${\ensuremath\mathfrak{h}}(X)$ and ${\ensuremath\mathfrak{h}}(Y)$ admit a Chow–Künneth decomposition as in Conjecture \[conj:murre\]. Then, with respect to the Chow–Künneth decomposition ${\ensuremath\mathfrak{h}}^n(X\times Y) = \bigoplus_{i+j = n}
{\ensuremath\mathfrak{h}}^{2d_X-i}(X)^\vee(-d_X)\otimes {\ensuremath\mathfrak{h}}^j(Y)$, $X\times Y$ satisfies Conjecture \[conj:murre\] if and only if $$\label{eq}
\operatorname{Hom}({\ensuremath\mathfrak{h}}^i(X),{\ensuremath\mathfrak{h}}^j(Y)(k))= 0\quad \text{for all } i<j-2k \text{ and for all
}
i>j+d_X-k.$$
This is formal: we have $$\begin{aligned}
\operatorname{Hom}(\mathds{1}(-k-d_X), {\ensuremath\mathfrak{h}}^{n}(X\times Y))
& = \bigoplus_{i-j=2d_{X}-n} \operatorname{Hom}(\mathds{1}(-k), {\ensuremath\mathfrak{h}}^j(Y)\otimes
{\ensuremath\mathfrak{h}}^i(X)^\vee)\\
& = \bigoplus_{i-j=2d_{X}-n}\operatorname{Hom}({\ensuremath\mathfrak{h}}^i(X),{\ensuremath\mathfrak{h}}^j(Y)(k)).
\end{aligned}$$
In other words, Murre’s conjecture implies that a motive of pure weight does not admit any non-trivial morphism to a motive of pure larger weight.
\[thm:product\] Let $X$ be a smooth projective irreducible variety of dimension $d_X$ over a field $k$.
(i) \[i\] The Chow motive of $X$ admits a decomposition $${\ensuremath\mathfrak{h}}(X) =
{\ensuremath\mathfrak{h}}^0(X) \oplus {\ensuremath\mathfrak{h}}^1(X) \oplus M \oplus {\ensuremath\mathfrak{h}}^{2d_X-1}(X) \oplus {\ensuremath\mathfrak{h}}^{2d_X}(X)$$ such that ${\ensuremath\mathrm{H}}^*({\ensuremath\mathfrak{h}}^i(X)) = {\ensuremath\mathrm{H}}^i(X)$; in particular, Conjecture \[conj:murre\] holds for curves and surfaces. Moreover, such a decomposition can be chosen such that
- ${\ensuremath\mathfrak{h}}^{2d_X}(X)(d_X) = {\ensuremath\mathfrak{h}}^0(X)^\vee \simeq {\ensuremath\mathfrak{h}}^0(X)$ and ${\ensuremath\mathfrak{h}}^{2{d_X}-1}(X)(d_X) = {\ensuremath\mathfrak{h}}^1(X)^\vee \simeq {\ensuremath\mathfrak{h}}^1(X)(1)$;
- ${\ensuremath\mathfrak{h}}^0(X)$ is the unit motive $\mathds 1$ and ${\ensuremath\mathfrak{h}}^1(X) \simeq
{\ensuremath\mathfrak{h}}^1(\operatorname{Pic}^0(X)_{\mathrm{red}})$;
- $\operatorname{Hom}(\mathds{1}(-i),{\ensuremath\mathfrak{h}}^1(X)) = 0$ for $i\neq
1$, and $\operatorname{Hom}(\mathds{1}(-1),{\ensuremath\mathfrak{h}}^1(X)) = \operatorname{Pic}^0(X)_{\mathrm{red}}(k)\otimes
{\ensuremath\mathds{Q}}$.
(ii) \[ii\]Equation holds in case $X$ and $Y$ are varieties of dimension $\leq 2$ endowed with a Chow–Künneth decomposition as in .
Item in the case of surfaces is the main result of [@murre-surfaces]. In fact, for any smooth projective variety $X$ of any dimension, ${\ensuremath\mathfrak{h}}^1(X)$ can be constructed as a direct summand of the motive of a smooth projective curve. As for , this was checked by Murre [@murre2] in the case one of $X$ and $Y$ has dimension $\leq 1$. Thanks to item and Proposition \[prop:formal\], it only remains to check that ${\ensuremath\mathrm{CH}}^l({\ensuremath\mathfrak{h}}^2(X) \otimes {\ensuremath\mathfrak{h}}^2(Y)) = 0$ for $l = 0,1$ for smooth projective surfaces $X$ and $Y$. For that purpose, we simply observe that for any choice of a Chow–Künneth decomposition (if it exists) on the motive of a smooth projective variety $Z$ we have ${\ensuremath\mathrm{CH}}^0(Z) = {\ensuremath\mathrm{CH}}^0({\ensuremath\mathfrak{h}}^0(Z))$ and ${\ensuremath\mathrm{CH}}^1(Z) =
{\ensuremath\mathrm{CH}}^1({\ensuremath\mathfrak{h}}^2(Z) \oplus {\ensuremath\mathfrak{h}}^1(Z))$. (Indeed, denote $\pi^i_Z$ the projectors corresponding to the Chow–Künneth decomposition of $Z$, then by definition $\pi^{2i}_Z$ acts as the identity on ${\ensuremath\mathrm{H}}^{2i}(Z)$ and hence on $\mathrm{im}({\ensuremath\mathrm{CH}}^i(Z) \to {\ensuremath\mathrm{H}}^{2i}(Z))$, and by Murre [@murre-surfaces] $\pi^1_Z$ acts as the identity on $\mathrm{ker}({\ensuremath\mathrm{CH}}^1(Z) \to {\ensuremath\mathrm{H}}^{2}(Z))$. Therefore $\pi^2_Z
+
\pi^1_Z$, which is a projector, acts as the identity on ${\ensuremath\mathrm{CH}}^1(Z)$.)
The following terminology will be convenient for our purpose. We say that a Chow motive $M$ is of *curve type* (or of *pure weight 1*) if it is isomorphic to a direct summand of a direct sum of motives of the form ${\ensuremath\mathfrak{h}}^1(C)$, where $C$ is a smooth projective curve defined over $k$. Motives of curve type form a thick additive subcategory[^6] and enjoy the following property, which is also shared by Tate motives:
\[prop:curve\] The full subcategory of motives whose objects are of curve type is abelian semi-simple. Moreover, the realization functor $M \mapsto {\ensuremath\mathrm{H}}^*(M)$ is conservative.
The first statement follows from the fact that this full subcategory of motives of curve type is equivalent to the category of abelian varieties up to isogeny, via the Jacobian construction; see [@andre Proposition 4.3.4.1]. The second statement follows from the first one together with the fact that ${\ensuremath\mathrm{H}}^{*}({\ensuremath\mathfrak{h}}^{1}(A))$ is a $2g$-dimensional vector space for an abelian variety $A$ of dimension $g$.
Proof of Theorem \[thm:huybrechts\]
-----------------------------------
First, we observe as in [@huybrechts-isogenous Section 2] that for twisted equivalences the yoga of Fourier–Mukai kernels, their action on Chow groups induced by Mukai vectors, and how they behave under convolutions works as in the untwisted case. Therefore, for ease of notation, we will only give a proof of Theorem \[thm:huybrechts\] in the untwisted case.
### Derived equivalences and motives, following Orlov. {#subsec:orlov}
In general, let $\Phi_{\mathcal{E}} : \mathrm{D}^b(X)
\stackrel{\sim}{\longrightarrow} \mathrm{D}^b(Y)$ be an exact equivalence with Fourier–Mukai kernel $\mathcal{E} \in \mathrm{D}^b(X\times Y)$ between the derived categories of two smooth projective $k$-varieties of dimension $d$. Its inverse can be described as $\Phi_{\mathcal{E}}^{-1} \simeq
\Phi_{\mathcal{E}^\vee\otimes p^*\omega_X[d]}\simeq
\Phi_{\mathcal{E}^\vee\otimes q^*\omega_Y[d]}$, where $\mathcal{E}^\vee$ is the derived dual of $\mathcal{E}$ and $p, q$ are the projections from $X\times Y$ to $X$ and $Y$ respectively. As observed by Orlov [@orlov], the Mukai vector $$v(\mathcal{E}) := \mathrm{ch}(\mathcal{E})\cdot \sqrt{\mathrm{td}(X\times
Y)} \in {\ensuremath\mathrm{CH}}^{*}(X\times Y)$$ induces a split injective morphism of motives ${\ensuremath\mathfrak{h}}(X) \longrightarrow \bigoplus_{i=-d}^{d} {\ensuremath\mathfrak{h}}(Y)(i)$ with left inverse given by $v(\mathcal{E}^\vee\otimes p^*\omega_X[d])$, *i.e.* $$\xymatrix{\mathrm{id} : \ {\ensuremath\mathfrak{h}}(X) \ar[rrr]^{v(\mathcal{E})\quad } &&&
\bigoplus_{i=-d}^{d} {\ensuremath\mathfrak{h}}(Y)(i) \ar[rrr]^{\qquad v(\mathcal{E}^*\otimes
p^*\omega_X[d])} && & {\ensuremath\mathfrak{h}}(X).
}$$ In particular, $v(\mathcal{E}^\vee\otimes p^*\omega_X[d]) \circ v(\mathcal{E})
=
\Delta_{X}$. In fact, as observed by Orlov [@orlov], the latter identity shows that $v(\mathcal E)$ seen as a morphism of ind-motives $ \bigoplus_{i\in
{\ensuremath\mathds{Z}}} {\ensuremath\mathfrak{h}}(X)(i) \to \bigoplus_{i\in {\ensuremath\mathds{Z}}} {\ensuremath\mathfrak{h}}(Y)(i)$ is an isomorphism with inverse given by $v(\mathcal{E}^\vee\otimes p^*\omega_X[d])$.
### The refined decomposition of the motive of surfaces, following Kahn–Murre–Pedrini {#subsec:kmp}
Let $S$ be a smooth projective surface over $k$. The motive ${\ensuremath\mathfrak{h}}(S)$ admits a Chow–Künneth decomposition as in Murre’s Theorem \[thm:product\]; in particular ${\ensuremath\mathfrak{h}}^0(X) = {\ensuremath\mathfrak{h}}^4(X)(2)^\vee$ is the unit motive $\mathds 1$ and ${\ensuremath\mathfrak{h}}^1(X) =
{\ensuremath\mathfrak{h}}^3(X)(2)^\vee$ is of curve type. Following Kahn–Murre–Pedrini [@kmp], the summand ${\ensuremath\mathfrak{h}}^2(S)$ admits a further decomposition $${\ensuremath\mathfrak{h}}^2(S) = {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S) \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)$$ defined as follows. Let $k^s$ be a separable closure of $k$ and let $E_1,\ldots,E_\rho$ be non-isotropic divisors in ${\ensuremath\mathrm{CH}}^1(S_{k^s})$ whose images in $\mathrm{NS}(X_{k^s})_{\ensuremath\mathds{Q}}$ form an orthogonal basis. Up to replacing each $E_i$ by $E_i - (\pi^1_S)_*E_i$, we can assume that $(\pi^1_S)_*E_i = 0$ for all $i$. Consider then the idempotent correspondence $$\pi^2_{\mathrm{alg},S} := \sum_{i=1}^\rho \frac{1}{\deg (E_i\cdot E_i)}E_i
\times
E_i.$$ Since $\pi^2_{\mathrm{alg},S}$ is the intersection form on $\mathrm{NS}(X_{k^s})_{\ensuremath\mathds{Q}}$, it is Galois-invariant, and hence does define an idempotent in ${\ensuremath\mathrm{CH}}^2(S\times
S)$. The motive $(S,\pi^2_{\mathrm{alg},S},0)$ is clearly isomorphic, after base-change to $k^s$, to the direct sum of $\rho$ copies of the Tate motive $\mathds{1}(-1)$. Moreover, it is easy to check that $\pi^2_{\mathrm{alg},S} $ is orthogonal to $\pi^i_S$ for $i\neq 2$ (use $(\pi^1_S)_*E_i = 0$). Equivalently, we have $\pi^2_{\mathrm{alg},S} \circ \pi_S^2 = \pi^2_S \circ
\pi^2_{\mathrm{alg},S} $, so that $(S,\pi^2_{\mathrm{alg},S},0)$ does define a direct summand of ${\ensuremath\mathfrak{h}}^2(S)$, denoted by ${\ensuremath\mathfrak{h}}^{2}_{\mathrm{alg}}(S)$. We then define $\pi^2_{\mathrm{tr},S} := \pi^2_S -
\pi^2_{\mathrm{alg},S} $ and ${\ensuremath\mathfrak{h}}^{2}_{\mathrm{tr}}(S):=(S,
\pi^2_{\mathrm{tr},S}, 0)$. It is then straightforward to check that such a decomposition satisfies $\operatorname{Hom}(\mathds{1}(-i),{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)) = 0$ for all $i\neq 2$. We note that ${}^t \pi^2_{\mathrm{alg},S} = \pi^2_{\mathrm{alg},S} $, and since ${}^t \pi_S^0 = \pi_S^4$ and ${}^t \pi_S^1 = \pi_S^3$ we also have ${}^t
\pi^2_{\mathrm{tr},S} = \pi^2_{\mathrm{tr},S}$. Moreover, although this won’t be of any use to us, we mention for comparison to [@huybrechts-derivedeq] that $\operatorname{Hom}(\mathds{1}(-2),{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)) $ coincides with the Albanese kernel.
More generally, the above refined decomposition can be performed for direct summand of motives of surfaces, *i.e.* for motives of the form $(S,p,0)$, where $S$ is a smooth projective $k$-surface and $p\in {\ensuremath\mathrm{CH}}^2(S\times S)$ is an idempotent. This will be used in the proof of Proposition \[prop:generalization\]. Indeed, by the above together with Theorem \[thm:product\], we have a decomposition $$\label{eq:dec}
{\ensuremath\mathfrak{h}}(S) = {\ensuremath\mathfrak{h}}^0(S) \oplus {\ensuremath\mathfrak{h}}^1(S) \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S) \oplus
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S) \oplus {\ensuremath\mathfrak{h}}^3(S) \oplus {\ensuremath\mathfrak{h}}^4(S),$$ where none of the direct summands admit a non-trivial morphism to another direct summand placed on its right. It follows that the morphism $p$, expressed with respect to the decomposition is upper-triangular. By [@vial-3-4 Lemma 3.1], the motive $M= (S,p,0)$ admits a weight decomposition $M=M^0\oplus M^1 \oplus M^2_{\mathrm{alg}} \oplus
M^2_{\mathrm{tr}} \oplus M^3 \oplus M^4$, where each factor is isomorphic to a direct summand of the corresponding factor in the decomposition . In particular, this decomposition of $M$ inherits the properties of the decomposition , *e.g.* $M^0$ and $M^4$ are of Tate type, $M^2_{\mathrm{alg}}$ becomes of Tate type after base-change to $k^s$ and $M^1$ and $M^3(1)$ are of curve type.
### A weight argument {#subsec:weight}
Thanks to Theorem \[thm:product\], $v(\mathcal{E})$ maps ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)$ possibly non-trivially only in summands of $$\label{eq:sum}
\bigoplus_{i=-2}^2
\big( {\ensuremath\mathfrak{h}}^0(S')(i) \oplus {\ensuremath\mathfrak{h}}^1(S')(i) \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S')(i) \oplus
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')(i) \oplus {\ensuremath\mathfrak{h}}^3(S')(i) \oplus {\ensuremath\mathfrak{h}}^4(S')(i)\big)$$ of weight $\leq
2$. Since $\operatorname{Hom}({\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S),\mathds{1}(-1)) = \operatorname{Hom}(\mathds{1}(1),
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)^\vee) = \operatorname{Hom}(\mathds{1}(-1), {\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)) =
0$, we see that ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$ is the only direct summand of weight $2$ in that admits a possibly non-trivial morphism from ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)$. Likewise, the only direct summand of of weight $\leq 2$ that maps possibly non-trivially in ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)$ via $v(\mathcal{E}^\vee\otimes p^*\omega_S)$ is ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$. It follows that the restriction of $v_2(\mathcal{E}) $ induces an isomorphism $$\pi_{\mathrm{tr},S'}^2\circ v_2(\mathcal{E}) \circ \pi_{\mathrm{tr},S}^2:
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)
\stackrel{\sim}{\longrightarrow}
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$$ with inverse $\pi_{\mathrm{tr},S}^2\circ
v_2(\mathcal{E}^\vee\otimes p^*\omega_S)\circ \pi_{\mathrm{tr}, S'}^2$; this fact will be used in the proof of Theorem \[thm:main\].
In a similar fashion, thanks to Theorem \[thm:product\] and having in mind that ${\ensuremath\mathfrak{h}}^1(S) \simeq {\ensuremath\mathfrak{h}}^3(S)(1)$ is the direct summand of the motive of a curve (and similarly for $S'$), $v(\mathcal{E})$ induces isomorphisms $${\ensuremath\mathfrak{h}}^0(S) \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S)(1) \oplus {\ensuremath\mathfrak{h}}^4(S)(2)
\stackrel{\sim}{\longrightarrow}
{\ensuremath\mathfrak{h}}^0(S') \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S')(1) \oplus {\ensuremath\mathfrak{h}}^4(S')(2)$$ and $${\ensuremath\mathfrak{h}}^1(S) \oplus {\ensuremath\mathfrak{h}}^3(S)(1) \stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}^1(S')
\oplus{\ensuremath\mathfrak{h}}^3(S')(1).$$ The first isomorphism yields an isomorphism ${\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S) \simeq
{\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S')$, while the second one yields, thanks to Theorem \[thm:product\], together with the semi-simplicity statement of Proposition \[prop:curve\], isomorphisms ${\ensuremath\mathfrak{h}}^1(S) \simeq
{\ensuremath\mathfrak{h}}^1(S')$ and ${\ensuremath\mathfrak{h}}^3(S) \simeq {\ensuremath\mathfrak{h}}^3(S')$. This finishes the proof of Theorem \[thm:huybrechts\].
A slight generalization to Theorem \[thm:huybrechts\] {#subsec:3-4}
-----------------------------------------------------
The content of this paragraph won’t be used in the proof of Theorem \[thm:main\]. Recall that Theorem \[thm:huybrechts\] fits more generally into the Orlov Conjecture \[conj:orlov\].
The method of proof of Theorem \[thm:huybrechts\] can be pushed through to establish the following:
\[prop:generalization\] Let $X$ and $Y$ be two smooth projective varieties of dimension 3 or 4 over a field $k$. Assume either of the following:
- $\dim X = 3$ and ${\ensuremath\mathrm{CH}}_0(X)$ is representable;
- $\dim X=4$, ${\ensuremath\mathrm{CH}}_0(X)$ and ${\ensuremath\mathrm{CH}}_0(Y)$ are both representable, and $X$ and $Y$ have same Picard rank.
Then $\mathrm{D}^b(X)\simeq \mathrm{D}^b(Y)$ implies that $ \mathfrak{h}(X)
\simeq
\mathfrak{h}(Y)$.
Here, we say that a smooth projective $k$-variety $X$ of dimension $d$ has *representable* ${\ensuremath\mathrm{CH}}_0$ if for a choice of universal domain (*i.e.*, algebraically closed field of infinite transcendence degree over its prime subfield) $\Omega$ containing $k$, there exists a smooth projective $\Omega$-curve $C$ and a correspondence $\gamma \in \operatorname{Hom}({\ensuremath\mathfrak{h}}(X_\Omega),{\ensuremath\mathfrak{h}}(C))$ such that $\gamma^*{\ensuremath\mathrm{CH}}_0(C) = {\ensuremath\mathrm{CH}}_0(X_\Omega)$. Examples of such varieties include varieties with maximally rationally connected quotient of dimension $\leq 1$, and in particular rationally connected varieties.
We start with the case of threefolds. By [@gg], $X$ admits a Chow–Künneth decomposition, where the even-degree summands are of Tate type, while the odd-degree summands are Tate twists of motives of curve type. The arguments of §\[subsec:orlov\] show that ${\ensuremath\mathfrak{h}}(Y)$ is a direct summand of $\bigoplus_{i=-3}^3 {\ensuremath\mathfrak{h}}(X)(i)$; in particular, by Kimura finite-dimensionality (or by Theorem \[thm:product\] together with [@vial-3-4 Lemma 3.1] as used in §\[subsec:kmp\]), ${\ensuremath\mathfrak{h}}(Y)$ has a Chow–Künneth decomposition with a similar property to that of $X$ (and hence has representable ${\ensuremath\mathrm{CH}}_0$). The arguments of §\[subsec:weight\] provide isomorphisms $${\ensuremath\mathfrak{h}}^0(X) \oplus {\ensuremath\mathfrak{h}}^2(X)(1) \oplus {\ensuremath\mathfrak{h}}^4(X)(2) \oplus {\ensuremath\mathfrak{h}}^6(X)(3) \simeq
{\ensuremath\mathfrak{h}}^0(Y)
\oplus {\ensuremath\mathfrak{h}}^2(Y)(1) \oplus {\ensuremath\mathfrak{h}}^4(Y)(2) \oplus {\ensuremath\mathfrak{h}}^6(Y)(3)$$ and $$\label{eq:odd}
{\ensuremath\mathfrak{h}}^1(X) \oplus {\ensuremath\mathfrak{h}}^3(X)(1) \oplus {\ensuremath\mathfrak{h}}^5(X)(2) \simeq {\ensuremath\mathfrak{h}}^1(Y) \oplus
{\ensuremath\mathfrak{h}}^3(Y)(1)
\oplus {\ensuremath\mathfrak{h}}^5(Y)(2),$$ Since the even-degree summands are of Tate type, and since $\dim
{\ensuremath\mathrm{H}}^{2i}(X) = \dim {\ensuremath\mathrm{H}}^{6-2i}(X)$ by Poincaré duality (and similarly for $Y$), we conclude that ${\ensuremath\mathfrak{h}}^{2i}(X) \simeq {\ensuremath\mathfrak{h}}^{2i}(Y)$ for all $i$. Now, a theorem of Popa–Schnell [@ps] says that two derived equivalent complex varieties have isogenous reduced Picard scheme (see [@honigs Appendix] for the case of $k$-varieties). It follows from Theorem \[thm:product\] that ${\ensuremath\mathfrak{h}}^1(X) \simeq {\ensuremath\mathfrak{h}}^1(Y)$, and then by duality that ${\ensuremath\mathfrak{h}}^5(X) \simeq {\ensuremath\mathfrak{h}}^5(Y)$. Since all terms of are of curve type, we deduce from the semi-simplicity statement of Proposition \[prop:curve\] that ${\ensuremath\mathfrak{h}}^3(X)\simeq {\ensuremath\mathfrak{h}}^3(Y)$. Alternately, from [@acmv], two derived equivalent threefolds have degree-wise isomorphic cohomology groups (the isomorphisms being induced by some algebraic correspondences); it then follows from the description of the motives of $X$ and $Y$ together with Proposition \[prop:curve\] that $X$ and $Y$ have isomorphic motives.
In the case of fourfolds, we first note by [@vial-abelian Theorem 3.11] that the motive of a fourfold $X$ with representable ${\ensuremath\mathrm{CH}}_0$ admits a decomposition of the form $${\ensuremath\mathfrak{h}}(X) \simeq (C,p,0) \oplus (S,q,1) \oplus (C,{}^tp,3),$$ for some curve $C$ and some surface $S$. It follows from the arguments of §\[subsec:kmp\] that ${\ensuremath\mathfrak{h}}(X)$ admits a Chow–Künneth decomposition such that ${\ensuremath\mathfrak{h}}^{2i+1}(X)(i)$ is of curve type for all $i$, ${\ensuremath\mathfrak{h}}^{2i}(X)$ is of Tate type for all $i\neq 2$, and ${\ensuremath\mathfrak{h}}^4(X)$ further decomposes as ${\ensuremath\mathfrak{h}}^4_{\mathrm{alg}}(X)
\oplus
{\ensuremath\mathfrak{h}}^4_{\mathrm{tr}}(X)$, with the property that ${\ensuremath\mathfrak{h}}^4_{\mathrm{alg}}(X)$ becomes, after base-change to $k^s$, a direct sum of Tate motives $\mathds{1}(-2)$ and ${\ensuremath\mathfrak{h}}^4_{\mathrm{tr}}(X)(1)$ is a direct summand of the motive of a surface with $\operatorname{Hom}({\ensuremath\mathfrak{h}}^4_{\mathrm{tr}}(X),\mathds{1}(-i))=0$ for $i\neq 3$. The arguments of §\[subsec:weight\] then provides isomorphisms
$${\ensuremath\mathfrak{h}}^0(X) \oplus {\ensuremath\mathfrak{h}}^2(X)(1) \oplus {\ensuremath\mathfrak{h}}_{\mathrm{alg}}^4(X)(2) \oplus
{\ensuremath\mathfrak{h}}^6(X)(3)\oplus
{\ensuremath\mathfrak{h}}^8(X)(4) \simeq {\ensuremath\mathfrak{h}}^0(Y) \oplus {\ensuremath\mathfrak{h}}^2(Y)(1) \oplus {\ensuremath\mathfrak{h}}^4_{\mathrm{alg}}(Y)(2)
\oplus
{\ensuremath\mathfrak{h}}^6(Y)(3)\oplus {\ensuremath\mathfrak{h}}^8(Y)(4),$$ $${\ensuremath\mathfrak{h}}^1(X) \oplus {\ensuremath\mathfrak{h}}^3(X)(1) \oplus {\ensuremath\mathfrak{h}}^5(X)(2) \oplus {\ensuremath\mathfrak{h}}^7(X)(3) \simeq
{\ensuremath\mathfrak{h}}^1(Y)
\oplus {\ensuremath\mathfrak{h}}^3(Y)(1) \oplus {\ensuremath\mathfrak{h}}^5(Y)(2) \oplus {\ensuremath\mathfrak{h}}^7(Y)(3)$$ $$\text{and} \quad {\ensuremath\mathfrak{h}}^4_{\mathrm{tr}}(X) \simeq {\ensuremath\mathfrak{h}}^4_{\mathrm{tr}}(Y).
\qquad$$
As in the case of threefolds, since the Picard numbers of $X$ and $Y$ agree, we conclude that ${\ensuremath\mathfrak{h}}^{2i}(X) \simeq {\ensuremath\mathfrak{h}}^{2i}(Y)$ for all $i\neq 2$ and that ${\ensuremath\mathfrak{h}}^4_{\mathrm{alg}}(X)
\simeq
{\ensuremath\mathfrak{h}}^4_{\mathrm{alg}}(Y)$, while by utilizing the Theorem of Popa–Schnell [@ps], we conclude from Theorem \[thm:product\] that ${\ensuremath\mathfrak{h}}^1(X) \simeq {\ensuremath\mathfrak{h}}^1(Y)$ and then by duality that ${\ensuremath\mathfrak{h}}^7(X)\simeq
{\ensuremath\mathfrak{h}}^7(Y)$. It follows by cancellation (Proposition \[prop:curve\]) that ${\ensuremath\mathfrak{h}}^3(X) \oplus
{\ensuremath\mathfrak{h}}^5(X)(1) \simeq {\ensuremath\mathfrak{h}}^3(Y)
\oplus
{\ensuremath\mathfrak{h}}^5(Y)(1)$. Since there is a Lefschetz isomorphism ${\ensuremath\mathrm{H}}^3(X)\simeq
{\ensuremath\mathrm{H}}^5(X)(1)$ and similarly for $Y$, we conclude (again from Proposition \[prop:curve\]) that ${\ensuremath\mathfrak{h}}^3(X)\simeq
{\ensuremath\mathfrak{h}}^3(Y)$ and ${\ensuremath\mathfrak{h}}^5(X)\simeq {\ensuremath\mathfrak{h}}^5(Y)$.
Motives of varieties as Frobenius algebra objects {#sec:Frob}
=================================================
Algebras and Frobenius algebras {#SS:alg}
-------------------------------
Let $\mathcal{C}$ be a symmetric monoidal category with tensor unit $\mathds{1}$. An *algebra object* in $\mathcal{C}$ is an object $M$ together with a unit morphism $\eta: \mathds{1}\to M$ and a multiplication morphism $\mu:
M\otimes M\to M$ satisfying the unit axiom $\mu\circ ({\ensuremath\mathrm{id}}\otimes
\eta)={\ensuremath\mathrm{id}}=\mu\circ (\eta\otimes {\ensuremath\mathrm{id}})$ and the associativity axiom $\mu\circ(\mu\otimes {\ensuremath\mathrm{id}})=\mu\circ ({\ensuremath\mathrm{id}}\otimes \mu)$. It is called *commutative* if moreover $\mu=\mu\circ c_{M, M}$ is satisfied, where $c_{M, M}$ is the commutativity constraint of the category $\mathcal{C}$. A *morphism of algebra objects* between two algebra objects $M$ and $N$ is a morphism $\phi: M\to N$ in $\mathcal{C}$ that preserves the multiplication $\mu$ and the unit $\eta$. We note that an algebra structure on an object $M$ of $\mathcal{C}$ induces naturally an algebra structure on the $n$-th tensor powers $M^{\otimes n}$ of $M$, and that a morphism $\phi: M \to N$ of algebra objects induces naturally a morphism of algebra objects $\phi^{\otimes n} : M^{\otimes
n} \to N^{\otimes n}$ which is an isomorphism if $\phi$ is.
If $\mathcal{C}$ is moreover rigid and possesses a $\otimes$-invertible object then we can speak of Frobenius algebra objects in $\mathcal{C}$:
\[def:FrobAlg\] Let $(\mathcal{C}, \otimes, \vee, \mathds{1})$ be a rigid symmetric monoidal category admitting a $\otimes$-invertible object denoted by $\mathds{1}(1)$. Let $d$ be an integer. A *degree*-$d$ *Frobenius algebra object* in $\mathcal{C}$ is the data of an object $M\in \mathcal{C}$ endowed with
- $\eta: \mathds{1}\to M$, a unit morphism;
- $\mu: M\otimes M\to M$, a multiplication morphism;
- $\lambda: M^{\vee}\stackrel{\sim}{\longrightarrow} M(d)$, an isomorphism, called the *Frobenius structure*;
satisfying the following axioms:
1. (Unit) $\mu\circ ({\ensuremath\mathrm{id}}\otimes \eta)={\ensuremath\mathrm{id}}=\mu\circ (\eta\otimes {\ensuremath\mathrm{id}})$;
2. (Associativity) $\mu\circ(\mu\otimes {\ensuremath\mathrm{id}})=\mu\circ ({\ensuremath\mathrm{id}}\otimes \mu)$;
3. (Frobenius condition) $({\ensuremath\mathrm{id}}\otimes \mu)\circ (\delta\otimes
{\ensuremath\mathrm{id}})=\delta\circ \mu=(\mu\otimes {\ensuremath\mathrm{id}})\circ ({\ensuremath\mathrm{id}}\otimes \delta)$,
where the *comultiplication* morphism $\delta: M\to M\otimes M(d)$ is defined by dualizing $\mu$ via the following commutative diagram: $$\xymatrix{
M^{\vee}\ar[r]^-{{}^{t}\mu} \ar[d]_{\lambda}^{\simeq} & M^{\vee}\otimes
M^{\vee}\ar[d]_{\lambda\otimes \lambda}^{\simeq}\\
M(d) \ar[r]^-{\delta(d)}& M(d)\otimes M(d)
}$$ We define also the *counit morphism* $\epsilon: M\to \mathds{1}(-d)$ by dualizing $\eta$ via the following diagram: $$\xymatrix{
M^{\vee}\ar[r]^{{}^{t}\eta}\ar[d]^{\lambda}_{\simeq}& \mathds{1}\\
M(d) \ar[ur]_{\epsilon(d)}&
}$$ We remark that $\epsilon$ and $\delta$ satisfy automatically the counit and coassociativity axioms.
A Frobenius algebra object $M$ is called *commutative* if the underlying algebra object is commutative: $\mu\circ c_{M,M}=\mu$. Commutativity is equivalent to the cocommutativity of $\delta$. The morphism $\beta=\epsilon\circ \mu: M\otimes M\to \mathds{1}(-d)$, called the *Frobenius pairing*, is also sometimes used. It is a symmetric pairing if $M$ is commutative.
In the case of Frobenius algebra objects of degree 0, the $\otimes$-invertible object $\mathds{1}(1)$ is not needed in the definition, and it is reduced to the usual notion of Frobenius algebra object in the literature (see for example [@Abrams], [@Kock]). In this sense, Definition \[def:FrobAlg\] generalizes the existing definition of Frobenius structure by allowing non-zero twists. We believe that our more flexible notion is necessary and adequate for more sophisticated tensor categories than that of vector spaces, such as the categories of Hodge structures, Galois representations, motives, *etc*.
\[rmk:MorIsom\] Morphisms of Frobenius algebra objects are defined in the natural way, that is, as morphisms $\phi: M\to N$ such that all the natural diagrams involving the structural morphisms are commutative. In particular, in order to admit non-trivial morphisms, the degrees of the Frobenius algebra objects $M$ and $N$ must coincide and the following diagram is then commutative: $$\xymatrix{
N^{\vee} \ar[r]^{{}^{t}\phi} \ar[d]^{\simeq}_{\lambda_{N}}&
M^{\vee}\ar[d]_{\simeq}^{\lambda_{M}}\\
N(d) & M(d)\ar[l]^{\phi(d)}
}$$ As a result, all morphisms between Frobenius algebra objects are in fact invertible. It is an exercise to show that an isomorphism $\phi: M\to N$ between two Frobenius algebra objects respects the Frobenius algebra structures if and only if it is compatible with the algebra structure (*i.e.* with $\mu$) and the Frobenius structure (*i.e.* with $\lambda$). This is proved in Proposition \[prop:RigidAlgIso\] in the case of Chow motives of smooth projective varieties. In addition, $\phi^{\otimes n} : M^{\otimes n} \to
N^{\otimes n}$ is naturally an isomorphism of Frobenius algebra objects, as is the dual ${}^t\phi : N^\vee \to M^\vee$.
We summarize the above discussion in the following
\[lemma:IsoFrobAlg\] Let $M, N$ be two Frobenius algebra objects of degree $d$. A morphism of algebra objects $\phi: M\to N$ is a morphism of Frobenius algebra objects if and only if it is an isomorphism and it is *orthogonal* in the sense that $\phi(d)^{-1}=\lambda_{M}\circ {}^{t}\phi\circ \lambda_{N}^{-1}$, or more succinctly, $\phi^{-1}={}^{t}\phi$.
The “if” part follows from the definition. The “only if” part is explained in Remark \[rmk:MorIsom\].
Now let us turn to important examples of Frobenius algebra objects.
\[ex:Cohomology\] Let $X$ be a connected compact orientable (real) manifold of dimension $d$. Then its cohomology group ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ is naturally a Frobenius algebra object of degree $d$ in the category of ${\ensuremath\mathds{Z}}$-graded ${\ensuremath\mathds{Q}}$-vector spaces (where morphisms are *degree-preserving* linear maps and the $\otimes$-invertible object is chosen to be ${\ensuremath\mathds{Q}}[1]$, the 1-dimensional vector space sitting in degree $-1$). The unit morphism $\eta: {\ensuremath\mathds{Q}}\to {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ is given by the fundamental class $[X]$; the multiplication morphism $\mu: {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ is the cup-product; the Frobenius structure comes from the Poincaré duality $$\lambda: {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})^{\vee}\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})[d]={\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathds{Q}}[d].$$ The induced comultiplication morphism $\delta: {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})[d]$ is the Gysin map for the diagonal embedding $X\hookrightarrow X\times X$; the counit morphism $\epsilon: {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\to
{\ensuremath\mathds{Q}}[-d]$ is the integration $\int_{X}$. The Frobenius condition is a classical exercise. Note that ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ is commutative, because the commutativity constraint in the category of graded vector spaces is the super one.
If instead we consider the cohomology group as merely an ungraded vector space, then it becomes a Frobenius algebra object of degree 0 (*i.e.* in the usual sense); this is one of the main examples in the literature.
\[ex:Hodge\] A pure (rational) Hodge structure is a finite-dimensional ${\ensuremath\mathds{Z}}$-graded ${\ensuremath\mathds{Q}}$-vector space $H=\oplus_{n\in Z}H^{(n)}$ such that each $H^{(n)}$ is given a Hodge structure of weight $n$. A morphism between two Hodge structures is required to preserve the weights. The category of pure Hodge structures is naturally a rigid symmetric monoidal category. The $\otimes$-invertible object is chosen to be ${\ensuremath\mathds{Q}}(1)$, which is the 1-dimensional vector space $(2\pi
i)\cdot
{\ensuremath\mathds{Q}}$ with Hodge structure purely of type $(-1, -1)$.
Let $X$ be a compact Kähler manifold of (complex) dimension $d$. Then ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ is naturally a commutative Frobenius algebra object of degree $d$ in the category of pure ${\ensuremath\mathds{Q}}$-Hodge structures. The structural morphisms are the same as in Example \[ex:Cohomology\] up to replacing $[d]$ by $(d)$. For instance, $\lambda: {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})^{\vee}\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})(d)$.
Our main examples of Frobenius algebra objects are the Chow motives of smooth projective varieties.
Frobenius algebra structure on the motives of varieties
-------------------------------------------------------
The category of rational Chow motives over a field $k$ is rigid and symmetric monoidal. We choose the $\otimes$-invertible object to be the Tate motive $\mathds{1}(1)$. Then for any smooth projective $k$-variety $X$ of dimension $d$, its Chow motive ${\ensuremath\mathfrak{h}}(X)$ is naturally a commutative Frobenius algebra object of degree $d$ in the category of Chow motives. Let us explain the structural morphisms in detail.
Let $\delta_X$ denote the class of the small diagonal $\{(x,x,x) : x\in X\}$ in ${\ensuremath\mathrm{CH}}_d(X\times
X\times X)$. Note that for $\alpha$ and $\beta$ in ${\ensuremath\mathrm{CH}}^*(X)$, we have $(\delta_X)_*(\alpha\times \beta) = \alpha\cdot \beta$, so that $\delta_X$ seen as an element of $\operatorname{Hom}({\ensuremath\mathfrak{h}}(X)\otimes {\ensuremath\mathfrak{h}}(X) , {\ensuremath\mathfrak{h}}(X))$ describes the intersection theory on the Chow ring of $X$, as well as the cup product of its cohomology ring. So it is natural to define the multiplication morphism $$\mu : {\ensuremath\mathfrak{h}}(X)
\otimes {\ensuremath\mathfrak{h}}(X) \longrightarrow {\ensuremath\mathfrak{h}}(X)$$ to be the one given by the small diagonal $\delta_{X}\in {\ensuremath\mathrm{CH}}^{2d}(X\times X\times X)=\operatorname{Hom}({\ensuremath\mathfrak{h}}(X)\otimes{\ensuremath\mathfrak{h}}(X), {\ensuremath\mathfrak{h}}(X))$; it can be checked to be commutative and associative. The unit morphism $\eta: \mathds{1}\to {\ensuremath\mathfrak{h}}(X)$ is again given by the fundamental class of $X$. The unit axiom is very easy to check.
The Frobenius structure is defined as the following canonical isomorphism, called the motivic Poincaré duality, given by the class of diagonal $\Delta_{X}\in {\ensuremath\mathrm{CH}}^{d}(X\times X)=\operatorname{Hom}({\ensuremath\mathfrak{h}}(X)^{\vee}, {\ensuremath\mathfrak{h}}(X)(d))$: $$\lambda: {\ensuremath\mathfrak{h}}(X)^{\vee}\stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}(X)(d).$$ One readily checks that the induced comultiplication morphism $$\delta: {\ensuremath\mathfrak{h}}(X)\to {\ensuremath\mathfrak{h}}(X)\otimes {\ensuremath\mathfrak{h}}(X)(d)$$ is given by the small diagonal $\delta_{X}\in {\ensuremath\mathrm{CH}}^{2d}(X\times X\times
X)=\operatorname{Hom}({\ensuremath\mathfrak{h}}(X), {\ensuremath\mathfrak{h}}(X)\otimes {\ensuremath\mathfrak{h}}(X)(d))$, while the counit morphism $$\epsilon:
{\ensuremath\mathfrak{h}}(X)\to \mathds{1}(-d)$$ is given by the fundamental class.
The following lemma proves that, endowed with these structural morphisms, ${\ensuremath\mathfrak{h}}(X)$ is indeed a Frobenius algebra object.
Notation is as above. We have an equality of endomorphisms of ${\ensuremath\mathfrak{h}}(X)\otimes
{\ensuremath\mathfrak{h}}(X)$: $$({\ensuremath\mathrm{id}}\otimes \mu)\circ (\delta\otimes {\ensuremath\mathrm{id}})=\delta\circ \mu=(\mu\otimes
{\ensuremath\mathrm{id}})\circ ({\ensuremath\mathrm{id}}\otimes \delta).$$
We only show $\delta\circ \mu=(\mu\otimes {\ensuremath\mathrm{id}})\circ ({\ensuremath\mathrm{id}}\otimes \delta)$, the other equality being similar. We have a commutative cartesian diagram without excess intersection: $$\xymatrix{
X \ar[r]^{\Delta}\ar[d]_{\Delta}& X\times X \ar[d]^{\Delta\times {\ensuremath\mathrm{id}}}\\
X\times X \ar[r]_-{{\ensuremath\mathrm{id}}\times \Delta}& X\times X\times X,
}$$ where $\Delta : X \to X\times X$ denotes the diagonal embedding. The base-change formula yields $$(\Delta\times {\ensuremath\mathrm{id}})^{*}\circ ({\ensuremath\mathrm{id}}\times
\Delta)_{*}=\Delta_{*}\circ\Delta^{*}$$ on Chow groups, hence also for Chow motives by Manin’s identity principle [@andre §4.3.1]. Now it suffices to notice that $\Delta_{*}$ is the comultiplication $\delta$ and $\Delta^{*}$ is the multiplication $\mu$.
In general, a tensor functor $F: \mathcal{C}\to \mathcal{C'}$ between two rigid symmetric monoidal categories sends a Frobenius algebra object in $\mathcal{C}$ to such an object in $\mathcal{C}'$. Example \[ex:Hodge\] is obtained by applying the Betti realization functor from the category of Chow motives to that of pure Hodge structures; Example \[ex:Cohomology\] (for Kähler manifolds) is obtained by further applying the forgetful functor (${\ensuremath\mathds{Q}}(1)$ is sent to ${\ensuremath\mathds{Q}}[2]$).
(Iso)morphisms of Chow motives as Frobenius algebra objects
-----------------------------------------------------------
The notion of morphisms between two algebra objects is the natural one. Let us spell it out for motives of varieties. A non-zero morphism $\Gamma : {\ensuremath\mathfrak{h}}(X) \rightarrow {\ensuremath\mathfrak{h}}(Y)$ between the motives of two smooth projective varieties over a field $k$ is said to *preserve the algebra structures* if the following diagram commutes[^7] $$\label{eq:diagmult}
\xymatrix{{\ensuremath\mathfrak{h}}(X)\otimes {\ensuremath\mathfrak{h}}(X) \ar[rr]^{\quad \delta_X} \ar[d]^{\Gamma\otimes
\Gamma} && {\ensuremath\mathfrak{h}}(X) \ar[d]^\Gamma \\
{\ensuremath\mathfrak{h}}(Y)\otimes {\ensuremath\mathfrak{h}}(Y) \ar[rr]^{\quad \delta_Y} && {\ensuremath\mathfrak{h}}(Y).
}$$ For example, if $f: Y \to X$ is a $k$-morphism, then $f^* : {\ensuremath\mathfrak{h}}(X) \to {\ensuremath\mathfrak{h}}(Y)$ is a morphism of algebra objects. Note that if $\Gamma : {\ensuremath\mathfrak{h}}(X) \rightarrow {\ensuremath\mathfrak{h}}(Y)$ preserves the algebra structures, then $\Gamma_* : {\ensuremath\mathrm{CH}}^*(X) \to {\ensuremath\mathrm{CH}}^*(Y)$ is a ${\ensuremath\mathds{Q}}$-algebra homomorphism. In fact, since in that case $\Gamma^{\otimes n} : {\ensuremath\mathfrak{h}}(X^n)
\rightarrow {\ensuremath\mathfrak{h}}(Y^n)$ also preserves the algebra structures for all $n>0$, $(\Gamma^{\otimes
n})_* : {\ensuremath\mathrm{CH}}^*(X^n) \rightarrow {\ensuremath\mathrm{CH}}^*(Y^n)$ is also a ${\ensuremath\mathds{Q}}$-algebra homomorphism. We say that the Chow motives of $X$ and $Y$ are *isomorphic as algebra objects* if there exists an isomorphism $\Gamma : {\ensuremath\mathfrak{h}}(X) \rightarrow
{\ensuremath\mathfrak{h}}(Y)$ that preserve the algebra structures. The following lemma is a formal consequence of the definition.
\[lem:formal\] Let $X$ and $Y$ be connected smooth projective varieties and let $\Gamma :
{\ensuremath\mathfrak{h}}(X)
\rightarrow {\ensuremath\mathfrak{h}}(Y)$ be a non-zero morphism that preserves the algebra structures.
(i) $\Gamma$ preserves the units: if $[X]$ is the fundamental class of $X$ in ${\ensuremath\mathrm{CH}}^0(X)$ and similarly for $Y$, then $$\Gamma_*[X] = [Y].$$
(ii) Suppose $X$ and $Y$ have same dimension and define $c$ to be the rational number such that $\Gamma^*[Y]=c\, [X]$; then $$(\Gamma \otimes
\Gamma)^* \Delta_Y= c\, \Delta_{X}.$$ In particular, $\Gamma$ is an isomorphism if and only if $c\neq 0$, and in this case, due to Lieberman’s formula[^8], the inverse of $\Gamma$ is equal to $\frac{1}{c}^t \Gamma$.
$(i)$ This is the analogue of the basic fact that a non-trivial homomorphism of unital algebras preserves the units. Concretely, the fundamental class of $X$ provides a morphism $1_X : \mathds{1} \to
{\ensuremath\mathfrak{h}}(X)$, and similarly for $Y$, and we need to show that $\Gamma\circ 1_X =
1_Y$. First, for dimension reasons we have $\Gamma\circ 1_X = \lambda\cdot 1_Y$ for some $\lambda \in {\ensuremath\mathds{Q}}$. Compose then the diagram with the morphism $1_X\otimes 1_X$; one obtains $\lambda^2 = \lambda$. If $\lambda =
0$, then by composing diagram with the morphism $1_X\otimes
\mathrm{id}_X$, we find that $\Gamma =0$. Hence $\lambda =1$ and we are done.
$(ii)$ The commutativity of provides the identity $\Gamma\circ \delta_X = \delta_Y\circ (\Gamma\otimes \Gamma)$. Letting the latter act contravariantly on $[Y]$ yields $$c\, \Delta_X = (\Gamma\otimes \Gamma)^*\Delta_Y = {}^t\Gamma\circ \Gamma,$$ where $c$ is the rational number such that $\Gamma^*[Y] = c\, [X]$ and where the second equality is Lieberman’s formula. Since we assume that $\Gamma$ is invertible, we get that $\Gamma^{-1}=\frac{1}{c}\, {}^t\Gamma$.
As is alluded to in Lemma \[lemma:IsoFrobAlg\], the notion of orthogonality is highly relevant when considering morphisms between Frobenius algebras. Let us recast it in the context of motives:
\[def:OrthIso\] Let $X$ and $Y$ be two smooth projective varieties of the same dimension and $\Gamma: {\ensuremath\mathfrak{h}}(X)\to {\ensuremath\mathfrak{h}}(Y)$ be an isomorphism between their Chow motives. Then by Lieberman’s formula we see that $\Gamma^{-1}={}^{t}\Gamma$ if and only if $(\Gamma\otimes \Gamma)_{*}\Delta_{X}=\Delta_{Y}$. In this case, $\Gamma$ is called an *orthogonal* isomorphism.
Finally, we can unravel the meaning of being isomorphic as Frobenius algebra objects for the motives of two varieties (and the same holds for Hodge morphisms between the cohomology algebras of smooth projective varieties of same dimension).
\[prop:RigidAlgIso\] Let $X$ and $Y$ be two smooth projective varieties of the same dimension and $\Gamma: {\ensuremath\mathfrak{h}}(X)\to {\ensuremath\mathfrak{h}}(Y)$ be a morphism between their motives. Then the following are equivalent:
1. $\Gamma$ is an isomorphism of Frobenius algebra objects.
2. $\Gamma$ is an algebra isomorphism and $\Gamma$ is *orthogonal*: that is, $\Gamma^{-1}={}^{t}\Gamma$ or equivalently, $(\Gamma\otimes \Gamma)_{*}\Delta_{X}=\Delta_{Y}$.
3. $\Gamma$ is an algebra isomorphism and $\deg(\Gamma)=1$: that is, $\deg (\Gamma_*[pt])=1$ or equivalently, $\Gamma^{*}[Y]=[X]$.
4. $\Gamma$ is an isomorphism and $(\Gamma\otimes
\Gamma)_{*}\Delta_{X}=\Delta_{Y}$ and $(\Gamma\otimes \Gamma\otimes
\Gamma)_{*}\delta_{X}=\delta_{Y}$.
The equivalence between $(i)$ and $(ii)$ is a special case of Lemma \[lemma:IsoFrobAlg\]. The equivalence between $(ii)$ and $(iii)$ can be read off Lemma \[lem:formal\]$(ii)$. For the equivalence between $(ii)$ and $(iv)$, one only needs to see that an orthogonal isomorphism ($\Gamma^{-1}={}^{t}\Gamma$) is an algebra morphism ($\Gamma \circ \delta_X = \delta_{Y} \circ (\Gamma \otimes
\Gamma)$) if and only if $(\Gamma\otimes \Gamma\otimes
\Gamma)_{*}\delta_{X}=\delta_{Y}$. But this again follows from Lieberman’s formula.
Derived equivalent K3 surfaces
==============================
The aim of this section is to prove Theorem \[thm:main\], Corollaries \[cor:Powers\] and \[cor:torelli\].
Proof of Theorem \[thm:main\] {#S:mainthm}
-----------------------------
The proof relies crucially on the Beauville–Voisin description of the algebra structure on the motive of K3 surfaces:
\[thm:bv\] Let $S$ be a K3 surface and let $o_S$ be the class of any point lying on a rational curve on $S$. Then, as cycle classes in ${\ensuremath\mathrm{CH}}_2(S\times S \times S)$, we have $$\label{eq:bv}
\delta_S = p_{12}^*\Delta_S\cdot p_3^*o_S + p_{13}^*\Delta_S \cdot p_2^*o_S
+
p_{23}^*\Delta_S\cdot p_1^*o_S - p_1^*o_S\cdot p_2^*o_S - p_1^*o_S\cdot
p_3^*o_S
- p_2^*o_S\cdot p_3^*o_S,$$ where $p_k : S\times S \times S \to S$ and $p_{ij}: S\times S \times S \to S
\times S$ denote the various projections.
Note that, for a K3 surface $S$, Theorem \[thm:bv\] implies that $\alpha\cdot \beta = \deg (\alpha\cdot\beta)\,o_S$ for all divisors $\alpha, \beta
\in
{\ensuremath\mathrm{CH}}^1(S)$, and that $c_2(S) = (\delta_S)_*\Delta_S = \chi(S)\,o_S = 24\, o_S
\in
{\ensuremath\mathrm{CH}}^2(S)$. (Of course, this is due originally to Beauville–Voisin [@bv].)
According to Proposition \[prop:RigidAlgIso\], in order to establish Theorem \[thm:main\], it is necessary and sufficient to produce a correspondence $\Gamma : {\ensuremath\mathfrak{h}}(S) \to
{\ensuremath\mathfrak{h}}(S')$ which is invertible and such that
1. $(\Gamma\otimes \Gamma)_*\Delta_S = \Delta_{S'}$, or equivalently $\Gamma^{-1} = {}^t\Gamma$;
2. $(\Gamma\otimes \Gamma\otimes \Gamma)_*\delta_S = \delta_{S'}$.
By the Beauville–Voisin Theorem \[thm:bv\], it is sufficient (in fact also necessary by looking at the contravariant action of $(ii)$ on $\Delta_{S'}$) to produce a correspondence $\Gamma : {\ensuremath\mathfrak{h}}(S) \to
{\ensuremath\mathfrak{h}}(S')$ which is invertible and such that
1. $(\Gamma\otimes \Gamma)_*\Delta_S = \Delta_{S'}$, or equivalently $\Gamma^{-1} = {}^t\Gamma$;
2. $\Gamma_*o_S = o_{S'}$.
We now proceed to the proof of Theorem \[thm:main\], *i.e.* to constructing an invertible correspondence satisfying $(i)$ and $(ii')$ above. Given a K3 surface $S$, we consider the refined Chow–Künneth decomposition of Kahn–Murre–Pedrini as described in §\[subsec:kmp\] given by $${\ensuremath\mathfrak{h}}(S) =
{\ensuremath\mathfrak{h}}^0(S) \oplus {\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S) \oplus
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S) \oplus {\ensuremath\mathfrak{h}}^4(S),$$ with $\pi^0_S = o_S\times S$ and $\pi^4_S = S\times o_S$, where $\pi_S^i$ denote the projectors on the corresponding direct summands and where $o_S$ denotes the Beauville–Voisin zero-cycle as in Theorem \[thm:bv\]. Moreover, the decomposition is such that ${}^t \pi^2_{\mathrm{alg},S} =
\pi^2_{\mathrm{alg},S} $ and ${}^t \pi^2_{\mathrm{tr},S} =
\pi^2_{\mathrm{tr},S}$.
Consider now two twisted derived equivalent K3 surfaces $S$ and $S'$. As in the proof of Theorem \[thm:huybrechts\], we only give a proof in the untwisted case, the twisted case being similar. We fix an exact linear equivalence $\Phi_{\mathcal{E}} : \mathrm{D}^b(S) \stackrel{\sim}{\longrightarrow}
\mathrm{D}^b(S')$ with Fourier–Mukai kernel $\mathcal{E} \in
\mathrm{D}^b(S\times S')$. The proof will proceed in two steps. First, we will construct an invertible correspondence $$\Gamma_{\mathrm{alg}} :
{\ensuremath\mathfrak{h}}_{\mathrm{alg}}(S) \to {\ensuremath\mathfrak{h}}_{\mathrm{alg}}(S') \quad \text{with} \
(\Gamma_{\mathrm{alg}})^{-1} = {}^t \Gamma_{\mathrm{alg}} \ \text{and} \
(\Gamma_{\mathrm{alg}})_* o_S = o_{S'},$$ where ${\ensuremath\mathfrak{h}}_{\mathrm{alg}}(S) = {\ensuremath\mathfrak{h}}^0(S)
\oplus {\ensuremath\mathfrak{h}}_{\mathrm{alg}}^2(S) \oplus {\ensuremath\mathfrak{h}}^4(S)$ (and similarly for $S'$) is the algebraic summand of the motive of $S$; second, we will construct an invertible correspondence on the transcendental summands of the motives of $S$ and $S'$: $$\Gamma_{\mathrm{tr}} : {\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S) \to
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S') \quad \text{with} \ (\Gamma_{\mathrm{tr}})^{-1} = {}^t
\Gamma_{\mathrm{tr}} \ \text{and} \ (\Gamma_{\mathrm{tr}})_* o_S = 0.$$ The correspondence $$\Gamma := \Gamma_{\mathrm{alg}} + \Gamma_{\mathrm{tr}} :
{\ensuremath\mathfrak{h}}(S) \to {\ensuremath\mathfrak{h}}(S')$$ will then provide the desired isomorphism of Frobenius algebra objects.
First, the numerical Grothendieck group $K_{0}^{\operatorname{num}}$ equipped with the Euler pairing is clearly a derived invariant. Using the Chern character isomorphism, we obtain an isometry between the quadratic spaces $\widetilde{\mathrm{NS}}(S_{k^s})_{\ensuremath\mathds{Q}}$ and $\widetilde{\mathrm{NS}}(S'_{k^s})_{\ensuremath\mathds{Q}}$, where $\widetilde{\mathrm{NS}}$ is the extended Néron–Severi group equipped with the Mukai pairing, hence is isometric to the (orthogonal) direct sum of the Néron–Severi lattice (endowed with the intersection pairing) and a copy of the hyperbolic plane. By Witt’s cancellation theorem, the Néron–Severi groups $\mathrm{NS}(S_{k^s})_{\ensuremath\mathds{Q}}$ and $\mathrm{NS}(S'_{k^s})_{\ensuremath\mathds{Q}}$ of two derived equivalent surfaces are isomorphic both as $\mathrm{Gal}(k)$-representations and as quadratic spaces; there exists therefore a correspondence $M = \pi^2_{\mathrm{alg},S'} \circ M \circ
\pi^2_{\mathrm{alg},S}$ in ${\ensuremath\mathrm{CH}}^2(S\times_k S')$ inducing an isometry $\mathrm{NS}(S_{k^s})_{\ensuremath\mathds{Q}}\simeq \mathrm{NS}(S'_{k^s})_{\ensuremath\mathds{Q}}$. This means that $M$ induces an isomorphism ${\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S) \stackrel{\sim}{\longrightarrow}
{\ensuremath\mathfrak{h}}^2_{\mathrm{alg}}(S')$ with inverse given by its transpose. It follows that $\Gamma_{\mathrm{alg}} := o_S \times S' + M + S\times o_{S'}$ induces an isomorphism ${\ensuremath\mathfrak{h}}_{\mathrm{alg}}(S)
\stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}_{\mathrm{alg}}(S')$ with inverse ${}^t\Gamma_{\mathrm{alg}}$. In addition, we have $(\Gamma_{\mathrm{alg}})_*
o_S = o_{S'}$.
Second, recall from §\[subsec:weight\] that $v_2(\mathcal{E})$ induces an isomorphism ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S) \stackrel{\sim}{\longrightarrow}
{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$ with inverse induced by $v_2(\mathcal{E}^\vee\otimes
p^*\omega_S)$. Since K3 surfaces have trivial first Chern class and trivial canonical bundle, it follows that the inverse of $v_2(\mathcal{E})$ is in fact its transpose. In other words, $\Gamma_{\mathrm{tr}}:= \pi^2_{\mathrm{tr},S'} \circ
v_2(\mathcal{E}) \circ
\pi^2_{\mathrm{tr},S}$ induces an isomorphism of Chow motives ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)
\stackrel{\sim}{\longrightarrow}{\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S')$ with inverse its transpose. Finally, we do have $(\pi^2_{\mathrm{tr}})_*o_S =0$ because of the orthogonality of $\pi^2_{\mathrm{tr},S}$ with $\pi^4_S$.
The required correspondences $\Gamma_{\mathrm{alg}}$ and $\Gamma_{\mathrm{tr}}$ have thus been constructed and this concludes the proof of Theorem \[thm:main\].
Proof of Corollary \[cor:Powers\]
---------------------------------
Let $S$ and $S'$ be two twisted derived equivalent K3 surfaces. Then due to Theorem \[thm:main\] their motives are isomorphic as Frobenius algebra objects. As is explained in §\[SS:alg\], isomorphisms of Frobenius algebra objects behave well with respect to (tensor) products, hence it suffices to see that for any $n\in {\ensuremath\mathds{Z}}_{>0}$ the Hilbert schemes of $n$ points $\operatorname{Hilb}^{n}(S)$ and $\operatorname{Hilb}^{n}(S')$ have isomorphic Chow motives as Frobenius algebra objects. To this end, we use the result of Fu–Tian [@FT] that describes the algebra object ${\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S))$ in terms of the algebra objects ${\ensuremath\mathfrak{h}}(S^{m})$ for $m\leq n$, together with some explicit combinatorial rules. More precisely, by [@FT Theorem 1.6 and Remark 1.7], for a K3 surface $S$, we have an isomorphism of algebra objects: $$\label{eqn:IsoFT}
\phi: {\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S))\simeq \left(\bigoplus_{g\in
\mathfrak{S}_{n}}{\ensuremath\mathfrak{h}}\left(S^{O(g)}\right), \star_{\operatorname{orb},
\operatorname{dt}}\right)^{\mathfrak{S}_{n}},$$ where $\mathfrak{S}_{n}$ is the symmetric group acting naturally on $S^{n}$; for a permutation $g$, $O(g)$ is its set of orbits in $\{1, \cdots, n\}$, $S^{O(g)}$ is canonically identified with the fixed locus $(S^{n})^{g}$, and finally $\star_{\operatorname{orb}, \operatorname{dt}}$ is the orbifold product with discrete torsion (see [@ftv; @FT]) defined as follows (let us omit the Tate twists for ease of notation): it is compatible with the $\mathfrak{S}_{n}$-grading, and for any $g, h \in \mathfrak{S}_{n}$, ${\ensuremath\mathfrak{h}}\left(S^{O(g)}\right)\otimes {\ensuremath\mathfrak{h}}\left(S^{O(h)}\right)\to
{\ensuremath\mathfrak{h}}\left(S^{O(g,h)}\right)$ is given by the pushforward via the diagonal inclusion $S^{O(g,h)}\hookrightarrow S^{O(g)}\times S^{O(h)}\times S^{O(gh)}$ of the cycle $\epsilon(g,h)c_{g,h}\in {\ensuremath\mathrm{CH}}(S^{O(g, h)})$, by [@FT Lemma 9.3]: $$\label{eqn:ObsCl}
c_{g,h}:=
\begin{cases}
0, & \text{ if } \exists\, t\in O(g,h) \text{ with } d_{g,h}(t)\geq 2\, ;\\
\prod_{t\in I}\left(24\operatorname{pr}_{t}^{*}(o_{S})\right), & \text{if } \forall t\in
O(g,h) \text{ has } d_{g,h}(t)=0 \text{ or } 1,
\end{cases}$$ where $\epsilon(g,h):=(-1)^{\frac{n-|O(g)|-|O(h)|+|O(gh)|}{2}}$, $O(g, h)$ is the set of orbits in $\{1, \cdots, n\}$ under the subgroup generated by $g$ and $h$; for any orbit $t\in O(g, h)$, $$d(g,h)(t):=\frac{2+|t|-|t/g|-|t/h|-|t/gh|}{2}$$ is the *graph defect* function [@FT Lemma 9.1] and $I:=\{t\in O(g,h)~\vert~ d_{g,h}(t)=1\}$ is the subset of orbits with graph defect 1.
As our isomorphism of algebra objects $\Gamma: {\ensuremath\mathfrak{h}}(S)\to {\ensuremath\mathfrak{h}}(S')$ satisfies $\Gamma_{*}(o_{S})=o_{S'}$, it is now clear from the above precise description that the right-hand side of for $S$ and for $S'$ are isomorphic algebra objects, and the isomorphism can be chosen orthogonal. As the morphism $\phi$ in satisfies $\phi^{-1}={}^{t}\phi$, we have ${\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S))$ and ${\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S'))$ are isomorphic Frobenius algebra objects. This completes the proof.
\[rmk:ChowRingIso\] It turns out that we do not need Theorem \[thm:main\] to show that two twisted derived equivalent K3 surfaces have isomorphic Chow rings. Indeed, Huybrechts’ result [@huybrechts-derivedeq] (generalized to the twisted case in [@huybrechts-isogenous]) provides a correspondence $\Gamma\in
{\ensuremath\mathrm{CH}}^{2}(S\times S')$ that induces an isomorphism of graded ${\ensuremath\mathds{Q}}$-vector spaces $\Gamma_{*}: {\ensuremath\mathrm{CH}}^{*}(S)\stackrel{\sim}{\longrightarrow}{\ensuremath\mathrm{CH}}^{*}(S')$ with the extra property of being isometric on the Néron–Severi spaces ${\ensuremath\mathrm{CH}}^{1}(S)\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{CH}}^{1}(S')$. Now thanks to the theorem of Beauville–Voisin [@bv] saying that the image of the intersection product of two divisors on a K3 surface is of dimension 1, this already implies that $\Gamma_{*}$ is actually an isomorphism of graded ${\ensuremath\mathds{Q}}$-algebras.
In contrast, in the situation of Corollary \[cor:Powers\], a derived equivalence between $D^{b}(S)$ and $D^{b}(S')$ does give rise to a derived equivalence between their powers and Hilbert schemes, thanks to Bridgeland–King–Reid [@BKR] and Haiman [@Haiman]. However, it is not at all clear for the authors how to produce an isomorphism of the Chow rings (or even the rational cohomology rings) of two derived equivalent holomorphic symplectic varieties starting from the Fourier–Mukai kernel; see Conjecture \[conj:HK\].
Proof of Corollary \[cor:torelli\] {#S:coro}
----------------------------------
The equivalence of ${(i)}$ and ${(ii)}$ is due to Huybrechts [@huybrechts-isogenous Corollary 1.4], while the implication $(ii) \Rightarrow (iii)$ is Theorem \[thm:main\]. We now prove the implication $(iii) \Rightarrow (i)$. Suppose that $\Gamma : {\ensuremath\mathfrak{h}}(S)
\rightarrow {\ensuremath\mathfrak{h}}(S')$ is an isomorphism that preserves the algebra structures. Let $c$ be the rational number such that $\Gamma^{*}[S']=c\,[S]$, or equivalently such that $\Gamma^{-1}=\frac{1}{c}{}^{t}\Gamma$ by Lemma \[lem:formal\]; then the following diagram is commutative:
$$\label{eq:diagmultcoho}
\xymatrix{{\ensuremath\mathrm{H}}^2(S)\otimes {\ensuremath\mathrm{H}}^2(S) \ar[rr]^{\quad \cup} \ar[d]^{(\Gamma\otimes
\Gamma)_*} && {\ensuremath\mathrm{H}}^4(S) \ar[d]^{\Gamma_*} \ar[rr]^{\deg} && {\ensuremath\mathds{Q}}\ar[d]^{\cdot
c}\\
{\ensuremath\mathrm{H}}^2(S')\otimes {\ensuremath\mathrm{H}}^2(S') \ar[rr]^{\quad \cup} && {\ensuremath\mathrm{H}}^4(S') \ar[rr]^{\deg} &&
{\ensuremath\mathds{Q}}.
}$$
The commutativity of the left-hand square of is implied directly by the assumption that $\Gamma$ preserves the algebra structures, while the commutativity of the right-hand square follows from the Poincaré dual of the identity $\Gamma^{*}[S']=c\,[S]$. If in addition $\Gamma$ preserves the Frobenius algebra structure, then $c=1$ by Proposition \[prop:RigidAlgIso\]. This means that $S$ and $S'$ are isogenous.
A motivic global Torelli theorem
--------------------------------
The aim of this section is to show that Lemma \[lem:formal\] directly allows to upgrade motivically the global Torelli theorem, without utilizing the decomposition of the diagonal of Beauville–Voisin (Theorem \[thm:bv\]). We denote ${\ensuremath\mathfrak{h}}(X)_{\ensuremath\mathds{Z}}$ the Chow motive of $X$ with integral coefficients.
\[thm:motglobtor\] Let $S$ and $S'$ be two complex projective K3 surfaces. The following statements are equivalent:
(i) $S$ and $S'$ are isomorphic;
(ii) ${\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Z}})$ and ${\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Z}})$ are Hodge isometric;
(iii) ${\ensuremath\mathfrak{h}}(S)_{\ensuremath\mathds{Z}}$ and ${\ensuremath\mathfrak{h}}(S')_{\ensuremath\mathds{Z}}$ are isomorphic as algebra objects.
The equivalence of items $(i)$ and $(ii)$ is the global Torelli theorem. The implication $(i)\Rightarrow (iii)$ is obvious. It remains to check that $(iii)
\Rightarrow (ii)$. Once it is observed that Lemma \[lem:formal\] holds with integral coefficients, we obtain the following commutative diagram (with $c\in {\ensuremath\mathds{Z}}$), which is similar to in the proof of Corollary \[cor:torelli\] $$\xymatrix{{\ensuremath\mathrm{H}}^2(S, {\ensuremath\mathds{Z}})\otimes {\ensuremath\mathrm{H}}^2(S, {\ensuremath\mathds{Z}}) \ar[rr]^{\quad \cup}
\ar[d]_{\simeq}^{(\Gamma\otimes
\Gamma)_*} && {\ensuremath\mathrm{H}}^4(S, {\ensuremath\mathds{Z}}) \ar[d]_{\simeq}^{\Gamma_*}
\ar[rr]_{\simeq}^{\deg}
&& {\ensuremath\mathds{Z}}\ar[d]_{\simeq}^{\cdot c}\\
{\ensuremath\mathrm{H}}^2(S', {\ensuremath\mathds{Z}})\otimes {\ensuremath\mathrm{H}}^2(S', {\ensuremath\mathds{Z}}) \ar[rr]^{\quad \cup} && {\ensuremath\mathrm{H}}^4(S', {\ensuremath\mathds{Z}})
\ar[rr]_{\simeq}^{\deg} &&
{\ensuremath\mathds{Z}}.
}$$ Therefore, there is an isometry of lattices between ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Z}})\otimes
\langle c\rangle$ and ${\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Z}})$, which implies that $c=1$.
Beyond K3 surfaces {#sect:MultOrlov}
==================
Orlov’s conjecture \[conj:orlov\] predicts that the Chow motives of two derived equivalent smooth projective varieties are isomorphic. Motivated by Theorem \[thm:main\], we raised the following question in the introduction:
\[ques:DeriveFrob\] When can we expect more strongly that a derived equivalence between two smooth projective varieties implies an isomorphism between their rational Chow motives as *Frobenius algebra objects*?
We make some remarks and speculations on this question in this section.
By Bondal–Orlov [@bo], two derived equivalent smooth projective varieties that are either Fano or with ample canonical bundle are isomorphic; in particular, their motives are isomorphic as Frobenius algebra objects. Similarly, Question \[ques:DeriveFrob\] also has a positive answer for curves, as they do not have non-isomorphic Fourier–Mukai partners [@HuybrechtsFMBook Corollary 5.46].
In general, one cannot expect in general a positive answer to Question \[ques:DeriveFrob\]. In fact, if ${\ensuremath\mathfrak{h}}(X)$ and ${\ensuremath\mathfrak{h}}(Y)$ are isomorphic as Frobenius algebra objects then by applying the Betti realization functor, their cohomology are isomorphic as Frobenius algebras, that is, due to Proposition \[prop:RigidAlgIso\], there is a (graded) isomorphism of ${\ensuremath\mathds{Q}}$-algebras ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})\to{\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$ sending the class of a point on $X$ to the class of a point on $Y$. However, as we will see below, this is not the case in general for derived equivalent varieties.
Calabi–Yau varieties
--------------------
\[ex:BC\] Borisov and Căldăraru [@bc] constructed derived equivalent (but non-birational) Calabi–Yau threefolds $X$ and $Y$ with the following properties: $\operatorname{Pic}(X)
={\ensuremath\mathds{Z}}H_X$ with $\deg(H_X^3)
= 14$ and $\operatorname{Pic}(Y) = {\ensuremath\mathds{Z}}H_Y$ with $\deg(H_Y^3) = 42$; hence there is no graded ${\ensuremath\mathds{Q}}$-algebra isomorphism between ${\ensuremath\mathrm{H}}^*(X,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(Y,{\ensuremath\mathds{Q}})$ that respects the point class. Therefore, ${\ensuremath\mathfrak{h}}(X)$ and ${\ensuremath\mathfrak{h}}(Y)$ are not isomorphic as Frobenius algebra objects. Nevertheless, thanks to the following proposition, ${\ensuremath\mathrm{H}}^*(X,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(Y,{\ensuremath\mathds{Q}})$ are Hodge isomorphic as graded ${\ensuremath\mathds{Q}}$-algebras and also as graded Frobenius algebras after extending the coefficients to $\mathbb
R$ [^9].
Let $X$ and $Y$ be two derived equivalent Calabi–Yau varieties of dimension $d\geq 3$. Suppose their Hodge numbers satisfy
- $h^{p,q}=0$ for all $p\neq q$ and $p+q\neq d$;
- $h^{p,p}=1$ for all $2p\neq d$ and $0\leq p \leq d$.
Then
1. There is a (graded) real Frobenius algebras isomorphism between ${\ensuremath\mathrm{H}}^{*}(X, \mathbb{R})$ and ${\ensuremath\mathrm{H}}^{*}(Y, \mathbb{R})$ preserving the real Hodge structures.
2. If $d$ is odd or $d$ is even and $s:= \frac{\deg(Y)}{\deg(X)}$ is a square in ${\ensuremath\mathds{Q}}$, then ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$ are isomorphic as graded ${\ensuremath\mathds{Q}}$-Hodge algebras. Here the degree is the top self-intersection number of the ample generator of the Picard group.
We first prove $(ii)$. Let $\mathcal{E}$ be the Fourier–Mukai kernel of the equivalence from ${\ensuremath\mathrm{D}}^{b}(X)$ to ${\ensuremath\mathrm{D}}^{b}(Y)$. By [@HuybrechtsBook Proposition 5.44], the correspondence given by the Mukai vector $v(\mathcal{E})\in {\ensuremath\mathrm{CH}}^{*}(X\times
Y)$ induces a ${\ensuremath\mathds{Z}}/2{\ensuremath\mathds{Z}}$-graded Hodge isometry $$\Phi^{{\ensuremath\mathrm{H}}}_{\mathcal{E}}:
{\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}}),$$ where both sides are equipped with the Mukai pairing. Note that as the varieties are Calabi–Yau, the Mukai pairing is simply given by the intersection pairing with some extra sign changes ([@HuybrechtsBook Definition 5.42]). The *transcendental cohomology* denoted by ${\ensuremath\mathrm{H}}^{*}_{\operatorname{tr}}(-,
{\ensuremath\mathds{Q}})$ is defined to be the orthogonal of all the Hodge classes; it is obviously preserved by $\Phi^{{\ensuremath\mathrm{H}}}_{\mathcal{E}}$. Thanks to our assumption on the Hodge numbers, the transcendental cohomology is concentrated in degree $d$. Therefore by restricting $\Phi_{\mathcal{E}}^{{\ensuremath\mathrm{H}}}$, we get a Hodge isometry $$\phi_{\operatorname{tr}}: {\ensuremath\mathrm{H}}^{d}_{\operatorname{tr}}(X,
{\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow}{\ensuremath\mathrm{H}}^{d}_{\operatorname{tr}}(Y, {\ensuremath\mathds{Q}}).$$ On the other hand, if $d$ is even, $\Phi_{\mathcal{E}}^{{\ensuremath\mathrm{H}}}$ also provides an isometry between the subalgebras of Hodge classes ${\ensuremath\mathrm{Hdg}}^{*}_{{\ensuremath\mathds{Q}}}(X)$ and ${\ensuremath\mathrm{Hdg}}^{*}_{{\ensuremath\mathds{Q}}}(Y)$. Since the quadratic space ${\ensuremath\mathrm{H}}^{0}\oplus\cdots\oplus {\ensuremath\mathrm{H}}^{d-2}\oplus {\ensuremath\mathrm{H}}^{d+2}\oplus
\cdots\oplus {\ensuremath\mathrm{H}}^{2d}$ equipped with the restriction of the Mukai pairing is isometric to $U^{\frac{d}{2}}\otimes {\ensuremath\mathds{Q}}$ for both $X$ and $Y$, the quadratic spaces ${\ensuremath\mathrm{Hdg}}^{d}_{{\ensuremath\mathds{Q}}}(X)$ and ${\ensuremath\mathrm{Hdg}}^{d}_{{\ensuremath\mathds{Q}}}(Y)$ are isometric by Witt cancellation theorem. Due to the assumption that $s:= \frac{\deg(Y)}{\deg(X)}$ is a square and to Witt’s theorem, we have an isometry $$\phi_{{\ensuremath\mathrm{Hdg}}}:
{\ensuremath\mathrm{Hdg}}^{d}_{{\ensuremath\mathds{Q}}}(X)\left(s\right) \longrightarrow {\ensuremath\mathrm{Hdg}}^{d}_{{\ensuremath\mathds{Q}}}(Y)$$ that sends $H_{X}^{\frac{d}{2}}$ to $H_{Y}^{\frac{d}{2}}$, where $H_{X}$ and $H_{Y}$ denote the ample generators of $\operatorname{Pic}(X)$ and $\operatorname{Pic}(Y)$, respectively.
Let us know try to define a graded Hodge algebra isomorphism $\psi: {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}}) \to {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$. Consider the following formulas with the numbers $a,
b$ to be determined later:
- $H^{i}_{X}\mapsto a^{i}\cdot H^{i}_{Y}$ for all $0\leq i\leq d$ and consequently $[{\ensuremath\mathrm{pt}}_{X}]\mapsto a^{d}\, s\cdot[{\ensuremath\mathrm{pt}}_{Y}]$, where $[{\ensuremath\mathrm{pt}}]$ is the class of a point;
- $a^{\frac{d}{2}}\cdot\phi_{{\ensuremath\mathrm{Hdg}}}: {\ensuremath\mathrm{Hdg}}_{{\ensuremath\mathds{Q}}}^{d}(X)\to
{\ensuremath\mathrm{Hdg}}^{d}_{{\ensuremath\mathds{Q}}}(Y)$;
- $b\cdot\phi_{{\ensuremath\mathrm{tr}}}:{\ensuremath\mathrm{H}}^{d}_{{\ensuremath\mathrm{tr}}}(X, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{d}_{{\ensuremath\mathrm{tr}}}(Y, {\ensuremath\mathds{Q}})$.
These formulas define an algebra isomorphism if and only if $b^{2}=a^{d}\, s$. This equation has non-zero rational solutions when $d$ is odd or $d$ is even and $s$ is a square in ${\ensuremath\mathds{Q}}$. Item $(ii)$ is therefore proved.
The proof of $(i)$ goes similarly as for $(ii)$ by replacing ${\ensuremath\mathds{Q}}$ by $\mathbb{R}$. Notice that the analogous assumption that $s$ is a square in $\mathbb{R}$ is automatically satisfied. So it is enough to see that there are always non-zero real solutions to the equation $b^{2}=a^{d}\, s=1$, where the last equality reflects the Frobenius condition.
Abelian varieties
-----------------
\[prop:AV\] Let $A$ and $B$ be two isogenous abelian varieties of dimension $g$. Then
1. ${\ensuremath\mathfrak{h}}(A)$ and ${\ensuremath\mathfrak{h}}(B)$ are isomorphic as algebra objects.
2. The following conditions are equivalent:
1. There is an isomorphism of Frobenius algebra objects between ${\ensuremath\mathfrak{h}}(A)$ and ${\ensuremath\mathfrak{h}}(B)$.
2. There is a graded Hodge isomorphism of Frobenius algebras between ${\ensuremath\mathrm{H}}^{*}(A, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{*}(B, {\ensuremath\mathds{Q}})$.
3. There exists an isogeny of degree $m^{2g}$ between $A$ and $B$ for some $m\in {\ensuremath\mathds{Z}}_{>0}$.
In the case that these equivalent conditions hold, the isomorphism in $(a)$, denoted by $\Gamma: {\ensuremath\mathfrak{h}}(A)\to {\ensuremath\mathfrak{h}}(B)$, can be chosen to respect moreover the motivic decomposition of Deninger–Murre [@dm] in the sense that $\Gamma\circ \pi_{A}^{i}=\pi_{B}^{i}\circ \Gamma$ for any $i$, where the $\pi^{i}$’s are the projectors corresponding to the decomposition.
3. ${\ensuremath\mathfrak{h}}(A)_{\mathbb{R}}$ and ${\ensuremath\mathfrak{h}}(B)_{\mathbb{R}}$ are isomorphic as Frobenius algebra objects in the category of Chow motives with real coefficients.
$(i)$ Consider any isogeny $f: B\to A$. Then $f^{*}: {\ensuremath\mathfrak{h}}(A)\to {\ensuremath\mathfrak{h}}(B)$ is an isomorphism of algebra objects with inverse given by $\frac{1}{\deg(f)}f_{*}$.\
$(ii)$ The implication $(a)\Longrightarrow (b)$ is obtained by applying the realization functor.\
$(b)\Longrightarrow (c)$. Let $\gamma: {\ensuremath\mathrm{H}}^{*}(A,
{\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow}{\ensuremath\mathrm{H}}^{*}(B, {\ensuremath\mathds{Q}})$ be a Frobenius algebra isomorphism preserving the Hodge structures, and let $\gamma_{i}: {\ensuremath\mathrm{H}}^{i}(A,
{\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{i}(B, {\ensuremath\mathds{Q}})$ be its $i$-th component, for all $0\leq i\leq 2g$. There exist a rational number $\lambda$ and an isogeny $f: B\to A$, such that $\gamma_{1}: {\ensuremath\mathrm{H}}^{1}(A, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{1}(B, {\ensuremath\mathds{Q}})$ is equal to $\frac{1}{\lambda}f^{*}|_{{\ensuremath\mathrm{H}}^{1}}$. As ${\ensuremath\mathrm{H}}^{*}(A, {\ensuremath\mathds{Q}})\cong
\bigwedge^{\bullet}
{\ensuremath\mathrm{H}}^{1}(A, {\ensuremath\mathds{Q}})$ as algebras and similarly for $B$, $\gamma$ is in fact determined by $\gamma_{1}$ in the following way: for any $i$, $\gamma_{i}=\wedge^{i}\gamma_{1}=\frac{1}{\lambda^{i}}f^{*}|_{{\ensuremath\mathrm{H}}^{i}}$. We compute that $${\ensuremath\mathrm{id}}={}^{t}\gamma\circ
\gamma=\left(\sum_{i}\frac{1}{\lambda^{i}}
f_{*}|_{{\ensuremath\mathrm{H}}^{2g-i}}\right)\circ
\left(\sum_{i}\frac{1}{\lambda^{i}}f^{*}|_{{\ensuremath\mathrm{H}}^{i}}\right)=\frac{1}{\lambda^{2g}}\deg(f)\cdot{\ensuremath\mathrm{id}}.$$ This yields that the isogeny $f$ is of degree $\lambda^{2g}$.\
$(c)\Longrightarrow (a)$ If there is an isogeny $f: B\to A$ of degree $m^{2g}$, then for any $0\leq
i\leq 2g$ consider the morphism $\Gamma_{i}:=\frac{1}{m^{i}}\pi^{i}_{B}\circ
f^{*}\circ \pi^{i}_{A}=\frac{1}{m^{i}} f^{*}\circ \pi^{i}_{A}$ from ${\ensuremath\mathfrak{h}}^{i}(A)$ to ${\ensuremath\mathfrak{h}}^{i}(B)$, which is an isomorphism with inverse $\Gamma_{i}^{-1}=\frac{1}{m^{2g-i}}\pi^{i}_{A}\circ f_{*}$. Here we use the motivic decomposition of Deninger–Murre [@dm] for abelian varieties ${\ensuremath\mathfrak{h}}(A)=\oplus_{i=0}^{2g}{\ensuremath\mathfrak{h}}^{i}(A)$, and $\pi^{i}$ is the projector corresponding to ${\ensuremath\mathfrak{h}}^{i}$. One readily checks that $\Gamma:=\sum_{i} \Gamma_{i}: {\ensuremath\mathfrak{h}}(A)\to {\ensuremath\mathfrak{h}}(B)$ is an isomorphism of algebra objects. Moreover, as $\pi^{i}={}^{t}\pi^{2g-i}$ for all $i$, we have that $\Gamma_{i}^{-1}={}^{t}\Gamma_{2g-i}$, hence $\Gamma^{-1}={}^{t}\Gamma$, that is, $\Gamma$ respects the Frobenius structures. Notice that by construction, $\Gamma$ respects the decomposition of Deninger–Murre.\
The proof of $(iii)$ is similar to the last part of the proof of $(ii)$. One only needs to notice that there is no obstruction to taking the $2g$-th root of a positive number in $\mathbb{R}$.
As a consequence, given two derived equivalent abelian varieties, in general there is no isomorphism of Frobenius algebra objects between their Chow motives (or their cohomology). Indeed, by Proposition \[prop:AV\]$(ii)$, the motives of two derived equivalent abelian varieties that cannot be related by an isogeny of degree the $2g$-th power of some positive integer are not isomorphic as Frobenius algebra objects. For instance, if one considers an abelian variety $A$ with Néron–Severi group generated by one ample line bundle $L$, then any isogeny between $A$ and $A^{\vee}$ is of degree $\chi(L)^{2}m^{4g}$ for some $m\in
{\ensuremath\mathds{Z}}_{>0}$. But in general, $\chi(L)$ is not a $g$-th power in ${\ensuremath\mathds{Z}}$. On the other hand, $A$ and $A^{\vee}$ are always derived equivalent by Mukai’s classical result [@MukaiAb].
Hyper-Kähler varieties
----------------------
One particularly interesting class of varieties for which we expect a positive answer consists of (projective) hyper-Kähler varieties; these constitute higher-dimensional generalizations of K3 surfaces. Note that by Huybrechts–Nieper-Wißkirchen [@HNW], any Fourier–Mukai partner of a hyper-Kähler variety remains hyper-Kähler.
\[conj:HK\] Let $X$ and $Y$ be two projective hyper-Kähler varieties. If there is an exact equivalence between triangulated categories $\operatorname{D}^b(X)$ and $\operatorname{D}^b(Y)$, then there exists an isomorphism of Chow motives ${\ensuremath\mathfrak{h}}(X)$ and ${\ensuremath\mathfrak{h}}(Y)$, as Frobenius algebra objects in the categories of Chow motives. In particular, their Chow rings as well as cohomology rings are isomorphic.
The following result is known to the experts; it answers the last part of Conjecture \[conj:HK\] for cohomology with complex coefficients.
Let $X$ and $Y$ be two derived equivalent projective hyper-Kähler varieties. Then their cohomology rings with complex coefficients are isomorphic as ${\ensuremath\mathds{C}}$-algebras.
[^10] As any exact equivalence $\operatorname{D}^{b}(X)\simeq
\operatorname{D}^{b}(Y)$ is given by Fourier–Mukai kernel, it lifts naturally to an equivalence of differential graded categories. Therefore we have an isomorphism of graded ${\ensuremath\mathds{C}}$-algebras between their Hochschild cohomology: $${\ensuremath\mathrm{HH}}^{*}(X)\simeq {\ensuremath\mathrm{HH}}^{*}(Y).$$ By a result of Calaque–Van den Bergh [@CVdB], which was also previously announced by Kontsevich, the Hochschild–Kostant–Rosenberg isomorphism twisted by the square root of the Todd genus gives rise to an isomorphism of ${\ensuremath\mathds{C}}$-algebras $${\ensuremath\mathrm{HH}}^{*}(X)\simeq \bigoplus_{i+j=*}{\ensuremath\mathrm{H}}^{i}(X, \bigwedge^{j}T_{X}).$$ Now the symplectic forms on $X$ induce an isomorphism between $T_{X}$ and $\Omega_{X}$, which yields isomorphisms of ${\ensuremath\mathds{C}}$-algebras: $$\bigoplus_{i+j=*}{\ensuremath\mathrm{H}}^{i}(X,
\bigwedge^{j}T_{X})\simeq \bigoplus_{i+j=*}{\ensuremath\mathrm{H}}^{i}(X, \Omega^{j}_{X})\simeq
{\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{C}}).$$ We can conclude by combining these isomorphisms.
There are not so many known examples of derived equivalent hyper-Kähler varieties. Let us test Conjecture \[conj:HK\] for the available ones.
Let $S$ and $S'$ be two derived equivalent K3 surfaces. Then for any $n\in
\mathbb N^{*}$, the $n$-th Hilbert schemes $\operatorname{Hilb}^{n}(S)$ and $\operatorname{Hilb}^{n}(S')$ are derived equivalent. Indeed, by combining the results of Bridgeland–King–Reid [@BKR] and Haiman [@Haiman], we have exact linear equivalences of triangulated categories: $${\ensuremath\mathrm{D}}^{b}(\operatorname{Hilb}^{n}(S))\simeq
{\ensuremath\mathrm{D}}^{b}(\mathfrak{S}_{n}\!-\!\operatorname{Hilb}(S^{n}))\simeq
{\ensuremath\mathrm{D}}^{b}_{\mathfrak
S_{n}}(S^{n}),$$ and similarly for $S'$; the Fourier–Mukai kernel $\mathcal{E}\boxtimes\cdots
\boxtimes \mathcal{E}$ induces an equivalence $${\ensuremath\mathrm{D}}^{b}_{\mathfrak
S_{n}}(S^{n})\simeq {\ensuremath\mathrm{D}}^{b}_{\mathfrak S_{n}}(S'^{n}),$$ where $\mathcal E\in
{\ensuremath\mathrm{D}}^{b}(S\times S')$ is the original Fourier–Mukai kernel inducing the equivalence between ${\ensuremath\mathrm{D}}^{b}(S)$ and ${\ensuremath\mathrm{D}}^{b}(S')$. We showed in Corollary \[cor:Powers\] that ${\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S))$ and ${\ensuremath\mathfrak{h}}(\operatorname{Hilb}^{n}(S'))$ are isomorphic as Frobenius algebra objects.
Conjecturally two birationally equivalent hyper-Kähler varieties are derived equivalent [@HuybrechtsFMBook Conjecture 6.24]. Thanks to the result of Rieß [@Riess], or rather its proof, we know that birational hyper-Kähler varieties have isomorphic Chow motives as Frobenius algebra objects, hence compatible with Conjecture \[conj:HK\]. There are by now some cases where the derived equivalence is known. The easiest example might be the so-called Mukai flop. Another instance of interest is as follows: given a projective K3 surface $S$ and a Mukai vector $v$, when the stability condition $\sigma$ varies in the chambers of the distinguished component $\operatorname{Stab}^{*}(S)$ of the manifold of stability conditions on ${\ensuremath\mathrm{D}}^{b}(S)$, the moduli spaces $M_{\sigma}(v)$ of $\sigma$-stable objects are all birational to each other, and their derived equivalence has been announced by Halpern-Leistner in [@DHL].
If one is willing to enlarge a bit the category of hyper-Kähler varieties to that of hyper-Kähler *orbifolds*[^11], Conjecture \[conj:HK\] is closely related to the so-called *motivic hyper-Kähler resolution conjecture* investigated in [@ftv] and [@FT]. Indeed, let $M$ be a projective holomorphic symplectic variety endowed with a faithful action of a finite group $G$ by symplectic automorphisms. The quotient stack $[M/G]$ is a hyper-Kähler (or rather symplectic) orbifold. If the main component of the $G$-invariant Hilbert scheme $X:=G\!-\!\operatorname{Hilb}(M)$ is a symplectic (or equivalently crepant) resolution of the singular variety $M/G$, then by Bridgeland–King–Reid [@BKR Corollary 1.3] there is an equivalence of derived categories ${\ensuremath\mathrm{D}}^{b}(X)\simeq {\ensuremath\mathrm{D}}^{b}([M/G])$. On the other hand, the motivic hyper-Kähler resolution conjecture [@ftv] predicts that the orbifold motive of $[M/G]$ endowed with the *orbifold product* is isomorphic to the motive of $X$ as algebra objects. In this sense, forgetting the Frobenius structure, we can obtain some evidences for the orbifold analogue of Conjecture \[conj:HK\]: for example between a K3 orbifold and its minimal resolution by [@FTsurface], between $[\ker(A^{n+1}\xrightarrow{+} A)/\mathfrak{S}_{n}]$ and the $n$-th generalized Kummer variety associated to an abelian surface $A$ by [@ftv], and between $[S^{n}/\mathfrak{S}_{n}]$ and the $n$-th Hilbert scheme of a K3 surface $S$ by [@FT]. In fact, the authors suspect that the motivic hyper-Kähler resolution conjecture can be stated more strongly as an isomorphism of Frobenius algebra objects with *complex* coefficients, and the proofs of our aforementioned results do confirm this stronger version.
Chern classes of Fourier–Mukai equivalences between K3 surfaces {#sec:v2}
===============================================================
The aim of this final section is to provide evidence for the fact that the Chern classes of Fourier–Mukai equivalences between two K3 surfaces $S$ and $S'$ define “distinguished” classes in the Chow ring of $S\times S'$, in the sense that they can be added to the Beauville–Voisin ring of $S\times S'$ and the resulting ring would still inject into cohomology *via* the cycle class map.
The Beauville–Voisin ring, and generalizations {#subsec:mck}
----------------------------------------------
Let $S$ be a K3 surface and define its Beauville–Voisin ring $R^*(S)$ to be the subring of ${\ensuremath\mathrm{CH}}^*(S)$ generated by divisors and Chern classes of the tangent bundle. By Beauville–Voisin’s Theorem \[thm:bv\], this ring has the property that it injects into cohomology via the cycle class map.
Let ${\ensuremath\mathfrak{h}}(S) = {\ensuremath\mathfrak{h}}^0(S)\oplus {\ensuremath\mathfrak{h}}^2(S) \oplus {\ensuremath\mathfrak{h}}^4(S)$ be the Chow–Künneth decomposition induced by $\pi^0_S = o_S\times S$, $\pi^4_S = S\times o_S$ and $\pi_S^2 =
\Delta_S - \pi_S^0 - \pi_S^4$. In [@sv Proposition 8.14] it was observed that the decomposition of the small diagonal is equivalent to the above Chow–Künneth decomposition being *multiplicative*, meaning that the multiplication morphism ${\ensuremath\mathfrak{h}}(S)
\otimes {\ensuremath\mathfrak{h}}(S) \to {\ensuremath\mathfrak{h}}(S)$ is compatible with the grading given by the Chow–Künneth decomposition.
The following (formal) facts about multiplicative Chow–Künneth decompositions will be used. Let $X$ and $Y$ be two smooth projective varieties, both having motive endowed with a multiplicative Chow–Künneth decomposition. Then [@sv Theorem 8.6] the *product Chow–Künneth decomposition* ${\ensuremath\mathfrak{h}}^n(X\times Y) =
\bigoplus_{i+j=n}
{\ensuremath\mathfrak{h}}^i(X)\otimes {\ensuremath\mathfrak{h}}^j(Y)$ is multiplicative. Moreover, if $p:X\times Y \to X$ denotes the projection, then $p^* : {\ensuremath\mathfrak{h}}(X) \to {\ensuremath\mathfrak{h}}(X\times Y)$ is graded (*i.e.* compatible with the Chow–Künneth decompositions) and $p_* :
{\ensuremath\mathfrak{h}}(X\times
Y)
\to {\ensuremath\mathfrak{h}}(X)$ shifts the gradings by $-2\dim Y$.
A Chow–Künneth decomposition on the motive of $X$ induces a bigrading on the Chow groups of $X$ given by $${\ensuremath\mathrm{CH}}^i(X)_{(j)} := {\ensuremath\mathrm{CH}}^i({\ensuremath\mathfrak{h}}^{2i-j}(X)),$$ which in case the Chow–Künneth decomposition is multiplicative satisfies $${\ensuremath\mathrm{CH}}^i(X)_{(j)} \cdot {\ensuremath\mathrm{CH}}^{i'}(X)_{(j')} \subseteq {\ensuremath\mathrm{CH}}^{i+i'}(X)_{(j+j')} .$$ Given smooth projective varieties endowed with multiplicative Chow–Künneth decompositions, the products of which are endowed with the product Chow–Künneth decompositions, we therefore see that ${\ensuremath\mathrm{CH}}^*(-)_{(0)}$ defines a subalgebra of ${\ensuremath\mathrm{CH}}^*(-)$ that is stable under pushforwards and pullbacks along projections, and stable under composition of correspondences belonging to ${\ensuremath\mathrm{CH}}^*(-\times
-)_{(0)}$.
Murre’s conjecture \[conj:murre\] and imply that ${\ensuremath\mathrm{CH}}^i(X)_{(0)} := {\ensuremath\mathrm{CH}}^i({\ensuremath\mathfrak{h}}^{2i}(X))$ injects in cohomology with image the Hodge classes for any choice of Chow–Künneth decomposition. (This is known unconditionally in the case $i
=0$ and $i = \dim
X$.) In particular, in the above situation of smooth projective varieties endowed with multiplicative Chow–Künneth decompositions, it is expected that the subalgebra ${\ensuremath\mathrm{CH}}^*(-)_{(0)}$ injects into cohomology with image the Hodge classes. In that sense, ${\ensuremath\mathrm{CH}}^*(-)_{(0)}$ is a maximal subalgebra of ${\ensuremath\mathrm{CH}}^*(-)$ with the property that it injects into cohomology *via* the cycle class map.
Adding the second Chern class of Fourier–Mukai equivalences to the BV ring
--------------------------------------------------------------------------
Recall the following theorem of Huybrechts [@huybrechts-JEMS Theorem 2] and Voisin [@voisin-ogrady Corollary 1.10].
\[thm:hv\] Let $\Phi_\mathcal{E} : \operatorname{D}^b(S)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S')$ be an exact linear equivalence between K3 surfaces with Fourier–Mukai kernel $\mathcal{E} \in
\operatorname{D}^b(S\times S')$. Then $v(\mathcal{E})$ preserves the Beauville–Voisin ring.
In light of the discussion in §\[subsec:mck\], it is natural to ask whether a more general statement could be true, namely:
\[ques:grade0\] Let $\Phi_\mathcal{E} : \operatorname{D}^b(S)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S')$ be an exact linear equivalence between K3 surfaces with Fourier–Mukai kernel $\mathcal{E} \in
\operatorname{D}^b(S\times S')$. Then does $v(\mathcal{E})$ belong to ${\ensuremath\mathrm{CH}}^*(S\times S')_{(0)}$?
For $i=0$ or $1$, the Mukai vectors $v_i(\mathcal E)$ obviously belong to ${\ensuremath\mathrm{CH}}^i(S\times S')_{(0)}$, since in those cases ${\ensuremath\mathrm{CH}}^i(S\times S')=
{\ensuremath\mathrm{CH}}^i(S\times S')_{(0)}$. In the case of $v_2(\mathcal{E})$, this can be deduced from Theorem \[thm:hv\]:
\[prop:v2\] Let $\Phi_\mathcal{E} : \operatorname{D}^b(S)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S')$ be an exact equivalence with Fourier–Mukai kernel $\mathcal{E} \in
\operatorname{D}^b(S\times S')$. Then $v_2(\mathcal{E})$ belongs to ${\ensuremath\mathrm{CH}}^2(S\times S')_{(0)}$.
Since ${\ensuremath\mathrm{CH}}^2(S\times S') = {\ensuremath\mathrm{CH}}^2(S\times S')_{(0)} \oplus {\ensuremath\mathrm{CH}}^2(S\times
S')_{(2)}$, it is enough to show that $v_2(\mathcal{E})_{(2)} = 0$. Let $\gamma$ be any cycle in ${\ensuremath\mathrm{CH}}^2(S\times S')$. On the one hand, we have $\gamma_{(2)} =
\pi^2_{\mathrm{tr},S'} \circ \gamma \circ \pi^4_S + \pi^0_{S'} \circ \gamma
\circ
\pi^2_{\mathrm{tr},S}$. On the other hand, we have $\gamma \circ \pi^4_S =
(p')^*\gamma_*
o_S$. Now setting $\gamma = v_2(\mathcal{E})$, Theorem \[thm:hv\] yields that $v_2(\mathcal E) \circ \pi^4_S$ is a multiple of $(p')^*o_{S'}$ and henceforth since $(\pi^2_{\mathrm{tr},S'})_*o_{S'} =0$ that $\pi^2_{\mathrm{tr},S'} \circ \gamma \circ \pi^4_S = 0$. Likewise, we have $\pi^0_{S'} \circ v_2(\mathcal E)
\circ
\pi^2_{\mathrm{tr},S} = 0$, and the proposition is established.
We deduce from §\[subsec:mck\] the following
\[thm:product-c2\] Let $\tilde{R}^*(S\times S')$ be the subring of ${\ensuremath\mathrm{CH}}^*(S\times S')$ generated by divisors, $p^*c_2(S)$, $(p')^*c_2(S')$ and $c_2(\mathcal{E})$, where $\mathcal{E}$ runs through the objects in $\mathrm{D}^b(S\times S')$ inducing exact linear equivalences $ \operatorname{D}^b(S)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S')$. The cycle class map $\tilde{R}^n(S\times S') \to {\ensuremath\mathrm{H}}^{2n}(S\times S',{\ensuremath\mathds{Q}})$ is injective for $n=3,4$. In particular, if $\Phi_{\mathcal{E}_{1}}, \Phi_{\mathcal{E}_{2}}
: \operatorname{D}^b(S) \stackrel{\sim}{\longrightarrow}
\operatorname{D}^b(S')$ are two exact linear equivalences with Fourier–Mukai kernels $\mathcal{E}_{1}, \mathcal{E}_{2} \in \operatorname{D}^b(S\times S')$, then $c_2(\mathcal{E}_{1})\cdot c_2(\mathcal{E}_{2}) \in {\ensuremath\mathds{Z}}[ o_S\times
o_{S'}]$.
Some speculations concerning Chern classes of twisted derived equivalent K3 surfaces
------------------------------------------------------------------------------------
It is natural to ask whether Theorem \[thm:hv\] extends to derived equivalences between twisted K3 surfaces:
\[ques:grade0twisted\] Let $\Phi_\mathcal{E} : \operatorname{D}^b(S,\alpha)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S',\alpha')$ be an exact equivalence between twisted K3 surfaces with Fourier–Mukai kernel $\mathcal{E} \in
\operatorname{D}^b(S\times S',\alpha^{-1} \boxtimes \alpha')$. Then does $v(\mathcal{E})$ preserve the Beauville–Voisin ring? More generally, does $v(\mathcal{E})$ belong to ${\ensuremath\mathrm{CH}}^*(S\times S')_{(0)}$?
We note that if $v(\mathcal{E})$ preserves the Beauville–Voisin ring, then the same argument as in the proof of Proposition \[prop:v2\] gives that $v_2(\mathcal{E})$ belongs to ${\ensuremath\mathrm{CH}}^2(S\times S')_{(0)}$.
Let us now define $E^*(S\times S')$ to be the subalgebra of ${\ensuremath\mathrm{CH}}^*(S\times S')$ generated by divisors, $p^*c_2(S)$, $(p')^*c_2(S')$, and the Chern classes of $\mathcal{E}$, where $\mathcal{E}$ runs through objects in $\operatorname{D}^b(S\times S',\alpha^{-1} \boxtimes \alpha')$ inducing exact equivalences $\Phi_\mathcal{E} : \operatorname{D}^b(S,\alpha)
\stackrel{\sim}{\longrightarrow} \operatorname{D}^b(S',\alpha')$ for some Brauer classes $\alpha \in \mathrm{Br}(S)$ and $\alpha' \in \mathrm{Br}(S')$. We then define $\tilde{E}^*(S\times S')$ to be the subalgebra of ${\ensuremath\mathrm{CH}}^*(S\times S')$ generated by cycles of the form $$\gamma_{n-1}\circ \cdots \circ \gamma_0,$$ where $\gamma_i \in E^*(S_i\times S_{i+1})$ for all $i$ for some K3 surfaces $S=S_0,S_1,\ldots, S_{n}=S'$. According to the discussion in §\[subsec:mck\], a positive answer to Question \[ques:grade0twisted\] would suggest that the following question should have a positive answer.
Does $\tilde{E}^*(S\times S')$ inject into cohomology via the cycle class map?
In particular, if ${\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}}) \simeq {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})$ is an isogeny, then the cycle class $v_2(\mathcal{E}_{n-1}) \circ \cdots \circ
v_2(\mathcal{E}_0)$ inducing the isogeny between $T(S)_{\ensuremath\mathds{Q}}:=
{\ensuremath\mathrm{H}}^*({\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S))$ and $T(S')_{\ensuremath\mathds{Q}}:= {\ensuremath\mathrm{H}}^*({\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S))$ (with $\mathcal{E}_0,\dots, \mathcal{E}_{n-1}$ as in ) should be canonically defined, *i.e.* should not depend on the choice of twisted derived equivalence between $S$ and $S'$ as in inducing the isogeny.
Non-isogenous K3 surfaces with isomorphic Hodge structures {#sect:Appendix}
==========================================================
Recall that two K3 surfaces $S$ and $S'$ are said to be *isogenous* if their second rational cohomology groups are Hodge isometric, that is, if there exists an isomorphism of Hodge structures $f : {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})
\stackrel{\simeq}{\longrightarrow} {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})$ making the following diagram commute: $$\xymatrix{{\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}}) \ar[rr]^{\qquad \cup} \ar[d]^{f\otimes
f} && {\ensuremath\mathrm{H}}^4(S,{\ensuremath\mathds{Q}})\ar[rr]^{\deg} && {\ensuremath\mathds{Q}}\ar@{=}[d]\\
{\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}}) \ar[rr]^{\qquad \cup} && {\ensuremath\mathrm{H}}^4(S',{\ensuremath\mathds{Q}})
\ar[rr]^{\deg} &&
{\ensuremath\mathds{Q}}.
}$$ We provide in this appendix infinite families of pairwise non-isogenous K3 surfaces with isomorphic rational Hodge structures, that is, for any two K3 surfaces $S$ and $S'$ belonging to the same family, we have ${\ensuremath\mathrm{H}}^{2}(S,
{\ensuremath\mathds{Q}})\simeq {\ensuremath\mathrm{H}}^{2}(S',
{\ensuremath\mathds{Q}})$ as ${\ensuremath\mathds{Q}}$-Hodge structures but there does not exist any isomorphism of ${\ensuremath\mathds{Q}}$-Hodge structures ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Q}})$ that is compatible with the intersection pairings given by $(\alpha,\beta) \mapsto \deg(\alpha\cup
\beta)$. By Mukai [@Mukai], such K3 surfaces are not derived equivalent to each other, and actually not even through a chain of twisted derived equivalences (*cf.* Huybrechts [@huybrechts-isogenous Theorem 0.1]).
In fact, our families have a stronger property: any two K3 surfaces $S$ and $S'$ in the same family have Hodge-isomorphic rational cohomology *algebras*, that is, for any two K3 surfaces $S$ and $S'$ of the family, we have a graded Hodge isomorphism $g :
{\ensuremath\mathrm{H}}^{*}(S, {\ensuremath\mathds{Q}})\stackrel{\simeq}{\longrightarrow} {\ensuremath\mathrm{H}}^{*}(S',
{\ensuremath\mathds{Q}})$ that respects the cup-product, *i.e.* such that the following diagram commutes: $$\xymatrix{{\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}}) \ar[rr]^{\qquad \cup} \ar[d]^{g\otimes
g} && {\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}}) \ar[d]^g \\
{\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}}) \ar[rr]^{\qquad \cup} && {\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}}).
}$$ We say that ${\ensuremath\mathrm{H}}^{*}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{*}(S',
{\ensuremath\mathds{Q}})$ are *Hodge algebra isomorphic*.
Motivation and statements
-------------------------
The aim of this appendix is to show that, for K3 surfaces, the notion of *isogeny* is strictly more restrictive than the notion of *Hodge algebra isomorphic*. In §\[SS:motivic\] this is upgraded motivically: we show that, for Chow motives of K3 surfaces, the notion of being *isomorphic as Frobenius algebra objects* is strictly more restrictive than the notion of being *isomorphic as algebra objects*, thereby justifying the somewhat technical condition $(iii)$ involving Frobenius algebra objects in the motivic Torelli statement of Corollary \[cor:torelli\].
First, we provide an infinite family of K3 surfaces whose rational cohomology rings are all Hodge algebra isomorphic (and, *a fortiori*, Hodge isomorphic), but they are pairwise non-isogenous. Precisely we have
\[thm:non-isogenousK3\] There exists an infinite family $\{S_i\}_{i \in {\ensuremath\mathds{Z}}_{>0}}$ of pairwise non-isogenous K3 surfaces such that, for all $j, k \in {\ensuremath\mathds{Z}}_{>0}$, ${\ensuremath\mathrm{H}}^*(S_j,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S_k,{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic. Moreover, such a family can be chosen to consist of K3 surfaces of maximal Picard rank 20.
We will present two proofs of this theorem, one in §\[SS:2\] and the other in §\[SS:1\]. Furthermore, we will show in Theorem \[thm:IsomChowMot\] that the Chow motives of K3 surfaces belonging to the family of K3 surfaces of Theorem \[thm:non-isogenousK3\] are all isomorphic as algebra objects.
Second, if the Picard number is maximal, a family of K3 surfaces as in Theorem \[thm:non-isogenousK3\] must have non-isometric Néron–Severi spaces (Lemma \[lemma:rho=20\]). In this sense, we can improve Theorem \[thm:non-isogenousK3\] in order to have in addition isometric ${\ensuremath\mathds{Q}}$-quadratic forms on the Néron–Severi spaces:
\[P:bothisonotisogenous\] There exists an infinite family $\{S_i\}_{i \in {\ensuremath\mathds{Z}}_{>0}}$ of pairwise non-isogenous K3 surfaces such that, for all $j\neq k \in {\ensuremath\mathds{Z}}_{>0}$, we have
- ${\ensuremath\mathrm{H}}^*(S_j,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S_k,{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic.
- ${\ensuremath\mathrm{H}}^2_{\operatorname{tr}}(S_j,{\ensuremath\mathds{Q}})\simeq {\ensuremath\mathrm{H}}^2_{\operatorname{tr}}(S_k,{\ensuremath\mathds{Q}})$, hence $\operatorname{NS}(S_{j})_{{\ensuremath\mathds{Q}}}\simeq \operatorname{NS}(S_{k})_{{\ensuremath\mathds{Q}}}$, as ${\ensuremath\mathds{Q}}$-quadratic spaces.
Moreover, such a family can be chosen to consist of K3 surfaces with transcendental lattice being any prescribed even lattice with square discriminant and of signature $(2, 2), (2, 4), (2, 6)$ or $(2, 8)$.
Furthermore, we will show in Proposition \[prop:IsoChowRing3\] that, assuming Conjecture \[conj:MGTT\], the K3 surfaces in such a family all have isomorphic Chow motives and isomorphic Chow rings, but their Chow motives are pairwise non-isomorphic as Frobenius algebra objects. This gives evidence that one cannot characterize the isogeny class of a complex K3 surface with its Chow ring.
\[R:BSV\] As mentioned to us by Chiara Camere, examples of a pair of non-isogenous K3 surfaces with isomorphic rational Hodge structures were constructed geometrically, via the so-called *Inose isogenies*[^12] [@Inose], by Boissier–Sarti–Veniani [@BSV]. Precisely, let $f$ be a symplectic automorphism of prime order $p$ of a K3 surface $S$ and let $S'$ be the minimal resolution of the quotient $S/\langle f \rangle$; the surface $S'$ is a K3 surface and, by definition, the rational map $S\dashrightarrow S'$ is a degree-$p$ Inose isogeny between these two K3 surfaces. On the one hand, we have isomorphisms of Hodge structures ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S,
{\ensuremath\mathds{Q}})={\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S,
{\ensuremath\mathds{Q}})^{f}\simeq{\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S/\langle f \rangle, {\ensuremath\mathds{Q}})\simeq
{\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$. On the other hand, [@BSV Theorem 1.1 and Corollary 1.2] provide many situations where ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$ are not isometric and consequently ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Q}})$ are not Hodge isometric. Note that this approach only produces finitely many non-isogenous K3 surfaces with isomorphic Hodge structures. The aim of this appendix is to show, via the surjectivity of the period map, that in fact one can produce an *infinite* family of such pairwise non-isogenous K3 surfaces.
Hodge isomorphic *vs.* Hodge algebra isomorphic
-----------------------------------------------
All the examples of non-isogenous K3 surfaces that we will consider will consist of K3 surfaces $S$ and $S'$ whose respective transcendental lattices $T$ and $T'$ become Hodge isometric after some twist, *i.e.* such that $T
\simeq T'(m)$ as Hodge lattices for some integer $m$. The aim of this paragraph is to show that the cohomology algebra of any two such K3 surfaces are Hodge algebra isomorphic; *cf.* Lemma \[lem:algiso\] below. This will reduce the proofs of Theorems \[thm:non-isogenousK3\] and \[P:bothisonotisogenous\] to showing that the K3 surfaces in the families have Hodge isomorphic transcendental cohomology groups.
First we state a general lemma based on the classical classification of quadratic forms over ${\ensuremath\mathds{Q}}$. This lemma will also be used in the proof of Theorem \[P:bothisonotisogenous\].
\[lemma:QuadForm\] Let $Q$ be a non-degenerate ${\ensuremath\mathds{Q}}$-quadratic form of even rank.
- If $\operatorname{disc}(Q) = 1$ and $\operatorname{rk}(Q) \equiv 0 \ \text{mod}\
4$, or if $\operatorname{disc}(Q) = -1$ and $\operatorname{rk}(Q) \equiv 2 \ \text{mod}\
4$, then for any $m\in {\ensuremath\mathds{Q}}_{>0}$, $Q$ and $Q(m)$ are isometric ${\ensuremath\mathds{Q}}$-quadratic forms.
- If $\operatorname{disc}(Q) = 1$ and $\operatorname{rk}(Q) \equiv 2 \ \text{mod}\
4$, or if $\operatorname{disc}(Q) = -1$ and $\operatorname{rk}(Q) \equiv 0 \ \text{mod}\
4$, then for any $m\in N{\ensuremath\mathds{Q}}(i)^{\times }$, $Q$ and $Q(m)$ are isometric ${\ensuremath\mathds{Q}}$-quadratic forms, where $N{\ensuremath\mathds{Q}}(i)^{\times }=\{x^{2}+y^{2} \mid (x, y)\neq
(0,
0)\in {\ensuremath\mathds{Q}}^{2}\}$ is the norm group of the field extension ${\ensuremath\mathds{Q}}(i)/{\ensuremath\mathds{Q}}$.
Obviously $Q$ and $Q(m)$ have the same rank and signature. As the rank of $Q$ is even, their discriminants are also the same (that is, only differ by a square). By the classification theory of quadratic forms over ${\ensuremath\mathds{Q}}$, we only need to check that for any prime number $\ell$, their $\varepsilon$-invariants are equal at all places $\ell$. Assume that $Q$ is equivalent to the diagonal form $\langle a_{1}, \ldots, a_{r}\rangle$ where $r$ is the rank of $Q$ and $a_{i}\in
{\ensuremath\mathds{Q}}$; its discriminant is thus given by $\operatorname{disc}(Q) =
\prod_{i=1}^{r}a_{i}$ and we have $$\begin{aligned}
\varepsilon_{\ell}(Q(m)) & := &\prod_{i<j}(a_{i}m,
a_{j}m)_{\ell}=\prod_{i<j}(a_{i}, a_{j})_{\ell}\left(\prod_{i=1}^{r}(a_{i},
m)_{\ell}\right)^{r-1}(m, m)_{\ell}^{r(r-1)/2}\\
& = &\varepsilon_{\ell}(Q) \big((\operatorname{disc}(Q)^{r-1}
(-1)^{r(r-1)/2},m\big)_\ell.
\end{aligned}$$ where $(a, b)_{\ell}$ is the Hilbert symbol of $a, b$ for the local field ${\ensuremath\mathds{Q}}_{\ell}$ and the last equality uses the identity $(m, m)_{\ell}=(m,
-1)_{\ell}$. One concludes by using the fact that $(m, -1)_{\ell} = 1$ for all prime numbers $\ell$ when $m\in N{\ensuremath\mathds{Q}}(i)^{\times }$.
\[lem:algiso\] Let $S$ and $S'$ be two complex K3 surfaces. The following statements are equivalent:
(i) There is a Hodge isometry ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}}) \simeq
{\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})(m)$, for some $m\in {\ensuremath\mathds{Q}}_{>0}$;
(ii) There is a Hodge isometry ${\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}}) \simeq {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})(m)$, for some $m\in {\ensuremath\mathds{Q}}_{>0}$;
(iii) The cohomology rings ${\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic.
$(i) \Rightarrow (ii)$. Clearly any choice of linear isomorphism between the orthogonal complements of ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$ provides a Hodge isomorphism ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}}) \simeq {\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Q}})$ that extends the Hodge isomorphism between ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$. Since the K3 lattice $\Lambda:=E_{8}(-1)^{\oplus 2}\oplus U^{\oplus 3}$ and its twist $\Lambda(m)$ are ${\ensuremath\mathds{Q}}$-isometric for all positive rational numbers $m$ (*cf.* Lemma \[lemma:QuadForm\]), the Hodge isometry ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}}) \simeq {\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S,
{\ensuremath\mathds{Q}}))(m)$ extends to a Hodge isometry ${\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}}) \simeq {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})(m)$ by Witt’s theorem, where the twist $(m)$ refers to the quadratic form.
$(ii) \Rightarrow (iii)$. Item $(ii)$ means that there is a Hodge class $\Gamma
\in
{\ensuremath\mathrm{H}}^4(S\times S',{\ensuremath\mathds{Q}})$ making the diagram $$\label{diag:degree}
\xymatrix{{\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}}) \ar[rr]^{\quad \cup}
\ar[d]^{(\Gamma\otimes
\Gamma)_*} && {\ensuremath\mathrm{H}}^4(S,{\ensuremath\mathds{Q}})
\ar[rr]^{\deg} && {\ensuremath\mathds{Q}}\ar[d]^{\cdot m}\\
{\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathrm{H}}^2(S',{\ensuremath\mathds{Q}}) \ar[rr]^{\quad \cup} && {\ensuremath\mathrm{H}}^4(S',{\ensuremath\mathds{Q}})
\ar[rr]^{\deg}
&&
{\ensuremath\mathds{Q}}}$$ commute. By imposing the $(4,0)$- and $(0,4)$-Künneth components of $\Gamma$ to be $[{\ensuremath\mathrm{pt}}]
\times [S']$ and $m\, [S]\times [{\ensuremath\mathrm{pt}}]$, respectively, we obtain that $\Gamma_*
:
{\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}}) \to {\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})$ is an isomorphism of algebras.
$(iii) \Rightarrow (ii)$. Let $\Gamma : {\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}}) \to {\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})$ be a Hodge algebra isomorphism. Then its restriction to ${\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})$ induces the commutative diagram , where $m$ is the rational number such that $\Gamma_*[{\ensuremath\mathrm{pt}}] = m\, [{\ensuremath\mathrm{pt}}]$.
$(ii)\Rightarrow (i)$. As any Hodge isomorphism must preserve the transcendental part, we see that the restriction of a Hodge isometry as in $(ii)$ gives a Hodge isometry ${\ensuremath\mathrm{H}}^{2}_{{\ensuremath\mathrm{tr}}}(S, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{2}_{{\ensuremath\mathrm{tr}}}(S', {\ensuremath\mathds{Q}})$.
First construction: an elementary approach {#SS:2}
------------------------------------------
Our first construction is for K3 surfaces of maximal Picard number $\rho=20$, and provides a proof of Theorem \[thm:non-isogenousK3\].
Given a K3 surface $S$ of Picard rank 20, the subspace ${\ensuremath\mathrm{H}}^{2,0}(S)\oplus
{\ensuremath\mathrm{H}}^{0,2}(S) \subset
{\ensuremath\mathrm{H}}^2(S,{\ensuremath\mathds{Q}})\otimes {\ensuremath\mathds{C}}$ is defined over ${\ensuremath\mathds{Q}}$, and consequently, up to multiplying by a real scalar, we can fix a generator $\sigma$ of the 1-dimensional space ${\ensuremath\mathrm{H}}^{2,0}(S)$ so that $\sigma+\bar\sigma$ is rational, i.e., lies in ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$.
Let $(-,-)$ denote the intersection pairing and set $v:=(\sigma, \bar\sigma) >
0$. Since $(\sigma+\bar\sigma, \sigma+\bar\sigma)=2(\sigma, \bar\sigma)=2v$, we see that $v\in {\ensuremath\mathds{Q}}_{>0}$. Since the real element $i(\sigma-\bar\sigma)$ spans the orthogonal complement of the rational line spanned by $\sigma + \bar{\sigma}$, the real line spanned by $i(\sigma-\bar\sigma)$ is also defined over ${\ensuremath\mathds{Q}}$. Since $i(\sigma-\bar\sigma)$ has norm $(i(\sigma-\bar\sigma), i(\sigma-\bar\sigma))=2(\sigma, \bar\sigma)=2v$, there exists a rational number $\alpha>0$ such that $i\sqrt{\alpha}(\sigma-\bar\sigma)\in {\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$.
Let $\{D_{1}, \ldots, D_{\rho}\}$ be an orthogonal basis of the Néron–Severi space $\operatorname{NS}(S)_{{\ensuremath\mathds{Q}}}$. Write $(D_{j},D_j)=2d_{j}$ with $d_{j}\in
{\ensuremath\mathds{Q}}$. We then have the following ${\ensuremath\mathds{Q}}$-basis of ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$: $$\{\sigma+\bar\sigma, i\sqrt{\alpha}(\sigma-\bar\sigma), D_{1}, \ldots,
D_{\rho}\}.$$
Let $V$ be the lattice ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Z}})$ equipped with the intersection pairing. We aim, via the (surjective) period map, to construct new K3 surfaces by constructing new Hodge structures on the lattice $V$. To this end, define a ${\ensuremath\mathds{Q}}$-linear map $\phi$ from the transcendental part ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ to $V_{{\ensuremath\mathds{Q}}}$ by the following formula $$\label{eqn:formula}
\begin{cases}
\phi(\sigma)=a\sigma+\bar b\bar\sigma+\sum_{j=1}^{\rho}c_{j}D_{j}\\
\phi(\bar\sigma)=b\sigma+\bar a\bar\sigma+\sum_{j=1}^{\rho}\bar c_{j}D_{j}
\end{cases}$$ with $a, b, c_{j}\in {\ensuremath\mathds{C}}$. The condition that the image of $\phi$ lies in $V_{{\ensuremath\mathds{Q}}}$ is equivalent to saying that $$\label{eq:Qalpha}
a, b, c_{j}\in {\ensuremath\mathds{Q}}(\sqrt{-\alpha}),$$ where ${\ensuremath\mathds{Q}}(\sqrt{-\alpha})$ is the imaginary quadratic extension of ${\ensuremath\mathds{Q}}$ with basis 1 and $i\sqrt{\alpha}$.
Consider the following subspaces of $V_{{\ensuremath\mathds{C}}}$: $$V^{2,0}:={\ensuremath\mathds{C}}\cdot\phi(\sigma), \quad V^{0,2}:={\ensuremath\mathds{C}}\cdot \phi(\bar\sigma), \quad
V^{1,1}:=(V^{0,2}+V^{2,0})^{\perp}.$$ By the global Torelli theorem [@PSS], they define a Hodge structure of a K3 surface $S'$ if and only if $$(\phi(\sigma), \phi(\sigma))=0 \text{ and } (\phi(\sigma), \phi(\bar\sigma))>0,$$ or equivalently, the following numerical conditions are satisfied: [^13] $$\label{eqn:constraints}
\begin{cases}
va\bar{b}+\sum_{j=1}^{\rho}c_{j}^{2}d_{j}=0\\
v(|a|^{2}+|b|^{2})+2\sum_{j=1}^{\rho}|c_{j}|^{2}d_{j}>0.
\end{cases}$$ By construction, there is an isomorphism between the ${\ensuremath\mathds{Q}}$-Hodge structures of the two K3 surfaces $S$ and $S'$: $$\widetilde \phi: {\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^{2}(S',
{\ensuremath\mathds{Q}}),$$ where $\widetilde\phi$ is given on ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ by $\phi$ and on $\operatorname{NS}(S)_{{\ensuremath\mathds{Q}}}$ by any isomorphism to $\operatorname{NS}(S')_{{\ensuremath\mathds{Q}}}$.
Note that there is *a priori* no reason for $\widetilde \phi$ to be an isometry. Our goal is actually to provide examples where no Hodge isometry exists. If there exists a Hodge isometry $$\psi: {\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})\to {\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Q}}),$$ then as ${\ensuremath\mathrm{H}}^{2,0}$ is 1-dimensional, there is a $\lambda\in {\ensuremath\mathds{C}}^{\times}$, such that $$\psi(\sigma)=\frac{1}{\lambda}\phi(\sigma)=\frac{1}{\lambda}\left(a\sigma+\bar
b\bar\sigma+\sum_{j=1}^{\rho}c_{j}D_{j}\right).$$ Since $\psi$ respects the ${\ensuremath\mathds{Q}}$-structures, we find that $\frac{a}{\lambda},
\frac{b}{\lambda}, \frac{c_{j}}{\lambda} \in {\ensuremath\mathds{Q}}(\sqrt{-\alpha})$, hence from that $\lambda\in {\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times$. The condition that $\psi$ is an isometry implies in particular that $$(\psi(\sigma), \psi(\bar\sigma))=(\sigma, \bar\sigma),$$ that is, $$|\lambda|^{2}=|a|^{2}+|b|^{2}+2\sum_{j=1}^{\rho}|c_{j}|^{2}\frac{d_{j}}{v}.$$
To summarize, any solution $a, b, c_{j}\in {\ensuremath\mathds{Q}}(\sqrt{-\alpha})$ of gives rise to a K3 surface $S'$ with ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$ isomorphic to ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$ as ${\ensuremath\mathds{Q}}$-Hodge structures and with ${\ensuremath\mathrm{H}}^{2}(S,
{\ensuremath\mathds{Q}})\otimes \langle m \rangle$ isometric to ${\ensuremath\mathrm{H}}^{2}(S', {\ensuremath\mathds{Q}})$ with $m := (\phi(\sigma),\phi(\bar{\sigma})) /
v$, but it would be isogenous to $S$ only if $$\label{eqn:Isometry}
|a|^{2}+|b|^{2}+2\sum_{j=1}^{\rho}|c_{j}|^{2}\frac{d_{j}}{v} \in
N{\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times,$$ where $N{\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times=\left\{|z|^{2} \mid z\in
{\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times\right\}=\left\{x^{2}+\alpha y^{2}\mid (x, y) \neq
(0,0) \in {\ensuremath\mathds{Q}}^2\right\}$ is the norm group of the extension ${\ensuremath\mathds{Q}}(\sqrt{-\alpha})/{\ensuremath\mathds{Q}}$. By the same argument, for two such K3 surfaces $S'$ and $S''$, corresponding to solutions $(a, b, c_{j})$ and $(a', b', c_{j}')$ of , if they are isogenous, then $$\label{eqn:Isometry2}
\frac{|a|^{2}+|b|^{2}+2\sum_{j=1}^{\rho}|c_{j}|^{2}d_{j}/v
}{|a'|^{2}+|b'|^{2}+2\sum_{j=1}^{\rho}|c'_{j}|^{2}d_{j}/v }\in
N{\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times.$$
By the previous discussion (together with Lemma \[lem:algiso\]), it is enough to provide infinitely many solutions of such that does not hold for each of them and such that does not hold for any two of them.
Let $S=\left(T_{0}^{4}+\cdots+T_{3}^{4}=0\right)\subset \mathbb{P}^{3}$ be the Fermat quartic surface. We know (see [@SD Appendix A]) that $S$ is the Kummer surface associated to an abelian surface $A$, which has a degree-two isogeny from $E\times E$, where $E\simeq {\ensuremath\mathds{C}}/{\ensuremath\mathds{Z}}\oplus {\ensuremath\mathds{Z}}\cdot i$ is the elliptic curve with $j$-invariant 1728.
$$\xymatrix{
&&& & S\ar[d]\\
E\times E\ar@{-->}[urrrr]^{\pi}\ar[rrr]_{\text{degree-2 isogeny}}&& &
A\ar[r]&
A/\{\pm 1\}
}$$
Consider the basis $\{e_{1}:=1, e_{2}:=i\}$ of ${\ensuremath\mathrm{H}}_{1}(E, {\ensuremath\mathds{Z}})$ and let $\{e_{1}^{*}, e_{2}^{*}\}$ be the dual basis of ${\ensuremath\mathrm{H}}^{1}(E, {\ensuremath\mathds{Z}})$. Denote by $z$ the holomorphic coordinate of ${\ensuremath\mathds{C}}$, so that ${\ensuremath\mathrm{H}}^{1,0}(E)$ is generated by $dz$. We see that $$\label{eqn:FormInBasis}
\begin{cases}
dz=e_{1}^{*}+i e_{2}^{*},\\
d\bar{z}=e_{1}^{*}-i e_{2}^{*},
\end{cases}$$ and that $\int_{E}dz\wedge d\bar{z}=-2i$.
Let $\sigma$ be the generator of ${\ensuremath\mathrm{H}}^{2,0}(S)$ such that $\pi^{*}(\sigma)=dz_{1}\wedge dz_{2}$ in ${\ensuremath\mathrm{H}}^{2,0}(E\times E)$. Using , one checks readily that $\sigma+\bar\sigma$ and $i\left(\sigma-\bar\sigma\right)$ belong to the rational lattice ${\ensuremath\mathrm{H}}^{2}(S,
{\ensuremath\mathds{Q}})$ because $$\begin{cases}
\pi^{*}(\sigma+\bar\sigma)=dz_{1}\wedge dz_{2}+d\bar{z}_{1}\wedge
d\bar{z}_{2}=2e_{1,1}^{*}\wedge e_{2,1}^{*}-2e_{1,2}^{*}\wedge e_{2,2}^{*};\\
\pi^{*}\left(i\sigma-i\bar\sigma\right)=idz_{1}\wedge
dz_{2}-id\bar{z}_{1}\wedge
d\bar{z}_{2}=-2e_{1,1}^{*}\wedge e_{2,2}^{*}-2e_{1,2}^{*}\wedge e_{2,1}^{*}.
\end{cases}$$ Hence $\{\sigma+\bar\sigma, i\left(\sigma-\bar\sigma\right)\}$ is a ${\ensuremath\mathds{Q}}$-basis of ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$; one can therefore take $\alpha=1$, and ${\ensuremath\mathds{Q}}(\sqrt{-\alpha})$ is simply ${\ensuremath\mathds{Q}}(i)$.
On the other hand, $$v=(\sigma,
\bar\sigma):=\int_{S}\sigma\wedge\bar\sigma=\frac{1}{\deg(\pi)}\int_{E\times
E}dz_{1}\wedge dz_{2}\wedge d\bar{z}_{1}\wedge
d\bar{z}_{2}=-\frac{1}{\deg(\pi)}\left(\int_{E}dz\wedge
d\bar{z}\right)^{2}=1.$$
Now take an orthogonal ${\ensuremath\mathds{Q}}$-basis $\{D_{1}, \cdots, D_{\rho}\}$ of $\operatorname{NS}(S)_{{\ensuremath\mathds{Q}}}$ such that $d_{1}=2$, $d_{2}=-1$; this is possible since the Néron–Severi lattice of $S$ is isomorphic to $E_{8}(-1)^{\oplus
2}\oplus U\oplus {\ensuremath\mathds{Z}}(-8)^{\oplus 2}$ (see [@SSvL]). We will only use solutions of with $c_{3}=\cdots=c_{\rho}=0$ (and remember that $\alpha=v=1$, $d_{1}=2$, $d_{2}=-1$): $$\begin{cases}
a\bar{b}+2c_{1}^{2}-c_{2}^{2}=0\\
|a|^{2}+|b|^{2}+4|c_{1}|^{2}-2|c_{2}|^{2}>0.
\end{cases}$$ For any integer $m>0$, we have a solution $$a=2m, \quad b=-\frac{1}{m}(1+i), \quad c_{1}=1, \quad c_{2}=1+i,$$ where
$|a|^{2}+|b|^{2}+4|c_{1}|^{2}-2|c_{2}|^{2}=\frac{2}{m^{2}}(2m^{4}+1)=|\frac{1+i}{m}|^{2}(2m^{4}+1)$. By the previous discussion, for our purpose, it suffices to produce an infinite sequence of positive integers $\{m_{j}\}_{j=1}^{\infty}$ such that for any $j$, $2m_{j}^{4}+1\notin N{\ensuremath\mathds{Q}}(i)^\times$ and for any $j\neq k$, $\frac{2m_{j}^{4}+1}{2m_{k}^{4}+1}\notin N{\ensuremath\mathds{Q}}(i)^\times$, where $N{\ensuremath\mathds{Q}}(i)=\{x^{2}+y^{2} \mid (x, y)\neq (0,0) \in {\ensuremath\mathds{Q}}^2\}.$
We conclude the proof of Theorem \[thm:non-isogenousK3\] by constructing such a sequence inductively by means of elementary arithmetics. This is the object of Lemma \[lem:el\] below.
\[lem:el\] Let $m_{1}:=1$ and for any $j$, $m_{j+1}:=\prod_{l=1}^{j}(2m_{l}^{4}+1)$. Then for any $j$, $2m_{j}^{4}+1\notin N{\ensuremath\mathds{Q}}(i)^\times$ and for any $j\neq k$, $\frac{2m_{j}^{4}+1}{2m_{k}^{4}+1}\notin N{\ensuremath\mathds{Q}}(i)^\times$.
A standard argument of infinite descent shows that a positive integer $n$ is not the sum of squares of two rational numbers, *i.e.*not in $N{\ensuremath\mathds{Q}}(i)^\times$, if and only if it is not the sum of squares of two integers. By the theorem of sum of two squares, the latter is equivalent to the condition that any prime divisor of $n$ with odd adic valuation is not congruent to 3 modulo 4.\
As the $m_{j}$’s are all odd, $2m_{j}^{4}+1\equiv 3 \mod 4$, hence is not the sum of two squares. On the other hand, by construction, for any $j\neq k$, $2m_{j}^{4}+1$ and $2m_{k}^{4}+1$ are coprime to each other, therefore their product admits a prime divisor $p$ congruent to 3 modulo 4 with odd adic valuation. Hence $\frac{2m_{j}^{4}+1}{2m_{k}^{4}+1}\notin N{\ensuremath\mathds{Q}}(i)^\times$.
The reason for requiring that $c_1$ and $c_2$ are not zero in the proof of Theorem \[thm:non-isogenousK3\] is the following. Let $\alpha \in {\ensuremath\mathds{Q}}_{>0}$, and consider a solution to with $c_2 = \cdots = c_\rho
=
0$, that is, a solution to $$\label{eqn:constraints3}
\begin{cases}
va\bar{b}+ c^{2}d=0\\
v(|a|^{2}+|b|^{2})+2|c|^{2}d>0,
\end{cases}$$ with $a,b,c \in {\ensuremath\mathds{Q}}(\sqrt{-\alpha})$, $v\in {\ensuremath\mathds{Q}}_{>0}$ and $d\in {\ensuremath\mathds{Z}}$. We observe that $ va\bar{b}+ c^{2}d=0$ implies that $|a||b| = \mp
|c|^2\frac{d}{v}$ is a non-negative rational number, where the sign depends on the sign of $d$. Recall from footnote \[fn:ab\] that $a$ and $b$ cannot be both zero; if one of them is zero, we may assume without loss of generality that it is $b$. Since $|a|^2$ is a rational number, this implies that $|b| = t |a|$ for some rational number $t \in {\ensuremath\mathds{Q}}_{\geq 0}$. It is then immediate to check that $$|a|^{2}+|b|^{2}+2|c|^{2}\frac{d}{v} = |a|^{2}+|b|^{2} \pm 2|a||b| =
|a|^2(1\pm t)^2.$$ In other words, the equation $ va\bar{b}+ c^{2}d=0$ (with $a$ and $b$ not both zero) forces the positivity of $|a|^{2}+|b|^{2}+2|c|^{2}d/v$, but also implies that it belongs to $N{\ensuremath\mathds{Q}}(\sqrt{-\alpha})^\times$. Therefore, the new K3 surface $S'$ obtained, via the global Torelli theorem, by imposing $c_2 = \cdots =
c_\rho = 0$ in must be isogenous to $S$.
Second construction: via Nikulin’s embedding {#SS:1}
--------------------------------------------
Our second construction is *via* Nikulin’s embedding theorem [@Nikulin] and provides proofs for Theorems \[thm:non-isogenousK3\] and \[P:bothisonotisogenous\].
### Hodge algebra isomorphic but non-isometric transcendental cohomology
This lattice-theoretic approach to show the existence of non-isogenous K3 surfaces with Hodge isomorphic second cohomology group was communicated to us by Benjamin Bakker, who attributes it to Huybrechts. Let $S$ be a projective K3 surface with Picard number $\rho$. Denote its transcendental lattice by $T:={\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Z}})$; it is an even lattice of signature $(2, 20-\rho)$. By Nikulin’s embedding theorem [@Nikulin Theorem 1.14.4] (see also [@HuybrechtsBook Theorem **14**.1.12, Corollary **14**.3.5]), when $12 \leq \rho\leq 20$, for any integer $m>0$, the lattice $T(m)$ admits a primitive embedding into the K3 lattice $\Lambda:=E_{8}(-1)^{\oplus 2}\oplus
U^{\oplus 3}$, unique up to $O(\Lambda)$. Now consider the (new) Hodge structure on $\Lambda$ given by declaring that the Hodge structure on $T(m)$ is the same as the one on $T$ and $T(m)^{\perp}$ is of type $(1, 1)$. By the surjectivity of the period map, there exists a K3 surface $S_{m}$, such that there is a Hodge isometry $T(m)\simeq {\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S_{m}, {\ensuremath\mathds{Z}})$. In particular, for all $m>0$, the Hodge structures ${\ensuremath\mathrm{H}}^{2}(S_{m}, {\ensuremath\mathds{Q}})$ are all isomorphic to ${\ensuremath\mathrm{H}}^{2}(S, {\ensuremath\mathds{Q}})$ and in fact, for all $m>0$, the cohomology algebras ${\ensuremath\mathrm{H}}^*(S_m,{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic due to Lemma \[lem:algiso\].
We now take $\rho=20$ and let $S$ be the Fermat quartic surface; its transcendental lattice $T$ is isomorphic to ${\ensuremath\mathds{Z}}(8)\oplus {\ensuremath\mathds{Z}}(8)$. In order to prove Theorem \[thm:non-isogenousK3\], it is enough to construct an infinite sequence of positive integers $\{m_{j}\}_{j=1}^{\infty}$, such that the lattices $T(m_{j})\simeq {\ensuremath\mathds{Z}}(8m_{j}) \oplus {\ensuremath\mathds{Z}}(8m_{j})$ are pairwise non-isometric over ${\ensuremath\mathds{Q}}$. However, it is easy to see that for $m, m'\in {\ensuremath\mathds{Z}}_{>0}$, the two ${\ensuremath\mathds{Q}}$-quadratic forms $ {\ensuremath\mathds{Q}}(8m) \oplus {\ensuremath\mathds{Q}}(8m)$ and $ {\ensuremath\mathds{Q}}(8m') \oplus {\ensuremath\mathds{Q}}(8m')$ are isometric if and only if $mm'$ belongs to $\{x^{2}+y^{2}\mid x, y\in {\ensuremath\mathds{Z}}\}$, which in turn is equivalent to the condition that any prime factor of $mm'$ congruent to 3 modulo 4 has even exponent, by the theorem of the sum of two squares. Therefore a desired sequence is easy to construct, for example, one can take $m_{j}$ to be the $j$-th prime number congruent to 3 modulo 4.
### Hodge algebra isomorphic and isometric but non-Hodge isometric transcendental cohomology
In the previous example, the K3 surfaces $S_{m_{j}}$ all have isomorphic ${\ensuremath\mathrm{H}}^{2}(-, {\ensuremath\mathds{Q}})$ as rational Hodge structures and their isogeny classes are distinguished from each other by the ${\ensuremath\mathds{Q}}$-quadratic forms on ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(-, {\ensuremath\mathds{Q}})$. We would like to go further and produce non-isogenous K3 surfaces with ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(-, {\ensuremath\mathds{Q}})$ both isomorphic as Hodge structures and isometric as ${\ensuremath\mathds{Q}}$-quadratic forms. Let us first remark that no such examples of K3 surfaces exist in the case of maximal Picard number $\rho=20$. Indeed, we have the following elementary result:
\[lemma:rho=20\] Let $S, S'$ be two K3 surfaces with maximal Picard number $\rho=20$.
- If ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Z}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Z}})$ are isometric lattices, then $S$ and $S'$ are isomorphic.
- If ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$ are isometric ${\ensuremath\mathds{Q}}$-quadratic forms, then $S$ and $S'$ are isogenous.
- If ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S', {\ensuremath\mathds{Q}})$ are Hodge isomorphic, then ${\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic.
We only prove the first two points; the third is left to the reader. Let $T$ be the quadratic space underlying their transcendental cohomologies. Due to the Hodge–Riemann bilinear relations, $T$ is positive definite. Choose an orthogonal basis $\{e_{1}, e_{2}\}$ of $T$ and let $d_{i}:= (e_{i}, e_{i})\in
{\ensuremath\mathds{Q}}_{>0}$. One observes that there are only two isotropic directions in $T\otimes
{\ensuremath\mathds{C}}$, namely $\sqrt{d_{1}}e_{1}\pm i\sqrt{d_{2}}e_{2}$, hence only two possible Hodge structures of K3 type on $T$. However, these two Hodge structures are Hodge isometric via the ${\ensuremath\mathds{Q}}$-linear transformation $e_{1}\mapsto e_{1}$; $e_{2}\mapsto -e_{2}$.
For K3 surfaces with $\rho=12, 14, 16, 18$, there are indeed examples of non-isogenous K3 surfaces with ${\ensuremath\mathrm{H}}_{\operatorname{tr}}^{2}(-, {\ensuremath\mathds{Q}})$ both isometric and isomorphic as rational Hodge structures, as is stated in Theorem \[P:bothisonotisogenous\].
Given any even lattice $T$ of signature $(2, 2), (2, 4), (2, 6)$ or $(2, 8)$ whose discriminant is a square, by Lemma \[lemma:QuadForm\], the ${\ensuremath\mathds{Q}}$-quadratic forms $T(m)\otimes {\ensuremath\mathds{Q}}$ and $T\otimes {\ensuremath\mathds{Q}}$ are isometric for any integer $m\in {\ensuremath\mathds{Z}}_{>0}$ which is the sum of two squares. On the other hand, a generic choice of an isotropic element $\sigma\in T\otimes {\ensuremath\mathds{C}}$ gives rise to an irreducible ${\ensuremath\mathds{Q}}$-Hodge structure on $T$, hence on all its twists $T(m)$, with minimal endomorphism algebra; i.e. $\operatorname{End}_{HS}(T_{{\ensuremath\mathds{Q}}})\simeq {\ensuremath\mathds{Q}}$. So for all $m\in {\ensuremath\mathds{Z}}_{>0}$ which is the sum of two squares, the twists $T(m)\otimes {\ensuremath\mathds{Q}}$ are all Hodge isomorphic and isometric. However, since for any such integers $m$ and $m'$, $\operatorname{Hom}_{HS}\left(T(m)\otimes {\ensuremath\mathds{Q}}, T(m')\otimes {\ensuremath\mathds{Q}}\right)={\ensuremath\mathds{Q}}$, we see that $T(m)\otimes {\ensuremath\mathds{Q}}$ and $T(m')\otimes {\ensuremath\mathds{Q}}$ are Hodge isometric if and only if $mm'$ is a square.
In order to realize the twists $ T(m)$ as transcendental lattices of K3 surfaces, we use Nikulin’s embedding theorem [@Nikulin Theorem 1.14.4] to get a primitive embedding of $T(m)$ into the K3 lattice $\Lambda$. For each $m\in {\ensuremath\mathds{Z}}_{>0}$, we can therefore construct a Hodge structure on $\Lambda$ by declaring that $T(m)$ carries the Hodge structure on $T$ and $T(m)^{\perp}$ is of type $(1, 1)$. By the surjectivity of the period map, for any $m \in {\ensuremath\mathds{Z}}_{>0}$, there exists a K3 surface $S_{m}$ with ${\ensuremath\mathrm{H}}^{2}_{\operatorname{tr}}(S_{m}, {\ensuremath\mathds{Q}})$ Hodge isometric to $T(m)$.
Now, thanks to Lemma \[lem:algiso\], it remains to construct an infinite sequence of positive integers $\{m_{j}\}_{j=1}^{\infty}$ which are sums of two squares, such that the product of any two different terms is not a square. This is easily achieved: for example, one can take $m_{j}$ to be the $j$-th prime number congruent to 1 modulo 4.
Consequences on motives {#SS:motivic}
-----------------------
### The general expectations
The following conjecture is a combination of the Hodge conjecture and of the conservativity conjecture (which itself is a consequence of the Kimura–O’Sullivan finite-dimensionality conjecture or of the Bloch–Beilinson conjectures).
\[conj:MGTT\] Two smooth projective varieties $X$ and $Y$ have isomorphic Chow motives if and only if their rational cohomologies are isomorphic as Hodge structures: $$\label{eqn:MotTorelli}
{\ensuremath\mathfrak{h}}(X)\simeq {\ensuremath\mathfrak{h}}(Y) \ \text{as Chow motives} \ \Longleftrightarrow {\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\simeq {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}}) \ \text{as graded Hodge structures}.$$
The implication $\Rightarrow$ holds unconditionally and is simply attained by applying the Betti realization functor. Regarding the implication $\Leftarrow$, the Hodge conjecture predicts that an isomorphism ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})\simeq
{\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$ of graded Hodge structures and its inverse are induced by the action of a correspondence. Hence the homological motives of $X$ and $Y$ are isomorphic. By conservativity, such an isomorphism lifts to rational equivalence, *i.e.* lifts to an isomorphism between the Chow motives of $X$ and $Y$.
Obviously, if $ {\ensuremath\mathfrak{h}}(X)\simeq {\ensuremath\mathfrak{h}}(Y) $ as (Frobenius) algebra objects in the category of Chow motives, then by realization ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^{*}(Y,
{\ensuremath\mathds{Q}}) $ are Hodge (Frobenius) algebra isomorphic. We would like to discuss to which extent the converse statement could be true. In general, this is not the case: consider for instance a complex K3 surface $S$; then the blow-up $S_1$ of $S$ at a point lying on a rational curve and the blow-up $S_2$ of $S$ at a very general point have Hodge isomorphic cohomology Frobenius algebras, but due to the Beauville–Voisin Theorem \[thm:bv\] we have $$1 = \operatorname{rk}\left({\ensuremath\mathrm{CH}}^1(S_1) \otimes {\ensuremath\mathrm{CH}}^1(S_1) \to
{\ensuremath\mathrm{CH}}^2(S_1)\right) \neq \operatorname{rk}\left({\ensuremath\mathrm{CH}}^1(S_2) \otimes {\ensuremath\mathrm{CH}}^1(S_2) \to
{\ensuremath\mathrm{CH}}^2(S_2)\right) =2,$$ in particular their Chow motives are not isomorphic as algebra objects.
However, in the case of hyper-Kähler varieties, one can expect:
\[conj:MGTT2\] Two smooth projective hyper-Kähler varieties $X$ and $Y$ have isomorphic Chow motives as (Frobenius) algebra objects if and only if their rational cohomology rings are Hodge (Frobenius) algebra isomorphic : $$\begin{aligned}
& {\ensuremath\mathfrak{h}}(X)\simeq {\ensuremath\mathfrak{h}}(Y) \ \text{as (Frobenius) algebra objects} \\
&\Longleftrightarrow {\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}})\ \text{and} \ {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}}) \ \text{are
Hodge (Frobenius) algebra isomorphic}.
\end{aligned}$$
We note that Corollary \[cor:torelli\] establishes this conjecture in the case of K3 surfaces and Frobenius algebra structures. In general, Conjecture \[conj:MGTT2\] is implied by the combination of the Hodge conjecture and of the “distinguished marking conjecture” for hyper-Kähler varieties [@fv Conjecture 2].
\[prop:conj\] Let $X$ and $Y$ be hyper-Kähler varieties of same dimension $d$. Assume:
- The Hodge conjecture in codimension $d$ for $X\times Y$;
- $X$ and $Y$ satisfy the “distinguished marking conjecture” [@fv Conjecture 2].
Then Conjecture \[conj:MGTT2\] holds for $X$ and $Y$.
By [@fv §3], the distinguished marking conjecture for $X$ and $Y$ provides for all non-negative integers $n$ and $m$ a section to the graded algebra epimorphism ${\ensuremath\mathrm{CH}}^*(X^n\times Y^m) \to \overline{{\ensuremath\mathrm{CH}}}^*(X^n\times Y^m)$ in such a way that these are compatible with push-forwards and pull-backs along projections. In addition, the images of the sections corresponding to ${\ensuremath\mathrm{CH}}^*(X^2) \to \overline{{\ensuremath\mathrm{CH}}}^*(X^2)$ and ${\ensuremath\mathrm{CH}}^*(Y^2) \to
\overline{{\ensuremath\mathrm{CH}}}^*(Y^2)$ contain the diagonals $\Delta_X$ and $\Delta_Y$, respectively. Here, $\overline{{\ensuremath\mathrm{CH}}}^*(-)$ denotes the Chow ring modulo numerical equivalence. In fact, since numerical and homological equivalence agree for abelian varieties, the same holds for $X$ and $Y$ (*via* their markings).
As before, the Hodge conjecture predicts that a Hodge isomorphism ${\ensuremath\mathrm{H}}^{*}(X,
{\ensuremath\mathds{Q}})\simeq {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$ and its inverse are induced by the action of an algebraic correspondence. We fix the isomorphism $\phi: {\ensuremath\mathfrak{h}}(X)
\stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}(Y)$ to be the correspondence that is the image of the Hodge class under the section to ${\ensuremath\mathrm{CH}}^*(X\times Y) \to
\overline{{\ensuremath\mathrm{CH}}}^*(X\times Y)$ inducing the Hodge isomorphism of (Frobenius) algebras ${\ensuremath\mathrm{H}}^{*}(X, {\ensuremath\mathds{Q}}) \stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^{*}(Y, {\ensuremath\mathds{Q}})$. Since the (Frobenius) algebra structure on the motives of varieties is simply described in terms of the rational equivalence class of the diagonal and of the small diagonal, the isomorphism $\phi$ provides, thanks to the compatibilities of the sections on the product of various powers of $X$ and $Y$, a morphism compatible with the (Frobenius) algebra structures.
Although we do not know how to establish Conjecture \[conj:MGTT2\] in general for K3 surfaces and algebra structures, we can still say something for K3 surfaces with a *Shioda–Inose structure*. Recall that a Shioda–Inose structure on a K3 surface $S$ consists of a Nikulin involution (that is, a symplectic involution) with rational quotient map $\pi :
S\dashrightarrow Y$ such that $Y$ is a Kummer surface and $\pi_*$ induces a Hodge isometry $T_S(2) \simeq T_Y$, where $T_S$ and $T_Y$ denote the transcendental lattices of $S$ and $Y$. If $S$ admits a Shioda–Inose structure, let $f: A\to Y$ be the quotient morphism from the complex abelian surface whose Kummer surface is $Y$. By [@morrison § 6], there is a Hodge isometry of transcendental lattices $T_S \simeq
T_A$, and $f^*\pi_*$ induces an isomorphism ${\ensuremath\mathrm{H}}_{\mathrm{tr}}^2(S,{\ensuremath\mathds{Q}})
\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^2_{\mathrm{tr}}(A,{\ensuremath\mathds{Q}})$ with inverse $\frac{1}{2} f_*\pi^*$.
\[prop:motivicglobalTorelli\] Let $S$ and $S'$ be two K3 surfaces with a Shioda–Inose structure (*e.g.* with Picard rank $\geq 19$, [@morrison Corollary 6.4]). The following conditions are equivalent.
(i) ${\ensuremath\mathrm{H}}^*(S,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S',{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic.
(ii) ${\ensuremath\mathfrak{h}}(S) \simeq {\ensuremath\mathfrak{h}}(S')$ as algebra objects in the category of rational Chow motives.
Let $S$ and $S'$ be two K3 surfaces with a Shioda–Inose structure. In a similar vein to Proposition \[prop:conj\], the proposition is a combination of the validity of the Hodge conjecture for $S\times S'$, together with the fact [@fv Proposition 5.12] that $S$ and $S'$ satisfy the distinguished marking conjecture of [@fv Conjecture 2]. The fact that the Hodge conjecture holds for $S\times S'$ reduces, *via* the correspondence-induced isomorphism ${\ensuremath\mathrm{H}}_{\mathrm{tr}}^2(S,{\ensuremath\mathds{Q}})
\stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{H}}^2_{\mathrm{tr}}(A,{\ensuremath\mathds{Q}})$ described above, to the fact that the Hodge conjecture holds for the product of any two abelian surfaces. The latter is proven in [@rm].
### Non-isogenous K3 surfaces with Chow motives isomorphic as algebra objects
By combining Theorem \[thm:non-isogenousK3\] with the fact [@morrison Corollary 6.4] that K3 surfaces of maximal Picard rank admit a Shioda–Inose structure, we can establish:
\[thm:IsomChowMot\] There exists an infinite family of K3 surfaces such that
- they are pairwise non-isogenous;
- their Chow motives are pairwise non-isomorphic as Frobenius algebra objects;
- their Chow motives are all isomorphic as algebra objects.
Let $\{S_i\}_{i \in {\ensuremath\mathds{Z}}_{>0}}$ be a family of pairwise non-isogenous K3 surfaces of maximal Picard rank such that ${\ensuremath\mathrm{H}}^*(S_j,{\ensuremath\mathds{Q}})$ and ${\ensuremath\mathrm{H}}^*(S_k,{\ensuremath\mathds{Q}})$ are Hodge algebra isomorphic for all $j,k \in {\ensuremath\mathds{Z}}$. Such a family of K3 surfaces exist thanks to Theorem \[thm:non-isogenousK3\]. By Corollary \[cor:torelli\] the Chow motives of these K3 surfaces are pairwise non-isomorphic as Frobenius algebra objects. The fact that the Chow motives of any two surfaces in the family are isomorphic as algebra objects is Proposition \[prop:motivicglobalTorelli\].
Finally, the following proposition gives evidence that the notion of “isogeny” for K3 surfaces is strictly more restrictive than the notion of “isomorphic Chow rings” (see Remark \[rmk:ChowRingIso\]):
\[prop:IsoChowRing3\] Assume that Conjecture \[conj:MGTT\] holds for K3 surfaces. Then there exists an infinite family $\{S_i\}_{i \in {\ensuremath\mathds{Z}}_{>0}}$ of pairwise non-isogenous K3 surfaces with the property that, for all $j,k \in {\ensuremath\mathds{Z}}_{>0}$, there exists an isomorphism ${\ensuremath\mathfrak{h}}(S_j)
\stackrel{\sim}{\longrightarrow}{\ensuremath\mathfrak{h}}(S_k)$ of Chow motives inducing a ring isomorphism ${\ensuremath\mathrm{CH}}^*(S_j) \stackrel{\sim}{\longrightarrow} {\ensuremath\mathrm{CH}}^*(S_k)$ such that the distinguished class $o_{S_j}$ is mapped to the distinguished class $o_{S_k}$.
Moreover, such a family can be chosen to consist of K3 surfaces with transcendental lattice being any prescribed even lattice with square discriminant and of signature $(2, 2), (2, 4), (2, 6)$ or $(2, 8)$.
We consider the infinite family constructed in Theorem \[P:bothisonotisogenous\]. For any $j\neq k$, $S_{j}$ and $S_{k}$ are not isogenous. On the other hand, Conjecture \[conj:MGTT\] implies that the Chow motives of $S_{j}$ and $S_{k}$ are isomorphic; in particular, by the same weight argument as in §\[subsec:weight\], there exists an isomorphism between their transcendental motives: $$\Gamma_{\operatorname{tr}}:
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{tr}}(S_{j})\simeq {\ensuremath\mathfrak{h}}^{2}_{\operatorname{tr}}(S_{k}).$$ As $\operatorname{NS}(S_{j})_{{\ensuremath\mathds{Q}}}$ and $\operatorname{NS}(S_{k})_{{\ensuremath\mathds{Q}}}$ are isometric by construction, there is an isomorphism between the algebraic part of their weight-2 motives $\Gamma^{2}_{\operatorname{alg}}:
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{alg}}(S_{j})\to{\ensuremath\mathfrak{h}}^{2}_{\operatorname{alg}}(S_{k})$ which induces the isometry between the Néron–Severi spaces. Combining them together, $\Gamma:=o_{S_{j}}\times
S_{k}+\Gamma_{\operatorname{alg}}^{2}+\Gamma_{\operatorname{tr}}+S_{j}\times
o_{S_{k}}$ yields an isomorphism between their Chow motives: $$\Gamma: {\ensuremath\mathfrak{h}}(S_{j})={\ensuremath\mathfrak{h}}^{0}(S_{j})\oplus
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{alg}}(S_{j})\oplus
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{tr}}(S_{j})\oplus
{\ensuremath\mathfrak{h}}^{4}(S_{j})\stackrel{\sim}{\longrightarrow} {\ensuremath\mathfrak{h}}(S_{k})={\ensuremath\mathfrak{h}}^{0}(S_{k})\oplus
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{alg}}(S_{k})\oplus
{\ensuremath\mathfrak{h}}^{2}_{\operatorname{tr}}(S_{k})\oplus
{\ensuremath\mathfrak{h}}^{4}(S_{k}),$$ with the extra property that it induces an isometry between the ${\ensuremath\mathds{Q}}$-quadratic spaces ${\ensuremath\mathrm{CH}}^{1}(S_{j})$ and ${\ensuremath\mathrm{CH}}^{1}(S_{k})$. Now as in Remark \[rmk:ChowRingIso\], according to the Beauville–Voisin theorem [@bv], the image of the intersection product ${\ensuremath\mathrm{CH}}^{1}(S_{j})\otimes
{\ensuremath\mathrm{CH}}^{1}(S_{j})\to{\ensuremath\mathrm{CH}}^{2}(S_{j})$ is 1-dimensional and similarly for $S_{k}$. This implies that $\Gamma$ induces an isomorphism of Chow rings with $\Gamma_*o_{S_j} = o_{S_k}$.
In Proposition \[prop:IsoChowRing3\], if one assumes Conjecture \[conj:MGTT2\] for K3 surfaces instead of Conjecture \[conj:MGTT\], then the K3 surfaces of the family have isomorphic Chow motives as algebra objects.
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[^1]: 2010 [*Mathematics Subject Classification:*]{} 14C25, 14C34, 14C15, 14F05, 14J28, 14J32.
[^2]: [*Key words and phrases.*]{} Motives, K3 surfaces, Torelli theorems, derived categories, cohomology ring.
[^3]: In this paper, Chow groups and motives are always with rational coefficients, except in Theorem \[thm:motglobtor\].
[^4]: Technically, one needs to further tensor the dual object by certain power of some tensor-invertible objects; this power is called the *degree* of this Frobenius structure.
[^5]: Manin’s identity principle only establishes that $ v_2(\mathcal{E}^\vee\otimes
p^*\omega_S)\circ
v_2(\mathcal{E})$ acts as the identity on ${\ensuremath\mathrm{CH}}^2({\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S))$ which implies that it is unipotent as an endomorphism of ${\ensuremath\mathfrak{h}}^2_{\mathrm{tr}}(S)$.
[^6]: It is however not stable under tensor products and it does not contain any Tate motives. The tensor category generated by the Tate motives and the motives of curve type consists of the so-called motives of *abelian type*.
[^7]: As explained in Lemma \[lem:formal\], a non-zero morphism between algebra objects that preserves the multiplication morphisms must also preserve the unit morphisms, and hence is a morphism of algebra objects in the sense of §\[SS:alg\].
[^8]: see e.g. [@andre §3.1.4], and [@vial-abelian Lemma 3.3] for a proof.
[^9]: The authors do not have examples of derived equivalent smooth projective varieties with non-isomorphic cohomology as ${\ensuremath\mathds{Q}}$-algebras or as $\mathbb{R}$-Frobenius algebras.
[^10]: It is not clear to the authors whether the isomorphism constructed in the proof preserves the Frobenius structure.
[^11]: Here by an *orbifold* we mean a smooth proper Deligne–Mumford stack with trivial generic stabilizer.
[^12]: Note that an Inose isogeny is not an isogeny in our sense.
[^13]: \[fn:ab\] We observe that these conditions imply that $a$ and $b$ cannot be both zero. Indeed, by the Hodge index theorem, one of the $d_i$’s is positive, say $d_1$, while all the others are negative. The triangular inequality applied to $c_1^2 d_1 = \sum_{j=2}^{\rho}c_{j}^{2}(-d_{j})$ yields $|c_{1}|^{2}d_{1} \leq \sum_{j=2}^{\rho}|c_{j}|^{2}(-d_{j})$, i.e., $\sum_{j=1}^{\rho}|c_{j}|^{2}d_{j}\leq 0$.
|
---
author:
- 'F. Cecconi'
- 'A. Puglisi'
- 'A. Sarracino'
- 'A. Vulpiani'
date: 'Received: date / Revised version: date'
title: 'Anomalous force-velocity relation of driven inertial tracers in steady laminar flows'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#sec:intro}
============
The study of the transport properties of inertial particles in fluids takes on a great importance in several fields, in engineering as well as in natural occurring settings: typical examples are pollutants and aerosols dispersion in the atmosphere and oceans [@Pollutant], optimization of mixing efficiency in different contexts, or the study of chemical [@Chem], biological [@CEN13] or physical interaction [@MEH16], with applications to the time scales of rain [@F02], in the sedimentation speed under gravity [@LCLV08], or to the planetesimal formation in the early Solar System [@B99].
Under external perturbations, the dynamics of tracer particles can be significantly modified, resulting in different behaviors which are not easily predicted from the properties of the unperturbed dynamics. In order to relate response functions and fluctuations in non-equilibrium conditions, relevant in the aforementioned cases, generalizations of the standard fluctuation-dissipation theorem have been derived in recent years [@CLZ07; @MPRV08; @LCSZ08; @LCSZ08b; @seifert; @BM13]. These approaches are generally valid in the small forcing limit, and a central problem remains the study of the motion in the presence of an external driving field $F$, in the nonlinear regime. In this case, the particles reach a stationary state characterized by a finite average velocity $v$ which non-trivially depends on the system parameters. The main point is then to understand the force-velocity relation $v(F)$, which contains relevant information on the system. These curves are strongly affected by the nature of the tracer/fluid interaction, and can show surprising nonlinear behaviors.
An example of such effects is provided by the so-called negative differential mobility (NDM), which means that $v(F)$, after a linear increase with the applied force, can show a non-monotonic behavior, attaining a (local) maximum for a given value of the force [@royce; @LF13; @BM14; @BIOSV14; @BSV15; @BIOSV16]. For larger intensity of the bias the velocity can show different behaviors, such as saturation or asymptotic linear growth, depending on the considered model. In other situations, one can also observe a more surprising feature: an *absolute* negative mobility (ANM), where the particle travels on average against the external field [@RERDRA05; @KMHLT06; @MKTLH07; @ERAR10]. In general, these non-linear behaviors are due to trapping mechanisms in the system, which lead to dynamical conditions such that an increase of the applied force can result in an increase of the trapping time, and, consequently, to a slowing down of the average tracer dynamics. Depending on the specific model, trapping can be due to the interaction of the tracer with the surrounding particles, to frustration in the system, to geometric constraints or to the coupling with underlying velocity fields.
Here we study the response to an external bias of inertial particles advected by steady (incompressible) cellular flows, in the nonlinear regime. In these systems, the presence of inertia induces a non-trivial deviation of the particle motion from the flow of the underlying fluid: the particles can remain trapped in regions close to upstream lines, yielding a slowing down of the dynamics, as observed in the context of gravitational settling [@M87; @RJM95; @F97; @A07], and typically leading to the phenomenon of preferential concentration [@BBCLMT07; @CCLT08]. In a recent work [@SCPV16], we have considered a model where, in addition to the cellular flow and the external force, the inertial tracer is subject to the action of a microscopic (thermal) noise. We have shown that a rich nonlinear behavior for the average particle velocity can be observed, as a function of the applied force, featuring both NDM and ANM. The latter was never observed in the standard systems studied in the literature. Here we present an extensive numerical investigation of the system studied in [@SCPV16], exploring a wide range of the model parameters and reconstructing the phase chart, where regions of NDM and ANM are identified.
The model {#sec:model}
=========
The dynamics of the inertial tracer in two dimensions, with spatial coordinates $(x,y)$ and velocities $(v_x,v_y)$, is described by the following equations $$\begin{aligned}
\dot{x}&=&v_x, \qquad \dot{y}=v_y \label{eq1} \\
\dot{v}_x&=&-\frac{1}{\tau}(v_x-U_x)+F +\sqrt{2D_0}\xi_x \label{eq2}, \\
\dot{v}_y&=&-\frac{1}{\tau}(v_y-U_y)+ \sqrt{2D_0}\xi_y, \label{eq3}
\label{model}\end{aligned}$$ where $\mathbf U = (U_x,U_y)$ is a divergenceless cellular flow defined by a stream-function $\psi$ as: $$U_x=\frac{\partial \psi(x,y)}{\partial y}, \qquad U_y=-
\frac{\partial \psi(x,y)}{\partial x}\;.
\label{eq:psi}$$ Here $\tau$ is the Stokes time, $F$ the external force, and $$\psi(x,y) = LU_0/2\pi \sin(2\pi x/L)\sin(2\pi y/L).$$ $\xi_x$ and $\xi_y$ are uncorrelated white noises with zero mean and unitary variance. A pictorial representation of the field (red arrows) and of a tracer’s trajectory (black arrows) is reported in Fig.\[fig:model\]. Measuring length and time in units of $L$ and $L/U_0$ respectively, and setting therefore $U_0=1$ and $L=1$, the typical time scale of the flow becomes $\tau^*=L/U_0=1$. Let us anticipate that the time scales ratio $\tau/\tau^*$ will play a central role in the behavior of the system. Another important parameter of our model is the microscopic thermal noise with diffusivity $D_0$, which guarantees ergodicity and can be expressed in terms of the temperature $T$ of the environment by the relation $D_0=T/\tau$.
In the following we will focus on the force-velocity relation, namely on the behavior of the stationary velocity $\langle v_x \rangle = \tau
F + \langle U_x[x(t),y(t)] \rangle$, where $\langle\cdots\rangle$ denotes averages over initial conditions and noise realizations. The numerical integration of the dynamical equations of the model is performed with a second-order Runge-Kutta algorithm [@honey], with a time step $\Delta t=10^{-2}$. Numerical results shown in the figures are averaged over about $10^{4}$ realizations. In our study, we will consider different regimes of the time scale ratio $\tau/\tau^*$, exploring the behavior of the force-velocity relation $\langle v_x\rangle(F)$ by varying the microscopic diffusivity $D_0$.
Force-velocity relation
=======================
Small Stokes time
-----------------
First, we consider the case $\tau\ll\tau^*$. Fig.\[fig:lowtau-VF\] shows $\langle v_x \rangle(F)$ for $\tau=10^{-2},10^{-1}$ and different values of $D_0$. Two linear regimes at small and large forces are well apparent. In the latter case, the simple behavior $\langle v_x\rangle(F)=\tau F$ is recovered. This is expected because at large forces the underlying velocity field is irrelevant. More surprisingly, in the range of intermediate forces a non-trivial nonlinear behavior takes place.
![Plot of $\langle v_x\rangle$ versus $F$ several values of $D_0$. Top panel refers to $\tau=10^{-2}$, while bottom panel refers to $\tau=10^{-1}$ Note the abrupt drops of the curves at the value $F=F^*= 1/\tau$.[]{data-label="fig:lowtau-VF"}](tau001.eps "fig:"){width=".8\columnwidth"} ![Plot of $\langle v_x\rangle$ versus $F$ several values of $D_0$. Top panel refers to $\tau=10^{-2}$, while bottom panel refers to $\tau=10^{-1}$ Note the abrupt drops of the curves at the value $F=F^*= 1/\tau$.[]{data-label="fig:lowtau-VF"}](tau01.eps "fig:"){width=".8\columnwidth"}
In particular, a non-monotonic behavior, corresponding to NDM, can be observed for small values of $D_0$. Note that the critical force value $F^*$ where the abrupt drop of velocity occurs is independent of the noise amplitude and scales with $\tau$ as $F^* = \tau^{-1}$. As $D_0$ is increased, the effect of the velocity field is averaged out and the force-velocity curve displays a simple monotonic behavior.
This scenario is illustrated by Fig.\[fig:traj\], showing trajectories of the tracer in the case $\tau=0.01$ and $D_0=10^{-5}$, for some values of the force $F\in [1,100]$ along with the streamlines of the effective flow obtained implementing a geometric singular perturbation approach in the Stokes time $\tau$ from the system with $\tau=0$. According to Fenichel [@FEN79], for small $\tau$, particle trajectories of the system are attracted by a two-dimensional slow manifold and the equations of motion along the manifold can be formally written as a perturbation series $$\begin{aligned}
\dot{x}&=& U_x(x,y) + F\tau + \sum_{n} \tau^{n}\;h_{n}(x,y) \label{eq:slow1}\\
\dot{y}&=& U_y(x,y) + \sum_{n} \tau^{n}\;k_{n}(x,y) \label{eq:slow2}\end{aligned}$$ where the terms $\{h_n,k_n\}$ are explicitly derived in Refs. [@RJM95; @BUR99]. Eqs. (\[eq:slow1\],\[eq:slow2\]), also referred to as [*inertial equations*]{}, have the advantage of reducing the dimensionality of the system from four to two, yet catching the correct asymptotic behavior of full-system trajectories. The drawback lies in the obvious technical difficulties to control the convergence and in dealing with truncation errors. Let us note that such a limit is singular and, since now we are in two dimensions, chaos cannot exist. In our case, retaining only the 4-th order terms in $\tau$ in Eqs. (\[eq:slow1\],\[eq:slow2\]) is sufficient to discuss the qualitative features of the trajectories on the slow manifold, see Fig.\[fig:traj\]. At small forces ($F=1,10,50$), the asymptotic trajectories (black dots) accumulate along the main streamlines of the effective velocity field, following the external force. For larger values ($F=70,90,100$), one observes an interesting phenomenon: the trajectories move towards upstream regions with respect to the external force, so that the inertial tracers slow their motion, decreasing the stationary average velocity.
Large Stokes time
-----------------
In the opposite regime, $\tau \gg\tau^*$, we find a behavior similar to that discussed above, with NDM at small $D_0$ and monotonic behavior for $D_0 \geq 10^{-3}$, see Fig. \[fig:bigTau-VF\]. However, the drop in the velocity is much milder in this case, and the minimum is reached for $F\approx 7\cdot 10^{-3}$. This behavior can be explained by the fact that, since inertia is larger in this case, the action of the velocity field is weaker and the effect of slowing down on the tracer is reduced.
![Force-velocity relation for $\tau=10$ and several values of $D_0$.[]{data-label="fig:bigTau-VF"}](tau10.eps){width=".8\columnwidth"}
Absolute negative mobility
--------------------------
If the relevant time scales of the tracer and of the underlying velocity field are comparable, $\tau \sim \tau^*$, a new surprising effect can be observed. In Fig. \[fig5\] we show $\langle
v_x \rangle(F)$ for $\tau=0.65$ (top panel) and $\tau=0.8$ (bottom panel), and different values of $D_0$. We observe a nonlinear complex behavior of the force-velocity relation and, in particular, we note that the average velocity can take on negative values, $\langle
v_x\rangle<0$ within the error bars, implying an absolute negative mobility of the tracer. This phenomenon occurs in a very narrow range of forces for small values of $D_0$ (see black dots in Fig. \[fig5\]), while for values of $D_0=10^{-3}$ negative mobility extends at small forces, even in the linear regime (see green diamonds). This effect is made possible by the non-equilibrium nature of our model due to the non-gradient form of the velocity field (see Eq. (\[eq:psi\])), which induces currents even in the absence of the external force.
In particular, our results seem to be consistent with the theoretical analysis presented in [@SER07], for an underdamped Brownian particle model in a one-dimensional periodic potential. Indeed, it is expected that in some regions of the parameter space, at fixed force, the tracer velocity follows the external force for low noise, but changes sign upon increasing the temperature.
![Top panel: force-velocity relation for $\tau=0.65$ and several values of $D_0$. The inset shows a zoom of the ANM region. Bottom panel: force-velocity curve for $\tau=0.8$ and several values of $D_0$. The large error bars at small forces are due to the fact that, in those cases, the inertial particles are constrained along few straight trajectories, with opposite mean velocity, strongly dependent on the initial conditions. This effect produces a large variance.[]{data-label="fig5"}](tau065.eps "fig:"){width=".8\columnwidth"} ![Top panel: force-velocity relation for $\tau=0.65$ and several values of $D_0$. The inset shows a zoom of the ANM region. Bottom panel: force-velocity curve for $\tau=0.8$ and several values of $D_0$. The large error bars at small forces are due to the fact that, in those cases, the inertial particles are constrained along few straight trajectories, with opposite mean velocity, strongly dependent on the initial conditions. This effect produces a large variance.[]{data-label="fig5"}](tau08.eps "fig:"){width=".8\columnwidth"}
A physical mechanism responsible for such a surprising effect has been proposed in [@SCPV16], based on a careful study of the tracer trajectories. This analysis showed that the motion of the tracer is realized along preferential “channels”, aligned downstream or upstream with respect to the force. Transitions between channels are induced by the subtle interplay between inertia and thermal noise, analogously to the mechanism described in [@MKTLH07] for a model of a driven inertial Brownian particle moving in a periodic potential and subject to a periodic forcing in one dimension. In our case, the external force can induce a bias in such transitions, yielding an average $\langle v_x \rangle
\neq 0$. In particular, we observed [@SCPV16] that in a specific range of forces the tracer can be biased to visit more frequently upstream channels, slowing its motion, and leading to NDM or even to ANM. An analogous mechanism has been described in [@SER07b].
Phase chart
-----------
Our extensive numerical study of the model is summarized in the phase chart reported in Fig. \[fig:phase\]. We identified three “phases”, corresponding to simple monotonic behavior (black dots), NDM (red squares) and ANM (blue triangles), for the velocity-force relation, in the parameter space $(\tau,D_0)$. For each couple of parameters, we performed numerical simulations studying the curve $\langle v_x\rangle(F)$ and focusing on the regions where non-linear behaviors occur.
![Phase chart in the parameter space of the model. Black dots identify the region where monotonic behavior of the force-velocity relation is observed, red squares regions where NDM takes place, and blue triangles regions where ANM occurs.[]{data-label="fig:phase"}](phase_new.pdf){width="1.\columnwidth"}
As expected, for large values of the microscopic diffusivity $D_0$, the system exhibits a simple behavior, because noise dominates over the effect of the underlying velocity field, and a monotonic behavior is observed. The same happens for large values of $\tau$, when again the underlying field can be neglected. NDM seems a typical phenomenon for small values of $\tau$ and $D_0$, where the different driving mechanisms acting on the tracer are comparable and coupled, leading to non-monotonic behaviors. Our study also unveils a narrow region where ANM can be observed, which occurs for values of $\tau\sim\tau^*$, where $\tau^*$ is the typical time scale of the underlying velocity field. For $\tau\to 0$, the occurrence of ANM can be excluded due to the no-go theorem discussed in [@SER07].
Conclusions
===========
We have investigated a model for the dynamics of an inertial tracer advected by a laminar velocity field, under the action of an external force and subject to thermal noise. This model can be useful to describe the transport properties of small particles (e.g. soil dust, man-made pollutants or swimming microorganisms [@CGCB17]) dispersed in fluids in different contexts, such as aerosol sedimentation, plankton concentration, or gravitational settling, with application in several areas of engineering, oceanography or meteorology.
We focused on the force-velocity relation of the tracer in the nonlinear forcing regime. The system shows a very rich phenomenology, featuring negative differential and absolute mobility, as summarized by the phase chart in the model parameter space. The emergence of these effects is due to the subtle coupling between the velocity field dynamics and the inertia of the tracer, and crucially depends on the amplitude of the microscopic thermal noise and on the time scale ratio between the tracer Stokes time and the characteristic time of the fluid. The presence of two non-equilibrium sources in the system, namely the non-gradient force and the external bias, and its combined action, make the model very rich: beyond the known trapping effects which result in a reduction of the tracer velocity upon increasing the applied force, in our model we also observe regions of negative mobility, where the particle travels against the external force. This phenomenon demands for experimental investigation in real systems.
Author contribution statement {#author-contribution-statement .unnumbered}
=============================
All authors contributed equally to the paper.
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|
---
author:
- 'A. Härter$^1$, A. Krükow$^1$, M. Deiß$^1$, B. Drews$^1$, E. Tiemann$^2$, & J. Hecker Denschlag$^1$'
bibliography:
- 'denseionization\_submission.bib'
title: 'Shedding Light on Three-Body Recombination in an Ultracold Atomic Gas'
---
While cold collisions of two atoms are understood to an excellent degree, the addition of a third collision partner drastically complicates the interaction dynamics. In the context of Bose-Einstein condensation in atomic gases, three-body recombination plays a crucial role[@Hess1983; @Burt1997; @Soding1999; @Esry1999] and it constitutes a current frontier of few-body physics[@Suno2009; @Wang2011; @Guevara2012]. However, the investigations focussed mainly on the atom loss rates established by the recombination events. Discussions of the final states populated in the recombination process were restricted to the special case of large two-body scattering lengths[@Fedichev1996; @Bedaque2000] and culminated in the prediction and observation of Efimov resonances[@Efimov1970; @Braaten2001; @Kraemer2006]. In the limit of large scattering lengths, recombination has been seen to predominantly yield molecules in the most weakly bound states$\,$[@Weber2003; @Jochim2003a]. However, in the more general case of a scattering length comparable to the van der Waals radius, the recombination products might depend on details of the interaction potential. In fact, ongoing theoretical studies using simplified models indicate that recombination does not necessarily always favor the most weakly bound state$\,$[@dIncao2013](see also$\,$[@Simoni2006]). In general, recombination processes are of fundamental interest in various physical systems[@Bates1962; @Hess1983; @Flower2007]. The control and tunability of ultracold atomic systems provide an experimental testbed for a detailed understanding of the nature of these processes.
Here, we demonstrate the probing of molecules with binding energies up to $h\times 750\:$GHz (where $h$ is Planck’s constant) generated via three-body recombination of ultracold thermal $^{87}$Rb atoms. We produce the atomic sample in an optical dipole trap located within a linear Paul trap. The recombination and detection process is illustrated in Fig.$\,$\[fig:1\]a-d. Following a recombination event, the created Rb$_2$ molecule can undergo resonance-enhanced multi-photon ionization (REMPI) by absorbing photons from the dipole trap laser at a wavelength around 1064.5$\:$nm. The ion is then captured in the Paul trap and detected essentially background-free with very high sensitivity on the single particle level. Fig.$\,$\[fig:1\]e shows a simplified scheme of the Rb$_2$ and Rb$_2^+$ potential energy curves. From weakly-bound molecular states three photons suffice to reach the molecular ionization threshold. An additional photon may dissociate the molecular ion. By scanning the frequency of the dipole trap laser by more than 60$\:$GHz we obtained a high resolution spectrum featuring more than 100 resonance peaks. This dense and complex spectrum contains information which vibrational, rotational and hyperfine levels of the Rb$_2$ molecule are populated. We present an analysis of these data and make a first assignment of the most prominent resonances. This assignment indicates that in the recombination events a broad range of levels is populated in terms of vibrational, rotational, electronic and nuclear spin quantum numbers.
Our experimental scheme to detect cold molecules makes use of the generally excellent detection efficiencies attainable for trapped ions. It is related to proven techniques where cold molecules in magneto-optical traps were photoionized from the singlet and triplet ground states[@Fioretti1998; @Gabbanini2000; @Lozeille2006; @Huang2006; @Salzmann2008; @Sullivan2011] (see also ref$\,$[@Mudrich2009]). Our method is novel as it introduces the usage of a hybrid atom-ion trap which significantly improves the detection sensitivity. We perform the following experimental sequence. A thermal atomic sample typically containing $N_\text{at}\approx 5 \times 10^5$ spin-polarized $^{87}$Rb atoms in the $|F=1,m_\textrm{F}=-1\rangle$ hyperfine state is prepared in a crossed optical dipole trap at a magnetic field of about 5$\:$G. The trap is positioned onto the nodal line of the radiofrequency field of a linear Paul trap. Along the axis of the Paul trap the centers of the atom and ion trap are separated by about $300\:\mu\textrm{m}$ to avoid unwanted atom-ion collisions (Fig.$\,$\[fig:1\]d). At atomic temperatures of about $700\:\textrm{nK}$ and peak densities $n_\text{0}\approx5\times
10^{13}\:\textrm{cm}^{-3}$ the total three-body recombination rate in the gas is $\Gamma_\text{rec}=L_3 n_\text{0}^2
N_\text{at}/3^{5/2}\approx 10\:$kHz. Here, the three-body loss rate coefficient $L_3$ was taken from ref$\,$[@Soding1999]. At the rate $\Gamma_\text{rec}$, pairs of Rb$_2$ molecules and Rb atoms are formed as final products of the reactions. Both atom and molecule would generally be lost from the shallow neutral particle trap due to the comparatively large kinetic energy they gain in the recombination event (in our case typically on the order of a few K $\times
k_\text{B}$ where $k_\text{B}$ is the Boltzmann constant). The molecule, however, can be state-selectively ionized in a REMPI process driven by the dipole trap laser. All of these molecular ions remain trapped in the deep Paul trap and are detected with single particle sensitivity (see Methods). In each experimental run, we hold the atomic sample for a time $\tau\approx10\:$s. After this time we measure the number of produced ions in the trap from which we derive (after averaging over tens of runs) the ion production rate $\Gamma_\textrm{ion}$ normalized to a cloud atom number of $10^6$ atoms.
As a consistency check of our assumption that Rb$_2$ molecules are ionized in the REMPI process, we verify the production of Rb$_2^+$ molecules. For this, we perform ion mass spectrometry in the Paul trap (see Methods). We detect primarily molecular Rb$_2^+$ ions, a good fraction of atomic Rb$^+$ ions but no Rb$_3^+$ ions. Our experiments show that Rb$^+$ ions are produced in light-assisted collisions of Rb$_2^+$ ions with Rb atoms on timescales below a few ms. Details of this dissociation mechanism are currently under investigation and will be discussed elsewhere.
Two pathways for the production of our neutral Rb$_2$ molecules come immediately to mind. One pathway is far-off-resonant photoassociation of two colliding Rb atoms (here with a detuning of about 500$\,$GHz$\times h$). This pathway can be ruled out using several arguments, the background of which will be discussed in more depth later. For one, we observe molecules with a parity that is incompatible with photoassociation of totally spin polarized ensembles. Furthermore, we observe a dependence of the ion production rate on light intensity that is too weak to explain photoassociation.
The second pathway is three-body recombination of Rb atoms. Indeed, by investigating the dependence of the ion production rate $\Gamma_\textrm{ion}$ (which is normalized to a cloud atom number of 10$^6$ atoms) on atomic density, we find the expected quadratic dependence (see Fig.$\,$\[fig:2\]). For this measurement the density was adjusted by varying the cloud atom number while keeping the light intensity of the dipole trap constant.
Next, we investigate the dependence of the ion production rate on the wavelength of the narrow-linewidth dipole trap laser (see Methods). We scan the wavelength over a range of about 0.3$\:$nm around 1064.5$\:$nm, corresponding to a frequency range of about 60$\:$GHz. Typical frequency step sizes are 50$\:$MHz or 100$\:$MHz. We obtain a rich spectrum of resonance lines which is shown in Fig.$\,$\[fig:3\]a. The quantity $\bar{\Gamma}_\textrm{ion}$ denotes the ion production rate normalized to the atom number of the cloud and to the square of the atomic peak density. We find strongly varying resonance strengths and at first sight fairly irregular frequency spacings. In the following we will argue that most resonance lines can be attributed to respective well-defined molecular levels (resolving vibrational, rotational and often even hyperfine structure) that have been populated in the recombination process. These levels are located in the triplet or singlet ground state, $a^3\Sigma_u^+$ and $X^1\Sigma_g^+$, respectively. The relatively dense distribution of these lines reflects that a fairly broad range of states is populated. A direct assignment of the observed resonances is challenging, as it hinges on the precise knowledge of the level structure of all the relevant ground and excited states. In the following we will access and understand the data step by step.
One feature of the spectrum that catches the eye is the narrow linewidth of many lines. For example, Fig.$\,$\[fig:3\]b shows a resonance of which the substructures have typical half-widths $\Delta\nu_r\approx50\:\textrm{MHz}$. This allows us to roughly estimate the maximal binding energy of the molecules involved. Since the velocity of the colliding ultracold atoms is extremely low, the kinetics of the recombination products is dominated by the released molecular binding energy $E_\textrm{b}$. Due to energy and momentum conservation the molecules will be expelled from the reaction with a molecular velocity $v_\textrm{Rb2}=
\sqrt{2 E_\textrm{b}/(3 m_\textrm{Rb2})}$ where $m_\textrm{Rb2}$ is the molecular mass. The molecular resonance frequency $\nu_0$ will then be Doppler-broadened with a half-width $\Delta \nu_\textrm{D}=\sqrt{3}\nu_0
v_\textrm{Rb2}/2c$. Here, $c$ is the speed of light. By comparing $\Delta \nu_\textrm{D}$ to the observed values of $\Delta\nu_r$ we estimate a maximal binding energy on the order of $E_\textrm{b,max}\approx h\times 2.5\:$THz. This simple analysis overestimates the value $E_\textrm{b,max}$ since it neglects the natural linewidth of the transition and possible saturation broadening. Still, it already strongly constrains the possible populated molecular levels that are observed in our experiment.
Next, we investigate the dependence of the ion production rate on laser intensity $I_\textrm{L}$. In our experimental setup, this measurement is rather involved because the laser driving the REMPI process also confines the atomic cloud. Thus, simply changing only the laser intensity would undesirably also change the density $n_\text{0}$ of the atoms. To prevent this from happening we keep $n_\text{0}$ constant ($n_\text{0}\approx 5\times 10^{13}\:\textrm{cm}^{-3}$) by adjusting the atom number and temperature appropriately. Due to these experimental complications we can only vary $I_\textrm{L}$ roughly by a factor of 2 (Fig.$\,$\[fig:4\]a). We set the laser frequency to the value of $\nu_\text{L}=\nu_\text{L}^0 \equiv 281610\:\textrm{GHz}$, on the tail of a large resonance (see Fig.$\,$3). The atomic temperatures in this measurement range between $500\:$nK and $1.1\:\mu$K, well above the critical temperatures for Bose-Einstein condensation. The atomic densities can therefore be described using a Maxwell-Boltzmann distribution. Assuming a simple power-law dependence of the form $\bar{\Gamma}_\textrm{ion}\propto I_\textrm{L}^\alpha$ we obtain the best fit using an exponent $\alpha=1.5(1)$ (solid green line in Fig.$\,$\[fig:4\]a). This fit is between a linear and a quadratic intensity dependence (dashed red and blue lines, respectively). Thus, at least two of the three transitions composing the ionization process are partially saturated at the typical intensities used.
In order to better circumvent possible density variations of the atomic cloud induced by changes in laser intensity, we employ a further method which enables us to vary the intensity with negligible effects on the atomic sample. We achieve this by keeping the time-averaged intensity $\langle
I_\textrm{L}\rangle$ constant and comparing the ion production rates within a continuous dipole trap and a “chopped" dipole trap in which the intensity is rapidly switched between $0$ and $2I_\textrm{L}$. In both cases the trap is operated at an intensity $\langle I_\textrm{L}\rangle\approx 15
\:\text{kW}/\text{cm}^2$. In the “chopped" configuration the intensity is switched at a frequency of $100\:\textrm{kHz}$ so that the atoms are exposed to the light for $5\:\mu\textrm{s}$ followed by $5\:\mu\textrm{s}$ without light. It should be noted that molecules formed in the “dark" period with sufficiently high kinetic energies may leave the central trapping region before the laser light is switched back on. They are then lost for our REMPI detection. Taking into account the molecular velocity and the transverse extensions of the laser beams we can estimate that this potential loss mechanism leads to errors of less than 30%, even at the highest binding energies relevant to this work ($E_\textrm{b}\approx h\times 750\:$GHz, see below). We did not observe evidence of such losses experimentally. Investigations were made by changing the chopping frequency. We define $R$ as ratio of the ion production rates in the “chopped" and the continuous trap configuration. Fig.$\,$\[fig:4\]b shows the results of these measurements for various laser frequencies $\nu_\text{L}$. We find a value $R\approx 1.5$ for off-resonant frequency settings $\nu_\text{L}- \nu_\text{L}^0<
0.4\:\textrm{GHz}$, in good agreement with the result presented in Fig.$\,$\[fig:4\]a. When scanning the laser onto resonance at $\nu_\text{L}- \nu_\text{L}^0 \approx 0.45\:$GHz (see also Fig.$\,$\[fig:3\]b) we obtain $R\approx 1$. This result indicates a linear intensity dependence of the REMPI process in the resonant case, which is explained by the saturation of two of the three molecular transitions involved. It is known that transitions into the ionization continuum (photon III, see Fig.$\,$1e) will not saturate under the present experimental conditions. This means that the excitation pathway via photon I and II must be saturated and therefore both close to resonance.
However, given the wavelength range of about 1064.5$\pm 0.15\,$nm, an inspection of the level structure shows that photon I can only resonantly drive three different transitions which connect vibrational levels in states $X$ and $a$ to vibrational levels in states $A, b$ and $c$ (see Fig.$\,$1e). Spectroscopic details for these transitions and the corresponding vibrational levels are given in the Methods section and in Fig.$\,$5. From recent spectroscopic studies[@Strauss2010; @Takekoshi2011; @Drozdova2012] and additional measurements in our lab[@Rbexp] the level structure of all relevant levels of the $X, a,
A, b,$ and $c$ states is well known. The absolute precision of most of the level energies is far better than 1$\:$GHz for low rotational quantum numbers $J$.
In the experimental data (Fig.$\,$\[fig:3\]a) the central region from $\nu_L - \nu^0_L = -6$ to 7$\:$GHz is marked by several prominent resonances that are significantly stronger than those observed throughout the rest of the spectrum. These resonance peaks can be explained by transitions from the $X$ ground state to $A$ and $b$ states. The prominence of these singlet transitions is explained by the near degeneracy of levels due to small hyperfine splittings. Indeed, by analyzing these strong resonances with regard to line splittings and intensities it was possible to consistently assign rotational ladders for total nuclear spin quantum numbers $I=1,2,3$ for the transition $X (v = 115)
\rightarrow A (v' = 68)$. The starting point of the rotational ladder for $I = 2$ was fixed by previous spectroscopic measurements[@Rbexp]. At frequencies $\nu_L - \nu^0_L \gtrsim
2\:$GHz additional strong lines appear that we attribute to the $X\:(v=109) \rightarrow b\:(v'=72)$ transition. The fact that we observe $X$ state molecules with $I = 1, 2, 3$ is interesting because for $I=1,3$ the total parity of the molecule is negative, while for $I=0,2$ it is positive. However, a two-body collision state of our spin polarized Rb atoms necessarily has positive total parity due to symmetry arguments and a photoassociation pathway would lead to ground state levels with positive parity. The observed production of molecules with negative total parity must then be a three-body collision effect.
We now consider the role of secondary atom-molecule collisions which would change the product distribution due to molecular relaxation. Two aspects are of importance: 1) depopulation of detected molecular levels and 2) population of detected molecular levels via relaxation from more weakly bound states. In our experiments reported here we detect molecules that are formed in states with binding energies on the order of hundreds of GHz. These molecules leave the reaction with kinetic energies of several K$\times k_\text{B}$. At these energies the rate coefficients for depopulating atom-molecule collisions are small (see e.g. ref.$\,$[@Simoni2006]) and the collision probability before the molecule is either ionized or has left the trap is below 1$\,$%.\
For the population processes, we can estimate an upper bound for rate coefficients by assuming recombination to occur only into the most weakly bound state with a binding energy of $24\,$MHz$\times
h$. In this case subsequent atom-molecule collision rates will be roughly comparable to those expected in the ultracold limit. At typical rate coefficients of $10^{-10}\,\text{cm}^3/\text{s}\;$ (see refs. [@Mukaiyama2004; @Staanum2006; @Zahzam2006; @Quemener2007]) and the atomic densities $n_\text{0}\sim1\times 10^{13}\:\textrm{cm}^{-3}$ used in the measurement shown in Fig.$\,$\[fig:2\], the collision probability before the molecule leaves the atom cloud is around 5$\,$%. This small probability grows linearly with density so that the density dependence of the ion production rate should show a significant cubic contribution if secondary collisions were involved (as expected for this effective four-body process). This is inconsistent with the data and thus indicates that the population that we detect is not significantly altered by secondary collisions.
We can roughly estimate the range of molecular rotation $J$ of the populated levels in the ground state. The strong isolated lines that we have assigned to the $X \: (v = 115) \rightarrow
A \: (v' = 68) $ transition are all contained within a relatively small spectral region ($|\nu_L - \nu^0_L| < 6\:$GHz) and are explained by rotational quantum numbers $J\leq 7$. Population of higher rotational quantum numbers would result in a continuation of the strong resonance lines stretching to transition frequencies beyond $\nu_L - \nu^0_L=10\:$GHz, which we do not observe. Similarly, if only rotational quantum numbers $J\leq 5$ were populated, a spectrum would result which does not have enough lines to explain the data. Thus, we can roughly set the limits on the molecular rotation to $J\leq 7$, a value that is also consistent with our observations of the spread of the transitions $X \rightarrow b$ and $a \rightarrow c$ (see Fig.$\,$\[fig:5\]). Finding quantum numbers as high as $J = 7$ is remarkable because the three-body collisions at $\mu$K temperatures clearly take place in an $s$-wave regime, i.e. at vanishing rotational angular momentum. Hence, one could expect to produce $X$ state molecules dominantly at $J = 0$, which, however, we do not observe.
Despite the limited spectral range covered by our measurements, we can already estimate the number of molecular vibrational levels populated in the recombination events. From the three states $X \: (v=109)$, $a\: (v=26)$ and $X\: (v = 115)$ that we can observe within our wavelength range, all deliver comparable signals in the spectrum of Fig.$\,$\[fig:3\]. This suggests that at least all vibrational states more weakly bound than $X \: (v=109)$ should be populated, a total of 38 vibrational levels (counting both singlet and triplet states). This is a significant fraction of the 169 existing levels of the $X$ and $a$ states, although restricted to a comparatively small range of binding energies.
In conclusion, we have carried out a first, detailed experimental study of the molecular reaction products after three-body recombination of ultracold Rb atoms. We use a high-power, narrow-linewidth laser to state-selectively ionize the produced molecules in a REMPI process. Subsequently, these ions are trapped in an ion trap and detected with very high sensitivity and negligible background. An analysis of the ionization spectrum allows us to identify population of several vibrational quantum levels indicating that the recombination events result in a fairly broad and uniform population distribution. We conjecture that dozens of vibrational levels are populated in total. Molecules are produced both in $X^1\Sigma_g^+$ as well as $a^3\Sigma_u^+$, with negative and positive total parity, various total nuclear spins and rotational quantum numbers reaching $J \leq 7$. Our work represents a first experimental step towards a detailed understanding on how the reaction channels in three-body recombination are populated. A full understanding will clearly require further experimental and theoretical efforts. On the experimental side the scanning range has to be increased and it could be advantageous to switch to a two-color REMPI scheme in the future. Such studies may finally pave the way to a comprehensive understanding of three-body recombination, which includes the details of the final products.\
Reaching beyond the scope of three-body recombination, the great sensitivity of our detection scheme has enabled us to state-selectively probe single molecules that are produced at rates of only a few Hz. We thereby demonstrate a novel scheme for precision molecular spectroscopy in extremely dilute ensembles.
Dipole trap and REMPI configuration.
------------------------------------
The crossed dipole trap is composed of a horizontal and a vertical beam focussed to beam waists of $\mathtt{\sim}90\:\mu\textrm{m}$ and $\mathtt{\sim}150\:\mu\textrm{m}$, respectively. It is positioned onto the nodal line of the radiofrequency field of the linear Paul trap with $\mu \text{m}$ precision. The two trap centers are separated by about $300\:\mu\textrm{m}$ along the axis of the Paul trap (see Fig. 1d). In a typical configuration, the trap frequencies of the dipole trap are $(175,230,80)\,\textrm{Hz}$ resulting in atom cloud radii of about $(6,7,16)\,\mu\textrm{m}$. The short-term frequency stability of the dipole trap laser source is on the order of $1\:\text{kHz}$ and it is stabilized against thermal drifts to achieve long-term stability of a few MHz. The two beams of the dipole trap are mutually detuned by $160\:\textrm{MHz}$ to avoid interference effects in the optical trap. Consequently, two frequencies are in principle available to drive the REMPI process. However, the intensity of the horizontal beam is 4 times larger than the one of the vertical beam and we have not directly observed a corresponding doubling of lines. Further details on the atom-ion apparatus are given in ref$\,$[@Smi2012].
Paul trap configuration.
------------------------
The linear Paul trap is driven at a radiofrequency of $4.17\:\textrm{MHz}$ and an amplitude of about $500\:\textrm{V}$ resulting in radial confinement with trap frequencies of $(\omega_\textrm{x,Ba},\omega_\textrm{y,Ba})=2\pi\times(220,230)\,\textrm{kHz}$ for a $^{138}\textrm{Ba}^+$ ion. Axial confinement is achieved by applying static voltages to two endcap electrodes yielding $\omega_\textrm{z,Ba}=2\pi\times40.2\:\textrm{kHz}$. The trap frequencies for “dark” Rb$_2^+$ and Rb$^+$ ions produced in the REMPI processes are $(m_\textrm{Ba}/m_\textrm{dark}\times\omega_\textrm{x,Ba},
m_\textrm{Ba}/m_\textrm{dark}\times\omega_\textrm{y,Ba},\sqrt{m_\textrm{Ba}/m_\textrm{dark}}\times\omega_\textrm{z,Ba})$ where $m_\textrm{Ba}$ and $m_\textrm{dark}$ denote the mass of the Ba$^+$ ion and the dark ion, respectively. The depth of the Paul trap depends on the ionic mass and exceeds $2\:\textrm{eV}$ for all ionic species relevant to this work.
Ion detection methods.
----------------------
We employ two methods to detect Rb$_2^+$ and Rb$^+$ ions both of which are not amenable to fluorescence detection. In the first of these methods we use a single trapped and laser-cooled $^{138}$Ba$^+$ ion as a probe. By recording its position and trapping frequencies in small ion strings with up to 4 ions we detect both the number and the masses of the ions following each REMPI process (see also[@Smi2010]). The second method is based on measuring the number of ions in the Paul trap by immersing them into an atom cloud and recording the ion-induced atom loss after a hold time of $2\:\textrm{s}$ (see also[@Harter2012]). During this detection scheme, we take care to suppress further generation of ions by working with small and dilute atomic clouds and by detuning the REMPI laser from resonance. Both methods are background-free in the sense that no ions are captured on timescales of days in the absence of the atom cloud. Further information on both detection methods is given in the Supplementary Information.
Spectroscopic details.
----------------------
Spin-orbit and effective spin-spin coupling in the $A$, $b$, and $c$ states lead to Hund’s case c coupling where the relevant levels of states $A$ and $b$ have $0_u$ symmetry while the levels of state $c^3\Sigma_g^+$ are grouped into $0_g^-$ and $1_g$ components. The level structure of the $0_{u}^+$ states is quite simple as it is dominated by rotational splittings. Typical rotational constants for the electronically excited states are on the order of 400$\:$MHz, for the weakly bound $X$ and $a$ states they are around 100-150$\:$MHz.
Figure 5 shows the relevant optical transitions between the $X, a$ states and the $A, b, c$ states in our experiment. For the given expected relative strengths of these transitions, we only consider Franck-Condon factors and the mixing of singlet and triplet states, while electronic transition moments are ignored. The colored arrows correspond to transitions with large enough Franck-Condon factors (typ. $10^{-2} ... 10^{-3})$ so that at laser powers of $\approx 10^4\,$W/cm$^2$ resonant transitions can be well saturated. Transitions marked with grey arrows can be neglected due to weak transition strengths, resulting from small Franck-Condon factors or dipole selection rules.
The authors would like to thank Stefan Schmid and Andreas Brunner for support during early stages of the experiment and Olivier Dulieu, Brett Esry, Jose d’Incao, William Stwalley, Ulrich Heinzmann, Jeremy Hutson, Pavel Soldan, Thomas Bergeman and Anastasia Drozdova for valuable information and fruitful discussions. This work was supported by the German Research Foundation DFG within the SFB/TRR21.
****
In this Supplementary Information we describe two methods that we employ to detect small numbers of Rb$_2^+$ and Rb$^+$ ions in our linear Paul trap.\
****
To implement our first ion detection method allowing mass-sensitive detection of “dark” ions we rely on the presence of a single “bright” ion in the trap. Information on additional ions can be extracted from its fluorescence position. When using this method, our experimental procedure begins with the loading of a single $^{138}$Ba$^+$-ion into our linear Paul trap. We laser-cool the ion and image its fluorescence light onto an electron-multiplying charge-coupled device camera. This enables us to determine the position of the trap center to better than $100\:\textrm{nm}$. The ion is confined at radial and axial trapping frequencies $\omega_\textrm{\,r,Ba} \approx 2\pi\times 220\:\textrm{kHz}$ and $\omega_\textrm{ax,Ba} \approx 2\pi\times
40.2\:\textrm{kHz}$ and typically remains trapped on timescales of days. Next, we prepare an ultracold atomic sample in the crossed dipole trap. At typical atomic temperatures of about $700\:$nK the atom cloud has radial and axial extensions of about $7\:\mu$m and $15\:\mu$m and is thus much smaller than the trapping volume of our Paul trap. To avoid atom-ion collisions we shift the Ba$^+$-ion by about $300\:\mu$m with respect to the atom cloud before the atomic sample arrives in the Paul trap. The shifting is performed along the axis of the trap by lowering the voltage on one of the endcap electrodes. Additionally, we completely extinguish all resonant laser light so that the atoms are only subjected to the light of the dipole trap. The atomic sample is moved into the center of the radial trapping potential of the Paul trap and is typically held at this position for a time $\tau_{\,\textrm{hold}}\approx 10\,\textrm{s}$. Despite the axial offset from the center of the Paul trap, the atom cloud at this position is fully localized within the trapping volume of the Paul trap. After the hold time the sample is detected using absorption imaging. Subsequently, the ion cooling beams are switched back on for fluorescence detection of the Ba$^+$-ion.\
The presence of a second ion in the trap leads to positional shifts of the $^{138}$Ba$^+$-ion by distances on the order of $10\:\mu$m (see Fig.$\,$\[fig:sup1\]). We make use of the mass-dependent trap frequencies of the Paul trap to gain information on the ion species trapped. In a two-ion Coulomb crystal composed of a Ba$^+$-ion and a dark ion, the axial center-of-mass frequency $\omega_\textrm{\,ax,2ion}$ shifts with respect to $\omega_\textrm{ax,Ba}$ depending on the mass of the dark ion $m_\textrm{dark}$ [@Morigi2001]. We measure $\omega_\textrm{\,ax,2ion}$ by modulating the trap drive at frequencies $\omega_\textrm{mod}$ and by monitoring the induced axial oscillation of the Ba$^+$-ion, visible as a blurring of the fluorescence signal. In this way, after each ion trapping event, we identify a resonance either at $\omega_\textrm{mod} \approx 2 \pi \times 44 \:\textrm{kHz}$ or $\omega_\textrm{mod} \approx 2 \pi \times 38 \:\textrm{kHz}$ corresponding to $m_\textrm{dark}=87\:\textrm{u}$ and $m_\textrm{dark}=174\:\textrm{u}$, respectively (see table \[table1\]).\
We have expanded this method for ion strings with up to four ions including the Ba$^+$-ion. For this purpose, we perform the following step-by-step analysis.
1. The position $x$ of the Ba$^+$-ion with respect to the trap center is detected. If $x\neq 0$, the value of $x$ allows us to directly determine the total number of ions in the string.
2. If $x=0$ we need to distinguish between a single Ba$^+$ ion and a three-ion string with Ba$^+$ at its center. This is done by modulating the trap drive at $\omega_\textrm{ax,Ba}$, thereby only exciting the Ba$^+$ ion if no further ions are present.
3. We destructively detect the Rb$^+$ ions by modulating the trap drive on a $5\:\text{kHz}$ wide band around $2\times \omega_\textrm{r,Rb}/(2\pi)=691\:\textrm{kHz}$. This selectively removes only Rb$^+$ ions from the string making use of the relatively weak inter-ionic coupling when exciting the ions radially.
4. Steps 1. and 2. are repeated to detect the number of remaining ions.
5. The Rb$_2^+$ ions are destructively detected via modulation around $2\times \omega_\textrm{r,Rb2}/(2\pi)=341\:\textrm{kHz}$.
Ion species $\omega_\textrm{\,ax,2ion}/2\pi$ \[kHz\] $\omega_\textrm{\,r}/2\pi$ \[kHz\]
------------------------------------ ------------------------------------------ ------------------------------------
$^{138}$Ba$^+$ and $^{138}$Ba$^+$ 40.2 220.0
$^{138}$Ba$^+$ and $^{87}$Rb$^+$ 44.0 345.3
$^{138}$Ba$^+$ and $^{87}$Rb$_2^+$ 37.7 170.7 \[table1\]
: [****]{}
\
****
We have also developed a second ion detection method that does not require an ion fluorescence signal. Instead, the trapped ions are detected via their interaction with an atomic sample. For this purpose, we produce a comparatively small atom cloud containing about $1\times 10^5$ atoms at a density of a few $10^{12}\:\textrm{cm}^{-3}$. In addition, we set the frequency of the dipole trap laser to an off-resonant value so that the production of additional ions during the ion probing procedure becomes extremely unlikely. We now fully overlap the ion and atom traps for an interaction time $\tau_\textrm{\footnotesize{int}}=2\,\textrm{s}$. By applying an external electric field of several V/m we set the ion excess micromotion energy to values on the order of tens of $k_\text{B}\times$mK [@Berkeland1998; @Harter2012]. Consequently, if ions are present in the trap, strong atom losses occur due to elastic atom-ion collisions. Fig.\[fig:sup2\] shows a histogram of the atom numbers of the probe atom samples consisting of the outcome of about 1,000 experimental runs. The histogram displays several peaks which can be assigned to the discrete number of ions in the trap. Up to five ions were trapped simultaneously and detected with high fidelity. The atom loss rate increases nonlinearly with ion number mainly because the interionic repulsion prevents the ions from all occupying the trap center where the atomic density is maximal. While ion detection method 2 does not distinguish ionic masses, it has advantages in terms of experimental stability and does not require the trapping of ions amenable to laser cooling or other fluorescence based detection techniques.
|
---
abstract: 'We present a complete next-to-leading order (NLO) QCD calculation to a heavy resonance production and decay into a top quark pair at the LHC, where the resonance could be either a Randall-Sundrum (RS) Kaluza-Klein (KK) graviton $G$ or an extra gauge boson $Z''$. The complete NLO QCD corrections can enhance the total cross sections by about $80\%- 100\%$ and $20\%- 40\%$ for the $G$ and the $Z''$, respectively, depending on the resonance mass. We also explore in detail the NLO corrections to the polar angle distributions of the top quark, and our results show that the shapes of the NLO distributions can be different from the leading order (LO) ones for the KK graviton. Moreover, we study the NLO corrections to the spin correlations of the top quark pair production via the above process, and find that the corrections are small.'
author:
- Jun Gao
- Chong Sheng Li
- Bo Hua Li
- Hua Xing Zhu
- 'C.-P.Yuan'
title: 'Next-to-leading order QCD corrections to a heavy resonance production and decay into top quark pair at the LHC'
---
introduction {#s1}
============
The top quark is the heaviest particle so far discovered, with a mass close to the electroweak symmetry breaking scale, and closely related to various new physics models beyond the standard model (SM). Thus it provides an effective probe for the electroweak symmetry breaking mechanism and the new physics beyond the SM through studying its production and decay at colliders. The Large Hadron Collider (LHC) is running now with a center of mass energy $\sqrt s =7{\rm\ TeV}$, and will collect 1 $\rm fb^{-1}$ experimental data during the initial run. After this initial state the LHC will turn to $\sqrt s =14{\rm \
TeV}$, with a design luminosity of $\sim 10\ {\rm fb^{-1}/yr}$ there will be $8\times 10^6$ top quark pairs and $3\times 10^6$ single top quarks produced yearly. As a result of all these, the precision measurement of the top quark properties, such as the mass, the production cross sections, the kinematic distributions, and the spin correlation effects, will be one of the prime tasks in the experiments at the LHC, and any deviations from the SM predictions will definitely be a hint for new physics beyond the SM.
To explore the connections between the new physics and the top quark, one possibility is to study the top quark pair invariant mass distribution and look for possible resonances since many new physics models predict the existence of a new resonance with a mass around ${\rm TeV}$, which can decay into a top quark pair, such as the Technicolor [@Hill:2002ap], Topcolor [@Hill:1991at], Little Higgs [@ArkaniHamed:2001nc], general $Z'$ models [@Langacker:2008yv; @Godfrey:2008vf], and Randall-Sundrum (RS) models [@Randall:1999ee]. In addition, in many such models the interaction between the heavy resonance and the top quark is enhanced as compared to the other fermions and the resonance will mainly decay into a top quark pair, for example, the Kaluza-Klein (KK) excitations of the graviton [@Fitzpatrick:2007qr], the gluon [@Gherghetta:2000qt] as well as the weak gauge bosons [@Agashe:2007ki] in the extended RS models. Once we have discovered such a resonance in the top quark pair invariant mass distribution, the next step is to measure its spin and couplings, and finally determine the underlying new physics dynamics, which have been studied in Refs. [@Barger:2006hm; @Frederix:2007gi]. It has been suggested in Refs. [@Barger:2006hm; @Frederix:2007gi] that it is possible to extract the spin and coupling information of the resonance from the top quark polar angle distributions and the spin correlations of the top quark pair. Those studies were carried out at the leading-order (LO) in QCD interactions. However, the next-to-leading order (NLO) QCD corrections may be large, for example, the NLO QCD corrections can enhance the cross sections of the single RS KK graviton or the $Z'$ production by about $70\%$ [@Mathews:2005bw; @Li:2006yv] and $20\%$ [@Ball:2007zza] respectively, so it is necessary to examine whether the QCD corrections will change the tree-level results and some conclusions of Refs. [@Barger:2006hm; @Frederix:2007gi] or not. In this paper we investigate the NLO QCD effects to a heavy resonance production and decay into a top quark pair, i.e., $pp\rightarrow X(color\
singlet)\rightarrow t\bar t$, at the LHC, where the $SU(3)_C$ color singlet state $X$ could be either a RS KK graviton $G$ or an extra gauge boson $Z'$.
The arrangement of this paper is as follows. Section \[s2\] is a brief review to the relevant models. In Sec. \[s3\] we show the details of the NLO calculations. Section \[s4\] contains the numerical results, and Sec. \[s5\] is a brief summary. The Appendix collects some analytic results at the LO.
the models {#s2}
==========
The RS KK graviton
------------------
In the RS model, a single extra dimension is compactified on a $\rm
S^1/Z_2$ orbifold with a radius $r$, which is not too large as compared to the Planck length. Two 3-branes, the Planck brane and the TeV brane, are located at the orbifold fixed points $\phi=0,\pi$, respectively, and the spacetime between the two 3-branes is simply a slice of a five-dimensional anti-de Sitter geometry. The five-dimensional warped metric is given by $$ds^2=e^{-2kr|\phi|}\eta_{\mu \nu}dx^{\mu}dx^{\nu}-r^2d\phi^2,$$ where $\phi$ is the five-dimensional coordinate, and $k \sim M_P$ is the curvature scale. By requiring $kr\sim 12$, one can suppress the Planck scale to $ M_P e^{-k\pi r}\sim O({\rm TeV})$ on the TeV brane, and then solve the gauge hierarchy problem. The gravity fields are treated as fluctuations under the background metric, and after expanding the gravity fields in the extra dimension we get infinite massive KK gravitons, which can interact with the SM fields [@Csaki:2004ay].
![Tree-level Feynman diagrams for the heavy resonance production and decay into a top quark pair.[]{data-label="f0"}](toppair3){width="70.00000%"}
The RS KK graviton can be produced through both the $gg$ fusion and the $q\bar q$ annihilation at the LO as shown in Fig. \[f0\]. The detailed Feynman rules of the graviton couplings can be found in Ref. [@Han:1998sg], and the propagator of the graviton in the unitary gauge in $n$ dimensions is given by [@Mathews:2005bw] $$P_G(k)={iB_{\mu\nu_,\rho\sigma}(k)\over k^2-m_X^2+im_X\Gamma_X},$$ with $$\begin{aligned}
B_{\mu\nu_,\rho\sigma}(k)&=&\left(g_{\mu\rho}-\frac{k_{\mu}k_{\rho}}{m_X^2}
\right)\left(g_{\nu\sigma}-\frac{k_{\nu}k_{\sigma}}{m_X^2}\right)+
\left(g_{\mu\sigma}-\frac{k_{\mu}k_{\sigma}}{m_X^2}
\right)\left(g_{\nu\rho}-\frac{k_{\nu}k_{\rho}}{m_X^2}\right)\nonumber
\\
&&-{2\over n-1}\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_X^2}
\right)\left(g_{\rho\sigma}-\frac{k_{\rho}k_{\sigma}}{m_X^2}\right),\end{aligned}$$ where $m_X$ and $\Gamma_X$ are the mass and the width of the heavy resonance, respectively.
The extra gauge boson $Z'$
--------------------------
The extra gauge boson $Z'$ could arise from an additional $U(1)'$ gauge symmetry [@Langacker:2008yv]. It could also be the KK excitation of the electroweak gauge bosons. It can only be produced through the $q\bar q$ annihilation at the LO, and its generic couplings to quarks are as follow $$Z'q\bar q \sim
\gamma_{\mu}\left(a_L\frac{1-\gamma_5}{2}+a_R\frac{1+\gamma_5}{2}\right).$$ We considered four combinations of $a_L$ and $a_R$, i.e., the pure vector coupling, $a_L=a_R=1$, the axial-vector coupling, $a_L=-a_R=-1$, the right-handed coupling, $a_L=0$, $a_R=1$, and the left-handed coupling, $a_L=1$, $a_R=0$, which are denoted by $Z'_1$, $Z'_2$, $Z'_3$ and $Z'_4$, respectively. The propagator of the $Z'$ in the unitary gauge is given by $$P_{Z'}(k)=\frac{i}{k^2-m_X^2+im_X\Gamma_X}\left(-g_{\mu\nu}+{k_{\mu}k_{\nu}\over
m_X^2}\right).$$
In our calculations of the process $pp\rightarrow X(color\
singlet)\rightarrow t\bar t$, what we mainly concern about are the ratios of the NLO results to the LO ones, so it is not necessary to specify the actual values of all the couplings. We simply assume the mass of the heavy resonance to be around ${\rm TeV}$ scale, which is not yet excluded by the current experiments. Besides, we only consider the narrow resonance cases and fix $\Gamma_X/m_X=1\%$ at both the LO and the NLO. We do not expect our conclusions to be largely modified even if $\Gamma_X/m_X$ increases to be at the order of 10%. Detailed discussions on the SM backgrounds and the discovery potential of the process can be found in Refs. [@Barger:2006hm; @Frederix:2007gi].
the NLO formalism {#s3}
=================
The complete NLO QCD corrections to the process $pp\rightarrow
X(color\ singlet)\rightarrow t\bar t$ can be factorized into two independent gauge invariant parts, i.e., the heavy resonance produced at the NLO with a subsequent decay at the LO, and produced at the LO with a subsequent decay at the NLO, similar to the cases studied in Ref. [@Campbell:2004ch]. The box diagrams, and the corresponding real correction diagrams, that connect the initial and the final states do not contribute to the squared matrix elements up to the NLO as the heavy resonance is a $SU(3)_C$ color singlet particle. This whole procedure can be illustrated as follows $$\begin{aligned}
|{\mathcal M}^{tree}_{2\rightarrow 2}|^2&=&|{\mathcal
M}^{tree}_{pro}|^2\otimes|{\mathcal M}^{tree}_{dec}|^2\otimes|P_X|^2, \nonumber \\
|{\mathcal M}^{real}_{2\rightarrow 3}|^2&=&\left\{|{\mathcal
M}^{tree}_{pro}|^2\otimes|{\mathcal M}^{real}_{dec}|^2+|{\mathcal
M}^{real}_{pro}|^2\otimes|{\mathcal M}^{tree}_{dec}|^2\right\}\otimes|P_X|^2, \nonumber \\
{\mathcal M}^{tree*}_{2\rightarrow 2}{\mathcal
M}^{loop}_{2\rightarrow 2}&=&\left\{|{\mathcal
M}^{tree}_{pro}|^2\otimes ({\mathcal M}^{tree*}_{dec}{\mathcal
M}^{loop}_{dec})+|{\mathcal M}^{tree}_{dec}|^2\otimes ({\mathcal
M}^{tree*}_{pro}{\mathcal M}^{loop}_{pro})\right\}\otimes|P_X|^2,\end{aligned}$$ we have suppressed the possible Lorentz indices here for simplicity.
We calculate the full squared matrix elements using the propagators ($P_X$) of the heavy resonance given in Sec. \[s2\], which can incorporate the full spin correlations between the production and decay processes in order to generate the correct kinematic distributions. We carry out all the QCD calculations in the ’t Hooft-Feynman gauge and use the dimension regularization scheme [@'tHooft:1972fi] (with the naive $\gamma_5$ prescription [@Chanowitz:1979zu]) in $n=4-2\epsilon$ dimensions to regularize all the divergences. The one-loop Feynman diagrams for the production and the decay of the heavy resonance are shown in Fig. \[f1\].
![Some one-loop Feynman diagrams for the production and the decay of the heavy resonances. Others not shown can be obtained by the exchange of the external quark or gluon lines.[]{data-label="f1"}](toppair1){width="70.00000%"}
Fig. \[f2\] shows the real correction Feynman diagrams for the production and the decay of the heavy resonance. The infrared divergences of the real corrections are extracted by using the two cutoff phase space slicing method [@Harris:2001sx]. Due to the limited space here we do not reproduce the details of the method.
![Some real correction Feynman diagrams for the production and the decay of the heavy resonances. Others not shown can be obtained by the exchange of the external quark or gluon lines.[]{data-label="f2"}](toppair2){width="70.00000%"}
Numerical results {#s4}
=================
In our numerical calculations we choose the input parameters to be $m_{top}=171{\rm\ GeV}$, $m_Z=91.188{\rm\ GeV}$, and $\alpha_s(m_Z)=0.118$ [@Amsler:2008zzb]. The running QCD coupling constant is evaluated at the three-loop order [@Amsler:2008zzb] with $n_f=5$, and the CTEQ6M (CTEQ6L1) parton distribution function (PDF) set [@Pumplin:2002vw] is used through the NLO (LO) calculations. We set both the renormalization and factorization scales equal to the mass of the heavy resonance, unless specified. Besides, in the two cutoff phase slicing method there are two arbitrary cutoff parameters, i.e., the soft cutoff $\delta_s$ and the collinear cutoff $\delta_c$. We have checked the cutoff dependence of all our numerical results, and found that the dependence is negligibly small for $\delta_s\leq1\times 10^{-3}$, so we choose $\delta_s=1\times10^{-3}$ and $\delta_c=\delta_s/50$ to obtain the numerical results presented below.
The total cross sections {#tot}
------------------------
In Fig. \[f3\] we show the NLO K factor, defined as the ratio of the NLO cross section $\sigma_{NLO}$ to the LO cross section $\sigma_{LO}$, as a function of the heavy resonance mass at the LHC with different center of mass energies. We can see that the total NLO QCD corrections can be large, which can enhance the total cross sections by about $80\%- 100\%$ and $20\%- 40\%$ for the $G$ and all four types of $Z'$ bosons, respectively, depending on the resonance mass. The NLO corrections from the production part are dominant, while the ones from the decay part are relatively small, but can still reach above ten percent in some regions. Our results of the NLO K factors of the production part agree with the ones given in Refs. [@Li:2006yv; @Ball:2007zza], where the total cross sections have been summed over the spins of the heavy resonance directly. In the following parts of our paper we will only show the results of the total NLO QCD corrections for simplicity. We further present the ratios of the total cross sections from the different channels for the graviton at both the LO and the NLO in Fig. \[f4\]. It can be seen that the contribution from the $gg$ channel is dominant at the low $m_X$ value region due to the large PDF of the gluon, and the contribution from the $q\bar q$ channel becomes important at the high $m_X$ value region since the PDF of the valence quark decreases more slowly than the gluon. And the NLO corrections can change the ratio of the contribution from the $q\bar q$ channel to the one from the $gg$ channel significantly.
![The NLO K factors as functions of the heavy resonance mass at the LHC, the solid and dotted lines correspond to including the total QCD corrections and the QCD corrections from the production part alone, respectively. The four groups of the curves from the top to the bottom correspond to the $G$, $Z'_2$, $Z'_3(Z'_4)$, and $Z'_1$, respectively.[]{data-label="f3"}](kfactor7 "fig:"){width="40.00000%"} ![The NLO K factors as functions of the heavy resonance mass at the LHC, the solid and dotted lines correspond to including the total QCD corrections and the QCD corrections from the production part alone, respectively. The four groups of the curves from the top to the bottom correspond to the $G$, $Z'_2$, $Z'_3(Z'_4)$, and $Z'_1$, respectively.[]{data-label="f3"}](kfactor14 "fig:"){width="40.00000%"}
![The ratios of the total cross sections from different channels for the graviton as functions of the graviton mass at both the LO and the NLO.[]{data-label="f4"}](ratio7 "fig:"){width="40.00000%"} ![The ratios of the total cross sections from different channels for the graviton as functions of the graviton mass at both the LO and the NLO.[]{data-label="f4"}](ratio14 "fig:"){width="40.00000%"}
The polar angle and invariant mass distributions
------------------------------------------------
It has been shown in Refs. [@Barger:2006hm; @Frederix:2007gi] that the polar angle distributions of the top quark are the key points to extract the spin and coupling information of the heavy resonance. The definition of this polar angle depends on the reference frame and axis chosen, here we considered two kinds of polar angles, one is the Collins-Soper angle $\theta_S$ [@Collins:1977iv], which is defined to be the angle between the top quark momentum and the axis that bisects the angle between the momentums of the incoming hadrons ($\vec{p}_A$ and $-\vec{p}_B$) in the $t\bar t$ rest frame, and for the $Z'$ we can define $\theta^*$ as the angle in the $t\bar t$ rest frame between the top quark momentum and the incident quark momentum which can be determined by the longitudinal boost direction of the $t\bar t$ rest frame at the LHC [@Barger:2006hm].
![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=800{\rm\ GeV}$.[]{data-label="f5"}](angleCM800 "fig:"){width="40.00000%"} ![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=800{\rm\ GeV}$.[]{data-label="f5"}](angleCS800 "fig:"){width="40.00000%"}
![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=1500{\rm\ GeV}$.[]{data-label="f6"}](angleCM1500 "fig:"){width="40.00000%"} ![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=1500{\rm\ GeV}$.[]{data-label="f6"}](angleCS1500 "fig:"){width="40.00000%"}
![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=3000{\rm\ GeV}$.[]{data-label="f7"}](angleCM3000 "fig:"){width="40.00000%"} ![The normalized top quark polar angle distributions at the LO and the ratios of the normalized NLO distributions to the LO ones, at the LHC with $\sqrt s=14{\rm\ TeV}$ for $m_X=3000{\rm\ GeV}$.[]{data-label="f7"}](angleCS3000 "fig:"){width="40.00000%"}
In Figs. \[f5\]-\[f7\] we show the normalized polar angle distributions of the top quark at the LO and the ratios of the normalized NLO distributions to the LO ones at the LHC with $\sqrt
s=14{\rm\ TeV}$. At the LO, we can use both the $\cos \theta_S$ and $\cos \theta^*$ distributions to distinguish the $Z'$ and the $G$ as their distributions have significantly different shapes. At the same time we can also use the $\cos \theta^*$ distribution to distinguish between the $Z'_{1,2}$ and the $Z'_{3,4}$ since the latter ones have a large forward-backward asymmetry. Furthermore, the differences of the polar angle distributions between the $Z'_1$ and the $Z'_2$ are very small for $m_X$ around $1{\rm\ TeV}$ or heavier, thus it is not possible to separate them through the polar angle distributions. We present the LO squared helicity amplitudes in the Appendix, which can explain the behavior of the LO polar angle distributions. After including the NLO corrections, we can see that for all the $Z'$ the changes of the distributions are negligibly small, which are no more then a few percent. But for the $G$, as the increasing of the resonance mass the NLO corrections can change the shapes of the distributions, for example, the NLO corrections make the distributions decrease more quickly at the both ends and can reach about 10% for $m_X=1500{\rm\ GeV}$. The corrections can be as large as 30%, and change the shapes of the distributions significantly for $m_X=3000{\rm\ GeV}$, which do not change the fact that the distributions of the $G$ and the $Z'$ are greatly different. Note that for a heavy enough resonance, the top quarks produced are highly boosted, which means the decay products of the top quark are close to each other and form a top jet. Recently, several methods based on the jet substructures have been proposed [@Kaplan:2008ie], which may be used to detect such a top jet efficiently, so it is possible to measure the NLO QCD effects to the polar angle distributions for a graviton with a mass of several ${\rm TeV}$ at the LHC. The reason that the NLO distributions for the graviton differ from the LO ones is that the NLO corrections change the ratio of the contributions from the $gg$ and $q\bar q$ channels, as shown in Sec. \[tot\], which have different shapes of distributions. As the resonance mass increases, the contributions from these two channels become comparable, so the changes are more significant. We further study the scale and PDF uncertainties of the NLO polar angle distributions for a graviton with $m_X=3000\ {\rm GeV}$. As shown in Fig. \[f8\], the uncertainty from the scale dependence is negligibly small, and the PDF uncertainty is within 10%, which is still small as compared to the NLO corrections. Here, we use two more PDF sets in the NLO calculations, i.e., the MRST2004nlo [@Martin:2004ir] and MSTW2008nlo [@Martin:2009bu] PDF sets.
![The scale and PDF uncertainties of the NLO polar angle distribution for the graviton at the LHC with $\sqrt s=14{\rm\ TeV}$ and $m_X=3000{\rm\ GeV}$.[]{data-label="f8"}](scaleCS "fig:"){width="40.00000%"} ![The scale and PDF uncertainties of the NLO polar angle distribution for the graviton at the LHC with $\sqrt s=14{\rm\ TeV}$ and $m_X=3000{\rm\ GeV}$.[]{data-label="f8"}](pdfCS "fig:"){width="40.00000%"}
![The normalized top quark pair invariant mass distributions at both the LO and the NLO at the LHC with $\sqrt s=14{\rm\ TeV}$.[]{data-label="f9"}](mass800 "fig:"){width="40.00000%"} ![The normalized top quark pair invariant mass distributions at both the LO and the NLO at the LHC with $\sqrt s=14{\rm\ TeV}$.[]{data-label="f9"}](mass1500 "fig:"){width="40.00000%"}
In Fig. \[f9\] we present the invariant mass distributions of the top quark pair including the NLO QCD corrections. At the LO they are just the Breit-Wigner distributions with a center value $m_X$ and a width $\Gamma_X$. While at the NLO the heavy resonance can decay into a top quark pair plus a hard gluon, so the NLO corrections increase the distributions in the lower invariant mass value region, and the changes of the distributions are more significant as the resonance mass increases. We also studied all the above distributions at the LHC with $\sqrt s= 7{\rm\ TeV}$, and the results are similar.
The spin correlations
---------------------
One of the unique features of the top quark is that it decays before the strong interaction can depolarize its spin. Thus, it is possible to extract the spin information of the produced top quark by studying the angular distributions of the decay products. For a spin up top quark (or a spin down anti-top quark), the decay angular distribution of the $i$th decay product is given by [@Mahlon:2000ze] $${1\over \Gamma_T}\frac{d\Gamma}{d(\cos\chi_i)}={1\over
2}\left(1+\alpha_i\cos\chi_i\right),$$ where $i$ could be quarks, $b, u, c, \bar{d}, \bar{s}$, or leptons, $\nu_l, \bar{l}$; $\chi_i$ is the angle between the $i$th decay product and the spin quantization axis in the top rest frame, and $\alpha_i$ are the correlation coefficients. For the charged leptons, $\alpha_l=1$ exactly, which means the charged leptons are maximally correlated with the top spin direction.
The spin correlations of the top quark pair also can be used for the identification of the heavy resonance, but the precision measurement of them is more difficult at the LHC. In order to study the spin correlations of the top quark pair production, the following double differential cross section is usually considered, $${1\over \sigma}\frac{d^2\sigma}{d(\cos\chi_i^+)d(\cos\chi_j^-)}=
{1\over4}\left(1-A\alpha_i\alpha_j\cos\chi_i^+\cos\chi_j^-
+b_+\alpha_i\cos\chi_i^+ + b_-\alpha_j\cos\chi_j^-\right),$$ neglecting the interference between the top spins we have $$\begin{aligned}
A&=&\frac{\sigma(t_{\uparrow}\bar t_{\uparrow}+t_{\downarrow}\bar
t_{\downarrow})-\sigma(t_{\uparrow}\bar
t_{\downarrow}+t_{\downarrow}\bar
t_{\uparrow})}{\sigma(t_{\uparrow}\bar
t_{\uparrow}+t_{\downarrow}\bar
t_{\downarrow})+\sigma(t_{\uparrow}\bar
t_{\downarrow}+t_{\downarrow}\bar t_{\uparrow})}, \nonumber\\
b_+&=&\frac{\sigma(t_{\uparrow}\bar t_{\uparrow}+t_{\uparrow}\bar
t_{\downarrow})-\sigma(t_{\downarrow}\bar
t_{\uparrow}+t_{\downarrow}\bar
t_{\downarrow})}{\sigma(t_{\uparrow}\bar
t_{\uparrow}+t_{\downarrow}\bar
t_{\downarrow})+\sigma(t_{\uparrow}\bar
t_{\downarrow}+t_{\downarrow}\bar t_{\uparrow})}, \nonumber\\
b_-&=&\frac{\sigma(t_{\uparrow}\bar
t_{\downarrow}+t_{\downarrow}\bar
t_{\downarrow})-\sigma(t_{\uparrow}\bar
t_{\uparrow}+t_{\downarrow}\bar
t_{\uparrow})}{\sigma(t_{\uparrow}\bar
t_{\uparrow}+t_{\downarrow}\bar
t_{\downarrow})+\sigma(t_{\uparrow}\bar
t_{\downarrow}+t_{\downarrow}\bar t_{\uparrow})}.\end{aligned}$$ In our following calculations we use the helicity basis in the $t\bar t$ center of mass frame, which means $\chi_i^+(\chi_j^-)$ is defined to be the angle between the $t(\bar t)$ direction in the $t\bar t$ center of mass frame and the corresponding decay product direction in the $t(\bar t)$ rest frame.
![The top quark pair spin correlation coefficients at the LO as functions of the heavy resonance mass at the LHC with $\sqrt s=14{\rm
\ TeV}$.[]{data-label="f10"}](spin14){width="70.00000%"}
In Fig. \[f10\] we show the LO results of the top quark pair spin correlation coefficients as functions of the heavy resonance mass at the LHC with $\sqrt s=14{\rm\ TeV}$. According to symmetry analysis we have $$\begin{aligned}
&&A(Z'_3)=A(Z'_4), \quad b_{\pm}(G,Z'_1,Z'_2)=0,\nonumber \\
&&b_+(Z'_3)=b_-(Z'_3)=-b_+(Z'_4)=-b_-(Z'_4),\end{aligned}$$ and at the LO for the axial vector $Z'_2$, $$A(Z'_2)=-1,$$ which can be seen from the helicity amplitudes in the Appendix. With the increasing heavy resonance mass, all the coefficients in Fig. \[f10\] approach $-1$ due to the fact that the cross sections for the $t\bar t$ with the same helicities vanish as the heavy resonance mass goes infinity. In Table \[t1\], we list some typical NLO results of those coefficients. We can see that the NLO QCD corrections are rather small, about $1\%-2\%$, and can be neglected at the LHC. We also investigate the cases for $\sqrt s=7{\rm\ TeV}$, and the results are almost the same at both the LO and the NLO.
\[0.9\]
--------------------------------------------------------------------------------------------------------------------------------------------
$\ $\ $resonance$\ $ $\ $$A$(LO)$\ $ $A$(NLO)$\ $ $\ $b_+$(NLO)$\ $ $\ $$b_-$(LO)$\ $ $b_-$(NLO)$\ $
$mass(GeV)$\ $ $$b_+$(LO)$\ $
---------------- ------------------- ----------------- -------------- ---------------- ---------------- ------------------- ----------------
$G$ -0.783 -0.778 0 0 0 0
$Z'_1$ -0.832 -0.826 0 0 0 0
$Z'_2$ -1.000 -0.986 0 0 0 0
$Z'_3$ -0.904 -0.896 0.947 0.943 0.947 0.943
$Z'_4$ -0.904 -0.896 -0.947 -0.943 -0.947 -0.943
$G$ -0.933 -0.913 0 0 0 0
$Z'_1$ -0.949 -0.933 0 0 0 0
$Z'_2$ -1.000 -0.980 0 0 0 0
$Z'_3$ -0.974 -0.955 0.986 0.977 0.986 0.977
$Z'_4$ -0.974 -0.955 -0.986 -0.977 -0.986 -0.977
--------------------------------------------------------------------------------------------------------------------------------------------
conclusions {#s5}
===========
We have calculated the complete NLO QCD corrections to a heavy resonance production and decay into a top quark pair at the LHC, where the resonance could be either a RS KK graviton $G$ or an extra gauge boson $Z'$. Our results show that the total NLO K factors can reach about $1.8- 2.0$ and $1.2- 1.4$ for the $G$ and all four types of $Z'$ bosons, respectively, depending on the resonance mass. And the NLO corrections from the production part are dominant, while the ones from the decay part are relatively small but can still reach above ten percent in some parameter regions. We also explore in detail the NLO corrections to the polar angle distributions of the top quark, and our results show that the NLO distributions are almost the same as the LO ones for all four types of $Z'$ bosons, while the shapes of the NLO distributions can be significantly different from the LO ones for the $G$, depending on the mass of the resonance. Moreover, the NLO corrections can also change the shapes of the top quark pair invariant mass distributions. Finally, we study the NLO corrections to the spin correlations of the top quark pair, and find that the corrections are negligibly small.
This work was supported in part by the National Natural Science Foundation of China, under Grants No.10721063, No.10975004 and No.10635030. C.P.Y was supported in part by the U.S. National Science Foundation under Grand No. PHY-0855561.
Appendix {#appendix .unnumbered}
========
In this appendix we give the individual nonvanishing LO squared helicity amplitudes for a heavy resonance production and decay into a top quark pair, $q\bar q\ (gg)\rightarrow X \rightarrow t\bar t$. For all the $Z'$ mediated processes, $$\begin{aligned}
\overline{|\mathcal{M}|^2}_{Z'_1}&=&\left\{\begin{array}{l}
(1-\beta^2)\sin^2(\theta){\mathcal A}, \ \ {\rm for\ helicities\ \{+-++\},\ \{+---\}},\\
\hspace{3.8cm} {\rm \{-+++\}\ and \ \{-+--\}} \\
4\sin^4(\theta/2){\mathcal A}, \hspace{1cm} {\rm for\ helicities\ \{+--+\}\ and\ \{-++-\}}\\
4\cos^4(\theta/2){\mathcal A}, \hspace{1cm} {\rm for\ helicities\ \{+-+-\}\ and\ \{-+-+\}}, \\
\end{array}\right.\\
\overline{|\mathcal{M}|^2}_{Z'_2}&=&\left\{\begin{array}{l}
4\beta^2\sin^4(\theta/2){\mathcal A}, \hspace{0.8cm} {\rm for\ helicities\ \{+--+\}\ and\ \{-++-\}}\\
4\beta^2\cos^4(\theta/2){\mathcal A}, \hspace{0.8cm} {\rm for\ helicities\ \{+-+-\}\ and\ \{-+-+\}}, \\
\end{array}\right.\\
\overline{|\mathcal{M}|^2}_{Z'_3}&=&\left\{\begin{array}{l}
(1-\beta^2)\sin^2(\theta){\mathcal A}/4, \ \ {\rm for\ helicities\ \{+-++\}\ and\ \{+---\}}\\
(1-\beta)^2\sin^4(\theta/2){\mathcal A}, \ \ {\rm for\ helicities\ \{+--+\}}\\
(1+\beta)^2\cos^4(\theta/2){\mathcal A}, \ \ {\rm for\ helicities\ \{+-+-\}}, \\
\end{array}\right.\\
\overline{|\mathcal{M}|^2}_{Z'_4}&=&\left\{\begin{array}{l}
(1-\beta^2)\sin^2(\theta){\mathcal A}/4, \ \ {\rm for\ helicities\ \{-+++\}\ and\ \{-+--\}}\\
(1-\beta)^2\sin^4(\theta/2){\mathcal A}, \ \ {\rm for\ helicities\ \{-++-\}}\\
(1+\beta)^2\cos^4(\theta/2){\mathcal A}, \ \ {\rm for\ helicities\ \{-+-+\}}, \\
\end{array}\right.\end{aligned}$$ and for the graviton mediated processes through the $q\bar q$ annihilation and the $gg$ fusion, $$\begin{aligned}
\overline{|\mathcal{M}|^2}_{G,q\bar q}&=&\left\{\begin{array}{l}
\beta^2(1-\beta^2)\sin^2(2\theta){\mathcal B}/64, \ \ {\rm for\ helicities\ \{+-++\},\ \{+---\}}, \\
\hspace{5cm} {\rm \{-+++\}\ and \ \{-+--\}} \\
\beta^2(1+2\cos(\theta))^2\sin^4(\theta/2){\mathcal B}/16, \ \ {\rm for\ helicities\ \{+--+\}},\\
\hspace{6.5cm} {\rm and\ \{-++-\}}\\
\beta^2(1-2\cos(\theta))^2\cos^4(\theta/2){\mathcal B}/16, \ \ {\rm for\ helicities\ \{+-+-\}},\\
\hspace{6.5cm} {\rm and\ \{-+-+\}},\\
\end{array}\right.\\
\overline{|\mathcal{M}|^2}_{G,gg}&=&\left\{\begin{array}{l}
3\beta^2(1-\beta^2)\sin^4(\theta){\mathcal B}/128, \ \ {\rm for\ helicities\ \{+-++\},\ \{+---\}}, \\
\hspace{5cm} {\rm \{-+++\}\ and \ \{-+--\}} \\
3\beta^2\sin^6(\theta/2)\cos^2(\theta/2){\mathcal B}/8, \ \ {\rm for\ helicities\ \{+--+\},\ and\ \{-++-\}} \\
3\beta^2\sin^2(\theta/2)\cos^6(\theta/2){\mathcal B}/8, \ \ {\rm for\ helicities\ \{+-+-\},\ and\ \{-+-+\}}, \\
\end{array}\right.\end{aligned}$$ with $${\mathcal A}=\frac{s^2}{(s-m_{Z'}^2)^2+m_{Z'}^2\Gamma_{Z'}^2},\ \
{\mathcal B}=\frac{s^4}{(s-m_G^2)^2+m_G^2\Gamma_G^2},$$ where $\theta$ is the polar angle between the momenta of the top quark and the light quark (or gluon) in the center of the mass frame of the top quark pair, $s$ is the square of the center of mass energy, $\beta \equiv \sqrt{1-4m_{top}^2/s}$, and the squared amplitudes have been summed and averaged over the color of the external particles.
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|
---
abstract: 'The Standard Model predicts a branching ratio for the decay mode $B_{s}\rightarrow\mu\mu$ of (3.32$\pm$0.32)$\times$10 $^{-9}$ while some SUSY models predict enhancements of up to 2 orders of magnitude. It is expected that at the end of its life the Tevatron will set an exclusion limit for this branching ratio of the order of 10 $^{-8}$, leaving one order of magnitude to explore. The efficient trigger, excellent vertex reconstruction and invariant mass resolution, and muon identification of the LHC[*b*]{} detector makes it well suited to observe a branching ratio in this range in the first years of running of the LHC. In this article an overview of the analysis that has been developed for the measurement of this branching ratio is presented. The event selection and the statistical tools used for the extraction of the branching ratio are discussed. Special emphasis is placed on the use of control channels for calibration and normalization in order to make the analysis as independent of simulation as possible. Finally, the expected performance in terms of exclusion and observation significance are given for a set of values of integrated luminosities.'
address: |
Universitat de Barcelona\
Avinguda Diagonal 647, 08028 Barcelona (Spain)\
E-mail: elopez$@$ecm.ub.es
author:
- ELÍAS LÓPEZ ASAMAR
- 'ON BEHALF OF THE LHC[*b*]{} COLLABORATION'
title: 'Prospect for measuring the branching ratio of $B_{s}\rightarrow\mu\mu$ at LHC[*b*]{}'
---
Introduction
============
In the Standard Model (SM), flavour changing neutral currents (FCNC) can be generated only through loop diagrams, resulting in low branching ratio (BR) predictions. Some models beyond SM can introduce contributions of size comparable to that of SM, yielding significantly different predictions for these branching ratios. This is the case of the decay $B_{s}\rightarrow\mu\mu$, suppressed by helicity, for which the SM predicts BR($B_{s}\rightarrow\mu\mu$) = (3.32$\pm$0.32)$\times$10 $^{-9}$ [@SM-prediction]. In the Minimal Supersymmetric extension of SM (MSSM) this quantity is proportional to tan$^6\beta$ [@MSSM-prediction], leading to enhancements of up to one order of magnitude in cases such as the Non-Universal Higgs Masses framework (NUHM) [@NUHM-prediction]. The current experimental limit for this BR is BR($B_{s}\rightarrow\mu\mu$) $<$ 47$\times$10$^{-9}$ at 90% C.L. [@CDF-measurement] (CDF collaboration, with 2 fb$^{-1}$), roughly a factor 15 above the SM prediction. This channel then offer the possibility of observing hints of Physics beyond SM, or in case of confirming the SM prediction, rejecting an important region of the parameter space of some of these models.
Experimental conditions
=======================
The LHC[*b*]{} detector [@TDR] is specially designed for studying $B$ meson decays produced in LHC from $pp$ collisions at a centre of mass energy of $\sqrt{s}$ = 14 TeV/c$^2$). The nominal integrated luminosity at the LHC[*b*]{} interaction point will be 2 fb$^{-1}$/year, resulting in $\sim$4$\times$10$^{11}$ $B$ meson pairs per year inside the acceptance of the detector. The performances most relevant for the measurement are the vertexing capabilities to identify the displaced vertices of $B$ decays (impact parameter (IP) resolution of 14+35/p$_T$ $\mu$m, p$_T$ standing for the transverse component of the momentum), the invariant mass resolution [@FlavLHC] ($\sim$20 MeV/c$^2$, compared to $\sim$35 MeV/c$^2$ at CMS, or $\sim$80 MeV/c$^2$ at ATLAS), and the muon identification [@muonID] (94% efficient with 1% pollution coming from mis-identified $\pi$/$K$).
Analysis strategy
=================
The first step of the analysis consist of a very loose selection that focus on efficiency for signal events, rather than on background rejection. This preliminary selection is based on the quality of the vertex, the invariant mass, the impact parameter (IP) of the muons, and the IP/$\sigma_{IP}$, decay length and momentum of the reconstructed $B$. It is 65% efficient over reconstructed signal events, keeping the bulk of the sensitive signal, and yields $\sim$40 signal events (assuming SM predictions) and $\sim$13$\times$10$^4$ background events per fb$^{-1}$ under nominal conditions, inside a mass window of 60 MeV/c$^2$ around the mass of the $B_s$. The remaining events are caracterized according to three discriminant variables [@bsmumu]:
- Invariant mass.
- Particle-ID likelihood (PIDL) which for both muons combines the likelihood of being muon and the likelihood of not being muon.
- Geometrical likelihood (GL) which combines the information of the distance of closest approach of the tracks of the two muons, the minimal IP/$\sigma_{IP}$ of the muons, the isolation of both muons, and the lifetime and impact parameter of the $B$ [@bsmumu]. Its distribution for signal and background is shown in Figure \[GL\].
![Distribution of geometrical likelihood for the signal (red line) and background coming from $B$ decays (green, filled), satisfying only the selection cuts (no trigger is applied).[]{data-label="GL"}](GL.png){width="50.00000%"}
Backgrounds are distributed differently from signal in the three-dimensional space spanned by the invariant mass and both likelihoods, thus defining regions of different sensitivity to the signal. Only backgrounds lying in sensitive regions can affect significantly the measurement. Defining the sensitive events as those having GL $>$ 0.5, the background composition of this subsample is shown in Table \[backgrounds\] [@bsmumu]. In this case, the signal-to-background ratio rises from 0.3% to 11%.
Channel Yield (2 fb$^{-1}$)
--------------------------------------------------------- ---------------------
$(b\rightarrow\mu,b\rightarrow\mu)$ 170 $\pm$ 90
$B_{c}^{+}\rightarrow J/\Psi (\mu\mu)\mu^{+}\nu_{\mu }$ $<$ 20 (90% C.L.)
$B\rightarrow h^+h^-$ mis-ID 8 $\pm$ 2
Signal 20 $\pm$ 2
: Sources of background for sensitive events (GL $>$ 0.5).[]{data-label="backgrounds"}
Calibration and normalization
=============================
Calibration and normalization procedures foreseen for this analysis rely entirely on data, in order to minimize dependence on simulations [@bsmumu]. Mass sidebands are used to calibrate background distributions, while control channels are used for calibration of signal properties:
- The particle-ID likelihood is calibrated using inclusive samples of $J$/$\Psi$($\mu\mu$) (for muon hypothesis) and $\Lambda$($pK$) (for non-muon hypothesis), which can be selected with high purity using only kinematical cuts.
- The geometrical likelihood and invariant mass are calibrated using $B\rightarrow h^+h^-$, which has the same kinematical properties as the signal. The trigger introduces strong biases in the distribution of the geometrical likelihood, which can be removed by using events triggered on particles not related with the signal ($\sim$7% of total $B\rightarrow h^+h^-$).
Control channels with known branching ratios allow to normalize the signal trough
$$N_S={\epsilon_S\over\epsilon_C}{f _{B,S}BR_S\over f _{B,C}BR_C}N_C \qquad .
\label{normalization}$$
Subindices $S$ and $C$ stand for signal and control channel respectively. The total efficiency $\epsilon$ can be split as $\epsilon_{rec/prod}\times\epsilon_{sel/rec}\times\epsilon_{trig/sel}$, where $\epsilon_{rec/prod}$ is the reconstruction efficiency on produced events, $\epsilon_{sel/rec}$ is the efficiency of the offline selection on reconstructed events, and $\epsilon_{trig/sel}$ is the efficiency of the trigger on offline-selected events. $B^0$ decays are aproppiate for calibration due to the accuracy in the measurements of their branching ratios. A proper choice of the control channel leads to cancellation of the effect of some sources of inefficiency:
- $B^+\rightarrow J/\Psi (\mu\mu )K^+$. The effect of the trigger is similar to that for the signal due to the $J$/$\Psi$ muons. The ratio of reconstruction efficiencies needs to take into account the different number of tracks in the final state between the signal and the control channel (two and three tracks respectively), and it is estimated through: $${ \epsilon _{rec} (2~tracks) \over \epsilon _{rec} (3~tracks) } \sim { \epsilon _{rec} (3~tracks) \over \epsilon _{rec} (4~tracks) }
\label{reco}$$ Using $B^0\rightarrow J/\Psi (\mu \mu)K^{*0}(\pi K)$ as the four-track decay channel the ratio between the left-hand side and the right-hand side is 92%.
- $B\rightarrow h^+h^-$. Has the same kinematic properties to that of signal, leading to cancellation of reconstruction and selection effects.
In both cases the ratio of trigger efficiencies is estimated using events triggered on particles not related with signal, which have a relatively small trigger bias on signal properties. The ratio of $B$ production fractions in Equation \[normalization\] is the main source of uncertainty of the measurement, as long as channels coming from $B^0$ decays are used for normalization. It introduces a systematic error of 13%.
Extraction of the branching ratio
=================================
The expected distributions of invariant mass and both particle-ID and geometrical likelihoods for signal and background will be obtained from the control channels and the mass sidebands respectively. These distributions will be combined together and compared with the measured ones, using the CL$_s$ method [@CLs]. In the absence of signal this analysis would reach the expected CDF limit with only an integrated luminosity of 0.1 fb$^{-1}$, and would reach the SM prediction with 2 fb$^{-1}$ [@bsmumu] (see Figure \[performance\]). In case of observing a signal, the 3$\sigma$ (5$\sigma$) measurement could be achieved with 3 fb$^{-1}$ (10 fb$^{-1}$) [@bsmumu] of data.
![Expected performance for exclusion (left) and measurement (right) of a given branching ratio as a function of the integrated luminosity. The expected limit set by CDF at the end of its life (8 fb$^{-1}$ of data) is also shown.[]{data-label="performance"}](performance.png){width="\textwidth"}
Conclusions
===========
The method developed for the measurement of the branching ratio of $B_{s}\rightarrow\mu\mu$ at LHC[*b*]{} makes an extensive use of the control channels in order to be independent of simulations. This analysis is expected to lead to relevant results even with early data, excluding the SM value at 90% CL with only 2 fb$^{-1}$ in absence of signal, or observing it with only 3 fb$^{-1}$, thus potentially being able to provide one of the first evidences of New Physics from the LHC.
[8]{} M. Blanke [*et. al.*]{}, arXiv:hep-ph/0604057 (2006). S .R. Choudhury and N. Gaur, Phys. Lett. [**B451**]{} (1999) 86. J. Ellis [*et. al.*]{}, arXiv:0709.0098 \[hep-ph\] (2007). T. Aaltonen [*et. al.*]{} \[CDF collaboration\], Phys. Rev, Lett. [**100**]{} (2008) 101802. The LHC[*b*]{} Collaboration, [*LHCb reoptimized design and perfomance*]{}, CERN/LHCC 2003-030. http://cern.ch/mlm/FlavLHC.html M. Gandelman, E. Polycarpo, LHC[*b*]{} 2005-099. D. Martinez, J. A. Hernando and F. Teubert, LHC[*b*]{} 2007-033. A. L. Read, CERN Yellow Report 2000-005.
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abstract: 'We investigate various ways to define an analogue of BGG category $\mathcal{O}$ for the non-semi-simple Takiff extension of the Lie algebra $\mathfrak{sl}_2$. We describe Gabriel quivers for blocks of these analogues of category $\mathcal{O}$ and prove extension fullness of one of them in the category of all modules.'
author:
- 'Volodymyr Mazorchuk and Christoffer S[ö]{}derberg'
title: 'Category $\mathcal{O}$ for Takiff $\mathfrak{sl}_2$'
---
Introduction and description of the results {#s1}
===========================================
The celebrated BGG category $\mathcal{O}$, introduced in [@BGG], is originally associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra. The definition of $\mathcal{O}$ is naturally generalized to all Lie algebras admitting some analogue of a triangular decomposition, see [@MP]. These include, in particular, Kac-Moody algebras and Virasoro algebra. Category $\mathcal{O}$ has a number of spectacular properties and applications to various areas of mathematics, see for example [@Hu] and references therein.
The paper [@DLMZ] took some first steps in trying to understand structure and properties of an analogue of category $\mathcal{O}$ in the case of a non-reductive finite dimensional Lie algebra. The investigation in [@DLMZ] focuses on category $\mathcal{O}$ for the so-called Schr[ö]{}dinger algebra, which is a central extension of the semi-direct product of $\mathfrak{sl}_2$ with its natural $2$-dimensional module. It turned out that, for the Schr[ö]{}dinger algebra, the behavior of blocks of category $\mathcal{O}$ with non-zero central charge is exactly the same as the behavior of blocks of category $\mathcal{O}$ for the algebra $\mathfrak{sl}_2$. At the same, block with zero central charge turned out to be significantly more difficult. For example, it was shown in [@DLMZ] that some blocks of $\mathcal{O}$ for the Schr[ö]{}dinger algebra have wild representation type, while all blocks of $\mathcal{O}$ for $\mathfrak{sl}_2$ have finite representation type.
In the present paper we look at a different non-reductive extension of the algebra $\mathfrak{sl}_2$, namely, the corresponding Takiff Lie algebra $\mathfrak{g}$ defined as the semi-direct product of $\mathfrak{sl}_2$ with the adjoint representation. Such Lie algebras were defined and studied by Takiff in [@Ta] with the primary interest coming from invariant theory. Alternatively, the Takiff Lie algebra $\mathfrak{g}$ can be described as the tensor product $\mathfrak{sl}_2\otimes_{\mathbb{C}}\big(\mathbb{C}[x]/(x^2)\big)$. The latter suggests an obvious generalization of the notion of a triangular decomposition for $\mathfrak{g}$ by tensoring the components of a triangular decomposition for $\mathfrak{sl}_2$ with $\mathbb{C}[x]/(x^2)$.
Having defined a triangular decomposition for $\mathfrak{g}$, we can define Verma modules and try to guess a definition for category $\mathcal{O}$. The latter turned out to be a subtle task as the most obvious definition of category $\mathcal{O}$ does not really work as expected, in particular, it does not contain Verma modules. This forced us to investigate two alternative definitions of category $\mathcal{O}$:
- the first one analogous to the definition of the so-called [*thick category $\mathcal{O}$*]{}, see, for example, [@So], in which the action of the Cartan subalgebra is only expected to be locally finite and not necessarily semi-simple as in the classical definition;
- and the second one given by the full subcategory of the thick category $\mathcal{O}$ from the first definition with the additional requirement that the Cartan subalgebra of $\mathfrak{sl}_2$ acts diagonalizably.
The results of this paper fall into the following three categories:
- We describe the linkage between simple object in both our versions of category $\mathcal{O}$ and in this way explicitly determine all (indecomposable) blocks, see Theorem \[thm9\].
- We determine the Gabriel quivers for all blocks, see Corollaries \[cor16\], \[cor23\], \[cor25\], \[cor33\].
- We prove that thick category $\mathcal{O}$ is extension full in the category of all $\mathfrak{g}$-modules, see Theorem \[thm6\].
For some of the blocks, we also obtain not only a Gabriel quiver, but also a fairly explicit description of the whole block, see Theorem \[thm15\]. Some of the results are unexpected and look rather surprizing. For example, the trivial $\mathfrak{g}$-module exhibits behavior different from the behaviour of all other simple finite dimensional $\mathfrak{g}$-modules, compare Lemma \[lem27-1\] and Proposition \[prop27\].
The paper is organized as follows: All preliminaries are collected in Section \[s2\], in particular, in this section we define all main protagonists of the paper and describe their basic properties. In Section \[s3\] we prove extension fullness of thick category $\mathcal{O}$ in the category of all $\mathfrak{g}$-modules. Section \[s4\] is devoted to the study of the decomposition of both $\mathcal{O}$ and its thick version $\widetilde{\mathcal{O}}$ into indecomposable blocks. As usual, generic Verma modules over $\mathfrak{g}$ are simple. In Section \[s6\] we describe the structure of those Verma modules that are not simple. Finally, in Section \[s5\], we compute first extensions between simple highest weight modules and in this way determine the Gabriel quivers of all block in $\mathcal{O}$ and $\widetilde{\mathcal{O}}$.
This paper is a revision, correction and extension of the master thesis [@So] of the second author written under the supervision of the first author.
**Acknowledgements:** This research was partially supported by the Swedish Research Council and G[ö]{}ran Gustafsson Stiftelse.
Takiff $\mathfrak{sl}_2$ and its modules {#s2}
========================================
Takiff $\mathfrak{sl}_2$ {#s2.1}
------------------------
In this paper we work over the field $\mathbb{C}$ of complex numbers. Consider the Lie algebra $\mathfrak{sl}_2$ with the standard basis $\{e,h,f\}$ and the Lie bracket $$[e,f]=h,\qquad [h,e]=2e,\qquad [h,f]=-2f.$$ Let $D:=\mathbb{C}[x]/(x^2)$ be the algebra of dual numbers. Consider the associated [*Takiff Lie algebra*]{} $\mathfrak{g}=\mathfrak{sl}_2\otimes_{\mathbb{C}}D$ with the Lie bracket $$[a\otimes x^i,b\otimes x^j]=[a,b]\otimes x^{i+j},$$ where $a,b\in \mathfrak{sl}_2$ and $i,j\in\{0,1\}$ with the Lie bracket on the right hand side being the $\mathfrak{sl}_2$-Lie bracket. Set $$\overline{e}:=e\otimes x,\qquad
\overline{f}:=f\otimes x,\qquad
\overline{h}:=h\otimes x.$$
Consider the standard triangular decomposition $$\mathfrak{sl}_2=\mathfrak{n}_-\oplus\mathfrak{h}\oplus\mathfrak{n}_+,$$ where $\mathfrak{n}_-$ is generated by $f$, $\mathfrak{h}$ is generated by $h$ and $\mathfrak{n}_+$ is generated by $e$. Let $\overline{\mathfrak{n}}_-$ be the subalgebra of $\mathfrak{g}$ generated by $e$ and $\overline{e}$, $\overline{\mathfrak{h}}$ the subalgebra of $\mathfrak{g}$ generated by $h$ and $\overline{h}$, and $\overline{\mathfrak{n}}_+$ be the subalgebra of $\mathfrak{g}$ generated by $f$ and $\overline{f}$. The we have the following [*triangular decomposition*]{} of $\mathfrak{g}$: $$\mathfrak{g}=\overline{\mathfrak{n}}_-\oplus\overline{\mathfrak{h}}\oplus\overline{\mathfrak{n}}_+.$$ We set $\mathfrak{b}:=\mathfrak{h}\oplus\mathfrak{n}_+$ and $\overline{\mathfrak{b}}:=\overline{\mathfrak{h}}\oplus
\overline{\mathfrak{n}}_+$.
For a Lie algebra $\mathfrak{a}$, we denote by $U(\mathfrak{a})$ the corresponding universal enveloping algebra.
The natural projection $\mathfrak{g}{\twoheadrightarrow}\mathfrak{sl}_2$ induced an inclusion of $\mathfrak{sl}_2$-Mod to $\mathfrak{g}$-Mod, which we denote by $\iota$.
By a direct calculation, it is easy to check that the [*Casimir element*]{} $$\label{eq4}
\mathtt{c}:=h\overline{h}+2\overline{h}+2f\overline{e}+2\overline{f}e$$ belongs to the center of $U(\mathfrak{g})$, see [@Mo Example 1.2].
(Generalized) weight modules {#s2.2}
----------------------------
A $\mathfrak{g}$-module $M$ is called a [*generalized weight module*]{} provided that the action of $U(\overline{\mathfrak{h}})$ on $M$ is locally finite. As $U(\overline{\mathfrak{h}})$ is just the polynomial algebra in $h$ and $\overline{h}$, every generalized weight module $M$ admits a decomposition $$M=\bigoplus_{\lambda\in\overline{\mathfrak{h}}^*}M^{\lambda},$$ where $M^{\lambda}$ denotes the set of all vectors in $M$ killed by some power of the maximal ideal $\mathbf{m}_{\lambda}$ of $U(\overline{\mathfrak{h}})$ corresponding to $\lambda$. We will say that a generalized weight module $M$ is a [*weight module*]{} provided that the action of $h$ on $M$ is diagonalizable. We will say that a weight module $M$ is a [*strong weight module*]{} provided that the action of $\overline{\mathfrak{h}}$ on $M$ is diagonalizable.
Note that submodules, quotients and extensions of generalized weight modules are generalized weight modules. Also submodules and quotients of (strong) weight modules are (strong) weight modules. From the commutation relations in $\mathfrak{g}$, for any generalized weight modules $M$ and any $\lambda\in\overline{\mathfrak{h}}^*$, we have $$\label{eq1}
\overline{\mathfrak{h}}M^{\lambda}\subset M^{\lambda},\qquad
\overline{\mathfrak{n}}_+M^{\lambda}\subset M^{\lambda+\alpha},\qquad
\overline{\mathfrak{n}}_-M^{\lambda}\subset M^{\lambda-\alpha},$$ where $\alpha\in \overline{\mathfrak{h}}^*$ is given by $\alpha(h)=2$, $\alpha(\overline{h})=0$.
If $M$ is a generalized weight module, then the set of all $\lambda$ for which $M^{\lambda}\neq 0$ is called the [*support*]{} of $M$ and denoted $\mathrm{supp}(M)$.
Verma modules {#s2.3}
-------------
For a fixed $\lambda\in \overline{\mathfrak{h}}^*$, we have the corresponding simple $U(\overline{\mathfrak{h}})$-module $\mathbb{C}_{\lambda}:=U(\overline{\mathfrak{h}})/\mathbf{m}_{\lambda}$. Setting $\overline{\mathfrak{n}}_+\mathbb{C}_{\lambda}=0$ defines on $\mathbb{C}_{\lambda}$ the structure of a $\overline{\mathfrak{b}}$-module. The $\mathfrak{g}$-module $$\Delta(\lambda):=
\mathrm{Ind}^{U(\mathfrak{g})}_{U(\overline{\mathfrak{b}})}\,
\big(\mathbb{C}_{\lambda}\big)\cong
U(\mathfrak{g})\bigotimes_{U(\overline{\mathfrak{b}})}\mathbb{C}_{\lambda}$$ is called the [*Verma module*]{} associated to $\lambda$. The standard argument, see [@Di Proposition 7.1.11], shows that $\Delta(\lambda)$ has a unique simple quotient. We denote this simple quotient of $\Delta(\lambda)$ by $L(\lambda)$. From the PBW Theorem and formula , it follows that $$\label{eq5}
\mathrm{supp}(\Delta(\lambda))=\{\lambda-n\alpha\,:\,n\in\mathbb{Z}_{\geq 0}\}.$$ In fact, from the PBW Theorem, it follows that, for $n\in \mathbb{Z}_{\geq 0}$, we have $$\label{eq2}
\dim\left(\Delta(\lambda)^{\lambda-n\alpha}\right)=n+1$$ as the elements $\{f^i\overline{f}^{n-i}v_{\lambda}\,:\, i=0,1,\dots,n\}$, where $v_{\lambda}$ denotes the canonical generator of $\Delta(\lambda)$, form a basis of $\Delta(\lambda)^{\lambda-n\alpha}$. The weight $\lambda$ is the [*highest weight*]{} of $\Delta(\lambda)$.
The following simplicity criterion for $\Delta(\lambda)$ can be deduced from the main result of [@Wi], however, we include a short proof for the sake of completeness.
\[prop1\] The module $\Delta(\lambda)$ is simple if and only if $\lambda(\overline{h})\neq 0$.
Let $v_{\lambda}$ be the canonical generator of $\Delta(\lambda)$. Assume first that $\lambda(\overline{h})=0$ and consider the element $w=\overline{f}v_{\lambda}$. Then we have $ew=\overline{e}w=0$ and hence, from the PBW Theorem and , it follows that the submodule $N$ in $\Delta(\lambda)$ generated by $w$ satisfies $N^{\lambda}=0$ and thus is a non-zero proper submodule. Therefore $\Delta(\lambda)$ is reducible in this case.
Now assume that $\lambda(\overline{h})\neq 0$. We need to show that any non-zero submodule $N$ of $\Delta(\lambda)$ contains $v_{\lambda}$. If $N^{\lambda}\neq 0$, then the fact that $v_{\lambda}\in N$ is clear. Assume now that $N^{\lambda}=0$ and let $n\in\mathbb{Z}_{>0}$ be minimal such that $N^{\lambda-n\alpha}\neq 0$. Let $w\in N^{\lambda-n\alpha}$ be a non-zero element. Using the PBW Theorem, we may write $$w=\sum_{i=0}^nc_if^i\overline{f}^{n-i}v_{\lambda}.$$ Denote the maximal value of $i$ such that $c_{i}\neq 0$ by $k$. Then it is easy to check that $(\overline{h}-\lambda(\overline{h}))^kw$ equals $\overline{f}^{n}v_{\lambda}$ up to a non-zero scalar. In particular, $N$ contains $\overline{f}^{n}v_{\lambda}$. But then it is easy to check that $e\overline{f}^{n}v_{\lambda}$ equals $\overline{f}^{n-1}v_{\lambda}$, up to a non-zero scalar. In particular, $N^{\lambda-n\alpha+\alpha}\neq 0$. The obtained contradiction proves that $N^{\lambda}\neq 0$ and the claim of the proposition follows.
For $\mu\in\mathfrak{h}^*$, we denote by $\Delta^{\mathfrak{sl}_2}(\mu)$ the $\mathfrak{sl}_2$-Verma module with highest weight $\mu$ and by $L^{\mathfrak{sl}_2}(\mu)$ the unique simple quotient of $\Delta^{\mathfrak{sl}_2}(\mu)$. From Proposition \[prop1\], we obtain the following corollary.
\[cor2\] For $\lambda\in\overline{\mathfrak{h}}^*$, we have $$L(\lambda)\cong
\begin{cases}
\Delta(\lambda),& \text{ if }\lambda(\overline{h})\neq 0;\\
\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}})),
&\text{ if }\lambda(\overline{h})= 0.
\end{cases}$$
If $\lambda(\overline{h})\neq 0$, then the claim is just a part of Proposition \[prop1\]. If $\lambda(\overline{h})=0$, then the unique up to scalar non-zero vector in $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}}))^{\lambda}$ generates a $\overline{\mathfrak{b}}$-submodule of $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}}))$ isomorphic to $\mathbb{C}_{\lambda}$. By adjunction, we obtain a non-zero homomorphism from $\Delta(\lambda)$ to $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}}))$ which must be surjective as the latter module is simple. Consequently, $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}}))$ must be isomorphic to $L(\lambda)$ by the definition of $L(\lambda)$.
(Thick) category $\mathcal{O}$ {#s2.4}
------------------------------
We define [*thick category $\mathcal{O}$*]{}, denoted $\widetilde{\mathcal{O}}$, as the full subcategory of the category of all finitely generated $\mathfrak{g}$-modules consisting of all $\mathfrak{g}$-modules the action of $U(\overline{\mathfrak{b}})$ on which is locally finite. Note that, by definition, all modules in $\mathcal{O}$ are generalized weight modules.
We define [*classical category $\mathcal{O}$*]{}, denoted $\mathcal{O}$, as the full subcategory of $\widetilde{\mathcal{O}}$ consisting of all weight modules. Finally, we define [*strong category $\mathcal{O}$*]{}, denoted $\underline{\mathcal{O}}$, as the full subcategory of $\mathcal{O}$ consisting of all strong weight modules.
As $U(\mathfrak{g})$ is noetherian, the categories $\widetilde{\mathcal{O}}$, $\mathcal{O}$ and $\underline{\mathcal{O}}$ are abelian categories closed under taking submodules, quotients and finite direct sums. Directly from the definition, it also follows that $\widetilde{\mathcal{O}}$ is closed under taking extensions, in particular, $\widetilde{\mathcal{O}}$ is a Serre subcategory of $\mathfrak{g}$-mod.
\[prop4\] For each $\lambda\in\overline{\mathfrak{h}}^*$, the module $\Delta(\lambda)$ belongs to both $\widetilde{\mathcal{O}}$ and $\mathcal{O}$. However, $\Delta(\lambda)$ does not belong to $\underline{\mathcal{O}}$.
That $\Delta(\lambda)\in \widetilde{\mathcal{O}}$ follows from . That $\Delta(\lambda)\in {\mathcal{O}}$ follows by combining the fact that $\Delta(\lambda)\in \widetilde{\mathcal{O}}$ and that the adjoint action of $h$ on $U(\mathfrak{g})$ is diagonalizable (implying that the action of $h$ on $\Delta(\lambda)$ is diagonalizable).
That $\Delta(\lambda)\not\in \underline{\mathcal{O}}$ follows from the fact that the matrix of the action of $\overline{h}$ in the basis $\{fv_{\lambda},\overline{f}v_{\lambda}\}$ of $\Delta(\lambda)^{\lambda-\alpha}$, where $v_{\lambda}$ is the canonical generator of $\Delta(\lambda)$, has the form $$\left(
\begin{array}{cc}
\lambda(\overline{h})&0\\
-2&\lambda(\overline{h})
\end{array}
\right)$$ and hence is not diagonalizable.
\[prop3\]
1. \[prop3.1\] The set $\{L(\lambda)\,:\,
\lambda\in\overline{\mathfrak{h}}^*\}$ is a complete and irredundant list of representatives of isomorphism classes of simple objects in $\widetilde{\mathcal{O}}$.
2. \[prop3.2\] The set $\{L(\lambda)\,:\,
\lambda\in\overline{\mathfrak{h}}^*\}$ is a complete and irredundant list of representatives of isomorphism classes of simple objects in ${\mathcal{O}}$.
3. \[prop3.3\] The set $\{\iota(L^{\mathfrak{sl}_2}(\mu))\,:\,
\mu\in{\mathfrak{h}}^*\}$ is a complete and irredundant list of representatives of isomorphism classes of simple objects in $\underline{\mathcal{O}}$.
Let $L$ be a simple module in $\widetilde{\mathcal{O}}$ and $v$ a non-zero element in $L$. Since the vector space $U(\overline{\mathfrak{b}})v$ is finite dimensional, it contains a non-zero element $w$ such that $\overline{\mathfrak{n}}_+w=0$, $hw=\lambda(h)w$ and $\overline{h}w=\lambda(\overline{h})w$, for some $\lambda\in \overline{\mathfrak{h}}^*$. Then $\mathbb{C}w$, is isomorphic, as a $\overline{\mathfrak{b}}$-module, to $\mathbb{C}_{\lambda}$. By Proposition \[prop4\], we have $\Delta(\lambda)\in \widetilde{\mathcal{O}}$. By adjunction, we obtain a non-zero homomorphism from $\Delta(\lambda)$ to $L$. This implies $L\cong L(\lambda)$ and proves claim . Claim is proved similarly.
To prove claim , we can use claim and hence just need to check, for which $\lambda\in \overline{\mathfrak{h}}^*$, the module $L(\lambda)$ belongs to $\underline{\mathcal{O}}$. If $\lambda(\overline{h})\neq 0$, then $L(\lambda)=\Delta(\lambda)$ by Corollary \[cor2\] and hence $L(\lambda)\not\in \underline{\mathcal{O}}$ by Proposition \[prop4\]. If $\lambda(\overline{h})=0$, then $L(\lambda)=\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}})$ by Corollary \[cor2\] and $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}})\in
\underline{\mathcal{O}}$ since the action of $\overline{h}$ on $\iota(L^{\mathfrak{sl}_2}(\lambda\vert_{\mathfrak{h}})$ is zero and thus diagonalizable. This completes the proof.
Proposition \[prop3\] has the following consequence.
\[cor5\] The functor $\iota$ induces an equivalence between the category $\mathcal{O}$ for $\mathfrak{sl}_2$ and the category $\underline{\mathcal{O}}$.
By construction, the functor $\iota$ is full and faithful and maps the category $\mathcal{O}$ for $\mathfrak{sl}_2$ to the category $\underline{\mathcal{O}}$. Hence, what we need to prove is that this restriction of $\iota$ is dense. By Proposition \[prop3\], $\iota$ hits all isomorphism classes of simple objects in $\underline{\mathcal{O}}$. In particular, $\overline{h}$ annihilates all simple objects in $\underline{\mathcal{O}}$. Since, by the definition of $\underline{\mathcal{O}}$, the action of $\overline{h}$ on any object in $\underline{\mathcal{O}}$ is semi-simple, it follows that $\overline{h}$ annihilates all objects in $\underline{\mathcal{O}}$.
Since the ideal of $\mathfrak{g}$ generated by $\overline{h}$ contains both $\overline{e}$ and $\overline{f}$, it follows that the latter two elements annihilate all object in $\underline{\mathcal{O}}$. This yields that every object in $\underline{\mathcal{O}}$ is, in fact, isomorphic to an object in the image of $\iota$. The claim follows.
Due to Corollary \[cor5\], the category $\underline{\mathcal{O}}$ is fairly well-understood, see e.g. [@Ma] for a very detailed description. Therefore, in what follows, we focus on studying the categories $\widetilde{\mathcal{O}}$ and $\mathcal{O}$.
Extension fullness of $\widetilde{\mathcal{O}}$ in $\mathfrak{g}$-Mod {#s3}
=====================================================================
The inclusion functor $\Phi:\widetilde{\mathcal{O}}\hookrightarrow \mathfrak{g}$-Mod is exact and hence induces, for each $M,N\in \widetilde{\mathcal{O}}$ and $i\geq 0$, homomorphisms $$\varphi_{M,N}^{(i)}:\mathrm{Ext}^i_{\widetilde{\mathcal{O}}}(M,N)\to
\mathrm{Ext}^i_{\mathfrak{g}\text{-}\mathrm{Mod}}(M,N)$$ of abelian groups. As $\widetilde{\mathcal{O}}$ is a full subcategory of $\mathfrak{g}$-Mod, all $\varphi_{M,N}^{(0)}$ are isomorphisms. As $\widetilde{\mathcal{O}}$ is a Serre subcategory of $\mathfrak{g}$-Mod, all $\varphi_{M,N}^{(1)}$ are isomorphisms. The main result of this section is the following statement.
\[thm6\] The category $\widetilde{\mathcal{O}}$ is [*extension full*]{} in $\mathfrak{g}$-Mod in the sense that all $\varphi_{M,N}^{(i)}$ are isomorphisms.
Theorem \[thm6\] is a generalization of [@CM2 Theorem 16] to our setup. We refer the reader to [@CM1; @CM2] for more details on extension full subcategories.
We follow the proof of [@CM1 Theorem 2]. Denote by $\hat{\tilde{\mathcal{O}}}$ the full subcategory of $\mathfrak{g}$-Mod consisting of all modules, the action of $U(\overline{\mathfrak{b}})$ on which is locally finite. The difference between $\hat{\tilde{\mathcal{O}}}$ and $\tilde{\mathcal{O}}$ is that, in the case of $\hat{\tilde{\mathcal{O}}}$, we drop the condition on modules to be finitely generated.
First we note that $\tilde{\mathcal{O}}$ is extension full in $\hat{\tilde{\mathcal{O}}}$. Indeed, if $M\in \hat{\tilde{\mathcal{O}}}$, $N\in \tilde{\mathcal{O}}$ and $\alpha:M\to N$ is a surjective homomorphism, we can use that $N$ is finitely generated to claim that $N$ is in the image of a finitely generated submodule of $M$. Therefore the fact that $\tilde{\mathcal{O}}$ is extension full in $\hat{\tilde{\mathcal{O}}}$ follows from [@CM1 Proposition 3] (applied in the situation $\mathcal{B}=\tilde{\mathcal{O}}$ and $\mathcal{A}=\hat{\tilde{\mathcal{O}}}$).
To complete the proof of the theorem, it remains to prove that $\hat{\tilde{\mathcal{O}}}$ is extension full in $\mathfrak{g}$-Mod. For a locally finite dimensional $U(\overline{\mathfrak{b}})$-module $V$, denote by $M(V)$ the induced module $\mathrm{Ind}_{U(\overline{\mathfrak{b}})}^{U(\mathfrak{g})}(V)$. Note that, by [@BM Theorem 6], the action of $U(\overline{\mathfrak{b}})$ on $M(V)$ is locally finite. The same computation as in the proof of [@CM1 Lemma 3] shows that, for any $V$ as above, any $N\in \hat{\tilde{\mathcal{O}}}$ and any $i\geq 0$, the natural map $$\mathrm{Ext}^i_{\hat{\widetilde{\mathcal{O}}}}(M(V),N)\to
\mathrm{Ext}^i_{\mathfrak{g}\text{-}\mathrm{Mod}}(M(V),N)$$ is an isomorphism. Therefore the extension fullness of $\hat{\tilde{\mathcal{O}}}$ in $\mathfrak{g}$-Mod follows from [@CM1 Proposition 1] (applied in the situation $\mathcal{A}=\mathfrak{g}$-Mod, $\mathcal{B}=\hat{\tilde{\mathcal{O}}}$ and $\mathcal{B}_0$ consisting of modules of the form $M(V)$, for $V$ as above). This completes the proof.
Description of blocks {#s4}
=====================
Characters and composition multiplicities {#s4.0}
-----------------------------------------
\[lem11\] Let $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$ and $M\in \mathcal{X}$. There exist $k\in \mathbb{Z}_{>0}$ and $\lambda_1,\lambda_2,\dots,\lambda_k\in \overline{\mathfrak{h}}^*$ such that $$\label{eq7}
\mathrm{supp}(M)\subset\bigcup_{i=1}^k\{\lambda_i-\mathbb{Z}_{\geq 0}\alpha\},$$ moreover, for each $\mu\in \mathrm{supp}(M)$, the space $M^{\mu}$ is finite dimensional.
If two modules $M_1$ and $M_2$ have the properties described in the formulation of the lemma, then any extension of $M_1$ and $M_2$ also has similar properties. By definition, $M$ is finitely generated, and hence, taking the first sentence into account, without loss of generality we may assume that $M$ is generated by one element $v\in M^{\nu}$, for some $\nu\in \overline{\mathfrak{h}}^*$.
The vector space $U(\overline{\mathfrak{b}})v$ is finite dimensional and $\overline{\mathfrak{h}}$-stable. Hence the $\overline{\mathfrak{h}}$-module $U(\overline{\mathfrak{b}})v$ has finite support, say $\{\lambda_1,\lambda_2,\dots,\lambda_k\}$. By the PBW Theorem, we have the decomposition $U(\mathfrak{g})=
U(\overline{\mathfrak{n}}_-)U(\overline{\mathfrak{b}})$. Hence $M=U(\overline{\mathfrak{n}}_-)\big(U(\overline{\mathfrak{b}})v\big)$, implying Formula . Moreover, since, considered as an adjoint $\overline{\mathfrak{h}}$-module, all generalized weight spaces of $U(\overline{\mathfrak{n}}_-)$ are finite dimensional, it follows that all $M^{\mu}$ are finite dimensional. This completes the proof.
For a finite subset $\boldsymbol{\mu}\subset \overline{\mathfrak{h}}^*$, set $$\overline{\boldsymbol{\mu}}=\bigcup_{\mu\in \boldsymbol{\mu}}
\{\mu-\mathbb{Z}_{\geq 0}\alpha\}.$$ We write $\boldsymbol{\mu}\preceq \boldsymbol{\nu}$ provided that $\overline{\boldsymbol{\mu}}\subset \overline{\boldsymbol{\nu}}$.
Consider the set $\mathbf{F}$ of all functions $\chi:\overline{\mathfrak{h}}^*\to\mathbb{Z}_{\geq 0}$ having the property that the [*support*]{} $\{\lambda\in \overline{\mathfrak{h}}^*\,:\,\chi(\lambda)\neq 0\}$ of $\chi$ belongs to $\overline{\boldsymbol{\mu}}$, for some $\boldsymbol{\mu}$ as above. The set $\mathbf{F}$ has the natural structure of an additive monoid with respect to the pointwise addition of functions. The neutral element of this monoid is the zero function.
Let $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$. Given $M\in \mathcal{X}$, we define the [*character*]{} $\mathbf{ch}(M)$ as the function from $\overline{\mathfrak{h}}^*$ to $\mathbb{Z}_{\geq 0}$ sending $\lambda$ to $\dim(M^{\lambda})$. By Lemma \[lem11\], we have $\mathbf{ch}(M)\in \mathbf{F}$. Clearly, characters are additive on short exact sequences, that is, for any short exact sequence $0\to K\to M\to N\to O$ in $\mathcal{X}$, we have $\mathbf{ch}(M)=\mathbf{ch}(K)+\mathbf{ch}(N)$.
\[prop12\] Let $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$.
1. \[prop12.2\] For any $M\in\mathcal{X}$, there are uniquely determined $\mathbf{k}_{\lambda}(M)\in \mathbb{Z}_{\geq 0}$, where $\lambda\in \overline{\mathfrak{h}}^*$, such that $$\mathbf{ch}(M)=\sum_{\lambda\in\overline{\mathfrak{h}}^*}\mathbf{k}_{\lambda}(M)
\mathbf{ch}(L(\lambda)).$$
2. \[prop12.1\] For every $\lambda\in \overline{\mathfrak{h}}^*$, the function $\mathbf{k}_{\lambda}:\mathrm{Ob}(\mathcal{X})\to\mathbb{Z}_{\geq 0}$ has the following properties:
1. \[prop12.1.1\] $\mathbf{k}_{\lambda}(L(\lambda))=1$;
2. \[prop12.1.2\] $\mathbf{k}_{\lambda}(L(\mu))=0$, if $\mu\neq\lambda$;
3. \[prop12.1.3\] $\mathbf{k}_{\lambda}(M)=0$, if $\lambda\not\in\mathrm{supp}(M)$;
4. \[prop12.1.4\] $\mathbf{k}_{\lambda}(M)$ is additive on short exact sequences.
Clearly, $\mathbf{k}_{\lambda}(M)=0$ if $\lambda\not\in\mathrm{supp}(M)$, and hence the sum in can be taken over $\mathrm{supp}(M)$ instead of the whole $ \overline{\mathfrak{h}}^*$.
Assume first that, for $M\in\mathcal{X}$, we have $$\mathbf{ch}(M)=\sum_{\lambda\in\mathrm{supp}(M)}a_{\lambda}
\mathbf{ch}(L(\lambda))=
\sum_{\lambda\in\mathrm{supp}(M)}b_{\lambda}
\mathbf{ch}(L(\lambda)),$$ where all $a_{\lambda}$ and $b_{\lambda}$ are in $\mathbb{Z}_{\geq 0}$. Assume that there is some $\lambda$ such that $a_{\lambda}\neq b_{\lambda}$. Let $X:=\{\lambda\,:\, a_{\lambda}>b_{\lambda}\}$ and $Y:=\mathrm{supp}(M)\setminus X$. Then we have $$\chi:=\sum_{\lambda\in X}(a_{\lambda}-b_{\lambda})
\mathbf{ch}(L(\lambda))=
\sum_{\mu\in Y}(b_{\mu}-a_{\mu})
\mathbf{ch}(L(\mu)).$$ By our assumptions, $\chi\in\mathbf{F}$ is non-zero. Then there exists $\nu\in \overline{\mathfrak{h}}^*$ such that $\chi(\nu)\neq 0$ but $\chi(\nu+m\alpha)=0$, for all $m\in\mathbb{Z}_{>0}$. If $\nu\in X$, then $\nu\not\in Y$ and from the property $\chi(\nu+m\alpha)=0$, for all $m\in\mathbb{Z}_{>0}$, we see that $\chi(\nu)\neq 0$ is not possible if we compute $\chi$ using the second expression. Similarly, if $\nu\in Y$, then $\nu\not\in X$ and we get that $\chi(\nu)\neq 0$ is not possible if we compute $\chi$ using the first expression. The obtained contradiction shows that, if a decomposition of the form as in exists, then it is unique.
Let us now prove existence of . If $M=0$, we set $\mathbf{k}_{\lambda}(M)=0$, for all $\lambda$. Let $M\in\mathcal{X}$ be non-zero and $\lambda\in\mathrm{supp}(M)$ be such that $\lambda+m\alpha\not\in\mathrm{supp}(M)$, for all $m\in\mathbb{Z}_{>0}$. Let $v\in M^{\lambda}$ be a non-zero element which is an eigenvector for both $h$ and $\overline{h}$ and set $K:=U(\mathfrak{g})v\subset M$ and $N:=M/K$. By adjunction, there is a non-zero epimorphism from $\Delta(\lambda)$ to $K$ sending the canonical generator of $\Delta(\lambda)$ to $v$. Let $K'$ denote the image, under this epimorphism, of the unique maximal submodule of $\Delta(\lambda)$. By construction, we have two short exact sequences: $$0\to K'\to K\to L(\lambda)\to 0\qquad\text{ and }\qquad
0\to K\to M\to N\to 0.$$
For each $\mu\in\mathrm{supp}(M)$, define $$\label{eq8}
\mathbf{k}_{\mu}(M)=
\begin{cases}
\mathbf{k}_{\mu}(K')+\mathbf{k}_{\mu}(N),& \text{ if }\mu\neq\lambda;\\
\mathbf{k}_{\mu}(K')+\mathbf{k}_{\mu}(N)+1,& \text{ if }\mu=\lambda.
\end{cases}$$ Note that $\mathrm{supp}(N)\subset \mathrm{supp}(M)$ and $\dim(N^{\lambda})<\dim(M^{\lambda})$, moreover, we also have $\mathrm{supp}(K')\subset \mathrm{supp}(M)$ and $\lambda\not\in \mathrm{supp}(K')$. Therefore, thanks to Lemma \[lem11\], Formula gives an iterative procedure which, after a finite number of iterations, completely determines $\mathbf{k}_{\mu}(M)$ such that holds by construction.
It remains to check that $\mathbf{k}_{\mu}(M)$ defined above have all the properties listed in . Properties - follow directly from the definition in the previous paragraph. Property follows from the equality in and the fact that characters are additive on short exact sequences.
The number $\mathbf{k}_{\mu}(M)$ will be called the [*composition multiplicity*]{} of $L(\mu)$ in $M$.
Some first extensions between simple objects {#s4.1}
--------------------------------------------
\[prop7\] Let $\lambda,\mu\in\overline{\mathfrak{h}}^*$ be such that $\lambda \neq \mu$ and $\lambda(\overline{h})\neq 0$. Then, for any $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$, we have $$\mathrm{Ext}^1_{\mathcal{X}} (L(\lambda),L(\mu))=
\mathrm{Ext}^1_{\mathcal{X}} (L(\mu),L(\lambda))=0 .$$
We prove that $\mathrm{Ext}^1_{\mathcal{X}} (L(\lambda),L(\mu))=0$, the second claim is similar. Assume that $$\label{eq3}
0\to L(\mu)\to M\to L(\lambda)\to 0$$ is a short exact sequence in $\mathcal{X}$. Note that $$\mathrm{supp}(M)=
\mathrm{supp}(L(\lambda))\bigcup
\mathrm{supp}(L(\mu))\subset
\{\lambda-\mathbb{Z}_{\geq 0}\alpha\}\bigcup
\{\mu-\mathbb{Z}_{\geq 0}\alpha\}$$ by . If $\mu\not\in\lambda+\mathbb{Z}\alpha$, then $M^{\lambda+\alpha}=0$ and hence $\overline{\mathbf{n}}_+M^{\lambda}=0$. By adjunction, this gives as a non-zero homomorphism from $\Delta(\lambda)=L(\lambda)$ to $M$ which splits . This implies the necessary claim in case $\mu\not\in\lambda+\mathbb{Z}\alpha$.
If $\mu\in\lambda+\mathbb{Z}\alpha$, then $\mu(\overline{h})=\lambda(\overline{h})\neq 0$. By applying the Casimir element $\mathtt{c}$, see , to the highest weight elements in $L(\lambda)$ and $L(\mu)$, we see that $\mathtt{c}$ acts as the scalar $\lambda(\overline{h})(\lambda(h)+2)$ on $L(\lambda)$ and as the scalar $\mu(\overline{h})(\mu(h)+2)$ on $L(\mu)$. As $\mu(\overline{h})=\lambda(\overline{h})\neq 0$ but $\mu\neq\lambda$, we obtain $$\lambda(\overline{h})(\lambda(h)+2)\neq \mu(\overline{h})(\mu(h)+2).$$ This means that $L(\lambda)$ and $L(\mu)$ have different central characters and hence splits. The claim of the proposition follows.
Following the proof of Proposition \[prop7\], we also obtain the following claim.
\[cor8\] Let $\lambda,\mu\in\overline{\mathfrak{h}}^*$ be such that $\lambda\not\in\mu+\mathbb{Z}\alpha$. Then, for any $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$, we have $$\mathrm{Ext}^1_{\mathcal{X}} (L(\lambda),L(\mu))=
\mathrm{Ext}^1_{\mathcal{X}} (L(\mu),L(\lambda))=0 .$$
Easy blocks {#s4.2}
-----------
Let $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$. Set $$\overline{\mathfrak{h}}^*_1:=
\{\lambda\in \overline{\mathfrak{h}}^*\,:\, \lambda(\overline{h})\neq 0\},\qquad
\overline{\mathfrak{h}}^*_0:=\overline{\mathfrak{h}}^*\setminus
\overline{\mathfrak{h}}^*_1.$$ For $\lambda\in \overline{\mathfrak{h}}^*_1$, denote by $\mathcal{X}(\lambda)$ the full subcategory of $\mathcal{X}$ consisting of all modules with support $\{\lambda-\mathbb{Z}_{\geq 0}\alpha\}$.
Difficult blocks {#s4.3}
----------------
For $\xi\in \overline{\mathfrak{h}}^*_0/\mathbb{Z}\alpha$, denote by $\mathcal{X}(\xi)$ the full subcategory of $\mathcal{X}$ consisting of all modules whose support is contained in $\xi$.
Block decomposition {#s4.4}
-------------------
\[thm9\] For $\mathcal{X}\in\{\mathcal{O},\widetilde{\mathcal{O}}\}$, we have a decomposition $$\label{eq6}
\mathcal{X}=
\bigoplus_{\lambda\in \overline{\mathfrak{h}}^*_1}\mathcal{X}(\lambda)
\oplus
\bigoplus_{\xi\in \overline{\mathfrak{h}}^*_0/\mathbb{Z}\alpha}
\mathcal{X}(\xi)$$ of $\mathcal{X}$ into a direct sum of indecomposable abelian subcategories (blocks).
Let $M\in \mathcal{X}$ be an indecomposable module. Then there is $\lambda\in \overline{\mathfrak{h}}^*$ such that $\mathrm{supp}(M)\subset \xi:=\lambda+\mathbb{Z}\alpha$. If $\lambda(\overline{h})=0$, then, by definition, $M\subset \mathcal{X}(\xi)$. If $\lambda(\overline{h})\neq 0$, then, by Proposition \[prop7\], all composition subquotients of $M$ are isomorphic to some $L(\mu)$, where $\mu\in \xi$. Therefore $M\in \mathcal{X}(\mu)$. This implies existence of the direct sum decomposition as in .
It remains to prove that all summands in the right hand side of are indecomposable. That each $\mathcal{X}(\lambda)$, where $\lambda\in \overline{\mathfrak{h}}^*_1$, is indecomposable, is clear as $\mathcal{X}(\lambda)$ contains, by construction, only one simple module, up to isomorphism.
Let us argue that each $\mathcal{X}(\xi)$, where $\xi\in \overline{\mathfrak{h}}^*_0/\mathbb{Z}\alpha$, is indecomposable. For this it is enough to show that, for every $\lambda\in\xi$, there is an indecomposable module $M\in \mathcal{X}(\xi)$ such that both $\mathbf{k}_{\lambda}(M)$ and $\mathbf{k}_{\lambda-\alpha}(M)$ are non-zero. Take $M=\Delta(\lambda)$. The module $\Delta(\lambda)$ is indecomposable as it has simple top. Moreover, $\mathbf{k}_{\lambda}(\Delta(\lambda))\neq 0$. Since $\lambda(\overline{h})=0$, we have $$e\overline{f}v_{\lambda}=\overline{e}\overline{f}v_{\lambda}=0
\quad\text{ and }\quad
h\overline{f}v_{\lambda}=(\lambda-\alpha)(h)\overline{f}v_{\lambda}.$$ Therefore, by adjunction, mapping $v_{\lambda-\alpha}$ to $\overline{f}v_{\lambda}$, extends to a non-zero homomorphism from $\Delta(\lambda-\alpha)$ to $\Delta(\lambda)$, implying that $\mathbf{k}_{\lambda-\alpha}(\Delta(\lambda))\neq 0$. The claim follows.
Structure of non-simple Verma modules {#s6}
=====================================
Easy case {#s6.1}
---------
\[prop51\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}_{\geq 0}$. Then there is a short exact sequence $$0\to \Delta(\lambda-\alpha)\to\Delta(\lambda)\to L(\lambda)\to 0.$$
Let $v_{\lambda}$ be the canonical generator of $\Delta(\lambda)$. From $\lambda(\overline{h})=0$, it follows that $e\overline{f}v_{\lambda}=\overline{e}\overline{f}v_{\lambda}=
\overline{h}\overline{f}v_{\lambda}=0$ and $h\overline{f}v_{\lambda}=(\lambda-\alpha)(h)v_{\lambda}$. Hence, by adjunction, there is a non-zero homomorphism $\Delta(\lambda-\alpha)\to\Delta(\lambda)$ sending $v_{\lambda-\alpha}$ to $\overline{f}v_{\lambda}$. By the PBW Theorem, this homomorphism is injective and the quotient $\Delta(\lambda)/\Delta(\lambda-\alpha)$ has a basis of the form $\{f^{i}v_{\lambda}\,:\,i\in \mathbb{Z}_{\geq 0}\}$.
Up to a positive integer, $e^{i}f^{i}v_{\lambda}$ is a multiple of $v_{\lambda}$ with the coefficient $\displaystyle\prod_{j=0}^{i-1}(\lambda(h)-j)$. As $\lambda(h)\notin\mathbb{Z}_{\geq 0}$, we obtain that the quotient $\Delta(\lambda)/\Delta(\lambda-\alpha)$ is a simple module and hence is isomorphic to $L(\lambda)$. The claim follows.
\[cor52\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}_{\geq 0}$. Then there is a filtration $$\dots\subset \Delta(\lambda-2\alpha)\subset \Delta(\lambda-\alpha)\subset \Delta(\lambda).$$ Moreover, all subquotients in this filtration are simple and we have $\displaystyle \bigcap_{i\in\mathbb{Z}_{\geq 0}}\Delta(\lambda-i\alpha)=0$.
The filtration given by Corollary \[cor52\] is the unique composition series of $\Delta(\lambda)$, in other words, under the assumptions of Proposition \[prop51\], $\Delta(\lambda)$ is a uniserial module.
Existence of such filtration and the claim that all subquotients in this filtration are simple follows directly from Proposition \[prop51\]. The claim that $\displaystyle \bigcap_{i\in\mathbb{Z}_{\geq 0}}\Delta(\lambda-i\alpha)=0$ follows from the fact that $\displaystyle \bigcap_{i\in\mathbb{Z}_{\geq 0}}\mathrm{supp}(\Delta(\lambda-i\alpha))
=\varnothing$, which, in turn, is a consequence of .
Difficult case {#s6.2}
--------------
\[lem57\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$. Then there are short exact sequences $$\label{eq57-1}
0\to \Delta(\lambda-\alpha)\to\Delta(\lambda)\to M\to 0$$ and $$\label{eq57-2}
0\to L(\lambda-(n+1)\alpha)\to M\to L(\lambda)\to 0.$$
Similarly to Proposition \[prop51\], the vector $\overline{f}v_{\lambda}$ generates a submodule of $\Delta(\lambda)$ isomorphic to $\Delta(\lambda-\alpha)$, giving the exact sequence , with $M=\Delta(\lambda)/\Delta(\lambda-\alpha)$. The module $M$ is isomorphic to a Verma module for $\mathfrak{sl}_2$ and has simple subquotients as described in , see [@Ma Theorem 3.16].
\[lem58\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$. Then the element ${f}^{n+1}v_{\lambda}$ generates a submodule $K_n$ of $\Delta(\lambda)$ such that the module $M_n:=\Delta(\lambda)/K_n$ is uniserial and has a filtration $$\label{eq58-1}
0=X_k\subset\dots \subset X_1\subset X_{0}=M_n,$$ where $k=\lceil\frac{n+1}{2}\rceil$ and $X_i/X_{i+1}\cong L(\lambda-i\alpha)$, for $i=0,1,\dots,k-1$.
We prove this statement by induction on $n$. The induction step moves $\lambda$ to $\lambda+\alpha$ and hence changes $n$ to $n+2$. Therefore we have two different cases for the basis of the induction.
[**Case 1: $n=0$.**]{} In this case $efv_{\lambda}=\overline{e}fv_{\lambda}=0$ and $\overline{h}fv_{\lambda}=\overline{f}v_{\lambda}$. From Lemma \[lem57\] we thus get $M\cong L(\lambda)$.
[**Case 2: $n=1$.**]{} In this case $ef^2v_{\lambda}=0$ and $\overline{e}f^2v_{\lambda}=2\overline{f}v_{\lambda}$. Again, from Lemma \[lem57\], we thus get $M\cong L(\lambda)$.
For the induction step, we note that $ef^nv_{\lambda}=0$. Consider $\Delta(\lambda-\alpha)$ as a submodule of $\Delta(\lambda)$ generated by the element $\overline{f}v_{\lambda}$. Note that the lowest weight of the module $L(\lambda-\alpha)$ is $-\lambda+\alpha$. The weight of $f^nv_{\lambda}$ is $-\lambda-\alpha$. As $\overline{e}$ has weight $\alpha$, it follows that $\overline{e}f^nv_{\lambda}$ belongs the submodule $K_{n-2}$ of $\Delta(\lambda-\alpha)$. In fact, it is easy to compute that $\overline{e}f^nv_{\lambda}$ equals $f^{n-1}\overline{f}v_{\lambda}$, up to a non-zero scalar. Therefore, by induction, we have a short exact sequence $$0\to M_{n-2}\to M_n\to L(\lambda)\to 0.$$ As $L(\lambda)$ is a unique simple top of $\Delta(\lambda)$, the module $L(\lambda)$ also must be a unique simple top of $M_n$. Now all necessary claims follow by induction.
As an immediate consequence of the above, we obtain:
\[cor59\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$. The Hasse diagram of the partially ordered, by inclusion, set of submodules of $\Delta(\lambda)$ of the form $\Delta(\lambda-i\alpha)$ and $K_i$ is as follows (here $k=\lceil\frac{n-1}{2}\rceil$): $$\xymatrix{
&\Delta(\lambda)\ar@{-}[dl]\ar@{-}[dr]&&&\\
K_n\ar@{-}[dr]&&\Delta(\lambda-\alpha)\ar@{-}[dl]\ar@{-}[dr]&&\\
&K_{n-2}\ar@{-}[dr]&&\dots\ar@{-}[dr]&\\
&&\dots\ar@{-}[dr]&&\Delta(\lambda-k\alpha)\ar@{-}[dl]\\
&&&K_{n-2k}\ar@{-}[dr]&\\
&&&&\Delta(\lambda-(k+1)\alpha)\ar@{-}[d]\\
&&&&\Delta(\lambda-(k+2)\alpha)\ar@{-}[d]\\
&&&&\dots
}$$
Gabriel quivers for all blocks {#s5}
==============================
Easy blocks {#s5.1}
-----------
\[thm15\] For $\lambda\in\overline{\mathfrak{h}}^*_1$, we have:
1. \[thm15.1\] The block $\mathcal{O}(\lambda)$ is equivalent to the category of finite dimensional $\mathbb{C}[[x]]$-modules.
2. \[thm15.2\] The block $\widetilde{\mathcal{O}}(\lambda)$ is equivalent to the category of finite dimensional $\mathbb{C}[[x,y]]$-modules.
Set $x:=\overline{h}-\lambda(\overline{h})$. Then, for any $M\in \mathcal{O}(\lambda)$, the finite dimensional vector space $M^{\lambda}$ is naturally a $\mathbb{C}[[x]]$-module. Moreover, the functor $F$ sending $M$ to $M^{\lambda}$ and the parabolic induction functor $G$ are, by the usual hom-tensor adjunction, a pair of adjoint functors between $\mathcal{O}(\lambda)$ and the category of finite dimensional $\mathbb{C}[[x]]$-modules. From the definitions, it follows immediately that they are each others quasi inverses, proving claim .
Claim is proved similarly, with $x:=\overline{h}-\lambda(\overline{h})$ and $y:={h}-\lambda({h})$.
Recall that the [*Gabriel quiver*]{} of a block is a directed graph whose
- vertices are isomorphism classes of simple objects in the block;
- the number of arrows from a vertex $L$ to a vertex $S$ equals the dimension of $\mathrm{Ext}^1(L,S)$.
As an immediate consequence of Theorem \[thm15\], we obtain:
\[cor16\] For $\lambda\in\overline{\mathfrak{h}}^*_1$, we have:
1. \[cor15.1\] The Gabriel quiver of $\mathcal{O}(\lambda)$ is: $\xymatrix{\bullet\ar@(ur,dr){}}$
2. \[cor15.2\] The Gabriel quiver of $\widetilde{\mathcal{O}}(\lambda)$ is: $\xymatrix{\bullet\ar@(ur,dr){}\ar@(ul,dl){}}$
Partial simple preserving duality {#s5.15}
---------------------------------
Denote by $\sigma$ the anti-involution of $\mathfrak{g}$ swapping $e$ with $f$, and $\overline{e}$ with $\overline{f}$. Note that $\sigma(h)=h$.
Let $\mathcal{X}\in\{\widetilde{\mathcal{O}},\mathcal{O}\}$. Denote by $\mathcal{X}_{\mathrm{fl}}$ the full subcategory of $\mathcal{X}$ consisting of modules of finite length.
For $M\in \mathcal{X}_{\mathrm{fl}}$, we can define on $$M^{\star}:=\bigoplus_{\lambda\in \overline{\mathfrak{h}}^*}
\mathrm{Hom}_{\mathbb{C}}(M^{\lambda},\mathbb{C})$$ the structure of a $\mathfrak{g}$-module via $(a\cdot f)(m):=f(\sigma(a)m)$. Then $M\mapsto M^{\star}$ is a contravariant and involutive self-equivalence of $\mathcal{X}_{\mathrm{fl}}$. From $\sigma(h)=h$, it follows that $\mathbf{ch}(M)=\mathbf{ch}(M^{\star})$. In particular, as simple modules in $\mathcal{X}$ are uniquely determined by their characters, it follows that $L(\lambda)^{\star}\cong L(\lambda)$, for all $\lambda\in \overline{\mathfrak{h}}^*$. In other words, the [*duality*]{} $\star$ is simple preserving.
\[cor19\] For all $\mathcal{X}\in\{\widetilde{\mathcal{O}},\mathcal{O}\}$ and $\lambda,\mu\in \overline{\mathfrak{h}}^*$, we have $$\mathrm{Ext}_{\mathcal{X}}^1(L(\lambda),L(\mu))\cong
\mathrm{Ext}_{\mathcal{X}}^1(L(\mu),L(\lambda)).$$
The left hand side of the equality is obtained from the right hand side by applying the simple preserving duality $\star$.
We note that $\star$ does not extend to the whole of $\mathcal{X}$ as $\star$ messes up the property of being finitely generated. For example, for an infinite length Verma module $\Delta(\lambda)\in \mathcal{X}$ as in Subsection \[s6.1\], the module $\Delta(\lambda)^{\star}$ is not finitely generated and hence does not belong to $\mathcal{X}$.
Difficult non-integral blocks {#s5.2}
-----------------------------
\[prop22\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}$. Then, for $\mu\in\lambda+\mathbb{Z}\alpha$, we have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\mu))\cong
\begin{cases}
\mathbb{C}, & \text{if }\mu=\lambda;\\
\mathbb{C},& \text{if }\mu=\lambda\pm\alpha;\\
0,& \text{otherwise}.
\end{cases}$$
By Corollary \[cor19\], without loss of generality we may assume that $\mu=\lambda-k\alpha$, for some $k\in\mathbb{Z}_{\geq 0}$. Let $$\label{eq22-1}
0\to L(\mu)\to M\to L(\lambda)\to 0$$ be a short exact sequence in $\mathcal{O}$.
Assume first that $k>0$ and that does not split. In this case $M$ must be generated by $M^{ \lambda}$ and hence, by adjunction, is a quotient of the Verma module $\Delta(\lambda)$. Under the assumptions $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}$, all submodules of $\Delta(\lambda)$ are described in Corollary \[cor52\]. Out of all possible quotients of $\Delta(\lambda)$, only the quotient $\Delta(\lambda)/\Delta(\lambda-2\alpha)$ has length two. This quotient has composition subquotients $L(\lambda)$ and $L(\lambda-\alpha)$. This implies that $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda-\alpha))\cong\mathbb{C}
\qquad\text{ and }\qquad
\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda-k\alpha))=0,
\text{ for }k>1.$$
It remains to compute $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))$. Consider a non-split short exact sequence in $\mathcal{O}$, with $\lambda=\mu$. The vector space $M^{\lambda}$ is, naturally, a $U(\overline{\mathfrak{h}})$-module. If this module were semi-simple, by adjunction there would exist two linearly independent homomorphisms from $\Delta(\lambda)$ to $M$ and hence would be split. Therefore $M^{\lambda}$ must be an indecomposable $U(\overline{\mathfrak{h}})$-module. As $h$ is supposed to act diagonalizably, such module $M^{\lambda}$ is unique, up to isomorphism. In particular, there is a basis $\{v,w\}$ of $M^{\lambda}$ such that the matrix of the action of $\overline{h}$ in this basis is $$\left(\begin{array}{cc}0&0\\1&0\end{array}\right).$$ Consider now the module $\displaystyle
\Delta(M^{\lambda}):=U(\mathfrak{g})
\bigotimes_{U(\overline{\mathfrak{b}})}M^{\lambda}$, where $\overline{\mathfrak{n}}_+M^{\lambda}=0$. By adjunction, $\Delta(M^{\lambda})$ surjects onto $M$. Hence, we just need to check how many submodules $K$ of $\Delta(M^{\lambda})$ have the property that $\Delta(M^{\lambda})/K$ has length two with both composition subquotients isomorphic to $L(\lambda)$. We claim that such submodule is unique, which implies that $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))\cong\mathbb{C}$. In fact, since $\mathbf{k}_{\lambda}(\Delta(M^{\lambda}))=2$ by construction, the uniqueness of $K$, provided that $K$ exists, is clear.
To prove existence, we consider the submodule $K$ of $\Delta(M^{\lambda})$ generated by $\overline{f}w$ and $\lambda(h)\overline{f}v-fw$ (note that $\lambda(h)\neq 0$ by our assumptions). It is easy to check that both these vectors are annihilated by $e$ and $\overline{e}$. The vector $\overline{f}w$ generates a submodule of $\Delta(M^{\lambda})$ isomorphic to $\Delta(\lambda-\alpha)$. The image of $\lambda(h)\overline{f}v-fw$ in the quotient $\Delta(M^{\lambda})/\Delta(\lambda)$ generates in this quotient a submodule isomorphic to $\Delta(\lambda-\alpha)$. Therefore, from Proposition \[prop51\] it follows that $\Delta(M^{\lambda})/K$ indeed has length two with both simple subquotients isomorphic to $L(\lambda)$. The claim follows.
As an immediate corollary from Proposition \[prop22\], we have:
\[cor23\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}$. Then, for $\xi:=\lambda+\mathbb{Z}\alpha$, the Gabriel quiver of $\mathcal{O}(\xi)$ has the form:
$$\xymatrix{
\dots\ar@/^1pc/[rr]&&
\lambda-\alpha \ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\lambda \ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\lambda+\alpha \ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\dots\ar@/^1pc/[ll]
}$$
Now we can proceed to $\widetilde{\mathcal{O}}$.
\[prop24\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}$. Then, for $\mu\in\lambda+\mathbb{Z}\alpha$, we have $$\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\mu))\cong
\begin{cases}
\mathbb{C}^2, & \text{if }\mu=\lambda;\\
\mathbb{C},& \text{if }\mu=\lambda\pm\alpha;\\
0,& \text{otherwise}.
\end{cases}$$
The case $\mu\neq\lambda$ is proved by exactly the same arguments as in Proposition \[prop22\]. The case $\mu=\lambda$ is also similar, but requires some small adjustments which we describe below.
Consider a non-split short exact sequence in $\widetilde{\mathcal{O}}$, with $\lambda=\mu$. The vector space $M^{\lambda}$ is an indecomposable $U(\overline{\mathfrak{h}})$-module of length two, namely, a self-extension of the simple $U(\overline{\mathfrak{h}})$-module $\mathbb{C}_{\lambda}$ corresponding to $\lambda$. The space of such self-extensions is two-dimensional (as $\overline{\mathfrak{h}}$ is two-dimensional). In fact, using the arguments as in the proof of Proposition \[prop22\], we can show that parabolic induction, followed by taking a canonical quotient, defines a surjective map from $\mathrm{Ext}^1_{U(\overline{\mathfrak{h}})}
(\mathbb{C}_{\lambda},\mathbb{C}_{\lambda})$ to $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))$ which sends isomorphic module to isomorphic and non-isomorphic modules to non-isomorphic (the latter claim is obvious by restricting the action to the generalized $\lambda$-weight space). This, clearly, implies the necessary claim. Here are the details.
There is a basis $\{v,w\}$ of $M^{\lambda}$ such that the matrices of the action of $h$ and $\overline{h}$ in this basis are $$\left(\begin{array}{cc}\lambda(h)&0\\p&\lambda(h)\end{array}\right)
\qquad\text{ and }\qquad
\left(\begin{array}{cc}0&0\\q&0\end{array}\right),$$ respectively, where $p$ and $q$ are complex numbers at least one of which is non-zero. Similarly to the proof of Proposition \[prop22\], one shows that the submodule $K$ of $\Delta(M^{\lambda})$ generated by $\overline{f}w$ and $-\frac{\lambda(h)}{q}\overline{f}v-fw$, in case $q\neq 0$, or $\overline{f}v$, in case $q=0$, is the unique submodule of $\Delta(M^{\lambda})$ such that $\Delta(M^{\lambda})/K$ is isomorphic to $M$. The claim follows.
As an immediate corollary from Proposition \[prop24\], we have:
\[cor25\] Assume that $\lambda\in\overline{\mathfrak{h}}^*$ is such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}$. Then, for $\xi:=\lambda+\mathbb{Z}\alpha$, the Gabriel quiver of $\widetilde{\mathcal{O}}(\xi)$ has the form:
$$\xymatrix{
\dots\ar@/^1pc/[rr]&&
\lambda-\alpha \ar@(ul,ur)[]\ar@(dl,dr)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\lambda \ar@(ul,ur)[]\ar@(dl,dr)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\lambda+\alpha \ar@(ul,ur)[]\ar@(dl,dr)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]&&
\dots\ar@/^1pc/[ll]
}$$
Other self-extensions of simples {#s5.3}
--------------------------------
\[cor29\] Let $\lambda\in\overline{\mathfrak{h}}^*$ be such that $\lambda(\overline{h})=0$ and $\lambda(h)\notin\mathbb{Z}_{\geq 0}$. Then we have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))\cong\mathbb{C}
\qquad\text{ and }\qquad
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\lambda))
\cong\mathbb{C}^2.$$
The follows directly from the corresponding parts in the proofs of Proposition \[prop22\] and Proposition \[prop24\].
\[lem27-1\] We have $$\mathrm{Ext}_{\mathcal{O}}^1(L(0),L(0))=
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(0),L(0))=0.$$
The elements $e$, $f$, $\overline{e}$ and $\overline{f}$ must annihilate any self-extension $M$ of $L(0)$ since $M^{\pm\alpha}=0$, for such $M$. As $h=[e,f]$ and $\overline{h}=[\overline{e},f]$, it follows that both $h$ and $\overline{h}$ must annihilate $M$ as well. Therefore $M$ splits.
\[prop27\] Let $\lambda\in\overline{\mathfrak{h}}^*$ be such that $\lambda(\overline{h})=0$ and $\lambda(h)\in\mathbb{Z}_{>0}$. Then we have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))\cong
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\lambda))
\cong\mathbb{C}.$$
By Weyl’s complete reducibility theorem, $h$ acts diagonalizably on any finite dimensional $\mathfrak{g}$-module. Hence any self-extension of $L(\lambda)$ lives in $\mathcal{O}$. Therefore it is enough to prove that $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))\cong\mathbb{C}$.
Let $M$ be a self-extension of $L_{\lambda}$. Then $M^{\lambda}$ is an $\mathbb{C}[\overline{h}]$-module and, similarly to Proposition \[prop22\], $M$ is indecomposable if and only if $M^{\lambda}$ is. As $\mathbb{C}[\overline{h}]$ is a polynomial algebra in one variable, this implies that $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))$ is at most one dimensional. To prove that $\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda))$ is exactly one-dimensional, it is enough to construct one non-split self-extension of $L(\lambda)$, which we do below.
Let $n:=\lambda(h)\in\mathbb{Z}_{\geq 0}$. By [@Ma Exercise 1.24], $L(\lambda)$ has a basis $\{v_{0},v_1,\dots,v_n\}$ such that $$ev_{i}=iv_{i-1},\quad
fv_{i}=(n-i)v_{i+1},\quad
hv_{i}=(n-2i)v_i,\quad\text{ for }\,\, i=0,1,\dots,n.$$ Take another copy $\overline{L(\lambda)}$ of $L(\lambda)$ with basis $\{\overline{v}_{0},\overline{v}_1,\dots,\overline{v}_n\}$ and similarly defined action. Consider $M=L(\lambda)\oplus\overline{L(\lambda)}$ and define $$\overline{e}v_{i}=i\overline{v}_{i-1},\quad
\overline{f}v_{i}=(n-i)\overline{v}_{i+1},\quad
\overline{h}v_{i}=(n-2i)\overline{v}_i,\quad\text{ for }\,\, i=0,1,\dots,n,$$ and $\overline{e}\overline{L(\lambda)}=
\overline{f}\overline{L(\lambda)}=\overline{h}\overline{L(\lambda)}=0$. It is straightforward that this defines on $M$ the structure of a $\mathfrak{g}$-module. As the action of $\overline{h}$ on $v_0$ is non-zero (here the condition $n>0$ is crucial!), the module $M$ is a non-split self-extension of $L(\lambda)$. This completes the proof.
Difficult integral blocks {#s5.4}
-------------------------
\[prop32\] Let $\lambda\in\overline{\mathfrak{h}}^*$ be such that $\lambda(\overline{h})= 0$.
1. \[prop32.1\] We have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda-\alpha))\cong
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\lambda-\alpha))\cong
\mathbb{C}.$$
2. \[prop32.2\] If $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$, then we have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda-(n+1)\alpha))\cong
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\lambda-(n+1)\alpha))
\cong\mathbb{C}.$$
3. \[prop32.3\] If $\lambda(h)\neq n\in\mathbb{Z}_{>0}$, then we have $$\mathrm{Ext}_{\mathcal{O}}^1(L(\lambda),L(\lambda-(n+1)\alpha))=
\mathrm{Ext}_{\widetilde{\mathcal{O}}}^1(L(\lambda),L(\lambda-(n+1)\alpha))
=0.$$
We start with claim . Assume that $$0\to L(\lambda-\alpha)\to M\to L(\lambda)\to 0$$ is a non-split short exact sequence. Then, similarly to Proposition \[prop22\], $M$ must be a quotient of $\Delta(\lambda)$. If $\lambda(h)\notin\mathbb{Z}_{\geq 0}$, then from Corollary \[cor52\] it follows that $\Delta(\lambda)$ has a unique quotient with correct composition subquotients. If $\lambda(h)\in\mathbb{Z}_{\geq 0}$, then from Lemma \[lem58\] it follows that $\Delta(\lambda)$ has a unique quotient with correct composition subquotients. This completes the proof of claim
We proceed with claim . Assume that $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$ and $$0\to L(\lambda-(n+1)\alpha)\to M\to L(\lambda)\to 0$$ is a non-split short exact sequence. Then, from Lemma \[lem57\] it follows that $\Delta(\lambda)$ has a unique quotient with correct composition subquotients. This completes the proof of claim .
Proposition \[prop51\], Lemma \[lem57\] and Lemma \[lem58\] imply that the only socle components possible in length two quotients of $\Delta(\lambda)$ are $\Delta(\lambda-\alpha)$ and $\Delta(\lambda-(n+1)\alpha)$, and the latter one is only possible under the additional assumption that $\lambda(h)=n\in\mathbb{Z}_{\geq 0}$. This implies claim and completes the proof.
Combining Proposition \[prop32\], Corollary \[cor29\], Lemma \[lem27-1\], Corollary \[cor19\] and Proposition \[prop27\], we obtain:
\[cor33\]
1. \[cor33.1\] The Gabriel quiver of ${\mathcal{O}}(\mathbb{Z}\alpha)$ is:
$$\xymatrix{
0\ar@/^1pc/[rr]\ar@/^1pc/[d]&&
\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
2\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\dots\ar@/^1pc/[ll]\\
\text{-}2\alpha\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[u]&&
\text{-}4\alpha\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\text{-}6\alpha\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\dots\ar@/^1pc/[ll]\\
}$$
2. [cor33.2]{} The Gabriel quiver of $\widetilde{\mathcal{O}}(\mathbb{Z}\alpha)$ is:
$$\xymatrix{
0\ar@/^1pc/[rr]\ar@/^1pc/[d]&&
\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
2\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\dots\ar@/^1pc/[ll]\\
\text{-}2\alpha\ar@(d,dr)[]\ar@(d,dl)[]
\ar@/^1pc/[rr]\ar@/^1pc/[u]&&
\text{-}4\alpha\ar@(d,dr)[]\ar@(d,dl)[]
\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\text{-}6\alpha\ar@(d,dr)[]\ar@(d,dl)[]
\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\dots\ar@/^1pc/[ll]\\
}$$
3. \[cor33.3\] The Gabriel quiver of ${\mathcal{O}}(\frac{1}{2}\alpha+\mathbb{Z}\alpha)$ is:
$$\xymatrix{
&&
\frac{1}{2}\alpha\ar@/^1pc/[rr]\ar@/^1pc/[d]\ar@/_2pc/[dll]\ar@(ul,ur)[]&&
\frac{1}{2}\alpha\text{$+$}\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\frac{1}{2}\alpha\text{$+$}2\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\dots\ar@/^1pc/[ll]\\
\frac{1}{2}\alpha\text{-}\alpha\ar@/^1pc/[rr]\ar@(dr,dl)[]\ar@/^1pc/[rru]&&
\frac{1}{2}\alpha\text{-}2\alpha
\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[u]\ar@/^1pc/[ll]&&
\frac{1}{2}\alpha\text{-}4\alpha
\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\frac{1}{2}\alpha\text{-}6\alpha
\ar@(dr,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\dots\ar@/^1pc/[ll]\\
}$$
4. \[cor36\] The Gabriel quiver of $\widetilde{\mathcal{O}}(\frac{1}{2}\alpha+\mathbb{Z}\alpha)$ is:
$$\xymatrix{
&&
\frac{1}{2}\alpha\ar@/^1pc/[rr]\ar@/^1pc/[d]\ar@/_2pc/[dll]\ar@(ul,ur)[]&&
\frac{1}{2}\alpha\text{$+$}\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\frac{1}{2}\alpha\text{$+$}2\alpha\ar@(ul,ur)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[d]&&
\dots\ar@/^1pc/[ll]\\
\frac{1}{2}\alpha\text{-}\alpha\ar@/^1pc/[rr]
\ar@(d,dr)[]\ar@(d,dl)[]\ar@/^1pc/[rru]&&
\frac{1}{2}\alpha\text{-}2\alpha
\ar@(d,dr)[]\ar@(d,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[u]\ar@/^1pc/[ll]&&
\frac{1}{2}\alpha\text{-}4\alpha
\ar@(d,dr)[]\ar@(d,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\frac{1}{2}\alpha\text{-}6\alpha
\ar@(d,dr)[]\ar@(d,dl)[]\ar@/^1pc/[rr]\ar@/^1pc/[ll]\ar@/^1pc/[u]&&
\dots\ar@/^1pc/[ll]\\
}$$
[999999]{} P. Batra, V. Mazorchuk. Blocks and modules for Whittaker pairs. J. Pure Appl. Algebra [**215**]{} (2011), no. 7, 1552–1568. I. Bernstein, I. Gelʹfand, S. Gelʹfand. A certain category of $\mathfrak{g}$-modules. Funkcional. Anal. i Prilozhen. [**10**]{} (1976), no. 2, 1–8. K. Coulembier, V. Mazorchuk. Extension fullness of the categories of Gelfand-Zeitlin and Whittaker modules. SIGMA Symmetry Integrability Geom. Methods Appl. [**11**]{} (2015), Paper 016, 17 pp. K. Coulembier, V. Mazorchuk. Some homological properties of category $\mathcal{O}$. III. Adv. Math. [**283**]{} (2015), 204–231. J. Dixmier. Enveloping algebras. Revised reprint of the 1977 translation. Graduate Studies in Mathematics, [**11**]{}. American Mathematical Society, Providence, RI, 1996. xx+379 pp. B. Dubsky, R. L[ü]{}, V. Mazorchuk, K. Zhao. Category $\mathcal{O}$ for the Schr[ö]{}dinger algebra. Linear Algebra Appl. [**460**]{} (2014), 17–50. J. Humphreys. Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$. Graduate Studies in Mathematics, [**94**]{}. American Mathematical Society, Providence, RI, 2008. xvi+289 pp. V. Mazorchuk. Lectures on $\mathfrak{sl}_2(\mathbb{C})$-modules. Imperial College Press, London, 2010. x+263 pp. A. Molev. Casimir elements for certain polynomial current Lie algebras. In: “Group 21, Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras," Vol. [**1**]{}, (H.-D. Doebner, W. Scherer, P. Nattermann, Eds). World Scientific, Singapore, 1997, 172–176. R. Moody, A. Pianzola. Lie algebras with triangular decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xxii+685 pp. C. S[ö]{}derberg. Category $\mathcal{O}$ for Takiff $\mathfrak{sl}_2$. Master Thesis. Department of Mathematics, Uppsala University, 2019. W. Soergel. [É]{}quivalences de certaines cat[é]{}gories de $\mathfrak{g}$-modules. C. R. Acad. Sci. Paris S[é]{}r. I Math. [**303**]{} (1986), no. 15, 725–728. S. Takiff. Rings of invariant polynomials for a class of Lie algebras. Transactions of the American Mathematical Society [**160**]{} (1971), 249–262. B. Wilson. Highest-weight theory for truncated current Lie algebras. J. Algebra [**336**]{} (2011), 1–27.
V. M.: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala,\
SWEDEN, email: [mazormath.uu.se]{}
C. S.: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala,\
SWEDEN, email: [christoffer.soderbergmath.uu.se]{}
|
---
author:
- 'Amir Gholami[^1]'
- 'Shashank Subramanian$^*$'
- Varun Shenoy
- Naveen Himthani
- |
\
Xiangyu Yue
- Sicheng Zhao
- Peter Jin
- George Biros
- Kurt Keutzer
bibliography:
- 'ref.bib'
title: A Novel Domain Adaptation Framework for Medical Image Segmentation
---
\[s:abstract\]
Introduction {#s:intro}
============
Methods {#s:methods}
=======
Results {#s:results}
=======
Conclusion {#s:conclusion}
==========
[^1]: Authors contributed equally
|
---
abstract: 'We rewrite the Klein-Gordon (KG) equation in an arbitrary space-time transforming it into a generalized Schrödinger equation. Then, we take the weak field limit and show that this equation has certain differences with the traditional Schrödinger equation plus a gravitational field. Thus, this procedure shows that the Schrödinger equation derived in a covariant manner is different from the traditional one. We study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. This particle is immersed in a gravitational field with the new Schrödinger equation. We study particular physical systems given examples for which we find their energy levels, effective potential and the wave function of the systems. The results contain the gravitational effects due to the curvature of the space-time. Finally, we discuss the possibility of the experimental verification of these effects in a laboratory using non-inertial reference frames.'
author:
- 'Omar Gallegos[^1]'
- 'Tonatiuh Matos[^2]'
bibliography:
- 'ref.bib'
date: February 2019
title: Weak gravitational quantum effects in boson particles
---
Introduction
============
In the last century, General Relativity (GR) and Quantum Mechanics (QM), the two pillars of modern physics, have been developed and verified independently with great precision, while quantum physics describes successfully the behavior of tiny particles, GR is very accurate for forces at cosmic scales. However, in some cases, the two theories produce incompatible results which give rise to different definitions for the same concept. We think the inconsistency between GR and QM owes to the concept of interaction between particles. In QM two particles interact when they exchange a virtual particle, while in GR the interaction is just due to the geometry of space-time. In this work we adopt the geometrical GR concept, instead of the exchange of virtual particles to test if a boson gas follows the Klein- Gordon equation in a curved space-time, and to test GR in quantum regime. These results could be useful either for laboratory particles as well as for the study of the quantum character of boson particles proposed as dark matter (see for example [@Matos:1998vk], [@Hui:2016ltb]).
An important problem in theoretical and fundamental physics is to have a Theory of Everything where the principal theories in physics, GR and QM, can be compatible. In the last decades, some theories have been proposed [@Loop],[@String] to this unification. Nevertheless, the experimental verification of these candidates and their theoretical problems are so far too complex. Several physicists have tried to test if gravity has a quantum nature with different proposals for experiments and observations[@Casimir][@entaglement], but some proposals are not feasible with the technology available to this day. We do not pretend to propose a new Theory of Everything, instead we want to give a new different way to measure the gravitational effect due to the curvature of space-time on quantum systems, specially on scalar particles. These results do not definitely prove if gravity is or not a quantum interaction but they present a closer path to answer this fundamental question.
Einstein’s Equivalence Principle (EEP), one of the foremost ideas for developing General Relativity, where the concept of inertia takes a different role from the one used in Newtonian mechanics. EEP states that experiments in a sufficiently small freely falling laboratory, over a sufficiently short time, give results that are indistinguishable from those of the same experiments in an inertial frame in the presence of a gravitational field. Hence, we use this principle to develop this work, studying some different examples of QM on an inertial frame immersed in a gravitational field. To measure these effects in a laboratory, we will place a quantum system on a non-inertial frame, hoping to get the same results both in the theoretical part and in the experimental one. Furthermore, we expect to obtain results on quantization as in QM. So, one of the objectives of this paper is to test the EEP in quantum scales, thus a proof of GR in a quantum scale.\
We would expect that there exists a regime where the quantum aspects of gravity are detectable. With a simple dimensional analysis we may have an idea where the gravitational effects on a quantum system are important, i.e. the physical scale where these effects can be observed. It is possible to compare two well-known quantities in QM and GR, which are the Bohr radius $r_{bohr}$ and the Schwarzschild radius $r_{sch}$. Both radii are defined as $r_{bohr}=\hbar/(\mu c \alpha)$ and $r_{sch}=2GM/c^2$ where $c$, $G$, $\alpha$ are the speed of light, the gravitational constant and the fine-structure constant respectively. Also $\hbar=h/2\pi$ is the Planck’s constant, while $M$ is the mass that produces the gravitational field, $\mu$ is the reduced mass of a couple of particles. In a case where the comparison between masses is such that $M>>m$, it leads us to the approximation $\mu\approx m$, where $m$ is the mass of a boson particle, which is inside a gravitational field generated by a source of mass $M$. Thereby, the following expression is obtained when both radii $r_{bohr}\sim r_{sch}$ are comparable $$\label{dimensional analysis}
M\sim\dfrac{\hbar c}{2G\alpha m}=\dfrac{m_{pl}^2}{2\alpha m}\approx\dfrac{3.25\times10^{-14}}{m}\text{kg}^2,$$ where $m_{pl}=\sqrt{\hbar c/G}$ is the Planck’s mass. This result was inspired by the micro black holes described in[@microBH] and it enables us to find the limit for measuring quantum gravitational effects. If we assume $m=m_e$ is the mass of an electron, we obtain that $M\sim 3.57\times10^{16}$kg is the mass for which an electron should feel a gravitational effect. On the other hand, the mass of Earth is $\sim 5\times 10^{24}$kg, namely, we should measure in Earth the quantum gravitational effects on an electron. If we consider now the mass of an ultralight scalar particle $\sim 10^{-22}eV/c^2$, the mass proposed in models of Scalar Field Dark Matter (SFDM)[@Matos:1998vk], [@Hui:2016ltb], we calculate from Eq.(\[dimensional analysis\]) that $M\sim 10^{12}M_\odot$ (the galaxies have a mass of this order). Thus, we can say, if dark matter is a scalar field with mass $m\sim 10^{-22}eV/c^2$ [@Matos:1998vk], [@Hui:2016ltb], the galaxy should present a gravitational quantum behavior. In this work, we study well-known examples of QM for scalar particles with corrections due to the curvature of space-time.
Generalized Schrödinger Equation
================================
In order to analyze the gravitational effect due to curvature of space-time in a quantum system, we focus on a scalar field following reference [@Matos:2016uxo], in which the KG covariant equation with an external potential is described $$\label{KG covariant}
\Box\Phi-\dfrac{d\mathscr{V}}{d\Phi^*}=0,$$ here $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu=\dfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu)$ is the D’Alembertian operator associated to an arbitrary metric $g_{\mu\nu}$, $\Phi=\Phi(t,\vec{x})$ is the scalar field, $\Phi^*=\Phi^*(t,\vec{x})$ is its conjugated complex and the scalar field potential is $\mathscr{V}=\mathscr{V}(\Phi,\Phi^*)$ endowed with an external potential $V$ just as it is shown in the following equation $$\label{potential}
\mathscr{V}=\left(\dfrac{m^2c^2}{\hbar^2}+\dfrac{\lambda n_0}{2}+\dfrac{2m}{\hbar^2}V\right)\Phi\Phi^*,$$ where $m$ is the mass of the scalar field and $n_0=n_0(t,\vec{x})$ is defined as the scalar field density, namely $n_0=|\Phi|^2$. The space-time is expanded in a 3+1 slices, such that the coordinate $t$ here is the parameter of evolution, the 3+1 metric then reads $$\label{metric}
\mathrm{d}s^2 = - N^2 c^2
\mathrm{d}t^2 + \gamma_{ij} \left(\mathrm{d}x^i + N^i c \, \mathrm{d}t \right) \left(
\mathrm{d}x^j + N^j c \, \mathrm{d}t \right) ,$$ $N$ represents the lapse function which measures the proper time of the observers traveling along the world line, $N^i$ is the shift vector that measures the displacement of the observers between the spatial slices and $\gamma_{ij}$ is the spatial metric.
The KG equation is a covariant equation whose origin lies in quantum field theory. We can obtain a Schrödinger equation, starting from equation (\[KG covariant\]) with the potential in Eq.(\[potential\]), using the metric (\[metric\]) and transforming the scalar field by $\Phi(t,\vec{x})=\Psi(t,\vec{x}) e^{-i\omega t}$. Following [@Matos:2016uxo] we obtain $$\begin{aligned}
i \nabla^0 \Psi - \frac{1}{2\omega} \Box_G \Psi
+ \frac{1}{2\omega} \left( \widetilde{m}^2 + \lambda n_0+V \right) \Psi &&
\nonumber\\
+\quad \frac{1}{2} \left( - \frac{\omega}{N^2}
+ i \, \Box_G \, t \right) \Psi &=& 0 ,
\label{eq:GP}\end{aligned}$$ being $\lambda$ the coupling parameter. Here $\omega=mc^2/\hbar$ is the characteristic frequency of the scalar field. The D’Alembertian operator $\Box_G=\nabla^\mu\nabla_\mu$ is associated to the metric (\[metric\]), and $\widetilde{m}$ stands for the mass in units where $c=\hbar=1$, i.e, $\widetilde{m}=m^2c^2/\hbar^
2$ (see [@Matos:2016uxo]). We can interpret the function $\Psi$ as a wave function analogously as in QM.
Beginning with Eq.(\[KG covariant\]) but now using the $\Psi$ variable, we obtain Eq.(\[eq:GP\]) using the 3+1 metric (\[metric\]) and the potential (\[potential\]). Equation (\[eq:GP\]) can be interpreted as the covariant generalization of the Schrödinger equation for any curved space-time, (see[@Chavanis:2016shp] and [@Matos:2016uxo]) where this equation in the weak field limit reduces to the standard Schrödinger one. Hereafter, we use the Newtonian geometry given by the Newtonian metric $$\label{Newtonian metric}
ds^2=-\left(1-\dfrac{2GM}{rc^2}\right)c^2dt^2+\left(1+\dfrac{2GM}{rc^2}\right)dx_idx^i,$$ Newtonian gravity is known to be valid when the gravitational fields are weak, that is $GM/rc^2<<1$. Using the Newtonian metric in Eq.(\[eq:GP\]) yields $$\label{KG-N equation}
\dfrac{\hbar^2}{2m}\Box_N\Psi+\left(1+\dfrac{2U}{c^2}\right)^{-1}\left(i\hbar\dfrac{\partial \Psi}{\partial t}+\dfrac{mc^2}{2}\Psi\right)-\dfrac{\hbar^2}{2m}V\Psi=0,$$ where $U=-GM/r$ is the gravitational potential, hither the D’Alembertian operator $\Box_N$ is related to the metric given by Eq.(\[Newtonian metric\]). Furthermore, we do not consider self-interaction, here the contribution of the term $\lambda$ is negligible. We take into account that the evolution of this function is small, thus the contribution of $\partial_0^2\Phi$ can be taken as null because the Newtonian potential fulfills $U/c^2<<1$. Additionally, one can ignore terms of equal or greater order than $\left(\dfrac{2U}{c^2}\right)^2$. With all this in mind, the linearization of the generalized Schrödinger equation (\[KG-N equation\]) gives rise to the following equation $$\label{KG-N motion equation}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V\Psi+mU\Psi +\left(\dfrac{2U}{c^2}V-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right)\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ Henceforth, we are going to work with the Laplacian operator $\nabla^2=\nabla\cdot\nabla$ in flat space, specially for the examples in the following chapters where we use spherical symmetry. In other words, equation (8) comes from the KG equation in a curved space-time immersed in a weak gravitational field, which is described by a Newtonian geometry.
Focusing on the comparison of the traditional Schrödinger equation plus a gravitational potential, we note that now there are two additional terms inside the parenthesis in Eq.(\[KG-N motion equation\]). These terms were not simply added in the Schrödinger equation, they appear from the covariant meaning of the equation. From a qualitative point of view, this means that the QM version of the interactions between particles fulfills the Schrödinger equation, while the GR version of these interactions fulfills the generalized Schrödinger Eq.(\[KG-N motion equation\]). This difference is the main idea of this work. In what follows, we calculate the quantum quantities for different external potentials of well known problems in QM and compare them with those corresponding to GR. For the subsequent chapters our equation of motion will be given by Eq.(\[KG-N motion equation\]).
With the goal of testing EEP in a quantum region, we are going to show in the subsequent chapters distinct calculations about well known examples in QM. In them, the correction terms due to the curvature of space-time which were introduced in the KG covariant equation derived in Eq.(\[KG-N motion equation\]) are going to appear. To compare the cases with and without a gravitational source, the extra terms in Eq.(\[KG-N motion equation\]) will be taken as perturbations in the Schrödinger equation, for this reason we decide to opt for the QM formalism, using perturbation theory to give a clearer comparison between the cases with and without gravitational field. The results of the gravitational field contribution have been well studied in the standard literature.
Free Particle
=============
Firstly, we consider a free scalar particle with mass $m$ under the influence of a gravitational field generated by a source of mass $M$. Using an external potential $V_{free}=0$ in the equation of motion (\[KG-N motion equation\]), we have that $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t},$$ we recall that the gravitational potential is defined by $U=-GM/r$ and that an extra term $-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi$ appears in the Schrödinger equation, although the treatment on these equations will be in the QM formalism. Using perturbation theory, from the previous equation we can write a principal Hamiltonian operator $\hat{H}_0$ and a perturbed Hamiltonian operator $\hat{H}_p$ in the following way $$\begin{aligned}
\label{H0 free particle}
&\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp free particle}
&\hat{H}_p\Psi=-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=\dfrac{4U}{c^2}(E_n^{(0)}-mU)\Psi.\end{aligned}$$ In general, to find the corrections to eigenvalues of energy due to a perturbation in QM, the following expression is used $$\label{energy correction}
E_n=E_n^{(0)}+\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}+\sum_{m\neq n}\dfrac{|\bra{\psi^{(0)}_m}\hat{H}_p\ket{\psi^{(0)}_n}|^2}{E_n^{(0)}-E_m^{(0)}}+...,$$ we can associate the Eq.(\[energy correction\]) as a series of contributions of higher order correction of energies $E_n=E_n^{(0)}+E_n^{(1)}+E_n^{(2)}+...$, where $E_n^{(j)}$ is the $j$th-order correction of the eigenvalues in the $n$th-state of energy, hence we can say that the first-order correction is given by $E_n^{(1)}=\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}$. The zero-order correction for eigenvalues of energy $E_n^{(0)}$ and the eigenfunctions are the well-known exact solutions of QM (without gravitational field). Equation (\[energy correction\]) gives the correction for a non-degenerated quantum system, though first-order correction is valid for both cases (degenerate and non degenerate systems). We do not present an expression for the second order correction in a degenerate system because the corrections we make are at most of first-order. Nevertheless, such a case can be found in standard QM textbooks.\
Going back to our example of a free scalar particle inside a gravitational field, the first order correction of energy using the perturbed Hamiltonian from the hydrogen atom-like problem in QM is $$\begin{aligned}
E_n&\approx E^{(0)}_n\left(1-\dfrac{4GM}{\rho_0c^2n^2}+...\right),
\\&=E^{(0)}_n\left(1+\dfrac{8E^{(0)}_n}{mc^2}+...\right)\notag,\end{aligned}$$ being $E^{(0)}_n$ the well known energy for the hydrogen atom, that is $$\label{energy atom gravitation}
E^{(0)}_n=-\dfrac{(GM)^2m^3}{2\hbar^2n^2}=-\dfrac{GMm}{2\rho_0n^2},$$ where $\rho_0=\hbar^2/(GMm^2)$ is the Bohr radius for the gravitational case. Thus, we expect that the gravitational field modifies the energy of a free particle in a quadratic level, suppressed by the rest energy of the boson particle. This result is surprising for an ultra-light dark matter boson [@Matos:1998vk][@Matos:2000ss], because this model postulates a boson particle with a mass of the order of $10^{-22}$eV$/c^2$. With such a mass the self-gravitation effects of the boson field are important in a system of particles, a feature that a heavy boson system does not present.
Isotropic Harmonic Oscillator
=============================
Another important example, not only in QM but in physics as a whole, is the study of the harmonic oscillator potential. With this in mind, we analyze the case of potential $V_{osc}$ for an isotropic harmonic oscillator. This analysis can be done in two ways. The first one is to start from Eq.(\[KG-N motion equation\]) with the principal potential given by the gravitational type $U=-GM/r$ and taking the isotropic harmonic oscillator $V_{osc}=\dfrac{1}{2}m\omega_0^2r^2$ as perturbation. The other way is to regard the principal Hamiltonian like as that of an isotropic harmonic oscillator, while the perturbation is taken from the gravitational potential. Therefore, equation (\[KG-N motion equation\]) with the potential $V_{osc}$, transforms into $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +\dfrac{1}{2}m\omega_0^2r^2\Psi+\left[\dfrac{2U}{c^2}\left(\dfrac{1}{2}m\omega_0^2r^2\right)-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right]\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$
Harmonic Oscillator inside a Gravitational Field
------------------------------------------------
We start with the first cases exposed previously, we suppose that one has an isotropic harmonic oscillator immersed in a gravitational field, that means, $V_{osc}<<U$. To solve this problem we can take the principal Hamiltonian operator $\hat{H}_0$ with the gravitational part and the perturbed Hamiltonian $\hat{H}_p$ in the following way $$\begin{aligned}
\label{principal H IHO in curved space}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp IHO in curved space}
\hat{H}_p&=\dfrac{1}{2}m\omega_0^2r^2-\dfrac{GMm\omega_0^2r}{c^2}+\dfrac{4GME^{(0)}_n}{rc^2}.\end{aligned}$$ We are interested in the wave function with spherical symmetry $\Psi=\Psi(t,r,\theta,\phi)$. Thus, we apply the separation of variables method for $\Psi=R_{nl}(r)Y_{lj}(\theta,\phi)\exp(-\dfrac{iE_n}{\hbar}t)$, where $Y_{lj}$ are the spherical harmonics and $R_{nl}(r)$ is the radial function for the hydrogen atom problem. Here $n,l$, and $j$ play the role of quantum numbers as in QM. The $\hat{H}_0$ Hamiltonian contains the well-known solutions of the eigenvalues from the hydrogen atom problem in terms of the recurrence relation for powers of $r$. It can be shown that the correction for first order of energy from Eq.(\[energy correction\]) is given by $$\begin{aligned}
E_n=&E^{(0)}_n\left[1-\dfrac{8E^{(0)}_n}{mc^2}-\dfrac{\omega_0^2\rho_0^2}{c^2}(3n^2-l(l+1))\right]
\\ &+\dfrac{1}{4}m\omega_0^2\rho_0^2\left[n^2(5n^2+1-3l(l+1))\right],\notag\end{aligned}$$ where $E^{(0)}_n$ is given by Eq.(\[energy atom gravitation\]). Note that if $\omega_0=0$, we return to the case of a free particle on a gravitational field. Observe that as in the previous case, the modifications due to the gravitational field are of second order, but now, an additional term is added which is proportional to the mass and the frequency $\omega_0$. In this case, the second term of the contributions of the perturbations of the gravitational potential due to the harmonic oscillator are negligible for ultra-light masses. Nevertheless, if it is a massive boson then the quadratic contributions are not important. In any case, for any boson mass there is a contribution of the harmonic oscillator that must be taken into account.
Gravitational Field inside Harmonic Oscillator
----------------------------------------------
On the other hand, assuming the gravitational field as a perturbation in an isotropic harmonic oscillator with spherical symmetry, the principal Hamiltonian operator $\hat{H}_0$ is given by $$\begin{aligned}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2\left[\left(r-\dfrac{GM}{c^2}\right)^2-\left(\dfrac{GM}{c^2}\right)^2\right]\Psi,\notag
\\&\approx-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2r^2-\Psi-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\Psi.\end{aligned}$$ Since the quantum system is far from the source, it is possible to do the previous approximation. For the perturbed Hamiltonian $\hat{H}_p$ we have $$\hat{H}_p=-\dfrac{GMm}{r}+\dfrac{4GME^{(0)}_n}{rc^2}.$$ The principal Hamiltonian is that of an isotropic harmonic oscillator with spherical symmetry whose eigenfunctions are well known from QM. These solutions are obtained after applying the separation of variables method, just as in the previous case $\Psi_{nklj}(t,r,\theta,\phi)=R_{kl}(r)Y_{lj}(\theta,\phi)\exp(-iE_nt)$. Here $R_{nl}(r)$ is the radial function for an isotropic harmonic oscillator $R_{kl}(r)=r^le^{-\gamma r^2}L^{(l+1/2)}_k(2\gamma r^2)$, with $\gamma=\dfrac{m\omega}{2\hbar}$ and $L^{p}_q(x)$ being the generalized associated Laguerre polynomials. The set $(n,k,l,j)$ become the quantum numbers for the isotropic harmonic oscillator with spherical symmetry from QM.Thus, the first order correction for the energy from Eq.(\[energy correction\]) is given by $$E_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2-GMm\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\left\langle \dfrac{1}{r}\right\rangle,$$ where $E^{(0)}_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)$ is a well known result from QM. Also note that there is a degeneration of these numbers since $n=2k+l$. A more general solution of $\left\langle \dfrac{1}{r}\right\rangle$ was found in [@Amrmstrong], that for the case we are discussing reduces to the expression $$\begin{aligned}
\label{iho correction}
\left\langle \dfrac{1}{r}\right\rangle=&\sqrt{\dfrac{m\omega_0}{\hbar}}\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!\times
\\&\sum_t\dfrac{(-1)^t}{[1/2(n-l-1)-t]!\Gamma(1/2(2l+1)+t+1)}\binom{-1/2}{t}^2\notag,\end{aligned}$$ where $\Gamma(z)$ is the Gamma function $\Gamma(z)=\int^\infty_0t^{z-1}\exp(-t)dt$. When only the first term is taken from the sum of Eq.(\[iho correction\]), we obtain the energy value $$\begin{aligned}
\label{correction oscillator}
E_n=&\hbar\omega\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\notag\\
&-GMm\sqrt{\dfrac{m\omega_0}{\hbar}}\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!.\end{aligned}$$
Note that if $G=0$, we return to the solution for an isotropic harmonic oscillator with spherical symmetry without gravitational field. It is peculiar that the Schwarzschild radius $r_{sch}$ is obtained in a very natural way.
The two cases we have analyzed in this section are extreme, and correspond to those where the gravitational field is much more intense than the harmonic oscillator and vice-versa. The case where the two potentials are comparable is much more complex and we leave it for a future work.
Infinite Spherical Well {#Infinite Spherical Potential G}
=======================
In this section we continue discussing well known examples from QM with the novelty of the presence of a gravitational field. We now deal with the problem of an infinite spherical well barrier $V_{inf}(r)$ that has two regions, this problem is commonly used in experimental verification or applications of QM. The barrier potential is given by $$V_{inf}(r)= \left\{
\begin{array}{ll}
0\ \ \ \text{if} \ \ \ 0<r<a, \\
\infty \ \ \ \text{otherwise}. \\
\end{array}
\right.$$ We are only interested in studying the region where $0<r<a$, since outside of this region the wave function is zero, thus the probability to find a particle in the $r>a$ is null. The equation of motion (\[KG-N motion equation\]) in the region of interest is $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi -\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ We can choose a principal Hamiltonian operator $\hat{H}_0$ from the motion equation, where $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi=E^{(0)}\Psi.$$ The zero-order correction of the energy is then $E^{(0)}_{ln}=\dfrac{\hbar^2}{2m}\dfrac{q_{ln}^2}{a^2}$. We can define the perturbed Hamiltonian $\hat{H}_p$ from the KG equation $$\begin{aligned}
\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}\right).\end{aligned}$$ The first-order correction of the energy $E^{(1)}$ can be calculated using the zero-order correction of the eigenfunction $\Psi_{nlk}^{(0)}(r,\theta,\phi)=A_{ln}j_l\left(\dfrac{q_{ln}}{a}r\right)Y_{lj}(\theta,\phi)$. Here $q_{ln}$ is n-th root of the Bessel spherical functions $j_l(x)$ and $A_{ln}$ is the constant of normalization $$A_{ln}^2=\dfrac{2}{a^3[j_{l+1}(q_{ln})]^2}.$$ Therefore $$\label{first order infinity sph energy}
E_{ln}^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle,
%$$ where, from perturbation theory we have $$\left\langle \dfrac{1}{r}\right\rangle=A_{ln}^2\int_0^{a}\abs{j_l\left(\dfrac{q_{ln}}{a}r\right)}^2rdr.$$ Unfortunately, it is not possible to solve the last integral with analytic methods, thus we will integrate it numerically. In general, there exists for each state $n=2l+1$ degeneration, thus for $l=0$, we have that $j_0(x)=\sin(x)/x$, whose n-th root is $q_{0n}=n\pi$ and the normalization’s constant is $A_{01}=\sqrt{2/a}$. Hence $$\left\langle \dfrac{1}{r}\right\rangle\approx\dfrac{2a}{\pi^2}(1.218),$$ Now, for $l=1$ the first excited state is three-fold degenerate, which means we need the first three roots of $j_1(x)=\dfrac{\sin(x)}{x^2}-\dfrac{\cos(x)}{x}$. The roots are $q_{11}\approx 4.49340$, $q_{12}\approx 7.72525$ and $q_{13}\approx 10.90412$. Therefore for $n=1$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{11}^2\left( \dfrac{a}{q_{11}}\right)^2 (0.4124),$$ for $n=2$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{12}^2\left( \dfrac{a}{q_{12}}\right)^2 (0.4590),$$ and for $n=3$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{13}^2\left( \dfrac{a}{q_{13}}\right)^2 (0.4778).$$ We can continue this process for the next excited states, as in the previous cases. If we make $G=0$, we return to solutions for QM without gravitational field.
Spherical Potential Barrier
===========================
The problem of square well potential with certain symmetry (spherical, cartesian or cylindrical) is important for some experiments of quantum systems. In this section, we study a square well barrier with spherical symmetry. Similarly as in the previous sections, we analyze the case of QM for a square well potential with space-time curvature. We use the Newtonian metric from (\[Newtonian metric\]), so the equation of motion (\[KG-N motion equation\]) transforms into $$\label{spherical KG motion eq}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +V_B\left(1+\dfrac{2U}{c^2}\right)\Psi-\dfrac{4U}{c^2}\dfrac{\hbar^2}{2m}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}$$ where again $U=-GM/r$ is the gravitational potential. We take the Laplacian operator $\nabla^2$ in spherical coordinates and the potential $V_B$ as $$V_B(r)= \left\{
\begin{array}{ll}
-U_0\ \ \ \text{if} \ \ \ r<a, \\
0 \ \ \ \ \ \ \ \text{if} \ \ \ r>a,
\end{array}
\right.$$ Thus, from Eq.(\[spherical KG motion eq\]) we can identify a principal Hamiltonian operator $\hat{H}_0$ $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V_B\Psi=E^{(0)}\Psi.
\label{perturbed sph barrier}$$ The perturbed Hamiltonian $\hat{H}_p$ can be defined from Eq.(\[spherical KG motion eq\]) $$\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)$$ In general, we can find the fist-order correction of energy from Eq.(\[energy correction\]) with the perturbed Hamiltonian operator in Eq.(\[perturbed sph barrier\]). This yields $$E^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle.$$ The case where $E<0$ shows the quantum nature of the system due to the fact that the energy spectrum is discrete. Potential $V_B$ defines naturally two regions, $Region\ I$ ($r<a$) and $Region\ II$ ($r>a$). Both regions without gravitational field are well-known. For $Region\ I$ the first-order correction of energy $E^{(1)}_{I}$ is given by $$E^{(1)}_I=-GMm\left(1+\dfrac{4E_{I}^{(0)}}{mc^2}+\dfrac{2U_0}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_I,$$ where the zero-order correction of the eigenvalue is $E_{I}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma_{ln})^2$ and the zero-order correction eigenfunction reads $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lp}(\theta,\phi)$. As usual, $Y_{lp}(\theta,\phi)$ are the spherical harmonic functions and $R_{ln}(r)=A_{ln}j_l(\gamma_{ln}r)$ is the radial solution, such that $j_l(x)$ are the spherical Bessel functions. Also $k_1=\gamma_{ln}$ is the n-th solution of the transcendental equation due to the boundary and continuity condition given by $$\label{trascendental eq}
\dfrac{1}{h_l^{(1)}(ik_2r)}\dfrac{dh_l^{(1)}(ik_2r)}{dr}|_{r=a}=\dfrac{1}{j_l(ik_1r)}\dfrac{dj_l(ik_1r)}{dr}|_{r=a}.$$ For $l=0$, it reduces to the transcendental equation $$-k_2=k_1\cot(k_1a).$$ In the limit $|E|<<U_0$, we return to the solution for the first root. When $l=0$ we recover the same result as in the last section, namely $k_1a\approx \pi/2$. In general for Region I, we need to integrate $$\left\langle \dfrac{1}{r}\right\rangle_I=A_{ln}^2\int^a_0\abs{j_l\left(\dfrac{\sigma_{ln}}{a}r\right)}^2rdr,$$ where $\sigma_{ln}=a\gamma_{ln}$. If we define the parameter $\eta=\sigma_{ln}/q_{ln}$ that compares the n-th solution of the transcendental equation with the n-th root, we see that when $\eta=1$ we return to case of Section \[Infinite Spherical Potential G\]. However, the solution of this integral should be calculated using numerical methods.
In $Region\ II$, the zero-order correction is given by $$E_{II}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma^*_{ln})^2=\dfrac{\hbar^2}{2M}\left(\dfrac{\sigma^*_{ln}}{a}\right)^2,$$ where we have introduced the new parameters $\sigma^*_{ln}=a\gamma^*_{ln}$, and $\eta^*=\gamma^*_{ln}/q_{ln}$. The first-order correction is similar to that of $Region\ I$ $$E^{(1)}_{II}=-GMm\left(1+\dfrac{4E_{II}^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_{II},$$ with the eigenfunction $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lj}(\theta,\phi)$. Nevertheless, the radial function has a new form $R_{ln}(r)=Bh_l^{(1)}(i\gamma_{ln}^*r)=Bj_l(i\gamma_{ln}^*r)+i Bn_l(i\gamma_{ln}^*r)$, where $B$ is a constant of normalization, $h_l^{(1)}(x)$ are the spherical Hankel functions of the first kind, $n_l(x)$ are the spherical Neumann function and $k_2=\gamma_{ln}^*$ is the solution of the transcendental equation (\[trascendental eq\]). Therefore $$\begin{aligned}
\left\langle \dfrac{1}{r}\right\rangle_{II}=&B^2\int_a^\infty\abs{h_l^{(1)}\left(i\gamma^*_{ln}\right)}^2rdr\notag\\
=&B^2\int_a^\infty\left(\abs{j_l(i\gamma_{ln}^*r)}^2- \abs{n_l(i\gamma_{ln}^*r)}^2\right) rdr,\end{aligned}$$ This is a general solution of this perturbation, and if we want to solve this integral, we need to set every parameter for a specific case. As expected, just like in the previous cases, if $G=0$ we return to the solutions without a gravitational field.
Conclusions
===========
Throughout the course of this work we studied the KG equation in a weak gravitational field for different external potentials with the objective of understanding the quantum effects of a boson gas on a gravitational field. Typical examples of QM were analyzed featuring the addition of space-time curvature (or gravitational effects). Starting from the most general equation for bosons in Quantum Field Theory in curved space-times, we found a generalized Schrödinger equation that is simply the KG covariant one. To solve the differential equation, we identified the principal and perturbed Hamiltonian operator in each case, and compared them with the well-known results in QM. Each example was worked out on an inertial frame. This is an important aspect to highlight due to the main idea of this work. We expect to obtain the same results when gravitational effects are measured on a quantum system in the laboratory. This quantum system (with scalar particles) will be on a non-inertial frame, where we predict that the Einstein’s equivalence principle gives us a correspondence between experimental and theoretical results.\
These results let us find the limit, where we could measure quantum gravitational effects, namely $2\alpha\, mM/m_{pl}^2\sim 1$. For this analysis we could be able to apply our results in micro black holes scale[@microBH]. For example, if we consider a mass of a scalar particle $m$ as the mass of the electron, we obtain that the gravitational mass $M$ that affects the electron by quantum gravitational effects is $\sim 10^{16}$kg. In the same way, a $M\sim 10^{13}$kg could affect a particle with a proton mass. If we think on the easiest non-inertial frame, we could think on a rotating system, for which there would not be a real mass $M$, but rather an effective mass given by angular frequency $\Omega$, where $\Omega^2=GM/r^3$. Using a detector to $r=1$ meter, the corresponding gravity can be reached with a rotational wheel spinning with an angular frequency of $\Omega\approx 1543$ rad/s=245 rev/s for $M\sim 10^{16}$kg, and $\Omega\approx 25$ rad/s=3.9 rev/s for $M\sim 10^{13}$kg. However, for a mass $M\sim M_\odot$, we would need an angular frequency of $\Omega\sim 10^{9}$rev/s, to obtain relativistic effects.\
Analyzing the typical experiments on a harmonic oscillator [@oscillator] and on an infinite well barrier [@pozo] for a particle with an electron mass, the correction due to the gravitational effects has, in both cases, a dominant term which is given by $GMm\left\langle \dfrac{1}{r}\right\rangle$. For the case of a harmonic oscillator with frequency $\omega_0\sim G$Hz and using the value for the mass $M\sim 10^{16}$kg. The dominant term of the first-order correction of the energy in this case is $\sim 10^{-5}E_{osc}^{(0)}$ from Eq.(\[correction oscillator\]), while other contributions in this equation are $<10^{-9}E^{(0)}_{osc}$. On the other hand, for a particle with an electron mass confined in an infinite well barrier of width $a\sim 10$nm, it is possible to obtain that $E_{inf}^{(0)}\sim 10$eV. The dominant term in Eq.(\[first order infinity sph energy\]) is $\sim 10^{-5}E_{inf}^{(0)}$ for the first-order correction of energy, and the other term in this correction is $ <10^{-20}E_{inf}^{(0)}$. Note that in either cases, the first-order corrections of energy have the same order if we compare them with their zero-order counterparts. With this analysis it is possible to conclude that we can measure in a laboratory the effects of the quantization of a weak gravitational field directly or using non-inertial systems.
Acknowledgments
===============
This work was partially supported by CONACyT México under grants CB-2011 No. 166212, CB-2014-01 No. 240512, Project No. 269652 and Fronteras Project 281; Xiuhcoatl and Abacus clusters at Cinvestav, IPN; I0101/131/07 C-234/07 of the Instituto Avanzado de Cosmología (IAC) collaboration (http:// www.iac.edu.mx). O.G. acknowledge financial support from CONACyT doctoral fellowship. Works of T.M. are partially supported by Conacyt through the Fondo Sectorial de Investigación para la Educación, grant CB-2014-1, No. 240512
[^1]: ogallegos@fis.cinvestav.mx
[^2]: tmatos@fis.cinvestav.mx
|
---
abstract: 'This brief report studies the behavior of entropy in two recent models of cosmic evolution by J. A. S. Lima, S. Basilakos, and F. E. M. Costa \[Phys. Rev. D , 103534 (2012)\], and J. A. S. Lima, S. Basilakos, and J. Solá \[arXiv:1209.2802\]. Both start with an initial de Sitter expansion, go through the conventional radiation and matter dominated eras to be followed by a final and everlasting de Sitter expansion. In spite of their outward similarities (from the observational viewpoint they are arbitrary close to the conventional Lambda cold dark matter model), they deeply differ in the physics behind them. Our study reveals that in both cases the Universe approaches thermodynamic equilibrium in the last de Sitter era in the sense that the entropy of the apparent horizon plus that of matter and radiation inside it increases and is concave. Accordingly, they are consistent with thermodynamics. Cosmological models that do not approach equilibrium at the last phase of their evolution appear in conflict with the second law of thermodynamics.'
author:
- 'José Pedro Mimoso [^1]'
- 'Diego Pavón[^2]'
title: Entropy evolution of universes with initial and final de Sitter eras
---
Introduction
============
As daily experience teaches us, macroscopic systems tend spontaneously to thermodynamic equilibrium. This constitutes the empirical basis of the second law of thermodynamics. The latter succinctly formalizes this by establishing that the entropy, $S$, of isolated systems never decreases, $S' \geq 0$, and that it is concave, $S'' < 0$, at least in the last leg of approaching equilibrium (see, e.g., [@callen]). The prime means derivative with respect the relevant variable. In our view, there is no apparent reason why this should not be applied to cosmic expansion. Before going any further we wish to remark that sometimes the second law is found formulated by stating just the above condition on $S'$ but not on $S''$. While this incomplete version of the law works well for many practical purposes it is not sufficient in general. Otherwise, one would witness systems with an always increasing entropy but never achieving equilibrium, something at stark variance with experience. \
In this paper we shall explore whether two recently proposed cosmological models, that start from an initial de Sitter expansion and go through the conventional radiation and matter eras to finally enter a never-ending de Sitter phase, present the right thermodynamic evolution of above. This is to say, we will see whether $S' \geq 0$ at all times and $S'' < 0$ at the transition from the matter era to the final de Sitter one. By $S$ we mean the entropy of the apparent horizon, $S_{h} = k_{B}\,
{\cal{A}}/ (4\,\ell_{pl}^{2})$, plus the entropy of the radiation, $S_{\gamma}$, and/or pressureless matter, $S_{m}$, inside it. As usual, ${\cal A}$ and $ \ell_{pl}$ denote the area of the horizon and Planck’s length, respectively. \
We shall consider the evolution of the entropy, first in the model of Lima, Basilakos and Costa [@ademir2012] and then in the model of Lima, Basilakos and Solá [@jsola1] (models I and II, respectively). Both assume a spatially flat Friedmann-Robertson-Walker metric, avoid -by construction- the horizon problem and the initial singularity of the big bang cosmology, evolve between an initial and a final de Sitter expansions (the latter being everlasting), and from the observational viewpoint they are very close to the conventional Lambda cold dark matter model. \
In spite of these marked coincidences the physics behind the models is deeply different. While model I rests on the production of particles induced by the gravitational field (and dispenses altogether with dark energy), model II assumes dark energy in the form of a cosmological constant that in reality varies with the Hubble factor in a manner prescribed by quantum field theory. \
We shall focus on the transitions from the initial de Sitter expansion to the radiation dominated era, and from the matter era to the final de Sitter expansion. As is well known, in the radiation and matter eras, as well as in the transition from one to another, both $S'$ and $S''$ are positive-definite quantities [@grg_nd1]. \
As usual, a zero subscript attached to any quantity indicates that it is to be evaluated at the present time.
Thermodynamic analysis of model I
=================================
In model I the present state of cosmic acceleration is achieved not by dark energy or as a result of modified gravity, but simply by the gravitationally induced production of particles [@ademir2012]. In this scenario the initial phase is a de Sitter expansion which, due to the creation of massless particles becomes unstable whence the Universe enters the conventional radiation dominated era. At this point -as demanded by conformal invariance [@parker1]- the production of these particles ceases whereby the radiation becomes subdominant and the Universe enters a stage dominated by pressureless matter (baryons and cold dark matter). Lastly, the negative creation pressure associated to the production of matter particles accelerates the expansion and ushers the Universe in a never-ending de Sitter era. The model is consistent with the observational tests, including the growth rate of cosmic structures [@ademir2012].
From de Sitter to radiation dominated expansion
-----------------------------------------------
The production of massless particles in the first de Sitter era induces a negative creation pressure related to the phenomenological rate of particle production, $\Gamma_{r}$, given by $p_{c}= -(1+w) \rho \Gamma_{r}/(3H)$ -see [@ilya] for more general treatments of the subject. Here, $\rho$ is the energy density of the fluid (radiation in this case), $w$ its equation of state parameter, and $H = \dot{a}/a$ the Hubble expansion rate. As a consequence, the evolution of the latter is governed by $$\dot{H} \, + \, \frac{3}{2}\, (1+w) H^{2} \, \left(1 \, - \,
\frac{\Gamma_{r}}{3H}\right) = 0 , \label{H1}$$ cf. equation (7) in [@ademir2012].
Because the rate must strongly decline when the Universe enters the radiation dominated era, it may be modeled as $ \Gamma
_{r}/(3H) = H/H_{I}$ with $H \leq H_{I}$, being $H_{I}$ the initial de Sitter expansion rate. In consequence, for $w = 1/3$ (thermal radiation) last equation reduces to $$\dot{H} \, + \, 2 H^{2} \left(1 \, - \, \frac{H}{H_{I}} \right) =
0 \, ,
\label{H2}$$ whose solution in terms of the scale factor reads $$H(a) = \frac{H_{I}}{1\, + \, D a^{2}} \,
\label{H3}$$ with $D$ a positive-definite integration constant. \
The area of the apparent horizon ${\cal A} = 4\pi
\tilde{r}^{2}_{\cal A}$, where $ \tilde{r}_{{\cal A}} =
\frac{1}{\sqrt{H^{2} + ka^{-2}}}\,$ is the radius [@bak-rey], trivially reduces, in the case under consideration (a spatially flat universe), to the Hubble length, $H^{-1}$. Accordingly, the entropy of the apparent horizon, $S_{h} = k_{B} \pi /(\ell_{pl} \,
H)^{2}$, as the Universe transits from de Sitter, $H = H_{I}$, to a radiation dominated expansion is simply $$S_{h} = \pi k_{B} \, \frac{(1\, +\, D a^{2})^{2}}{(\ell_{pl} \,
H_{I})^{2}} \, . \label{shor2}$$ It is readily seen that $S_{h}$ is a growing, $S'_{h}
>0$, and convex, $S''_{h} > 0$, function of the scale factor (the prime stands for $d/da$).
In its turn, the evolution of the entropy of the radiation fluid inside the horizon can be determined with the help of Gibbs equation [@callen] $$T_{\gamma}\, dS_{\gamma} = d\left(\rho_{\gamma}\, \frac{4 \pi}{3}
\tilde{r}^{3}_{{\cal A}}\right) \, + \, p_{\gamma} \, d
\left(\frac{4 \pi}{3} \, \tilde{r}^{3}_{{\cal A}}\right) \, ,
\label{gibbs1}$$ where $$\rho_{\gamma} = \rho_{I} \left[1 \, + \, \lambda^{2}
\left(\frac{a}{a_{*}} \right)^{2} \right]^{-2} \,
,
\label{rhogamma}$$ $\rho_{I}$ is the critical energy density of the initial de Sitter phase, $\lambda^{2} = D a_{*}^{2}$, and $p_{\gamma} =
\rho_{\gamma}/3$. In its turn, $a_{*}$ denotes the scale factor at the transition from de Sitter to the beginning of the standard radiation epoch. \
Likewise, the dependence of the radiation temperature on the scale factor is given by $$T_{\gamma}= T_{I} \left[1 \, + \, \lambda^{2}
\left(\frac{a}{a_{*}} \right)^{2} \right]^{-1/2}
\label{Tgamma}$$ -cfr. Eq. (13) in [@ademir2012]. \
In consequence, $$T_{\gamma} \, S'_{\gamma} = \frac{4\pi}{3}\, \frac{\rho_{I}\,
D}{H_{I}^{3}}\, a > 0 \, . \label{gibbs2}$$ Accordingly, $S'_{h} \, + \, S'_{\gamma} \geq 0$, i.e., the total entropy -which encompasses the horizon entropy plus the entropy of the fluid in contact with it- does not decrease. In other words, the generalized second law (GSL), first formulated for black holes and their environment [@jakob] and later on extended to the case of cosmic horizons [@extended], is satisfied. \
Let us now discern the sign of $S''_{h} \, + \,
S''_{\gamma}$. As we have already seen, $S''_{h} > 0$. As for $S''_{\gamma}$, we insert $T_{\gamma} = T_{I} \, (1\, + \,
Da^{2})^{-1/2}$ into Eq.(\[gibbs2\]) and obtain $$S'_{\gamma} = \frac{4 \pi}{3} \, \frac{\rho_{I}\, D}{H^{3}_{I}\,
T_{I}}\,a (1 \, + \, D a^{2})^{1/2} \, .
\label{Sprimegamma}$$ Thus, $$S''_{\gamma} = \frac{4 \pi}{3} \, \frac{\rho_{I} D}{H^{3}_{I}\,
T_{I}} \, \left[\frac{1 \, + \, 2D\, a^{2}}{(1 \, + \, D \,
a^{2})^{1/2}}\right] > 0 \, .
\label{sgammapprime}$$ Therefore, $S''_{h} \, + \, S''_{\gamma} > 0$; that is to say, in the transition from the initial de Sitter expansion to radiation domination, the total entropy is a convex function of the scale factor. If it were concave, the Universe could have attained a state of thermodynamic equilibrium and would have not left it unless forced by some “external agent". The initial de Sitter expansion ($H = H_{I}$, and no particles) was a state of equilibrium, but only a metastable one for the Universe was obliged to leave it by the production of particles which acted as an external agent.
From matter domination to the final de Sitter expansion
-------------------------------------------------------
The Hubble function of spatially flat Lambda cold dark matter models obeys $$\dot{H} \, + \, \frac{3}{2} H^{2} \, \left[1\, - \,
\left(\frac{H_{\infty}}{H}\right)^{2} \right] = 0 \, ,
\label{dotH1}$$ where $H_{\infty} = \,$ constant denotes its asymptotic value at the far future. \
In a dust filled universe ($w = 0$) with production rate $\Gamma_{dm} \leq 3H$ of pressureless matter the Hubble factor obeys $$\dot{H} \, + \, \frac{3}{2}\, H^{2}\, \left(1 \, - \,
\frac{\Gamma_{dm}}{3H} \right) = 0 \, .\label{dotH2}$$ Comparison with the previous equation leads to $\Gamma_{dm}/(3H) = (H_{\infty}/H)^{2}\, $, i.e., $\Gamma_{dm}
\propto H^{-1}$. \
Thus, $$H^{2} = H_{0}^{2} \, [\tilde{\Omega}_{m} \, a^{-3} \, + \,
\tilde{\Omega}_{\Lambda}] \, ,
\label{Hsquare}$$ where $\tilde{\Omega}_{\Lambda} = (H_{\infty}/H_{0})^{2}
= 1 \, - \, \tilde{\Omega}_{m} = \,$ constant $ >0$. \
Recalling that $S_{h} = k_{B}\, {\cal{A}}/
(4\,\ell_{pl}^{2})$, it follows $S'_{h} = -2 \pi k_{B} \,
H'/(\ell_{pl}^{2} \, H^{3})$ with $$H' = - \frac{3}{2}\, H_{0} \, \frac{(1 \, - \,
\tilde{\Omega}_{\Lambda})\, a^{-4}}{[(1 \, - \,
\tilde{\Omega}_{\Lambda})\, a^{-3}\, + \, \tilde{\Omega}_{\Lambda}
]^{1/2}} \, .
\label{Hprime}$$ Then $$S'_{h} = 6 \frac{\pi k_{B}}{\ell_{pl}^{2}} \, \frac{(1 \, - \,
\tilde{\Omega}_{\Lambda})\, a^{-4}}{H_{0}^{2}\, [(1 \, - \,
\tilde{\Omega}_{\Lambda})\, a^{-3}\, + \, \tilde{\Omega}_{\Lambda}
]^{2}} > 0 \, . \label{shorprime2}$$ \
As for the entropy of dust matter, it suffices to realize that every single particle contributes to the entropy inside the horizon by a constant bit, say $k_{B}$. Then, $S_{m} =
k_{B} \frac{4 \pi}{3} \tilde{r}^{3}_{{\cal A}}\, n$, where the number density of dust particles obeys the conservation equation $
n' = (n/(aH)) [\Gamma_{dm} \, - \, 3H] < 0 $ with $\Gamma_{dm} =
3H_{0}^{2}\,\tilde{\Omega}_{\Lambda}/H > 0$. \
Thus, $$S'_{m} = \frac{4 \pi}{3} k_{B}\, \frac{n}{H^{4}}
\left[\frac{\Gamma_{dm}\, - \, 3H}{a} \, - \, 3 H'\right] \, .
\label{smprime}$$ Since $\Gamma_{dm} - 3H < 0 $ and $H' < 0$ the sign of $S'_{m}$ is undecided at this stage. To ascertain it consider the square parenthesis in (\[smprime\]) and multiply it by $aH/3$. One obtains $$\frac{aH}{3}\, \left[\frac{\Gamma_{dm}\, - \, 3H}{a} \, - \, 3
H'\right] = \textstyle{1\over{2}} H_{0}^{2}\, (1\, - \,
\tilde{\Omega}_{\Lambda})\, a^{-3} > 0. \label{oneobtains}$$ In consequence, $S'_{m} > 0$ and the GSL, $S'_{h} \, +
\, S'_{m} \geq 0$, is satisfied also in this case. \
Let us now consider the sign of $S''_{h} \, + \,
S''_{m}$ in the limit $a \rightarrow \infty$. From $S'_{h} = -
\frac{2 \pi k_{B}}{\ell_{pl}^{2}} \, (H'/H^{3})\, $ it follows, $$S''_{h} = - \frac{2 \pi k_{B}}{\ell_{pl}^{2}} \frac{1}{H^{4}}\, [H
H'' \, - \, 3H'^{2} ]. \label{Shorpprime2}$$ In virtue of (\[Hprime\]) we get $$H \, H'' = \frac{3}{2}H_{0}^{2}\, \left\{
\frac{4(1-\tilde{\Omega}_{\Lambda})[(1-\tilde{\Omega}_{\Lambda})a^{-3}+
\tilde{\Omega}_{\Lambda}]a^{-5} \, + \,
(3/2)(1-\tilde{\Omega}_{\Lambda})^{2} \,
a^{-8}}{(1-\tilde{\Omega}_{\Lambda})a^{-3} \, + \,
\tilde{\Omega}_{\Lambda}} \right\} \, , \label{hhpprime}$$ whence $$HH'' \, - \, 3H'^{2} = \frac{3}{2}\, H_{0}^{2} \,
\left\{\frac{4(1-\tilde{\Omega}_{\Lambda})[(1-\tilde{\Omega}_{\Lambda})a^{-3}+
\tilde{\Omega}_{\Lambda}]a^{-5}}{(1-\tilde{\Omega}_{\Lambda})a^{-3}
\, + \, \tilde{\Omega}_{\Lambda}}\right\} > 0 \, .
\label{hhpprime}$$ Thereby, in view of (\[Shorpprime2\]) we get $S''_{h}
<0$. \
As for the sign of $S''_{m}$ it suffices to recall, on the one hand, Eq. (\[smprime\]) and realize that $\Gamma_{dm}(a
\rightarrow \infty) =
3H_{0}^{2}\tilde{\Omega}_{\Lambda}/H_{\infty} = 3H_{\infty}$ and that $H'(a \rightarrow \infty) \rightarrow 0$; then, $S'_{m}(a
\rightarrow \infty) = 0$. And, on the other hand, that $S'_{m} (a
< \infty) > 0$. Taken together they imply that $S'_{m}$ tends to zero from below, i.e., that $S''_{m}(a \rightarrow \infty) <0$. \
Altogether, when $a \rightarrow \infty$ one has $S''_{h}\, + \, S''_{m} <0$, as expected. Put another way, in the phenomenological model of Lima [*et al.*]{} [@ademir2012] the Universe behaves as an ordinary macroscopic system [@grg_nd2]; i.e., it eventually tends to thermodynamic equilibrium, in this case characterized by a never-ending de Sitter expansion era with $H_{\infty} = H_{0}\, \sqrt{\tilde{\Omega}_{\Lambda}} < H_{0}$.
Thermodynamic analysis of model II
==================================
The model of Ref. [@jsola1] is based on the assumption that in quantum field theory in curved spacetime the cosmological constant is a parameter that runs with the Hubble rate in a specified manner [@parker2; @jsola2]. As in the previous model, the vacuum decays into radiation and nonrelativistic particles while the Universe expands from de Sitter to de Sitter through the intermediate eras of radiation and matter domination. However, as said above, the physics of both models differ drastically from one another. While model I dispenses altogether with dark energy, model II assumes dark energy in the form of cosmological term, $\Lambda$, that evolves with expansion. \
According to model II, the running of $\Lambda$ is given by the sum of even powers of the Hubble expansion rate, $$\Lambda(H) = c_{0} \, + \, 3 \nu H^{2} + \, 3 \alpha
\frac{H^{4}}{H_{I}^{2}} \, ,
\label{Lambda(H)1}$$ where $c_{0}$, $\alpha$ and $\nu$ are constants of the model. The absolute value of the latter is constrained by observation as $|\nu| \sim 10^{-3}$. At early times the last term dominates, and at late times ($H \ll H_{I}$) it becomes negligible whereby (\[Lambda(H)1\]) reduces to $$\Lambda(H) = \Lambda_{0} \, +\, 3 \nu (H^{2}\, - \, H_{0}^{2}) \,
\label{Lambda(H)2}$$ with $\Lambda_{0} = c_{0} \, + \, 3 \nu H_{0}^{2}$. \
At the early universe, integration of the field equations gives for the Hubble function, the energy density of radiation and of vacuum, the following expressions [@jsola1] $$H(a) = \sqrt{\frac{1-\nu}{\alpha}} \, \frac{H_{I}}{\sqrt{D\, a^{3
\beta}+1}} \qquad \; \; (\beta = (1-\nu)(1+w))\, ,
\label{H(a)running1}$$ $$\rho_{\gamma} = \rho_{I} \frac{(1-\nu)^{2}}{\alpha} \frac{D\, a^{3
\beta}}{[D\, a^{3 \beta}\, +\, 1]^{2}}\, , \; \; {\rm and} \; \;
\rho_{\Lambda} = \frac{\Lambda}{8 \pi G} = \rho_{I} \frac{1 -
\nu}{\alpha} \, \frac{\nu \, D\, a^{3 \beta}\, +\, 1}{[D\, a^{3
\beta}\, + \, 1]^{2}}\, ,\label{rhorunning1}$$ where $D (>0)$ is an integration constant. \
As in model I, at the transition from de Sitter to the radiation era, the first and second derivatives of $S_{h}$ and $S_{\gamma}$ (with respect to the scale factor) are all positive. Thus, at this transition the GSL is fulfilled but neither the radiation era nor the subsequent matter era correspond to equilibrium states since at them $S'' > 0$. \
By integration of the field equations at late times it is seen that the transition between the stages of matter domination to the second (and final) de Sitter expansion is characterized by $$H(a) = \frac{H_{0}}{\sqrt{1-\nu}}\, \sqrt{(1-\Omega_{\Lambda 0})\,
a^{-3(1-\nu)}\, + \, \Omega_{\Lambda 0}- \nu}\, ,
\label{H(a)running2}$$ $$\rho_{m} = \rho_{m 0}\, a^{-3(1-\nu)}\, \quad {\rm and} \quad
\rho_{\Lambda}(a) = \rho_{\Lambda 0} \, + \, \frac{\nu}{1-\nu}\,
\rho_{m0} \, \left[a^{-3 (1-\nu)}\, - \, 1 \right]\, ,
\label{rhorunning2}$$ where $\Omega_{\Lambda 0}= 8\pi G \rho_{\Lambda
0}(3H_{0}^{2})^{-1}$. From the pair of equations (\[rhorunning2\]) we learn that dust particles are created out of the vacuum at the rate $\Gamma_{dm} = \nu H$. \
As in the previous model, $S_{h}' >0$ and $S_{h}'' <0$. However at variance with it, the matter entropy, $S_{m} = k_{B}
\frac{4 \pi}{3} \tilde{r}^{3}_{{\cal A}}\, n \propto H^{-3}\, n$, decreases with expansion and is convex. This is so because, in this case, the rate of particle production, $\Gamma_{dm}$, goes down and cannot compensate for the rate of dilution caused by cosmic expansion. Nevertheless, as it can be easily checked, $S'_{h}$ and $S''_{h}$ dominate over $S'_{m}$ and $S''_{m}$, respectively, as $a \rightarrow \infty$. Thus, as in model I, the total entropy results a growing and concave function of the scale factor, at least at the far future stage. Hence, the Universe gets asymptotically closer and closer to thermodynamic equilibrium.
Discussion and concluding remarks
=================================
The second law of thermodynamics constrains the evolution of macroscopic systems; thus far, all attempts to disprove it by means of “counterexamples" have failed. While it seems reasonable to expect it to be obeyed also by the Universe as a whole, a proof of this on first principles is still lacking. However, persuasive arguments based on the Hubble history, suggesting that, indeed, the Universe behaves as any ordinary macroscopic thermodynamic system (i.e., that it tends to a maximum entropy state), were recently given in [@grg_nd2]. On the other hand, in view of the strong connection between gravitation and thermodynamics -see e.g., [@tedj; @padm]- it would be shocking that the Universe behaved otherwise. In this spirit we have considered models [@ademir2012] and [@jsola1], each of them covering the whole cosmic evolution (i.e., the two well-known eras of radiation and matter dominance sandwiched between an initial and a final de Sitter expansions), and consistent with recent observational data. In both models, the entropy, as a function of the scale factor, never decreases and is concave at least at the last stage of evolution, signaling that the Universe is finally approaching thermodynamic equilibrium. \
In principle, the initial de Sitter eras should be stable ($H$ and $S$ are constants when $t \rightarrow - \infty$) but owing to particle production, which can be viewed as “external" agent acting on the otherwise isolated system, an instability sets in. Once the Universe gets separated from thermodynamic equilibrium it reacts trying to restore it -as ordinary systems do-, only that at lower energy scale. This is finally achieved at the last de Sitter expansion. Given that two de Sitter expansions cannot directly follow one another an intermediate phase (comprised by the radiation and matter eras) is necessary in between.\
As is well known, irreversible particle production, as is the case in models I and II, implies generation of entropy (see e.g., [@ilya], [@ademir1996]); something rather natural because the new born particles necessarily increase the volume of the phase space. Our analysis takes this into account in an implicit and straightforward manner via the $\Gamma$ rates of particle production. These quantities modify the corresponding expressions for the Hubble factor and hence $S'$ and $S''$. For instance, setting $\Gamma_{r}$ to zero in Eq. (\[H1\]) (which would kill model I) leads to $D = 0$ and therefore to $S'_{\gamma}
= S''_{\gamma} = 0$ (Eqs. (\[Sprimegamma\]) and (\[sgammapprime\]), respectively). Likewise if, in the same model, one sets $\Gamma_{dm}$ to zero, then $S'_{m}$ (Eq. (\[smprime\])) decreases. Analogous statements can be made about model II if the parameter $\nu$ (that enters the corresponding $\Gamma$ rates) is forced to vanish (again, this would kill the model).\
When quantum corrections to Bekenstein-Hawking entropy law are taken into account, the entropy of black hole horizons generalizes to $ S_{h} = k_{B} \, \left[
\frac{{\cal{A}}}{4\,\ell_{pl}^{2}} \, - \, \frac{1}{2} \, \ln
\left(\frac{{\cal{A}}}{\ell_{pl}^{2}}\right)\right]$ plus higher order terms [@meissner; @ghosh]. Assuming this also applies to the cosmic apparent horizon, one may wonder up to what extent this may modify our findings. The answer is that the modifications are negligible whereby our results are robust against quantum corrections to the Bekenstein-Hawking entropy. We illustrate this point by noting that the expression for $S'_{h}$ of model I in the transition from the initial de Sitter regime to radiation domination presents now the overall multiplying factor $\left\{1\,
- \, \frac{\ell_{pl}^{2}\, H_{I}^{2}}{8\pi
(1+Da^{2})^{2}}\right\}$. In this expression the second term is negligible on account of the quantity $\ell_{pl}^{2}$ in the numerator. It is noteworthy that the imposing of the condition $S'_{h} >0$ leads to $\frac{\ell_{pl}^{2}\, H_{I}^{2}}{8 \pi
(1+Da^{2})^{2}} < 1$. When this inequality is evaluated in the limit $a \rightarrow 0$, the upper bound on the square of the initial Hubble factor, $H_{I}^{2} < 8 \pi/\ell_{pl}^{2}$ follows. Thus, the nice and convincing result, that the initial Hubble factor cannot be arbitrarily large (its square, not much larger than Planck’s curvature) arises straightforwardly from the quantum corrected Bekenstein-Hawking entropy law.\
Likewise, a study of $S'_{h}$ of model II in the transition from de initial Sitter expansion to radiation domination leads, in the same limit of very small $a$, to the upper bound $ H_{I}^{2} < 2\pi \alpha/[(1-\nu) \ell_{pl}^{2}]$, i.e., to essentially identical result on the maximum permissible value of $H_{I}$. It is remarkable that in spite of being models I and II so internally different, they share this bound.\
We conclude that models I and II show consistency with thermodynamics, and that their overall behavior (in particular, the reason why they evolve precisely to de Sitter in the long run) can be most easily understood from the thermodynamic perspective. Further, these results remain valid also if quantum corrections to Bekenstein-Hawking entropy law are incorporated. \
It would be interesting to explore the possible connection of the second law when applied to expanding universes with the “cosmic no-hair conjecture" [@gibbons]. Loosely speaking, the latter asserts that “all expanding-universe models with positive cosmological constant asymptotically approach the de Sitter solution" [@bob]. There is an ample body of literature on this -see e.g., [@jdbarrow] and [@cotsakis] and references therein. In the light of the above we may venture to speculate that the said conjecture and the tendency to thermodynamic equilibrium at late times are closely interrelated. Nevertheless, this is by no means the last word as the subject calls for further study. \
Before closing, note that the particle production does not vanish in the long run ($\Gamma_{dm}(a \rightarrow \infty) = 3
H_{\infty}$, and $\Gamma_{dm}(a \rightarrow \infty) =
\frac{\nu}{1-\nu}H_{0}\sqrt{\Omega_{\Lambda 0} - \nu}$ in models I and II, respectively). Then, the question arises as to whether it will be strong enough to bring instability on the second (an in principle, final) de Sitter expansion. Our tentative answer is in the negative; the reason being that in both cases the expansions tend to strictly de Sitter ($H = {\rm constant} >0$). However, a definitive response requires far more consideration and it lies beyond the scope of this work. At any rate, if instability sets in again, one should expect that the whole story repeats itself anew though at a much lower energy.
[99]{} H. Callen, [*Thermodynamics*]{} (John Wiley, New York, 1960). J.A.S. Lima, S. Basilakos, and F.E.M. Costa, Phys. Rev. D , 103534 (2012). J.A.S. Lima, S. Basilakos, and J. Solá, Mon. Not. R. Astron. Soc. (in the press), arXiv: 1209.2802. N. Radicella and D. Pavón, Gen. Relativ. Gravit. , 685 (2012). I. Prigogine, Gen. Relativ. Gravit. , 767 (1989); M.O. Calvão, J.A.S. Lima, and I. Waga, Phys. Lett. A , 223 (1992); J.A.S. Lima, A.S.M. Germano, Phys. Lett. A , 373 (1992); W. Zimdahl, D.J. Schwarz, A. B. Balakin, and D. Pavón, Phys. Rev. D , 063501 (2001). L.E. Parker, Phys. Rev. Lett. , 562 (1968). D. Bak and S.J. Rey, Classical Quantum Gravity , L83 (2000). J.D. Bekenstein, Phys. Rev. D , 2333 (1973); [*ibid*]{} , 3292 (1974). G. Gibbons and S.W. Hawking, Phys. Rev. D , 2738 (1977); P.C.W. Davies, Classical Quantum Gravity , L225 (1987); [*ibid*]{} , 1349 (1988); D. Pavón, Classical Quantum Gravit. , 487 (1990). N. Radicella and D. Pavón, Gen. Relativ. Gravit. , 63 (2013). L.E. Parker and D.J. Toms, [*Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity*]{} (Cambridge University Press, Cambridge, 2009). J. Solá, J. Phys. Conf. Ser. , 012033 (2011). T. Jacobson, Phys. Rev. Letters , 1260 (1995). T. Padmanbhan, Phys. Rep. , 49 (2005). J.A.S. Lima, Phys. Rev. D , 2571 (1996). K.A. Meissner, Class. Quantum Grav. , 5245 (2004). A. Ghosh and P. Mitra, Phys. Rev. D , 027502 (2005). G. Gibbons and S.W. Hawking, Phys. Rev. D , 2738 (1977). R.M. Wald, Phys. Rev. D , 2118 (1983). J.D. Barrow and G. Götz, Phys. Lett. B , 228 (1989). S. Cotsakis and G. Flessas, Phys. Lett. B , 69 (1993).
[^1]: E-mail: jpmimoso@cii.fc.ul.pt
[^2]: E-mail: diego.pavon@uab.es
|
---
abstract: 'Following Rosen’s quantization rules, two of the Authors (CC and FF) recently described the Schwarzschild black hole (BH) formed after the gravitational collapse of a pressureless “star of dust” in terms of a “gravitational hydrogen atom”. Here we generalize this approach to the gravitational collapse of a charged object, namely, to the geometry of a Reissner-Nordstrom BH (RNBH) and calculate the gravitational potential, the Schrödinger equation and the exact solutions of the energy levels of the gravitational collapse. By using the concept of *BH effective state*, previously introduced by one of us (CC), we describe the quantum gravitational potential, the mass spectrum and the energy spectrum for the extremal RNBH. The area spectrum derived from the mass spectrum finds agreement with a previous result by Bekenstein. The stability of these solutions, described with the Majorana approach to the Archaic Universe scenario, show the existence of oscillatory regimes or exponential damping for the evolution of a small perturbation from a stable state.'
author:
- 'C. Corda'
- 'F. Feleppa'
- 'F. Tamburini'
title: 'On the quantization of the extremal Reissner-Nordstrom black hole'
---
Introduction
============
Black holes are at all effects theoretical and conceptual laboratories where one discusses, tests and tries to understand the fundamental problems and potential contradictions that arise in the various attempts made to unify Einstein’s general theory of relativity with quantum mechanics. In a previous paper [@key-1], two of us suggested a new model of quantum BH based on a mathematical analogy to that of the hydrogen atom obtained by using the same quantization approach proposed in 1993 by the historical collaborator of Einstein, Nathan Rosen [@key-2]. In this view, BHs should behave as regular quantum systems with a discrete energy spectrum, finding similarities with Bekenstein’s results [@key-3]. The quantum properties of BHs are ruled by a Schrödinger’s equation with a wave function representation that could play an important role in the solution of the famous BH information paradox too [@key-4]. This approach, even if it presents some mathematical analogies that recall the behavior of the hydrogen atom, is different from that adopted in [@key-5], where the evolution of the BH microstates are described by a perturbative quantum field theory defined in a Schwarzschild metric background and a cut-off is required to replace the transverse coordinates by a lattice.
In this paper, we extend the results of [@key-1] to the class of extremal RNBHs, starting from the gravitational collapse of a set of dust particles with electrostatic charge. Also in this case, the study of the collapse of a simple set of dust particles can be a first ad valid approach to face the problem as it presents the advantage of finding and putting in evidence some fundamental properties (or analogies) that can be found also in other models characterized by, e.g., more articulated descriptions in terms of quantum fields that require a more complicated interpretation. In any case, this method, even if simplified, has a good validity as, up to now, there is not a unique and valid theory of quantum gravity that suggests a specific model to use.
In Rosen’s approach to the Schwarzschild solution, the ground state can be interpreted in terms of phenomenology at Planck scales, with the caveat that the Planck mass found as the ground state can be interpreted in terms of energy fluctuations, graviton exchange equivalent to wormhole connections avoiding the presence of a singularity in a flat background [@key-6]. This means that when the BH evaporates leaves a flat spacetime where quantum fluctuations occur, in agreement with the standard scenario.
Thanks to the simplicity of the model used here, within Rosen’s framework, we find the gravitational potential, the Schrödinger’s equation and the solution for the energy levels’ collapse that lead to a RNBH by focalizing our attention to the extremal case. Also here we find that this simple model obeys the rules of an hydrogen atom and the extremal RNBH mass spectrum is found as an exact solution. From the mass spectrum, the area spectrum is also obtained. What is really interesting is that, in any case, one finds that the energy levels, the evolution of the quantum microstates and the evolution of the perturbations of any energy level found obey a mathematical structure similar to that of the hydrogen atom as it was in the early days of quantum mechanics; this is essentially due to the spherical symmetry of the system and the solutions may vary from model to model depending on the mathematical shape of the function that describes the fields and acts as potential in the Schrödinger equation. The advantage of analyzing such an oversimplified model is that it can give some interesting hints to the understanding of the phenomenology at the Planck’s scales.
Application of Rosen’s quantization approach to the gravitational collapse of a charged BH
==========================================================================================
The gravitational collapse is one of the most important problems in general relativity. In particular, a complete description of the gravitational collapse of a charged perfect fluid in the class of Friedmann universe models can be found in [@key-7]. To describe the interior of the collapsing star one can use the well-known Friedmann-Lemaitre-Robertson-Walker (FLRW) line element that can be written, in Planck units ($G=c=k_{B}=\hbar=1/4\pi\epsilon_{0}=1$), as in [@key-8; @key-9]: $$ds^{2}=d\tau^{2}-a^{2}(t)[d\chi^{2}-\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta d\phi^{2})],$$ where $a(t)$ is the scale factor. Notice that, by setting the value for $\sin^{2}\chi$, one chooses the case of a spacetime with positive curvature, which corresponds to a gas sphere whose dynamics starts at rest at a given finite radius. On the other hand, the external geometry is described by the Reissner-Nordstrom line-element [@key-7] $$ds^{2}=Zdt^{2}-\frac{1}{Z}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})),$$ where $$Z(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.\label{eq:3}$$ In Eq. ($\ref{eq:3}$), the quantity $M$ represents the total mass of the collapsing star and the parameter $Q$ is its electrostatic charge. The internal homogeneity and isotropy of the FLRW line-element break up at the star’s surface that occurs at a certain given radius $\chi=\chi_{0}$. Thus, one has to consider a range of values for the parameter $\chi$ defined in the interval $0\le\chi\le\chi_{0}$, with $\chi_{0}<\frac{\pi}{2}$ during the collapse[@key-8], with the requirement that the interior FLRW geometry matches the exterior Reissner-Nordstrom geometry. Such a matching is given by setting [@key-7] $$r=a\sin\chi
\label{eq:4}$$ and $$M=\frac{a}{2}\sin\chi+\frac{Q^{2}}{2a\sin\chi}+\frac{a\dot{a}}{2}\sin\chi^{3}-\frac{a}{2}\sin\chi\cos\chi^{2}.\label{eq:5}$$
In the following we will apply the Rosen’s quantization approach [@key-2], who described a closed, homogeneous and isotropic universe, to that of a collapsing star leading to the formation of a RNBH, and discuss in detail the results.
By solving the Einstein-Maxwell field equations, one can find a system analogous to the Friedmann equations with the additional terms derived from the electromagnetic stress-energy tensor [@key-10]: $$\begin{aligned}
3\left(\frac{1+\dot{a}^{2}}{a^{2}}\right)&=&4\pi\frac{E_{0}^{2}}{a^{4}}+8\pi\rho,\label{eq:6}
\\
-2\frac{\ddot{a}}{a}-\left(\frac{1+\dot{a}^{2}}{a^{2}}\right)&=&-4\pi\frac{E_{0}^{2}}{a^{4}}+8\pi\rho,
\\
-2\frac{\ddot{a}}{a}-\left(\frac{1+\dot{a}^{2}}{a^{2}}\right)&=&4\pi\frac{E_{0}^{2}}{a^{4}},\label{eq:8}\end{aligned}$$ having set the cosmological constant term $\Lambda=0$ and $E_{0}^{2}\equiv\frac{Q}{\left(a\sin\chi\right)^{2}}$.
As it turns out, we can deal with these equations by using a similar procedure as in the standard case [@key-7]. Equation ($\ref{eq:6}$) becomes equivalent to $$\frac{a}{2}+\frac{a\dot{a}^{2}}{2}-\frac{2\pi}{3}\frac{E_{0}^{2}}{a}=\frac{4\pi}{3}\rho a^{3}.\label{eq:9}$$ After taking the time derivative of both sides and using Eq.(\[eq:8\]), we find that $$\rho=\frac{3M_{*}}{4\pi a^{3}}+\frac{E_{0}}{a^{4}},\label{eq:10}$$ where $M_{*}$ is an integration constant. In the absence of electromagnetic field (i.e. for $E_{0}=0$) we find, as in [@key-1], that $$\rho=\frac{3a_{0}}{8\pi a^{3}}.$$ Hence, we can deduce that the integration constant is $$M_{*}=\frac{a_{0}}{2}.$$ Finally, we obtain the expression of the density as $$\rho=\frac{3a_{0}}{8\pi a^{3}}+\frac{E_{0}}{a^{4}}.$$ Using Eq. ($\ref{eq:9}$) and Eq. ($\ref{eq:10}$), and multiplying by $M/2$, one obtains $$\frac{1}{2}M\dot{a}^{2}-\left(\frac{2M\pi E_{0}^{2}}{a^{2}}+\frac{Ma_{0}}{2a}\right)=-\frac{M}{2},$$ which can be interpreted as an energy equation for a particle in one-dimensional motion having coordinate $a$: $$E=T+V,$$ where the kinetic energy is $$T=\frac{1}{2}M\dot{a}^{2},$$ and the potential energy is $$V(a)=-\left(\frac{2M\pi E_{0}^{2}}{a^{2}}+\frac{Ma_{0}}{2a}\right).\label{eq:17}$$ Thus, the total energy is $$E=-\frac{M}{2}.\label{eq:18}$$ From the Friedmann-like equation already discussed one obtains the equation of motion of the particle with momentum given by $$P=M\dot{a},$$ and associated Hamiltonian $$\label{eq:20}
\mathcal{H}=\frac{P^{2}}{2M}+V.$$
Up to now, we faced the problem of the gravitational collapse from the classical point of view only. In order to discuss it also from the quantum point of view, we then need to define a wave-function $$\Psi\equiv\Psi(a,\tau).\label{eq: 21}$$ Thus, in correspondence of the classical equation (\[eq:20\]), one gets the traditional Schrödinger equation $$i\frac{\partial\Psi}{\partial t}=-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+V\Psi.\label{eq:22}$$ For a stationary state with energy $E$ one obtains $$\Psi=\Psi(a)e^{-iE\tau},$$ and equation ($\ref{eq:22}$) becomes $$-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+V\Psi=E\Psi.\label{eq:24}$$ Besides, by considering Eq. ($\ref{eq:17}$), equation ($\ref{eq:24}$) can be written as $$-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+\left(-\frac{2\pi ME_{0}^{2}}{a^{2}}-\frac{Ma_{0}}{2a}\right)\Psi=E\Psi.\label{eq:25}$$ Now, by setting $$A=-2\pi ME_{0}^{2},\hspace{1cm}B=\frac{Ma_{0}}{2},$$ we can write $$-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial a^{2}}+\left(\frac{A}{a^{2}}-\frac{B}{a}\right)\Psi=E\Psi.$$ We note that this potential has been deeply studied in [@key-11], where the authors found the energy spectrum that in our case is $$\begin{aligned}
E_{n}&=&-\frac{2MB^{2}}{\left(2n-1+\sqrt{1+8A}\right)^{2}} \nonumber
\\
&=& -\frac{M^{3}a_{0}^{2}}{2\left(2n-1+\sqrt{1-16\pi ME_{0}^{2}}\right)^{2}},\label{eq:28}\end{aligned}$$ where $n$ is the principal quantum number. It should be noted that, when $E_{0}=0$, we recover the result which has been found in [@key-1]: $$E_{n}=-\frac{M^{3}a_{0}^{2}}{8n^{2}}.\label{eq:29}$$ Following Rosen’s approach, then one inserts Eq. ($\ref{eq:18}$) into Eq. ($\ref{eq:28}$), obtaining $$-\frac{2M^{3}a_{0}^{2}}{4(2n-1+\sqrt{1-M16\pi E_{0}^{2}})^{2}}=-\frac{M}{2}.\label{eq: 30}$$ The last equation admits only two acceptable solutions, namely $$\begin{aligned}
M_{1n}=&-\frac{\sqrt{4a_{0}^{2}-8a_{0}n\beta+4a_{0}\beta+\beta^{2}}}{2a_{0}^{2}}\nonumber \\
&+\frac{4a_{0}n-2a_{0}-\beta}{2a_{0}^{2}},\label{eq: 31}\end{aligned}$$ and $$\begin{aligned}
M_{2n}=&\frac{\sqrt{4a_{0}^{2}-8a_{0}n\beta+4a_{0}\beta+\beta^{2}}}{2a_{0}^{2}}\nonumber \\
&+\frac{4a_{0}n-2a_{0}-\beta}{2a_{0}^{2}},\label{eq:32}\end{aligned}$$ where $\beta=16\pi E_{0}^{2}$. If $E_{0}=0$ (i.e. $\beta=0$), then one obtains $$M_{1n}=\frac{4a_{0}(n-1)}{2a_{0}^{2}}=\frac{2(n-1)}{a_{0}}, \label{eq:33}$$ and $$M_{2n}=\frac{4a_{0}n}{2a_{0}^{2}}=\frac{2n}{a_{0}}.\label{eq:34}$$ The second solution, $M_{2n}$, recovers the previous result presented in [@key-1], where the ground state of the collapsing star is found by setting $n=1$. Now, from Eq. ($\ref{eq:33}$), for $n=1$, we obtain a mass spectrum equal to zero, that is unacceptable from a physical point of view. Therefore, one has to consider as solution only the value $M_{2n}$ that we will write from now as $M_{n}$ for the sake of simplicity. On the other hand, by using Eq. ($\ref{eq:18}$), one finds the energy levels of the collapsing star as $$\begin{aligned}
E_{n}=&-\frac{\sqrt{a_{0}^{2}-2a_{0}n\beta+a_{0}\beta+(\beta/2)^{2}}}{2a_{0}^{2}}\nonumber \\
&+\frac{2a_{0}n-a_{0}-\beta/2}{2a_{0}^{2}}.\label{eq:35}\end{aligned}$$ Eq. ($\ref{eq:32}$) represents the mass spectrum of the collapsing star, while Eq. ($\ref{eq:35}$) represents its energy spectrum, where the gravitational energy, which is given by Eq. (18), is included.
Energy spectrum and ground state for the extremal RNBH
======================================================
Let us consider the case of a completely collapsed star, i.e. a RNBH. Setting $\chi=\pi/2$, and evaluating equations ($\ref{eq:4}$) and ($\ref{eq:5}$) at $\tau=0$, we obtain $$r=a\;\;r_{i}=a_{0}=M+\sqrt{M^{2}-Q^{2}},\label{eq: 36}$$ having considered the initial velocity of the collapse equal to zero. Moreover, now it is also $E_{0}=\frac{Q}{a_{0}^{2}}$. For the sake of simplicity, here we will discuss the extremal case, that means $M=Q.$ Thus, Eq. (36) becomes $$r=a\;\;r_{i}=a_{0}=M,\label{eq: 37}$$ with $E_{0}=1/M$. Now it is possible to find the potential energy of an extremal RNBH, its Schrödinger equation, the inert mass spectrum and the energy spectrum.
The potential energy is found from Eq. ($\ref{eq:17}$), ($\ref{eq:25}$) and ($\ref{eq:28}$): $$V(r)=-\left(\frac{2\pi}{Mr^{2}}+\frac{M^{2}}{2r}\right).\label{eq: 38}$$ The Schrödinger equation is $$-\frac{1}{2M}\frac{\partial^{2}\Psi}{\partial r^{2}}-\left(\frac{2\pi}{Mr^{2}}+\frac{M^{2}}{2r}\right)\Psi=E\Psi,\label{eq: 39}$$ and the energy spectrum is $$E_{n}=-\frac{M^{5}}{2\left(2n-1+\sqrt{1-\frac{16\pi}{M}}\right)^{2}}.\label{eq: 40}$$ The mass spectrum of an extremal RNBH can be obtained from the energy spectrum by recalling that $E=-M/2$. One then gets $$M_{n}^{2}=2n-1+\sqrt{1-\frac{16\pi}{M_{n}}}.
\label{eq:41}$$ For large values of the principal quantum number $n$ of the extremal RNBH, one gets an approximated solution of Eq. (\[eq:41\]), that is $$M_{n}=\sqrt{2n}.\label{eq: 42}$$ This permits to write down the energy levels of the extremal RNBH as $$E_{n}=-\frac{\sqrt{2n}}{2}.\label{eq: 43}$$ Actually, a final further correction is needed. In fact, based on the absorptions of external particles, the extremal RNBH mass changes during the jumps from one energy level to another. The RNBH mass increases for energy absorptions. Therefore, one must also include this dynamical behavior to properly describe the properties of the extremal RNBH.
To do that, one can introduce the *RNBH effective state* as in [@key-12]. Let us consider the emission of Hawking quanta or the absorption of external particles. Starting from the seminal work by Parikh and Wilczek [@key-13], one of us (CC) introduced the concept of *BH effective temperature*, *effective mass* and *effective charge* [@key-14]. In the case of an extremal RNBH the mass equals the electrostatic charge; to describe it, the use of the BH *effective mass* concept is sufficient. More precisely, if one considers the case of absorptions, where $M$ is the initial extremal RNBH mass before the absorption and $M+\omega$ is the final extremal RNBH mass after the absorption of an external particle having mass-energy $\omega$, the extremal RNBH *effective mass* can be introduced as in [@key-12], viz., $$M_{E}(\omega)\equiv M+\frac{\omega}{2}.
\label{eq: 44}$$ The effective mass is defined as an averaged quantity [@key-1; @key-12; @key-14] and actually represents the average of the initial and final BH masses [@key-1; @key-12; @key-14] before/after the absorption. In the present case, this averaged quantity represents the extremal RNBH mass *during* the expansion. Therefore, in order to take the dynamical behavior of an extremal RNBH into account, in the case of increasing mass, one must replace the extremal RNBH mass $M$ with the extremal RNBH effective mass, obtaining a potential $$V(r)=-\left(\frac{2\pi}{M_{E}r^{2}}+\frac{M_{E}^{2}}{2r}\right).\label{eq: 46}$$ The RNBH Schrödinger equation becomes $$-\frac{1}{2M_{E}}\frac{\partial^{2}\Psi}{\partial r^{2}}-\left(\frac{2\pi}{M_{E}r^{2}}+\frac{M_{E}^{2}}{2r}\right)\Psi=E\Psi.\label{eq: 47}$$ The introduction of the extremal RNBH effective mass in the dynamical equations can be rigorously justified by using Hawking’s periodicity argument [@key-15]; see [@key-1; @key-16] for a deeper insight in the mathematical approach.
Now, from the quantum point of view, we want to obtain the energy eigenvalues describing absorptions phenomena that start just after the formation of the extremal RNBH, i.e. from the ideal case of a RNBH having null mass. This is obtained by replacing $M\rightarrow0$ and $\omega\rightarrow M$ in Eq. (\[eq: 44\]), obtaining $$M_{E}\equiv\frac{M}{2}.\label{eq: 48}$$ This permits to write down the final equations for the extremal RNBH mass and energy spectra as $$M_{n}=2\sqrt{2n}\label{eq: 49}$$ and $$E_{n}=-\sqrt{2n},\label{eq: 50}$$ respectively. Now, one recalls that the relationship between area, mass and charge of the RNBH is [@key-3] $$A=4\pi\left(M+\sqrt{M^{2}-Q^{2}}\right)^{2},
\label{eq: Bekenstein}$$ that becomes, for the extremal RNBH, $$A=4\pi M^{2}.
\label{eq: 52}$$
Consider now an extremal RNBH which is excited at the level $n$; if one assumes that a neighboring particle is captured by the extremal RNBH causing a transition from the state with $n$ to the state with $n+1$, then the variation of the extremal RNBH area becomes $$\Delta A_{n\rightarrow n+1}\equiv A_{n+1}-A_{n}
\label{eq: absorbed}$$ and, by combining Eqs. (\[eq: 49\]) and (\[eq: 52\]), one immediately finds the quantum of area to be $$\Delta A_{n\rightarrow n+1}=4\pi\left(M_{n+1}^{2}-M_{n}^{2}\right)=16\pi.\label{eq: 54}$$
A similar case was analysed by Bekenstein [@key-3], who considered approximately constant the RNBH mass during the transition, obtaining $$\Delta A=8\pi M\epsilon,
\label{eq: area quantum Bekenstein}$$ where the quantity $\epsilon$ is the total energy of the absorbed particle. In Bekenstein’s approximation it is $$M\simeq M_{n}\simeq M_{n+1}\simeq2\sqrt{2n}\simeq2\sqrt{2n+1},\label{eq: 56}$$ while, from Eq. (\[eq: 50\]), one gets $$\epsilon=\sqrt{2n+1}-\sqrt{2n}.\label{eq: 57}$$ Thus, combining Eqs. (\[eq: area quantum Bekenstein\]), (\[eq: 56\]) and (\[eq: 57\]), one obtains $$\Delta A\simeq16\pi.\label{eq: 58}$$ This shows that our result describing the quantum of area of the extremal RNBH is consistent with the results found by Bekenstein [@key-3].
Stability of the solutions with respect to a perturbation
=========================================================
The mathematical analogy of the gravitational collapse leading to an extremal RNBH with that of an hydrogen atom, presents additional advantages that can characterize the stability of the solutions, already found, with respect to an external perturbation. This is obtained starting from Rosen’s approach to cosmology and from that in [@key-17]. The Schrödinger equation ruling the gravitational collapse of an extremal RNBH, as in the Archaic universe scenario [@key-18], can admit the existence of oscillatory solutions induced by the perturbation or an exponential suppression of the perturbation. This behavior is observed also in universes represented by a dust model with cosmological constant expressed in terms of Bessel functions.
Consider, after simple algebra, the general mathematical form of Eq. (\[eq: 47\]). The generic Schrödinger equation for the collapse of a RNBH becomes $$\psi^{\prime\prime}+P\ \psi=0,$$ where $'$ represents the derivative with respect to the parameter $r$, and the function $P$ is defined as $$P=\frac{4\pi}{Mr^{2}}+\frac{M^{3}}{r}-\frac{M}{2}.$$ We can study the evolution of the solutions of Eq. (\[eq: 47\]) by adopting the approach by Ettore Majorana in his unpublished notes [@key-19] for the study of the stability and scattering of the hydrogen-like atom as in [@key-18]. Thus one can write $$\psi=u\exp{\left\{ i\int\frac{k_{1}}{u^{2}}dx\right\} }.$$ The equation of motion becomes $$u^{\prime\prime}-\frac{k_{1}}{u^{2}}+P\ u=0,$$ with the general solution $$\psi=u_{1}[A^{*}\exp(I)+B^{*}\exp(-I)],$$ where $$I \equiv i\int\frac{1}{u_{1}^{2}}dx,$$ and $A^{*}$ and $B^{*}$ are two coefficients.
Let us now suppose that the perturbation is small and slowly varying, namely, $|P'/P|\ll1$. Then we can set, without loosing in generality, $u=P^{-1/4}$ and $P>0$, that is obtained in the following interval: $$\frac{2M^{2}-\sqrt{M^{4}+\frac{16\pi}{M}}}{4}<r<\frac{2M^{2}+\sqrt{M^{4}+\frac{16\pi}{M}}}{4}$$ After having set the values of $A^{*}$ and $B^{*}$ from the initial conditions assumed in the gravitational collapse, and considering $$\begin{aligned}
P' & = & -\frac{8\pi M^{4}}{r^{3}}-\frac{M^{3}}{r^{2}}\\
P'' & = & \frac{24\pi}{r^{4}}+\frac{2M}{r^{3}}, \end{aligned}$$ we clearly find oscillatory solutions [@key-19]. By defining $$I(P) \equiv \int\sqrt{P}\left(1-\frac{P'^{2}}{32P^{3}}\right)da,$$ these solutions can be written as $$\begin{aligned}
\psi=&\frac{1}{\sqrt[4]{P}}\left(1+\frac{PP''-5/4P'^{2}}{16P^{3}}\right)\nonumber \\
&\times \left\{ \left.\begin{array}{c}
\sin\\
\cos
\end{array}\right.\left[-\frac{P'}{8P^{3/2}}+I(P)\right]\right\} ,\end{aligned}$$ which means that a small and slow (in $r$) perturbation induces an oscillatory regime in the RNBH state. Being the coordinate $r$ positive, this condition holds if $r>\sqrt[4]{\frac{4\pi}{M}}$. This quantity is related to the turnaround point $r_{0}=\frac{M}{2}$ where the infall velocity slows all the way to zero between the two horizons of the extremal RNBH and the oscillation may occur when the coordinate takes values $r>\sqrt[4]{4\pi}$.
Another type of solution is found when $P<0$. By setting $P_{1}=-P>0$, Majorana found the following solution for a small perturbation in $r$: $$\begin{aligned}
\psi=&\frac{1}{\sqrt[4]{P_{1}}}\left(1-\frac{P_{1}P_{1}''-5/4P_{1}'^{2}}{16P_{1}^{3}}\right)\nonumber \\
&\times \exp\left\{\pm\frac{P_{1}'}{8P_{1}^{3/2}}+I(P_1)\right\}\end{aligned}$$ In this case the perturbation is suddenly exponentially damped, confirming the stability of the solutions found so far.
Conclusion remarks
==================
We discussed the properties and behavior of the gravitational collapse of a charged object forming a Riessner-Nordström black hole down to the quantum level, by calculating the gravitational potential, the Schrödinger equation and the exact solutions of the energy levels of the gravitational collapse. By using the concept of *BH effective state* [@key-1], the quantum gravitational potential, the mass spectrum and the energy spectrum for the extremal RNBH have been found. From the mass spectrum, the area spectrum has been derived too and this area spectrum is consistent with the previous result by Bekenstein [@key-3].
Finally, the stability of these solutions were described with the Majorana approach [@key-19] to the Archaic Universe scenario [@key-18], finding the existence of oscillatory regimes or exponential damping for the evolution of a small perturbation from a stable state.
Acknowledgments
===============
One of the authors (F.T.) gratefully acknowledges ZKM and Peter Weibel for the financial support.
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---
abstract: 'We study theoretically the $BD\bar{D}$ and $BDD$ systems to see if they allow for possible bound or resonant states. The three-body interaction is evaluated implementing the Fixed Center Approximation to the Faddeev equations which considers the interaction of a $D$ or $\bar{D}$ particle with the components of a $BD$ cluster, previously proved to form a bound state. We find an $I(J^P)=1/2(0^-)$ bound state for the $BD\bar{D}$ system at an energy around $8925-8985$ MeV within uncertainties, which would correspond to a bottom–hidden-charm meson. In contrast, the $BDD$ system, which would be bottom–double-charm and hence manifestly exotic, we have found hints of a bound state in the energy region $8935-8985$ MeV, but the results are not stable under the uncertainties of the model, and we cannot assure, neither rule out, the possibility of a $BDD$ three-body state.'
author:
- 'J. M. Dias'
- 'V. R. Debastiani'
- 'L. Roca'
- 'S. Sakai'
- 'E. Oset'
title: 'On the binding of the $BD\bar{D}$ and $BDD$ systems'
---
Introduction
============
The traditional field of few body, which has basically concentrated on few nucleon systems [@Alt; @Fonseca; @Epelbaum] or nucleons and hyperons [@Hiyama] is gradually giving rise to less conventional systems. Systems with two mesons and one baryon were studied in [@19] with the surprising result that the $1/2^+$ low lying excited baryons could be reproduced with this picture. Similar conclusions were found in [@22; @23]. Systems of three mesons were also studied and many known resonances could be described within this picture [@24; @25; @26].
The jump to the charm sector was done with the study of the $DNN$ system in Ref. [@30] and $NDK$, $\bar{K}DN$, $ND\bar{D}$ molecules were also studied in Ref. [@31]. The charm sector with three mesons was initiated with the description of the $Y(4260)$ as a resonant state of $J/\psi K\bar{K}$ [@MartinezTorres]. A more complete list of works along those lines can be found in Ref. [@DKK]. In this latter work the $DKK$ and $DK\bar{K}$ systems were studied and the Fixed Center Approximation to the Faddeev equations (FCA) was used. The FCA assumes that there is a cluster of two particles, in this case the $DK$, which forms the $D_{s0}^*(2317)$ [@34; @36; @37; @38; @42], and the third particle rescatters multiply with the two particles of the cluster. Direct comparison of the results for the $NK\bar{K}$ system with the FCA [@Xie], with a variational method [@Jido], or full Faddeev equations [@alberKan], confirms the accuracy of the FCA to deal with these problems when we have one couple that clearly binds, like in this case where the $\bar{K}N$ gives rise to the $\Lambda(1405)$. The application of the FCA to obtain the $K^-d$ scattering length [@Kamalov] also leads to results comparable to those obtained using a different field theoretical approach [@Rusetsky]. In the present case the $DKK$, $DK\bar{K}$ systems are substituted by $BDD$ and $BD\bar{D}$ and there are clear analogies also in the results.
The $DK$ system is bound, and the analogous $BD$ system was also found to be bound in the study of Ref. [@SakaiRoca], where a state of $I=0$ and $J^P=0^+$ was found around $7100$ MeV, with binding energy between $20-50$ MeV, which would be the analogous of the $D_{s0}^*(2317)$ as a bound $DK$ system. The second $D$ or $\bar{D}$ can then scatter with the $BD$ components of the cluster and lead eventually to more binding, giving rise to a three-body molecule. In the first case the $DB$ interaction will be attractive but the $DD$ is mostly repulsive and it is unclear what will prevail. In the second case the $\bar{D}D$ will be attractive but the $\bar{D}B$ is attractive in $I=0$ and repulsive in $I=1$, and again it is uncertain what will happen. This is analogous to the $DKK$ and $DK\bar{K}$ systems where we had a similar behaviour, the $K$ playing the role of the $D$ and the $D$ playing the role of $B$. Our final result will reveal that the $BD\bar{D}$ system clearly binds while the case for the $BDD$ is uncertain.
Formalism
=========
The bulk of the formalism to implement the FCA to evaluate the three-body interaction in the $BD\bar{D}$ and $BDD$ systems is analogous to the $DK\bar{K}$ and $DKK$ case of Ref. [@DKK]. Therefore in the present section we just show the differences and modifications for the present case and refer to Ref. [@DKK] for further details on the formalism.
Let us address first the $BD\bar{D}$ interaction. We need to consider the isospin doublets $(B^+,B^0)$, $(\bar{B}^0,-B^-)$, $(D^+,-D^0)$, $(\bar{D}^0,D^-)$ and the $|BD,~I=0\rangle$ state given by $$\label{BDI0}
|BD,~I=0\rangle = -\frac{1}{\sqrt{2}}(B^+D^0 +B^0D^+)$$ We then have to consider the interaction with a $\bar{D}$ to account for the $BD\bar{D}$ dynamics. Considering the possible intermediate steps in the multiple scattering, we need the following channels contributing to the three-body interaction:
---------------------- ---------------------- -----------------------
$1)D^-[B^+D^0]\quad$ $2)D^-[B^0D^+]\quad$ $3)\bar{D}^0[B^0D^0]$
$4)[B^+D^0]D^-\quad$ $5)[B^0D^+]D^-\quad$ $6)[B^0D^0]\bar{D}^0$
---------------------- ---------------------- -----------------------
The difference between the configurations $1)$, $2)$, $3)$ and $4)$, $5)$, $6)$, respectively, is that in the former ones the $\bar{D}$ outside the cluster interacts with the $B$ inside the cluster while in the later ones it interacts with the $D$. Then we define the partition functions $T_{ij}$ which sum all possible diagrams that begin with configuration $i)$ and finish with configuration $j)$, following an analogous scheme to the one proposed in Ref. [@Sekihara]. We show in Fig. \[diagT11\] the diagrams that contribute to $T_{11}$.
![Multiple scattering diagrams that go in the construction of the partition function $T_{11}$.[]{data-label="diagT11"}](BbarBD_Diagrams.eps){width="\textwidth"}
We then have $$\label{T11}
T_{11}^{\rm FCA} (s) = t_1 + t_1 \,G_0 \,T_{41}^{\rm FCA} + t_2 \,G_0 \,T_{61}^{\rm FCA},$$ where $s$ is the total three-body center-of-mas energy; $t_1$, $t_2$ are defined later in Eq. and $G_0$ is the $\bar{D}$ propagator modulated by the $BD$ wave function (see details in Eqs. $(2)-(5)$ of Ref. [@DKK], substituting $D \rightarrow B$, $K \rightarrow D$ and $D_{s0}^*(2317)$ by the $BD(7100)$ molecule). We can write the equivalent equations for the other $T_{ij}$ partitions and obtain the set of algebraic equations. $$\label{Tij}
T_{ij}^{\rm FCA} (s) = V_{ij}^{\rm FCA}(s) +\sum\limits_{l=1}^{6}
\tilde{V}^{\rm FCA}_{il}(s)\,G_0(s)\,T^{\rm FCA}_{lj}(s)\, ,$$ which solution is: $$\label{Tij-1}
T_{ij}^{\rm FCA}(s)=\sum\limits_{l=1}^{6}\Big[\,1-
\tilde{V}^{\rm FCA}(s)\,G_0(s)\,\Big]^{-1}_{il}\,V_{lj}^{\rm FCA}(s)\, ,$$ with $$\label{VBDDbar}
V^{\rm FCA} =
\left (
\begin{array}{@{\,}cccccc@{\,}}
t_{1} & 0 & t_{2} & 0 & 0 & 0 \\
0 & t_{3} & 0 & 0 & 0 & 0 \\
t_{2} & 0 & t_{4} & 0 & 0 & 0 \\
0 & 0 & 0 & t_{5} & 0 & 0 \\
0 & 0 & 0 & 0 & t_{6} & t_{7} \\
0 & 0 & 0 & 0 & t_{7} & t_{8} \\
\end{array}
\right ) ,
\qquad
\tilde{V}^{\rm FCA} =
\left (
\begin{array}{@{\,}cccccc@{\,}}
0 & 0 & 0 & t_{1} & 0 & t_{2} \\
0 & 0 & 0 & 0 & t_{3} & 0 \\
0 & 0 & 0 & t_{2} & 0 & t_{4} \\
t_{5} & 0 & 0 & 0 & 0 & 0 \\
0 & t_{6} & t_{7} & 0 & 0 & 0 \\
0 & t_{7} & t_{8} & 0 & 0 & 0 \\
\end{array}
\right ) ,$$ and the amplitudes $t_i$ are given by $$\begin{aligned}
\label{tiBDDbar}
\begin{tabular}{l l}
$t_{1}=t_{B^+D^-\to B^+D^-}\,, \quad$ & $t_{5}=t_{D^0D^-\to D^0D^-}\,,$ \\
$t_{2}=t_{B^+D^-\to B^0\bar{D}^0}\,, \quad$ & $t_{6}=t_{D^+D^-\to D^+D^-}\,,$\\
$t_{3}=t_{B^0D^-\to B^0D^-}\,, \quad$ & $t_7=t_{D^+D^-\to D^0\bar{D}^0}\,,$ \\
$t_{4}=t_{B^0\bar{D}^0\to B^0\bar{D}^0}\,, \quad$ & $t_8=t_{D^0\bar{D}^0\to D^0\bar{D}^0}\, .$\\
\end{tabular}\end{aligned}$$ The amplitudes in Eq. (\[tiBDDbar\]) are the $B\bar{D}$ and $D\bar{D}$ unitarized scattering amplitudes, taken from Refs. [@SakaiRoca] and [@37; @Gamermann:2008jh] respectively. Note that, as explained in Ref. [@37], there is, with respect to [@SakaiRoca; @37; @Gamermann:2008jh], an extra normalization factor $M_{BD}/M_i$ with $M_{BD}$ the mass of the bound state found in the $BD$ system [@SakaiRoca] and $M_i$ the mass of the particle of the cluster involved. This is introduced for convenience to use the Mandl-Shaw [@MandlShaw:2010] normalization for external $\bar{D} (D)$ and $[BD]$ states. For the evaluation of $D\bar{D}$ interaction in Ref. [@Gamermann:2008jh], from where the $X(3700)$ resonance was dynamically obtained, other meson-meson channels were considered, like $\pi\pi$, $\eta\eta$, $K\bar K$, $D_s \bar D_s$ and $\eta \eta_c$, but which turn out to have much less influence than the $D\bar{D}$ channel. Hence, we can neglect in the present work all the channels except the $D\bar{D}$. However, if we do this we cannot get a width for the $X(3700)$ resonance since the $D\bar{D}$ threshold is far above the position of that resonance. Therefore, in order to get also the width of the $X(3700)$ obtained in Ref. [@Gamermann:2008jh], which was 36 MeV, we have included the $\eta\eta$ channel in addition to the $D\bar{D}$ but with a renormalized value of the $\eta\eta\to\ D\bar D$ potential such as to reproduce the 36 MeV width.
On the other hand, for the case $BDD$ case we can proceed in an analogous way and again we get Eq. but with $$\label{VBDD}
V^{\rm FCA} =
\left (
\begin{array}{@{\,}cccccc@{\,}}
\bar{t}_{1} & 0 & 0 & 0 & 0 & 0 \\
0 & \bar{t}_{2} & \bar{t}_{3} & 0 & 0 & 0 \\
0 & \bar{t}_{3} & \bar{t}_{4} & 0 & 0 & 0 \\
0 & 0 & 0 & \bar{t}_{5} & 0 & \bar{t}_{5} \\
0 & 0 & 0 & 0 & \bar{t}_{6} & 0 \\
0 & 0 & 0 & \bar{t}_{5} & 0 & \bar{t}_{5} \\
\end{array}
\right ) ,
\quad
\tilde{V}^{\rm FCA} =
\left (
\begin{array}{@{\,}cccccc@{\,}}
0 & 0 & 0 & \bar{t}_{1} & 0 & 0 \\
0 & 0 & 0 & 0 & \bar{t}_{2} & \bar{t}_{3} \\
0 & 0 & 0 & 0 & \bar{t}_{3} & \bar{t}_{4} \\
\bar{t}_{5} & 0 & \bar{t}_{5} & 0 & 0 & 0 \\
0 & \bar{t}_{6} & 0 & 0 & 0 & 0 \\
\bar{t}_{5} & 0 & \bar{t}_{5} & 0 & 0 & 0 \\
\end{array}
\right ) ,$$ and $$\begin{aligned}
\label{tiBDD}
\begin{tabular}{l l}
$\bar{t}_{1}=t_{D^+B^+\to D^+B^+}\,, \quad$ & $\bar{t}_{4}=t_{D^0B^+\to D^0B^+}\,,$ \\
$\bar{t}_{2}=t_{D^+B^0\to D^+B^0}\,, \quad$ & $\bar{t}_{5}=t_{D^+D^0\to D^+D^0}\,,$ \\
$\bar{t}_{3}=t_{D^+B^0\to D^0B^+}\,, \quad$ & $\bar{t}_{6}=t_{D^+D^+\to D^+D^+}\,,$ \\
\end{tabular}\end{aligned}$$ where the $DD$ amplitudes are taken from Ref. [@37]. We use isospin symmetry to build the $I=1$ amplitude of $DD$, including a factor $2$ in the interaction of $D^+D^0 \to D^+D^0$ given in Ref. [@37], while the $I=0$ amplitude vanishes. One should be careful to include a factor $1/2$ in the kernel to account for the normalization of identical particles, which later has to be restored multiplying $t_{DD}^{I=1}$ by $2$.
Finally, the $T_{BD\bar{D}(D)}$ three-body scattering amplitude in isospin $1/2$ in terms of the amplitudes in Eq. (\[Tij-1\]) is given by $$\begin{aligned}
\label{Ttotal}
\nonumber T_{BD\bar{D}(D)}=&\frac{1}{2}\Big(
T^{\rm FCA}_{11}+T^{\rm FCA}_{12}+T^{\rm FCA}_{14}+T^{\rm FCA}_{15}
+T^{\rm FCA}_{21}+T^{\rm FCA}_{22}+T^{\rm FCA}_{24}+T^{\rm FCA}_{25}\\
&+T^{\rm FCA}_{41}+T^{\rm FCA}_{42}+ T^{\rm FCA}_{44}+T^{\rm FCA}_{45}
+T^{\rm FCA}_{51}+T^{\rm FCA}_{52}+T^{\rm FCA}_{54}+T^{\rm FCA}_{55}
\Big)\,.\end{aligned}$$
Results
=======
We have two main sources of uncertainty in our model. The first one is the cutoff used to regularize the $BD$ and $B\bar D$ loop functions needed for the evaluation of the unitarized scattering amplitudes [@SakaiRoca] of Eqs. (\[tiBDDbar\]) and (\[tiBDD\]). This is carried out in Ref. [@SakaiRoca] and in the present work by using a three-momentum cutoff within the range $q_{\rm max}=400-600$ MeV. The second source of uncertainty is the prescription used to evaluate the center-of-mass energy ($\sqrt{s_{3i}}$) of the projectile, particle $(3)$ ($\bar{D}$ or $D$) and one of the particles in the cluster, $(1)$ or $(2)$ ($B$ or $D$). These energies are the argument entering the amplitudes in Eqs. (\[tiBDDbar\]) and (\[tiBDD\]). Two prescriptions were given in Ref. [@DKK] (see Eqs. $(19-22)$ of that reference) and we consider also both of them in the present work. The first prescription (I) is standard in works implementing the FCA scheme and was used in Refs. [@40; @51], and the second prescription (II) takes into account the sharing of the binding energy between the three particles and was introduced in Ref. [@DKK]. We will consider the differences obtained changing the value of $q_{\rm max}$ and implementing both prescriptions for $\sqrt{s_{3i}}$ as an estimation of the uncertainty of our results.
In Fig. \[T2BDDbar\] we show the results for $|T_{BD\bar{D}}|^2$ in terms of $\sqrt{s}$, the overall CM energy of the $BD\bar{D}$ three-body system, using both prescriptions for $\sqrt{s_{3i}}$ and $q_{\rm max}=600$ MeV.
![$|T_{BD\bar{D}}|^2$ with prescriptions I, II for $\sqrt{s_{\bar{D}B}}$, $\sqrt{s_{\bar{D}D}}$ and $q_{\rm max}=600$ MeV with and without considering width (from $\eta \eta$ channel) for the $X(3700)$ through the $D\bar D$ interaction. The two curves with $\eta \eta$ channel were multiplied by a factor $10^4$ for comparison.[]{data-label="T2BDDbar"}](T2BDDbar.eps){width="80.00000%"}
We show results with and without considering the width of the $X(3700)$, through the inclusion or not of the $\eta\eta$ channel in the $D\bar D$ interaction, as explained below Eq. (\[tiBDDbar\]). If we do not consider this width, we find, both for prescription I and II, a neat and narrow peak below the threshold corresponding to the $BD$ cluster resonant mass ($7093$ MeV) $+$ the $\bar{D}$ mass which equals $8962$ MeV. This peak corresponds, thus, to a three-body bound state. The peaks appear at $8926$ MeV (prescription I) and $8944$ MeV (prescription II). If we include the $\eta\eta $ channel in the $D\bar D$ interaction such as to get the right $X(3700)$ width, the position of the states found barely increases by less than 5 Mev but a width for the three-body state is obtained of about 10 MeV.
The results with $q_{\rm max}=400$ MeV are similar with peaks at the energies $8977$ MeV and $8983$ MeV, respectively, compared with the new threshold of the $BD$ cluster ($7129$ MeV) $+$ $\bar{D}$ mass which equals $8998$ MeV, and once again the $BD\bar{D}$ system binds. The different values obtained with the different prescriptions and cutoffs provide a value for the $BD\bar{D}$ bound state in the range $8925-8985$ MeV, with a dispersion in the results that can be considered as the uncertainty of our calculation.
It is interesting to see the origin of this binding. The cluster $BD$ is bound thanks to the attractive $BD$ interaction in $I=0$. The $B\bar{D}$ is attractive in $I=0$ but repulsive in $I=1$ [@SakaiRoca]. Similarly the $D\bar{D}$ is attractive in isospin $I=0$, generating a narrow bound state (the $X(3700)$) around $3720$ MeV [@37; @Gamermann:2008jh], and in $I=1$ it is also attractive, but very weakly. In Eq. one can write the $t_i$ amplitudes in terms of the isospin amplitudes and see that both $I=0$ and $I=1$ $B\bar{D}$ and $D\bar{D}$ amplitudes participate in the process. In order to illustrate the importance of the most attractive components we remove the $I=0$ part of the $B\bar{D}$ amplitudes or the $I=0$ part of the $D\bar{D}$ ones. If we remove the $I=0$ part of the $B\bar{D}$ interaction the peak disappears for both prescriptions and for the two cutoffs. If we only remove the $I=0$ part of the $D\bar{D}$ interaction, then the peak only disappears in prescription II with cutoff $400$ MeV; in all the other cases a peak still remains, although the binding becomes smaller. We conclude that, while the $D\bar{D}$ attraction helps in the building up of the three-body bound state, the main source of binding is the $I=0$ component of the $B\bar{D}$ interaction.
As for the $BDD$ system, which would lead to a manifestly exotic meson with two charm quarks and a bottom antiquark, the amplitudes that we obtain using prescription I show a clear narrow bound state with both cutoffs for the $BD$ cluster, as can be seen in Fig. \[T2BDDpresc1\]. For $q_{\rm max}=400$ MeV the peak appears at $8985$ MeV, below the corresponding threshold of $BD$ cluster + $D$ ($8998$ MeV), while for $q_{\rm max}=600$ MeV the peak appears at $8936$ MeV, again below its threshold ($8962$ MeV).
However, when we switch to prescription II the bound state disappears, as can be seen in Fig. \[T2BDDpresc2\]. In this case, we get similar structures to the one that was found in the $DKK$ interaction [@DKK].
In order to further explore the possibility of binding in the $BDD$ system, we have tried another potential to describe the $DD$ interaction, in analogy to the ones of $KK$ from Ref. [@DKK], with an extra factor $1/2$ to account for the absence of the $\phi$ exchange, based on the local hidden gauge approach [@Bando; @Meissner; @Nagahiro]. With this interaction a structure similar to a resonant state is found below threshold for prescription I, but we have noticed that it is too sensitive to changes in the cutoff values. On the other hand, the results using prescription II still show structures similar to the one found in the $DKK$ system [@DKK], which cannot be clearly related to a three-body bound state or resonance.
It is interesting to notice that in analogy to the $DKK$ system, we have a very attractive interaction of the external $D$ with the $B$ of the cluster (the same responsible for the binding of the cluster itself), confronted with the repulsive interaction of $DD$. The three-body binding seems to depend on a very delicate equilibrium between these two interactions, and in the particular case of the $BDD$ system we could not arrive to a decisive conclusion which would be stable under the model uncertainties.
Conclusions {#conclusions .unnumbered}
===========
We have studied the $BDD$ and $BD\bar{D}$ systems using dynamical models for the $BD$, $B\bar{D}$ and $DD$, $D\bar{D}$ interaction, which have been tested in previous works. Given the strong binding of the $BD$ pair, we use the Fixed Center Approximation (FCA) to the Faddeev equations to evaluate the three-body interaction by considering the multiple rescattering of the external $D$ or $\bar D$ meson with the components of the $BD$ cluster. This scheme has proved its reliability in many other cases where two of the particles of the three-body system are strongly clusterized.
We obtained that the $BD\bar{D}$ system is bound and we get an energy of the $BD\bar{D}$ system of about $8925-8985$ MeV considering uncertainties. This result is quite stable under changes in the model that we implement to determine the uncertainty. Our study reveals that the $I=0$ $B\bar{D}$ and $D\bar{D}$ amplitudes, which are attractive, are the main reason for the binding of the $BD\bar{D}$ system, in particular the attraction of the $B\bar{D}$ pair.
As for the $BDD$ system, which would be manifestly exotic, we have found some clues of a bound state in the energy region $8935-8985$ MeV, but the results where not stable under the theoretical uncertainties of the model. Therefore, we cannot assure, neither rule out, the possibility of a three-body state in the $BDD$ interaction.
From our results of the $BD\bar{D}$ study, we expect that a bottom mesonic resonance with quantum numbers $I(J^P)=1/2(0^-)$ and mass around $8925-8985$ MeV could be experimentally found in future studies in hadron facilities.
Acknowledgments {#acknowledgments .unnumbered}
===============
J. M. Dias would like to thank the Brazilian funding agency FAPESP for the financial support. V. R. Debastiani wishes to acknowledge the support from the Programa Santiago Grisolia of Generalitat Valenciana (Exp. GRISOLIA/2015/005). This work is also partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under the contract number FIS2014-57026-REDT, FIS2014-51948-C2-1-P, and FIS2014-51948-C2-2-P, and the Generalitat Valenciana in the program Prometeo II-2014/068.
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---
abstract: 'We recently discovered a yellow supergiant (YSG) in the Small Magellanic Cloud (SMC) with a heliocentric radial velocity of $\sim300$ km s$^{-1}$ which is much larger than expected for a star in its location in the SMC. This is the first runaway YSG ever discovered and only the second evolved runaway star discovered in a different galaxy than the Milky Way. We classify the star as G5-8 I, and use de-reddened broad-band colors with model atmospheres to determine an effective temperature of $4700\pm 250$K, consistent with what is expected from its spectral type. The star’s luminosity is then $\log L/L_\odot \sim 4.2 \pm 0.1$, consistent with it being a $\sim$ 30Myr 9$M_\odot$ star according to the Geneva evolution models. The star is currently located in the outer portion of the SMC’s body, but if the star’s transverse peculiar velocity is similar to its peculiar radial velocity, in 10 Myr the star would have moved 1.6$^\circ$ across the disk of the SMC, and could easily have been born in one of the SMC’s star-forming regions. Based on its large radial velocity, we suggest it originated in a binary system where the primary exploded as a supernovae thus flinging the runaway star out into space. Such stars may provide an important mechanism for the dispersal of heavier elements in galaxies given the large percentage of massive stars that are runaways. In the future we hope to look into additional evolved runaway stars that were discovered as part of our other past surveys.'
author:
- 'Kathryn F. Neugent, Philip Massey, Nidia Morrell, Brian Skiff, and Cyril Georgy'
title: |
A Runaway Yellow Supergiant Star in the\
Small Magellanic Cloud
---
Discovery
=========
Neugent et al. (2010) conducted a radial velocity study of yellow stars seen in the direction of the Small Magellanic Cloud (SMC) in order to identify its yellow supergiant (YSG) population. As shown in Figure \[fig:VelR\], the observed radial velocities are either clustered around 0 km s$^{-1}$ (as expected for foreground yellow dwarfs), or around the SMC’s heliocentric radial velocity of 158 km s$^{-1}$ (Richter et al. 1987) as indicated by the black line and expected for SMC yellow supergiants. One star, J01020100-7122208, however, has a heliocentric radial velocity of around 300 km s$^{-1}$, 140 km s$^{-1}$ greater than expected. Neugent et al. (2010) don’t explicitly comment on this star; at the time we believed it to be a likely short-period binary. However, we have now completed additional observations that rule out this explanation. Instead, this star is the first runaway YSG discovered and the second evolved runaway star discovered in another galaxy.
The concept of runaway massive stars has been around for sixty years. Blaauw (1956a, 1956b) discovered that some OB stars stars have much higher space velocities than other stars in their surroundings. Zwicky (1957) was the first to hypothesize that such stars should exist due to supernovae explosions causing the secondary in a binary system to be shot off into space. However, Gies & Bolton (1986) showed that the close binary frequency of runaway OB stars was the same as for non-runaways, suggesting that the binary ejection mechanism was not the primary source of runaway stars. There have since been other theories as to how such runaway stars exist including the effects of different dynamical interactions (Leonard & Duncan 1990) or interactions with massive black holes (Capuzzo-Dolcetta & Fragione 2015; Fragione & Capuzzo-Dolcetta 2016). Runaway stars aren’t rare, with as many as 50% of OB stars considered runaways with peculiar (difference between observed and expected) radial velocities larger than 40 km s$^{-1}$ (Gies & Bolton 1986).
Given that such a large fraction of OB stars are runaways, it is surprising that there are very few evolved massive stars known to be runaways. Three Galactic red supergiants (RSGs), Betelgeuse, $\mu$ Cep and IRC-10414, are thought to be runaways based upon the presence of bow shocks (Noriega-Crespo et al. 1997, Cox et al. 2012, and Gvaramadze et al. 2013). Evans & Massey (2015) recently identified a runaway RSG in the Andromeda Galaxy directly from its peculiar velocity. The star we discuss here is the first known runaway YSG anywhere and is only the second known evolved massive star runaway of any kind found in another galaxy. In Section 2 we’ll discuss our observations and how we calculated the radial velocities. In Section 3, we’ll give an overview of the physical properties of this runaway YSG. In Section 4 we’ll discuss our results and conclusions.
Observations, Reductions, and Radial Velocity Calculation
=========================================================
As mentioned above, we originally believed that the star’s abnormally large radial velocity was due to binary motion. Once we entertained the idea of it being a runaway star we obtained three additional spectra to investigate this possibility.
Our first (discovery) spectrum was obtained using Hydra on the CTIO 4-m Blanco telescope on (UT) 2009 Oct 9. The second spectrum was obtained on (UT) 2017 Aug 16 using the Echelle on the du Pont 2.1-m telescope on Las Campanas in Chile. The spectrum was under-exposed in the blue, as our goal was simply to check on the radial velocity from the Ca II triplet in the far red. Subsequently (2017 Dec 31) we obtained a high signal-to-noise spectrum with MagE on the Las Campanas Baade 6.5-m Magellan telescope for the purposes of spectral classification and radial velocity measurement, plus a shorter exposure the following night simply to confirm the radial velocity. The wavelength ranges, spectral resolution ($\Delta \lambda/\lambda$), and exposure times are summarized in Table \[tab:obs\]. To help with spectral classification we also observed six bright ($V\sim 10-12$) yellow supergiants in the SMC and LMC, ranging in type from F0 to G8.
The full calibration and reduction details for the Hydra spectrum are given in Neugent et al. (2010). For the MagE spectra, we obtained bias and flat-field exposures during the day, and each program exposure was followed immediately by a comparison arc exposure. We used Jack Baldwin’s [*mtools*]{} routines in IRAF[^1] for the spectral extractions; these routines are available from the Las Campanas website. The standard IRAF echelle reduction tasks were then used for wavelength calibration and flux calibration using spectrophotometric standards. For the duPont data, Milky flats were obtained with the afternoon sky and a diffusor in front of the slit, and the reductions are all performed with IRAF, without intervention of the ’mtools’ package.
We measured the radial velocities using two methods: first, by fitting a Gaussian to each of the Ca[ii]{} triplet lines ($\lambda 8498$, $\lambda 8542$, $\lambda 8662$), and secondly by cross-correlating the Hydra radial velocity standard stars from Neugent et al. (2010) against the new spectra in the region around the Ca[ii]{} triplet. The two methods agreed to within 1 km s$^{-1}$. We give the average heliocentric radial velocities (HRV) of the two methods in Table \[tab:obs\]. All four values are consistent with each other, essentially ruling out the possibility that the large peculiar radial velocity is due to motion in a close binary system. A small difference is seen in the values between various telescopes, which we believe is due to minor offsets in the instrumental velocity zero-points. To investigate this further, we measured the radial velocities of the six MagE spectral standards, five of which have published radial velocities from other sources. As is shown in Table \[tab:stds\], we find that the differences ranged from -14.0 km s$^{-1}$ to +18.6 km s$^{-1}$, consistent with our assertion that our measurements do not indicate any significant radial velocity variations for our runaway.
For consistency we also remeasured the velocity from our original Hydra spectrum. Much to our chagrin, we discovered that we had originally misapplied the heliocentric correction. This affects all of the individual radial velocities given in Table 1 of that paper by $\sim$11.0 km s$^{-1}$, with the previously published values too large. We take the opportunity to reissue the table here, as Table \[tab:allobs\]. Note, however, that this does not affect anything else in that paper, as our assignment of membership to the SMC was based upon the distribution of velocities, and those were completely accurate in a relative sense, since the heliocentric corrections for all stars were the same. We also took the opportunity to update the table with revised spectral types that were subsequently published.
Physical Properties
===================
In this section we use our spectral information and photometry to determine the star’s effective temperature and luminosity, allowing us to place the star on the H-R diagram. We can then use evolution models to determine other physical properties, such as approximate mass and age.
Spectral Classification
-----------------------
In Figure \[fig:type\] we compare the spectrum of our star to six spectral standards, Sk 105 (F0 Ia), Sk 55 (F3 Iab), HD 271182 (F8 0), HD 269953 (G0 0), HD 269723 (G4 0), HD 268757 (G8 0). The two early F stars are SMC members classified by Ardenberg & Maurice (1977); the others are LMC members classified by Keenan & McNeil (1989), and considered to be MK standards. (The luminosity class “0" is one notch above the Ia designation.) We see immediately that our runaway is of G-type, based both upon the abundance of metal lines (despite the lower metallicity of the SMC compared to that of the LMC), and the weakness of the hydrogen lines. The strength of the G-band argues that the star is closer to type G8 than to G0; we assign a type of G5-8 I. For G stars, the strengths of the Ca II H and K lines provide a crude luminosity indicator, developing wide wings at high luminosities (Kaler 2011), and it is clear from Figure \[fig:type\] that these lines in our runaway are comparable to those of the supergiant standards.
Effective Temperature
---------------------
The uncertainties in the effective temperature scale for late-type supergiants are nicely discussed in the classic review by Böhm-Vitense (1981). A typical temperature for a G8 I star is 4570-90 K, while a G3 I star would be 4980 K (Böhm-Viense 1981, Cox 2000). Thus the expected temperature range for our star would be 4500-4800 K, with the caveat that the relationship between spectral type and effective temperature has only been established for solar metallicity stars. Since the metallicity of the SMC is roughly one-quarter solar, we expect that the effective temperature will be slightly cooler, probably by $\sim100$K, based on our previous modeling experience of RSGs (see e.g., Levesque et al. 2006).
We can refine our estimate of the effective temperature by instead using broad-band photometry and appealing to stellar atmosphere models of the appropriate metallicity. The major uncertainty is what to assume for the reddening. The observed colors are included in Table \[tab:allobs\]. According to Cox (2000), the intrinsic $(B-V)_0$ of a G5 I star is 1.02, while that of a G8 I is 1.14. The observed $B-V$ color is 1.15, suggesting a color excess in the range $E(B-V)=0.01-0.14$. However, we know that the foreground reddening towards the SMC corresponds to an $E(B-V)=0.04$, while the typical color excess of SMC OB stars is $E(B-V)=0.09$ (Massey et al. 2007). Let us consider then three different values for the reddening, $E(B-V)=0.04$ (minimum, only foreground), 0.09 (typical of OB stars), and 0.14. For the other colors, we adopt the relations between color excesses given by Schlegel et al. (1998), e.g., $E(U-B)=0.72E(B-V)$, $E(V-K)=2.95E(B-V)$, and $E(J-K)=0.54E(J-K)$.
In Table \[tab:temps\] we list the deredded colors, along with the corresponding effective temperatures derived from the updated ATLAS9 models (Castelli & Kurucz 2003) for a metallicity of \[-0.5\], corresponding to 0.3$\times$ solar, appropriate for the SMC[^2]. The range of reddening we consider changes the derived temperature by only 100 K. There is a larger difference (500 K) depending upon which color index we use[^3]. We adopt an average temperature 4700 +/- 250, in good agreement with the temperature we obtained via the spectral type. Note that the two methods are essentially independent of each other.
We adopt an absolute visual magnitude of the star $M_V=-5.4\pm0.2$, based upon a distance of the SMC of 59 kpc (van den Bergh 2000). The uncertainty represents the difference in the correction for extinction based upon the amount of reddening assumed, with $A_V=3.1E(B-V)$. The bolometric correction is then $-0.4\pm0.2$ based on the ATLAS9 models, with the error corresponding to the uncertainty in temperature. Thus the star’s luminosity is $\log L/L_\odot=4.2\pm0.1$. We summarize the physical properties in Table \[tab:physprop\].
H-R Diagram
-----------
In Figure \[fig:HRD\] we show the star’s location in the H-R diagram and include the latest version of the Geneva evolution tracks. These have been interpolated to $z=0.004$ (appropriate for the SMC) using the grid of models from Georgy et al. (2013). We see that the luminosity and effective temperature of our runaway corresponds to a star with an initial mass of about 9$M_\odot$. The tracks were computed using the assumption of an initial equatorial rotational velocity that is 40% of the critical break-up speed, which is probably the most representative (see discussion in Ekström et al. 2012 and Georgy et al. 2013). However, we show by the dotted 9 $M_\odot$ track the effect of a 1.5$\times$ lower rotational velocity. The age associated with this stage is about 30 Myr, regardless of the initial assumptions. It must be noted, however, that the expulsion mechanism behind this star becoming a runaway could have altered its evolutionary background and thus the age and mass we are able to predict.
Discussion and Conclusions
==========================
We briefly considered the possibility that this was was a foreground object: a very metal poor star out in the Galactic halo, for instance, could have a very high space velocity. Once we obtained the blue/optical spectrum described here, though, we could rule out this interpretation, as such a metal-poor object would show almost no spectral features other than the Balmer lines, the G-band, and H and K, while our optical spectrum discussed above shows numerous metal lines. The remote possibility that this was a foreground dwarf with an extraordinary radial velocity could also be dismissed from the wide wings of of the H and K lines, indicating high luminosity, as discussed in Section 3.
The spatial location of the runway YSG is shown in Figure \[fig:location\]. It is not located within the central portion of the SMC and is instead out on the edge of the galaxy, far from any regions of current star formation. We posed the following question: If the star’s tangential motion is the same as its peculiar radial velocity, how far would it move during its lifetime? We used 10 Myr as a ball-park estimate for the time involved, since we do not know how long it has been traveling (the star’s age is 30 Myr); in that period of time it would have moved 1.6 degrees! This distance is shown in Figure \[fig:location\] as the large red circle. It clearly encompasses the central portion of the SMC suggesting that this star could have been ejected from the body of the galaxy as an unevolved OB star. Note, however, that the kinematics of the SMC is quite complicated. Stanimirovic et al. (2004) found that there is a velocity gradient throughout the galaxy with the northeast quadrant (close to where this star is located) rotating up to 10 km s$^{-1}$ faster than the SMC’s average heliocentric radial velocity. Thus, it is possible that this star’s peculiar velocity is smaller than expected, but by no more than 10 km s$^{-1}$ (see Figure 3 in Stanimirovic et al. 2004).
We suspect that the YSG began in a binary system and was flung out into space when the primary star went supernova. Gies & Bolton (1986) concluded that supernova explosions are not the primary mechanism behind runaway stars but most runaway stars have peculiar velocities on the order of 40-80 km s$^{-1}$ and this runaway’s is much higher. As noted by Evans & Massey (2015), only two Galactic O stars are known to have peculiar velocities $>$ 100 km s$^{-1}$, and all three runaway RSGs in the Milky Way have peculiar velocities $<$ 70 km s$^{-1}$. We believe that the high peculiar velocity of this YSG points to it originating due to a supernova explosion following the original suggestion of Zwicky (1957) and Blaauw (1956a, 1956b).
Many runaway stars are found using their bow shocks, or a presence of an infrared excess (Gvaramadze et al. 2010). We estimated that in this case the bow shock would be around 3.0$\arcsec$ away from the star. We first looked for any sign of the bow shock in MIPS SAGE-SMC data (Gordon et al. 2011) but sadly the resolution was too poor. We then obtained H$\alpha$ imaging of the star using the Slope 1-m telescope at Las Campanas but again we saw nothing. Finally, we looked at MCELS data (Smith et al. 2000) to no avail. If there is a bow shock for this runaway star, we were not able to find it.
Another interesting point to make is that these runaway stars can act as dispersing mechanism for enriched elements; for example, this YSG will eventually explode as a supernova far away from where it was created. Since such a large fraction of OB stars are runaways, this must occur on a fairly regular basis. Thus, the process of runaway stars acts as a mechanism for dispersing elements throughout the galaxy. Although massive stars are well known to be the major source of many of the heavier elements (Maeder 1981), runaways have been largely ignored in favor of diffusion and similar processes as a means of dispersal (see, e.g., Pipino & Matteucci 2009). Recent evidence, however, suggests that our own Solar System may have been created by a runaway Wolf-Rayet star (Tatischeff & Sereville 2010)!
As future work, we hope to look into other potential YSG runaways that we found as part of a similar project in the Large Magellanic Cloud (LMC). One star in particular, J04530398-6937285 has a radial velocity of 373 km s$^{-1}$, much higher than expected, and warrants further investigation[^4].
We would like to thank Maria Drout for help collecting the discovery spectra for this star as part of the larger project to identify all YSGs in the SMC. We would also like to thank Sean Points from the MCELS team who helped us try and find evidence of a bow shock. We very much appreciate the continued support of our work from the Arizona Time Allocation Committee. Finally, we would like to acknowledge the NSF grant AST-1612874.
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[l l l l l l l l]{} Date & HJD & Telescope/Instrument & Wavelength & Resolution & Exp. time (s) & HRV (km s$^{-1}$)\
09 Oct 2009 & 2455113.67 & CTIO 4-m/Hydra & 7300-9050Å& 3000 & 900 & 296.4\
16 Aug 2017 & 2457981.87 & DuPont 2.5-m/Echelle & 3500-9050Å & 45000 & 1800 & 298.0\
31 Dec 2017 & 2458118.59 & Baade 6.5/MagE & 3200Å-1$\mu$m & 4100 & 1800 & 305.1\
1 Jan 2018 & 2458119.58 & Baade 6.5/MagE & 3200Å-1$\mu$m & 4100 & 600 & 305.7\
[l l l l l r]{} & Type & MagE\* & Published\* & Ref & (MagE - Published)\*\
Sk 55 & F3 Iab & 168.4 & 159.9 & 1 & 8.5\
HD271182 & F8 0 & 309.9 & 323.9 & 1 & -14.0\
HD269953 & G0 0 & 259.0 & 240.4 & 1 & 18.6\
HD269723 & G4 0 & 323.4 & 332.0 & 2 & -8.6\
HDE268757 &G8 0 & 284.8 & 271.6 & 1 & 13.2
[l r r r c r c c r r c c c c l]{} J00342570-7322334 & 0 34 25.69 & -73 22 33.5 & 11.6 & 0.25 & 12.79 & & 3 & -9.98 & 100.2 & 3 & & & &\
J00342883-7328192 & 0 34 28.82 & -73 28 19.3 & 10.72 & 0.59 & 13.03 & 0.97 & 4 & 91.12 & 109.7 & 2 & & G6IV & 12 &\
J00344094-7319366 & 0 34 40.94 & -73 19 36.7 & 9.18 & 0.69 & 11.73 & & 3 & 38.02 & 106.5 & 3 & & & &\
J00345529-7213044 & 0 34 55.30 & -72 13 04.4 & 11.58 & 0.31 & 12.93 & & 3 & -10.29 & 56 & 3 & & & &\
J00345992-7339156 & 0 34 59.91 & -73 39 15.8 & 10.66 & 0.33 & 12.13 & & 3 & 9.92 & 77.3 & 3 & & F8 & 6 & Flo 72\
J00350326-7332586 & 0 35 03.25 & -73 32 58.8 & 10.77 & 0.8 & 13.8 & 1.54 & 4 & 134.82 & 75.4 & 1 & & G6Ib/II & 12 & SkKM 2\
J00351576-7340108 & 0 35 15.75 & -73 40 10.9 & 10.25 & 0.47 & 12.13 & 0.85 & 4 & -9.58 & 115.1 & 3 & & K2V & 12 &\
J00353473-7341121 & 0 35 34.74 & -73 41 12.3 & 11.04 & 0.37 & 12.55 & & 3 & -16.38 & 77.8 & 3 & & & &\
J00353564-7205134 & 0 35 35.64 & -72 05 13.5 & 11.28 & 0.34 & 12.7 & & 3 & 2.51 & 63.3 & 3 & & & &\
J00354135-7210010 & 0 35 41.38 & -72 10 01.0 & 11.51 & 0.31 & 12.79 & & 3 & 18.31 & 77.1 & 3 & & & &\
J00361882-7216201 & 0 36 18.86 & -72 16 20.1 & 10.54 & 0.77 & 13.38 & 1.16 & 4 & -13.89 & 71.6 & 3 & & & &\
J00362691-7230502 & 0 36 26.94 & -72 30 50.2 & 10.34 & 0.33 & 11.69 & & 3 & -12.99 & 97.8 & 3 & & F7 & 6 & Flo 83\
J00364059-7319091 & 0 36 40.59 & -73 19 09.2 & 10.06 & 0.6 & 12.3 & & 3 & 15.92 & 107.7 & 3 & & K0IV & 12 &\
J00365948-7338435 & 0 36 59.49 & -73 38 43.5 & 10.99 & 0.45 & 12.83 & 0.82 & 4 & 14.72 & 87.6 & 3 & & & &\
J00370536-7326159 & 0 37 05.36 & -73 26 16.0 & 10.46 & 0.78 & 13.51 & 1.55 & 4 & 122.32 & 90.4 & 2 & & & & SkKM 6\
J00371053-7349353 & 0 37 10.53 & -73 49 35.4 & 11.22 & 0.06 & 11.53 & & 3 & -44.48 & 14 & 3 & & A2 & 1 & HD 3542\
J00375257-7201070 & 0 37 52.59 & -72 01 07.2 & 11.69 & 0.35 & 13.24 & 0.59 & 4 & 19.41 & 91.7 & 3 & & & &\
J00381017-7311321 & 0 38 10.16 & -73 11 32.2 & 8.98 & 0.3 & 10.21 & & 3 & 6.82 & 82.1 & 3 & & F0 & 9 & HD 3659; Flo 96\
J00382510-7400464 & 0 38 25.08 & -74 00 46.5 & 11.22 & 0.38 & 12.86 & & 3 & -33.43 & 30 & 3 & & & &\
J00382659-7359394 & 0 38 26.57 & -73 59 39.6 & 6.23 & 0.92 & 9.49 & & 3 & -10.03 & 85.8 & 3 & & K5 & 5 & CPD-74 54\
J00383391-7409574 & 0 38 33.91 & -74 09 57.4 & 9.97 & 0.67 & 12.58 & & 3 & 1.77 & 43.4 & 3 & & & &\
J00390248-7357293 & 0 39 02.47 & -73 57 29.3 & 11.32 & 0.3 & 12.56 & 0.65 & 4 & -3.93 & 44 & 3 & & & &\
J00390455-7332038 & 0 39 04.56 & -73 32 03.8 & 10.77 & 0.33 & 12.36 & & 3 & -21.78 & 81.7 & 3 & & & &\
J00391218-7256366 & 0 39 12.18 & -72 56 36.7 & 9.52 & 0.89 & 13.03 & 1.56 & 2 & 130.28 & 41.7 & 1 & & & & SkKM 8\
J00391298-7222186 & 0 39 12.99 & -72 22 18.8 & 11.14 & 0.36 & 12.73 & 0.8 & 5 & 47.31 & 94.8 & 3 & & & &\
J00391532-7237084 & 0 39 15.33 & -72 37 08.4 & 11.31 & 0.41 & 13.02 & & 3 & 7.68 & 69.5 & 3 & & & &\
J00391534-7219542 & 0 39 15.35 & -72 19 54.4 & 11.04 & 0.37 & 12.53 & 0.39 & 4 & 13.51 & 92.2 & 3 & & & &\
J00391791-7301226 & 0 39 17.92 & -73 01 22.6 & 9.24 & 0.52 & 11.16 & & 3 & 29.28 & 126.6 & 3 & & G7V & 12 &\
J00393161-7233027 & 0 39 31.60 & -72 33 02.8 & 10.4 & 0.48 & 12.44 & 0.77 & 4 & -14.79 & 98.6 & 3 & & & &\
J00393530-7157347 & 0 39 35.32 & -71 57 34.8 & 10.97 & 0.28 & 12.28 & & 3 & -4.19 & 78.2 & 3 & & & &\
J00393667-7406537 & 0 39 36.66 & -74 06 53.7 & 11.37 & 0.46 & 13.27 & & 3 & 16.97 & 34.7 & 3 & & & &\
J00400920-7234423 & 0 40 09.21 & -72 34 42.5 & 10.86 & 0.21 & 11.85 & & 3 & -3.15 & 68.4 & 3 & & F0 & 6 & Flo 110\
J00400936-7305417 & 0 40 09.35 & -73 05 41.8 & 10.05 & 0.62 & 12.46 & & 3 & 25.38 & 49.9 & 3 & & K1IV & 12 &\
J00401981-7334106 & 0 40 19.80 & -73 34 10.5 & 10.94 & 0.32 & 12.41 & 0.58 & 1 & 15.02 & 88 & 3 & & G0III/IV & 22 & Flo 111\
J00405279-7415045 & 0 40 52.80 & -74 15 04.5 & 10.73 & 0.61 & 13.09 & & 3 & -34.73 & 48.4 & 3 & & G8III/IV & 22 &\
J00410308-7407593 & 0 41 03.07 & -74 07 59.4 & 10.8 & 0.41 & 12.39 & 0.69 & 4 & -27.03 & 104.7 & 3 & & & &\
J00410383-7200377 & 0 41 03.82 & -72 00 38.0 & 10.62 & 0.28 & 11.91 & & 3 & 25.41 & 120.4 & 3 & & F7V & 6 & Flo 117\
J00411604-7232167 & 0 41 16.05 & -72 32 16.8 & 8.66 & 0.8 & 11.63 & & 3 & 203.58 & 68.9 & 1 & & G6Iab & 12 &\
J00413588-7410375 & 0 41 35.88 & -74 10 37.5 & 6.47 & 0.69 & 9.01 & & 3 & -8.13 & 133.3 & 3 & & K1III & 9 & HD 4067\
J00414144-7334317 & 0 41 41.47 & -73 34 31.7 & 8.09 & 0.84 & 11.17 & 1.32 & 1 & -10.48 & 120.1 & 3 & & & &\
J00414223-7216541 & 0 41 42.25 & -72 16 54.2 & 11.22 & 0.35 & 12.75 & & 3 & -10.59 & 104.7 & 3 & & & &\
J00414344-7343239 & 0 41 43.45 & -73 43 23.9 & 9.49 & 0.58 & 12.09 & 1.33 & 1 & 97.42 & 99.4 & 2 & & G0Iab & 12 & HV 821; RMC 1\
J00414355-7329119 & 0 41 43.57 & -73 29 11.9 & 9.04 & 0.64 & 11.45 & 1.04 & 1 & 46.02 & 97.3 & 3 & & K2IV & 12 &\
J00420472-7203063 & 0 42 04.72 & -72 03 06.5 & 10.49 & 0.28 & 11.7 & & 3 & 21.11 & 78.6 & 3 & & F5I: & 6 & Flo 131\
J00423897-7355507 & 0 42 38.96 & -73 55 50.6 & 11.25 & 0.32 & 12.59 & 0.57 & 1 & 10.67 & 80.2 & 3 & & & &\
J00424953-7338083 & 0 42 49.54 & -73 38 08.2 & 10.21 & 0.56 & 12.46 & 0.88 & 1 & -6.68 & 139.1 & 3 & & G8IV & 12 &\
J00430535-7328062 & 0 43 05.36 & -73 28 06.1 & 11.11 & 0.58 & 13.3 & 0.87 & 1 & -2.76 & 29.8 & 3 & & & &\
J00430758-7327500 & 0 43 07.57 & -73 27 49.8 & 11.04 & 0.34 & 12.52 & 0.61 & 1 & 17.62 & 85.1 & 3 & & & &\
J00431164-7323109 a & 0 43 11.64 & -73 23 10.8 & 11.5 & 0.18 & 11.98 & 0.04 & 2 & 79.33 & 51.5 & 1 & & B8Ia+ & 7 & Sk 3; AzV 2; RMC 2\
J00431458-7330139 & 0 43 14.58 & -73 30 13.8 & 11.19 & 0.31 & 12.58 & 0.52 & 1 & 68.63 & 78.8 & 3 & & & &\
J00431497-7337064 & 0 43 14.96 & -73 37 06.3 & 9.34 & 0.68 & 11.85 & 1.06 & 1 & -15.48 & 103.9 & 3 & & K1III & 12 & Flo 146\
J00431668-7400527 & 0 43 16.66 & -74 00 52.5 & 11.01 & 0.41 & 12.72 & 0.76 & 1 & 53.27 & 81.7 & 3 & & & &\
J00431939-7209434 & 0 43 19.39 & -72 09 43.5 & 11.46 & 0.41 & 13.25 & & 3 & 10.83 & 74.4 & 3 & G8V & & &\
J00433304-7137483 & 0 43 33.04 & -71 37 48.3 & 11.78 & 0.29 & 13.21 & & 3 & -12.83 & 34.2 & 3 & G2V & & &\
J00433619-7217166 & 0 43 36.20 & -72 17 16.6 & 11.11 & 0.58 & 13.36 & 1 & 4 & -23.17 & 59.1 & 3 & G8I & & &\
J00434441-7401260 & 0 43 44.39 & -74 01 25.9 & 10.56 & 0.91 & 14.01 & 1.42 & 1 & 135.57 & 62.6 & 1 & & & &\
J00434661-7133027 & 0 43 46.60 & -71 33 02.7 & 10.26 & 0.7 & 12.93 & & 3 & -35.33 & 72.5 & 3 & & & &\
J00440978-7325464 & 0 44 09.78 & -73 25 46.5 & 9.83 & 0.68 & 12.5 & 1.18 & 1 & 49.24 & 91 & 3 & & K2.5IV & 12 &\
J00441154-7211122 & 0 44 11.55 & -72 11 12.3 & 10.52 & 0.72 & 13.21 & & 3 & 70.23 & 84.7 & 3 & G8III & & &\
J00442211-7134588 & 0 44 22.11 & -71 34 58.7 & 11.58 & 0.18 & 12.53 & & 3 & 30.77 & 66.6 & 3 & & & &\
J00442921-7214555 & 0 44 29.21 & -72 14 55.4 & 8.77 & 0.26 & 9.95 & & 3 & 1.13 & 74.1 & 3 & F5II & F3V & 9 & HD 4344; Flo 159\
J00443797-7342507 & 0 44 37.96 & -73 42 50.8 & 11.15 & 0.56 & 13.39 & 1 & 1 & 34.07 & 27.9 & 3 & & & &\
J00444592-7154502 & 0 44 45.92 & -71 54 50.2 & 11.46 & 0.42 & 13.14 & & 3 & -14.77 & 50.9 & 3 & G8V & & &\
J00450014-7313452 & 0 45 00.16 & -73 13 45.2 & 6.35 & 0.91 & 9.58 & & 3 & 12.04 & 82.3 & 3 & & K0 & 1 & HD 4420\
J00450023-7222303 & 0 45 00.26 & -72 22 30.3 & 11.52 & 0.38 & 13.1 & 0.56 & 4 & -11.07 & 39.4 & 3 & G2I & & &\
J00450023-7310065 & 0 45 00.24 & -73 10 06.5 & 10.52 & 0.63 & 12.97 & 1.03 & 1 & 40.84 & 46.5 & 3 & & K0II/III & 22 & AzV 23F\
J00451179-7307093 & 0 45 11.80 & -73 07 09.2 & 9.16 & 0.7 & 11.71 & 1.03 & 1 & -21.74 & 84.7 & 3 & & G3II/III & 22 &\
J00451547-7415010 & 0 45 15.48 & -74 15 00.9 & 11.67 & 0.31 & 13.16 & 0.59 & 5 & -11.93 & 100.8 & 3 & & & &\
J00451741-7415422 & 0 45 17.40 & -74 15 42.1 & 11.1 & 0.61 & 13.42 & 0.89 & 4 & 31.37 & 95 & 3 & & & &\
J00451949-7144466 & 0 45 19.50 & -71 44 46.6 & 10.68 & 0.65 & 13.08 & & 3 & -5.83 & 112.3 & 3 & & & &\
J00452017-7202576 & 0 45 20.17 & -72 02 57.6 & 11.67 & 0.24 & 12.86 & & 3 & 1.53 & 73 & 3 & F5V & & &\
J00454118-7156104 & 0 45 41.19 & -71 56 10.5 & 10.96 & 0.7 & 13.53 & 1.07 & 4 & 13.73 & 61.3 & 3 & G8V & & &\
J00454793-7144342 & 0 45 47.93 & -71 44 34.2 & 11.35 & 0.33 & 12.77 & & 3 & -3.73 & 108.3 & 3 & & & &\
J00460240-7321000 & 0 46 02.39 & -73 21 00.0 & 10.86 & 0.75 & 13.77 & 1.37 & 1 & 133.94 & 62.9 & 1 & & G5Iab & 12 &\
J00462301-7146568 a & 0 46 23.02 & -71 46 56.7 & 6.07 & 0.86 & 9.19 & & 3 & 1.15 & 101.9 & 3 & K2III & K2III & 9 & HD 4577\
J00462423-7320084 & 0 46 24.23 & -73 20 08.4 & 10.18 & 0.23 & 11.3 & & 3 & -2.66 & 65.4 & 3 & & F9III & 12 & Flo 181\
J00462977-7158227 & 0 46 29.77 & -71 58 22.6 & 11.09 & 0.46 & 13.15 & 0.92 & 5 & -29.77 & 80 & 3 & K0III & & &\
J00463746-7316142 a & 0 46 37.48 & -73 16 14.2 & 11.21 & 0.34 & 12.75 & 0.71 & 1 & 28.3 & 102.6 & 3 & G2V & & &\
J00464324-7126321 & 0 46 43.25 & -71 26 32.2 & 10.33 & 0.47 & 12.25 & & 3 & 9.57 & 87 & 3 & & & &\
J00464964-7120125 & 0 46 49.65 & -71 20 12.7 & 8.44 & 0.78 & 11.23 & & 3 & 6.17 & 88.9 & 3 & & & &\
J00464984-7313525 a & 0 46 49.85 & -73 13 52.5 & 10.21 & 0.91 & 13.58 & 1.19 & 1 & 106.8 & 70.1 & 2 & & K0Iab/Ib & 22 &\
J00465308-7242518 & 0 46 53.11 & -72 42 51.7 & 10.23 & 0.53 & 12.04 & 0.63 & 2 & 141.48 & 38.1 & 1 & & G0Iab & 12 & HV 824; cepheid\
J00465795-7132158 & 0 46 57.95 & -71 32 15.9 & 11 & 0.42 & 12.67 & & 3 & -21.93 & 94.8 & 3 & & & &\
J00465844-7216256 & 0 46 58.44 & -72 16 25.7 & 9.89 & 0.88 & 13.16 & 1.25 & 2 & 153.53 & 105 & 1 & F8 & & & SkKM 15\
J00470092-7322137 a & 0 47 00.92 & -73 22 13.6 & 11.61 & 0.37 & 13.22 & 0.51 & 1 & 166.8 & 26.7 & 1 & A5:I: & & &\
J00470493-7249319 & 0 47 04.93 & -72 49 31.8 & 8.59 & 0.26 & 9.78 & 0.46 & 2 & -5.32 & 25.7 & 3 & & F9IV/V & 12 & HD 4651\
J00470584-7311315 a & 0 47 05.87 & -73 11 31.6 & 10.06 & 0.84 & 13.2 & 1.31 & 1 & 135.7 & 71.8 & 1 & & & & SkKM 18\
J00470689-7241144 & 0 47 06.89 & -72 41 14.4 & 8.57 & 0.71 & 11.45 & & 3 & 12.78 & 76.6 & 3 & & K3III/IV & 22 &\
J00470869-7314119 a & 0 47 08.69 & -73 14 11.9 & 9.94 & 0.9 & 13 & 0.69 & 1 & 112.3 & 77.2 & 2 & & G6Ia+Bne & 22 & LHA 115-S 7; RMC 14F\
J00471122-7130154 & 0 47 11.22 & -71 30 15.5 & 11.37 & 0.34 & 13.05 & 0.5 & 4 & 2.97 & 100.9 & 3 & & & &\
J00471355-7208364 & 0 47 13.55 & -72 08 36.5 & 9.83 & 0.6 & 12.18 & & 3 & 66.93 & 115.8 & 3 & G8III & & &\
J00472498-7137576 & 0 47 24.98 & -71 37 57.6 & 11.13 & 0.48 & 13.01 & 0.62 & 5 & 36.87 & 81.4 & 3 & & & &\
J00473213-7407291 & 0 47 32.13 & -74 07 29.2 & 12.61 & 0.02 & 12.97 & & 3 & 126.47 & 24.7 & 1 & & A2II & 3 & 2dFS 0655\
J00473438-7140232 & 0 47 34.37 & -71 40 23.2 & 12.03 & 0.19 & 13.02 & & 3 & 71.27 & 31.3 & 1 & & & &\
J00473872-7307488 & 0 47 38.74 & -73 07 48.8 & 12.17 & 0.02 & 12.24 & -0.07 & 2 & 143.7 & 48.8 & 1 & B2I & B5Ia & 7 & AzV 22; Sk 15\
J00473889-7322539 a & 0 47 38.90 & -73 22 53.8 & 11.96 & 0.06 & 12.18 & 0.09 & 1 & 166.2 & 43.2 & 1 & B2I & B3Ia & 7 & AzV 23; Sk 17\
J00474278-7130575 & 0 47 42.78 & -71 30 57.6 & 10.05 & 0.59 & 12.28 & & 3 & -30.93 & 121.8 & 3 & & & &\
J00474745-7355289 & 0 47 47.45 & -73 55 29.0 & 8.47 & 0.6 & 10.66 & & 3 & 4.07 & 99.9 & 3 & & & &\
J00474903-7218195 & 0 47 49.04 & -72 18 19.8 & 9.9 & 0.59 & 12.29 & 0.94 & 4 & 50.23 & 114.7 & 3 & K0III & & &\
J00475033-7314227 & 0 47 50.35 & -73 14 22.7 & 11.63 & 0.09 & 12.14 & 0.14 & 1 & 112.46 & 55.5 & 1 & B8I & A2I & 8 & AzV 27; Sk 19\
J00475150-7144545 & 0 47 51.49 & -71 44 54.5 & 11.16 & 0.37 & 12.71 & & 3 & -4.13 & 44.4 & 3 & & & &\
J00480629-7306379 a & 0 48 06.31 & -73 06 38.0 & 11.5 & 0.21 & 12.47 & 0.27 & 1 & 104.9 & 58.7 & 1 & B8:I: & A0 & 14 & AzV 31; Sk 20\
J00480954-7221283 & 0 48 09.55 & -72 21 28.6 & 10.54 & 0.92 & 13.93 & 1.46 & 1 & 152.93 & 63.9 & 1 & & G8Iab & 12 & SkKM 26\
J00482593-7146264 & 0 48 25.94 & -71 46 26.3 & 11.02 & 0.57 & 13.31 & 0.98 & 5 & 33.17 & 98.2 & 3 & & & &\
J00482632-7209508 & 0 48 26.33 & -72 09 50.9 & 10.89 & 0.33 & 12.38 & 0.61 & 1 & -20.47 & 79.3 & 3 & G0V & & &\
J00482768-7300412 & 0 48 27.69 & -73 00 41.2 & 12.39 & 0.04 & 12.79 & 0.12 & 1 & 107.36 & 65.2 & 1 & B9:V: & A0Ia & 13 & AzV 38; Sk 24\
J00483182-7259116 & 0 48 31.83 & -72 59 11.5 & 10.23 & 0.86 & 13.63 & 1.52 & 1 & 121.42 & 113.8 & 2 & & G7Ib & 12 &\
J00483517-7222102 a & 0 48 35.18 & -72 22 10.3 & 10.09 & 0.88 & 13.58 & 1.65 & 1 & 176.69 & 77.8 & 1 & G2? & G8Iab & 12 & SkKM 32\
J00483708-7347222 & 0 48 37.10 & -73 47 22.3 & 11.25 & 0.28 & 12.57 & 0.53 & 1 & 1.46 & 40 & 3 & & & &\
J00484483-7320190 & 0 48 44.86 & -73 20 18.9 & 8.03 & 0.73 & 10.75 & & 3 & 3.36 & 124 & 3 & K2III & K2III/IV & 22 & Flo 208\
J00484542-7255274 & 0 48 45.44 & -72 55 27.4 & 10.63 & 0.86 & 14.11 & 1.51 & 1 & 136.62 & 73.3 & 1 & & G8Iab & 12 &\
J00484775-7304569 a & 0 48 47.75 & -73 04 57.0 & 10.41 & 0.54 & 12.6 & 0.9 & 1 & 32.2 & 110.5 & 3 & G5V & G6III & 12 &\
J00484835-7239005 & 0 48 48.37 & -72 39 00.6 & 10.62 & 0.89 & 13.94 & 1.45 & 1 & 136.49 & 66.3 & 1 & & & & SkKM 34\
J00485433-7332057 & 0 48 54.36 & -73 32 05.8 & 11.5 & 0.36 & 12.96 & 0.58 & 1 & -9.86 & 66.4 & 3 & & & &\
J00490129-7230266 & 0 49 01.32 & -72 30 26.7 & 10.25 & 0.24 & 11.29 & & 3 & 31.59 & 40.3 & 3 & A8I & A3V & 6 & Flo 211\
J00490294-7150393 & 0 49 02.96 & -71 50 39.2 & 9.59 & 0.71 & 12.34 & & 3 & -28.57 & 90 & 3 & K2:III & & &\
J00490296-7321409 & 0 49 02.97 & -73 21 40.9 & 11.05 & 0.03 & 11.02 & -0.04 & 2 & 137.36 & 66.7 & 1 & B2I & B5Ia & 7 & HD 4862; Sk 27; RMC 5\
J00490802-7217461 a & 0 49 08.02 & -72 17 46.3 & 8.97 & 0.55 & 11.18 & & 3 & 29.16 & 121.7 & 3 & K0V & K0V & 12 & Flo 213\
J00490961-7219017 & 0 49 09.62 & -72 19 01.8 & 9.45 & 0.67 & 12 & 1.12 & 1 & -36.31 & 106.7 & 3 & K2III & K0III & 12 &\
J00491181-7241346 & 0 49 11.83 & -72 41 34.7 & 10.05 & 0.87 & 13.45 & 1.58 & 1 & 140.99 & 68.7 & 1 & G2I & & & SkKM 40\
J00491408-7204322 & 0 49 14.10 & -72 04 32.4 & 9.63 & 0.91 & 13.2 & 1.65 & 1 & 155.03 & 89.6 & 1 & G8III & & & SkKM 42\
J00492430-7326449 a & 0 49 24.32 & -73 26 45.0 & 9.89 & 0.42 & 11.63 & & 3 & 30 & 120.3 & 3 & G5V & & &\
J00493436-7217233 a & 0 49 34.37 & -72 17 23.5 & 11.38 & 0.43 & 13.15 & 0.66 & 1 & 109.36 & 77 & 3 & G0 & G0 & 3 & 2dFS 0733\
J00494543-7252477 & 0 49 45.44 & -72 52 47.7 & 12.32 & 0.06 & 12.77 & 0.18 & 1 & 144.12 & 48.9 & 1 & A0:I: & A0I & 8 & AzV 53\
J00494643-7227462 & 0 49 46.42 & -72 27 46.3 & 12.26 & 0.05 & 12.66 & 0.11 & 1 & 198.69 & 42.2 & 1 & A3III & & &\
J00495125-7255452 & 0 49 51.27 & -72 55 45.2 & 11.02 & 0.06 & 11.16 & 0 & 2 & 135.92 & 55.2 & 1 & B2I & B2.5Ia & 7 & AzV 56; Sk 31\
J00495574-7302508 & 0 49 55.74 & -73 02 50.8 & 10.38 & 0.82 & 13.53 & 1.21 & 1 & 141.56 & 68.2 & 1 & & & & SkKM 50\
J00495575-7353183 & 0 49 55.75 & -73 53 18.3 & 10.64 & 0.58 & 12.88 & 0.87 & 1 & 2.86 & 34.3 & 3 & & K2V & 12 &\
J00495705-7158295 & 0 49 57.04 & -71 58 29.6 & 11.44 & 0.46 & 13.36 & 0.86 & 4 & 25.93 & 93.2 & 3 & G8III & & &\
J00500252-7245241 & 0 50 02.52 & -72 45 24.2 & 10.9 & 0.3 & 12.2 & 0.51 & 1 & 16.89 & 84.4 & 3 & G0I & F9III & 12 &\
J00500608-7307452 a & 0 50 06.08 & -73 07 45.3 & 10.46 & 0.14 & 11.02 & 0.13 & 2 & 173.6 & 64.1 & 1 & B5I & B8Ia+ & 7 & HD 4976; Sk 33; RMC 6\
J00501135-7211229 & 0 50 11.35 & -72 11 23.0 & 10.14 & 0.87 & 13.49 & 1.55 & 1 & 119.93 & 106.5 & 2 & G5? & & & SkKM 53\
J00501239-7309580 a & 0 50 12.38 & -73 09 58.1 & 11.09 & 0.39 & 12.73 & 0.67 & 1 & 1.2 & 73.6 & 3 & G5V & & &\
J00501256-7129348 & 0 50 12.56 & -71 29 34.9 & 8.26 & 0.32 & 9.54 & 0.55 & 2 & 22.47 & 124.6 & 3 & & F7V & 9 & HD 4969\
J00501967-7336112 a & 0 50 19.63 & -73 36 11.2 & 6.85 & 0.91 & 10.18 & & 3 & 55.5 & 110 & 3 & & K3V & 6 & Flo 229; HD 5003\
J00502124-7306094 & 0 50 21.25 & -73 06 09.5 & 10.37 & 0.85 & 13.69 & 1.53 & 1 & 135.76 & 76.6 & 1 & & & &\
J00502177-7126485 & 0 50 21.78 & -71 26 48.6 & 11.31 & 0.29 & 12.57 & 0.38 & 5 & 1.87 & 112.9 & 3 & & & &\
J00502480-7200011 & 0 50 24.80 & -72 00 01.3 & 9.89 & 0.87 & 13.37 & 1.62 & 4 & 163.13 & 84 & 1 & & & & SkKM 55\
J00502879-7245092 a & 0 50 28.79 & -72 45 09.2 & 9.88 & 0.46 & 11.63 & 0.63 & 1 & 119.09 & 91.8 & 2 & & G0Ib & 15 & RMC 7; cepheid\
J00503106-7252207 & 0 50 31.07 & -72 52 20.7 & 11.27 & 0.41 & 12.9 & 0.6 & 1 & 150.12 & 41.4 & 1 & & O5.5I(f) & 28 & Sk 38\
J00503158-7328425 a & 0 50 31.58 & -73 28 42.6 & 10.79 & 0.1 & 11.18 & 0.11 & 2 & 169.3 & 69.8 & 1 & A2I & B9Ia+ & 7 & HD 5030; Sk 39; RMC 8\
J00503368-7315249 & 0 50 33.68 & -73 15 24.9 & 10.05 & 0.44 & 11.94 & 0.78 & 1 & 167.04 & 57.1 & 1 & & & &\
J00503655-7229108 & 0 50 36.55 & -72 29 10.8 & 11.46 & 0.25 & 12.64 & 0.52 & 1 & 19.59 & 96.5 & 3 & F8V & & &\
J00503839-7328182 a & 0 50 38.40 & -73 28 18.2 & 11.03 & 0.04 & 11.04 & -0.02 & 2 & 165.1 & 59.2 & 1 & B2:I: & B1Ia+ & 12 & HD 5045; AzV 78; RMC 9\
J00504262-7236481 & 0 50 42.60 & -72 36 48.1 & 10.57 & 0.63 & 12.98 & 0.98 & 1 & 11.99 & 79.8 & 3 & & G7IV & 12 &\
J00504717-7242576 a & 0 50 47.16 & -72 42 57.6 & 8.92 & 0.92 & 12.66 & 1.69 & 1 & 136.09 & 67 & 1 & & K1Ia/Iab & 22 & SkKM 58\
J00504958-7241541 & 0 50 49.58 & -72 41 54.1 & 10.03 & 0.9 & 13.49 & 1.47 & 1 & 113.99 & 87.3 & 2 & & G8Ia/Iab & 22 & SkKM 62\
J00505473-7337239 & 0 50 54.73 & -73 37 24.0 & 11.28 & 0.25 & 12.53 & 0.48 & 1 & -23.54 & 79.9 & 3 & & A7V & 6 & Flo 237\
J00505814-7138174 & 0 50 58.13 & -71 38 17.4 & 10 & 0.66 & 12.55 & & 3 & 13.17 & 87.3 & 3 & & & &\
J00505824-7241343 & 0 50 58.25 & -72 41 34.2 & 10.29 & 0.35 & 11.87 & & 3 & -15.22 & 52.3 & 3 & & G0V & 12 &\
J00510268-7236448 & 0 51 02.69 & -72 36 44.9 & 11.2 & 0.37 & 12.7 & 0.6 & 1 & 7.89 & 94.6 & 3 & G0V & & &\
J00511191-7228177 & 0 51 11.91 & -72 28 17.8 & 12.06 & 0.12 & 12.54 & 0.08 & 1 & 141.09 & 39.5 & 1 & A2I & A2I & 8 & AzV 90; Sk 44\
J00511365-7233016 & 0 51 13.65 & -72 33 01.6 & 10.73 & 0.81 & 13.87 & 1.41 & 1 & 144.89 & 92.1 & 1 & & G8Ib & 12 & SkKM 68\
J00511592-7135136 & 0 51 15.92 & -71 35 13.6 & 10.26 & 0.9 & 13.8 & 1.46 & 5 & 153.97 & 92.8 & 1 & & & &\
J00511737-7133589 & 0 51 17.38 & -71 33 58.8 & 9.93 & 0.41 & 11.52 & & 3 & 36.27 & 123.2 & 3 & & & &\
J00511824-7243246 & 0 51 18.25 & -72 43 24.7 & 10.59 & 0.86 & 13.77 & 1.33 & 1 & 150.79 & 108.7 & 1 & & & &\
J00513037-7316098 & 0 51 30.37 & -73 16 09.9 & 11.76 & 0.07 & 12.13 & 0.07 & 2 & 159.26 & 56.9 & 1 & A0I & B9Ia & 7 & AzV 101; Sk 47\
J00513209-7255507 & 0 51 32.09 & -72 55 50.7 & 11.01 & 0.45 & 12.73 & 0.59 & 1 & 142.46 & 58.8 & 1 & A5:I? & & &\
J00513554-7219580 & 0 51 35.52 & -72 19 58.1 & 8.97 & 0.68 & 11.51 & 1.09 & 1 & 32.99 & 115 & 3 & G8III & K0III & 12 &\
J00513658-7355459 & 0 51 36.56 & -73 55 45.9 & 8.57 & 0.72 & 11.15 & 1.14 & 1 & 59.76 & 86.4 & 3 & & K0III & 12 & Flo 246\
J00513764-7225596 & 0 51 37.64 & -72 25 59.6 & 9.57 & 0.88 & 12.97 & 1.64 & 1 & 155.59 & 84.1 & 1 & & G8.5Iab & 12 & SkKM 76\
J00513784-7240061 & 0 51 37.85 & -72 40 06.2 & 10.67 & 0.56 & 12.83 & 0.74 & 1 & 145.69 & 49.4 & 1 & F8V & & &\
J00513860-7131570 & 0 51 38.61 & -71 31 57.1 & 7.38 & 0.69 & 10.02 & & 3 & -18.33 & 93.3 & 3 & & K0 & 9 & HD 5115\
J00515197-7244135 & 0 51 51.98 & -72 44 13.5 & 11.72 & 0.04 & 12.13 & 0.08 & 2 & 147.09 & 57.9 & 1 & A3Ia & A0Ia & 13 & AzV 110; Sk 49\
J00515354-7223510 & 0 51 53.54 & -72 23 51.0 & 10.45 & 0.63 & 12.88 & 1 & 1 & -23.61 & 99 & 3 & G5V & & &\
J00515545-7248584 & 0 51 55.47 & -72 48 58.3 & 11.09 & 0.45 & 12.66 & 0.55 & 1 & -27.41 & 71.3 & 3 & G0IV & F0 & 11 & AzV 30F\
J00515983-7245596 & 0 51 59.84 & -72 45 59.7 & 12.22 & 0.15 & 12.8 & 0.11 & 1 & 118.29 & 40.7 & 1 & A3I & A1 & 11 & AzV 32F\
J00515998-7303332 & 0 51 59.98 & -73 03 33.3 & 10.65 & 0.86 & 13.72 & 1.15 & 1 & 150.86 & 110.3 & 1 & & & &\
J00520744-7333007 & 0 52 07.46 & -73 33 00.8 & 11.05 & 0.4 & 12.63 & 0.75 & 1 & 55.16 & 101.4 & 3 & & & &\
J00520803-7216491 a & 0 52 08.03 & -72 16 49.3 & 10.61 & 0.89 & 13.96 & 1.43 & 1 & 154.83 & 75.8 & 1 & & K1Ib & 12 & SkKM 83\
J00521105-7320365 & 0 52 11.05 & -73 20 36.6 & 11.85 & 0.17 & 12.53 & 0.19 & 1 & 143.36 & 44.6 & 1 & A2:I & & &\
J00521331-7334478 & 0 52 13.32 & -73 34 48.0 & 11.59 & 0.32 & 12.92 & 0.57 & 1 & 39.96 & 73.3 & 3 & & & &\
J00521343-7253209 & 0 52 13.43 & -72 53 21.0 & 8.55 & 0.67 & 11.03 & 1.07 & 1 & 16.76 & 95.5 & 3 & K0III & K1III/IV & 22 & Flo 255\
J00521905-7309229 & 0 52 19.06 & -73 09 23.2 & 10.74 & 0.77 & 13.26 & 0.66 & 1 & 138.26 & 72.9 & 1 & & G0Iab & 12 &\
J00522498-7241029 & 0 52 24.98 & -72 41 03.2 & 12.4 & 0.1 & 12.69 & 0.03 & 1 & 150.09 & 45.2 & 1 & B5I & B8I & 8 & AzV 122\
J00522757-7318563 & 0 52 27.57 & -73 18 56.4 & 12.46 & 0.1 & 12.54 & 0.03 & 1 & 160.56 & 48.1 & 1 & B2I & B3Ia & 7 & AzV 125; Sk 52\
J00523079-7302581 & 0 52 30.80 & -73 02 58.2 & 9.39 & 0.64 & 11.81 & 1.02 & 1 & 50.46 & 106 & 3 & K0III & K0III & 12 &\
J00523081-7313144 & 0 52 30.82 & -73 13 14.6 & 11.44 & 0.25 & 12.42 & 0.27 & 1 & 127.86 & 41.3 & 1 & A2:I & F2I & 8 & AzV 127\
J00523150-7211374 a & 0 52 31.49 & -72 11 37.5 & 9.41 & 0.89 & 12.98 & 1.65 & 1 & 160.13 & 85.3 & 1 & & G8Iab & 12 & SkKM 91\
J00523564-7251053 & 0 52 35.65 & -72 51 05.3 & 9.74 & 0.85 & 13.04 & 1.5 & 1 & 129.19 & 80.5 & 1 & K0:III: & & & SkKM 94\
J00524734-7320299 & 0 52 47.37 & -73 20 30.0 & 11.38 & 0.37 & 12.95 & 0.67 & 1 & 0.66 & 73.9 & 3 & G2V & & &\
J00525020-7213199 a & 0 52 50.19 & -72 13 20.0 & 10.41 & 0.89 & 13.76 & 1.5 & 1 & 151.23 & 94.2 & 1 & & & &\
J00525121-7306535 & 0 52 51.23 & -73 06 53.6 & 10.42 & 0.12 & 10.95 & 0.13 & 2 & 137.86 & 61.3 & 1 & A2I & A0Ia & 10 & HD 5277; Sk 54; RMC 10\
J00525704-7212000 a & 0 52 57.05 & -72 12 00.0 & 10.79 & 0.54 & 12.88 & 0.82 & 1 & 32.93 & 109.1 & 3 & K0V & & &\
J00530146-7223591 & 0 53 01.46 & -72 23 59.2 & 10.51 & 0.83 & 13.61 & 1.27 & 1 & 150.39 & 93 & 1 & & G2 & 3 & 2dFS 0905\
J00530165-7230305 & 0 53 01.65 & -72 30 30.6 & 10.67 & 0.79 & 13.68 & 1.35 & 1 & 138.29 & 77.3 & 1 & & & & SkKM 99\
J00530229-7154161 & 0 53 02.29 & -71 54 16.1 & 11.29 & 0.38 & 12.92 & & 3 & 10.36 & 86.4 & 3 & & & &\
J00530489-7238000 & 0 53 04.89 & -72 38 00.1 & 10.57 & 0.06 & 10.85 & 0.04 & 2 & 152.44 & 48.2 & 1 & & B8Ia+ & 7 & HD 5291; Sk 56; RMC 11\
J00530773-7307095 & 0 53 07.74 & -73 07 09.6 & 12.45 & 0.02 & 12.65 & 0.06 & 1 & 153.76 & 52.2 & 1 & B8:I: & & &\
J00530894-7229386 & 0 53 08.94 & -72 29 38.7 & 8.45 & 0.91 & 11.91 & 1.77 & 1 & 133.29 & 75.2 & 1 & G5I & K3Ia & 12 & SkKM 104\
J00530985-7313248 & 0 53 09.86 & -73 13 24.9 & 9.61 & 0.31 & 11.01 & & 3 & 8.76 & 105.3 & 3 & G0V & F9III & 12 & Flo 270\
J00531104-7257039 & 0 53 11.07 & -72 57 03.9 & 9.58 & 0.69 & 12.19 & 1.18 & 1 & 1.46 & 104.2 & 3 & K1III & K1III/IV & 22 &\
J00531572-7222029 & 0 53 15.72 & -72 22 03.0 & 12 & 0.19 & 12.79 & 0.21 & 1 & 130.29 & 38.2 & 1 & A5I & F5I? & 8 & AzV 142\
J00531596-7315325 & 0 53 15.96 & -73 15 32.6 & 12.19 & 0.14 & 12.82 & 0.19 & 1 & 158.86 & 50.6 & 1 & A3:II: & A5 & 11 & AzV 37F\
J00532140-7217166 a & 0 53 21.41 & -72 17 16.7 & 10.98 & 0.35 & 12.35 & 0.5 & 1 & 168.43 & 36.2 & 1 & F0I & & &\
J00532362-7247013 a & 0 53 23.61 & -72 47 01.4 & 10.12 & 0.82 & 13.32 & 1.21 & 1 & 155.52 & 88.3 & 1 & & G8Iab & 12 & SkKM 108\
J00532650-7251595 a & 0 53 26.51 & -72 51 59.6 & 10.24 & 0.71 & 12.85 & 1.11 & 1 & 24.22 & 87.9 & 3 & G8I & K1III & 12 &\
J00533093-7231577 & 0 53 30.92 & -72 31 57.9 & 10.2 & 0.89 & 13.61 & 1.56 & 1 & 136.99 & 73.7 & 1 & & & &\
J00533726-7223557 & 0 53 37.26 & -72 23 55.8 & 12.22 & 0.19 & 12.99 & 0.23 & 1 & 106.49 & 35.6 & 3 & A5I & & &\
J00533784-7232393 & 0 53 37.82 & -72 32 39.4 & 11.56 & 0.29 & 12.7 & 0.34 & 1 & 167.89 & 29.7 & 1 & F0I & & &\
J00534149-7352097 & 0 53 41.51 & -73 52 09.9 & 7.19 & 0.3 & 8.44 & 0.51 & 2 & 12.36 & 81.6 & 3 & & F6IV & 9 & Flo 279; HD 5370\
J00534262-7217135 & 0 53 42.65 & -72 17 13.7 & 10.28 & 0.41 & 11.89 & 0.59 & 1 & 143.99 & 77.6 & 1 & G2:I & F2Ib & 15 & RMC 12\
J00534575-7253389 & 0 53 45.76 & -72 53 39.0 & 9.62 & 0.81 & 12.94 & 1.57 & 1 & 141.64 & 67.7 & 1 & & G6Iab & 12 & SkKM 111\
J00535656-7254396 & 0 53 56.56 & -72 54 39.6 & 10.49 & 0.9 & 13.63 & 1.12 & 1 & 135.64 & 56.2 & 1 & & G8II & 12 & SkKM 118\
J00535761-7228404 & 0 53 57.59 & -72 28 40.5 & 12.58 & 0.02 & 12.62 & 0.01 & 1 & 142.79 & 46.5 & 1 & B8I & B9I & 8 & AzV 150\
J00540272-7315318 & 0 54 02.71 & -73 15 32.0 & 12.17 & 0.21 & 12.92 & 0.2 & 1 & 116.66 & 35.8 & 1 & & & &\
J00540321-7231445 & 0 54 03.21 & -72 31 44.7 & 11.17 & 0.14 & 11.87 & 0.17 & 2 & 117.54 & 41.8 & 1 & & A3I & 14 & AzV 152; Sk 58\
J00540520-7426579 & 0 54 05.20 & -74 26 57.9 & 8 & 0.62 & 10.35 & & 3 & -6.41 & 56.5 & 3 & & K1 & 6 & Flo 290\
J00541781-7322539 & 0 54 17.82 & -73 22 54.1 & 9.67 & 0.67 & 12.2 & 1.02 & 1 & 27.56 & 151.9 & 3 & G8 & K1III/IV & 22 &\
J00542237-7328093 & 0 54 22.39 & -73 28 09.3 & 10.64 & 0.29 & 11.84 & 0.49 & 1 & 5.83 & 28.2 & 3 & F5? & & &\
J00542687-7252596 & 0 54 26.88 & -72 52 59.8 & 9.49 & 0.84 & 12.84 & 1.68 & 1 & 150.84 & 53.6 & 1 & & G7.5Iab & 12 & SkKM 123\
J00543871-7303076 & 0 54 38.71 & -73 03 07.8 & 10.83 & 0.23 & 12.9 & & 3 & -18.04 & 69.7 & 3 & F0III & F9V & 12 & AzV 47F\
J00543893-7221199 & 0 54 38.96 & -72 21 19.8 & 9.8 & 0.91 & 13.32 & 1.58 & 1 & 145.39 & 83.3 & 1 & & G8Ia/Iab & 22 & SkKM 127\
J00544146-7214139 a & 0 54 41.47 & -72 14 13.9 & 12.73 & 0.07 & 12.96 & 0.05 & 1 & 149.43 & 48.6 & 1 & B8I & & &\
J00544483-7325111 & 0 54 44.86 & -73 25 11.2 & 8.53 & 0.71 & 11.18 & 1.27 & 1 & 35.46 & 80.4 & 3 & K0III: & K0.5III & 12 &\
J00545086-7341274 & 0 54 50.86 & -73 41 27.6 & 8.78 & 0.76 & 11.5 & 1.18 & 1 & 39.53 & 90.6 & 3 & K0III & K1III/IV & 22 &\
J00545150-7322073 & 0 54 51.53 & -73 22 07.5 & 11.07 & 0.34 & 12.55 & 0.64 & 1 & 19.66 & 105.1 & 3 & G0:V & & &\
J00545294-7247208 a & 0 54 52.95 & -72 47 21.0 & 11.08 & 0.4 & 12.76 & 0.68 & 1 & 42.02 & 83.5 & 3 & G2V & & &\
J00545445-7309033 & 0 54 54.45 & -73 09 03.6 & 10.29 & 0.7 & 13.12 & 1.38 & 1 & 154.46 & 99.4 & 1 & & G5Ia/Iab & 22 & SkKM 129\
J00545575-7342543 a & 0 54 55.77 & -73 42 54.4 & 8.47 & 0.35 & 9.96 & & 3 & 29.39 & 81.7 & 3 & G8V & G1V & 12 & Flo 304; HD 5498\
J00545763-7304401 & 0 54 57.63 & -73 04 40.3 & 11.38 & 0.33 & 12.68 & 0.46 & 1 & 172.56 & 34.9 & 1 & A5I & & &\
J00550182-7206498 & 0 55 01.82 & -72 06 49.8 & 8.2 & 0.59 & 10.58 & & 3 & 7.16 & 110.4 & 3 & & G & 6 & Flo 302\
J00550377-7300367 & 0 55 03.77 & -73 00 36.9 & 8.86 & 0.87 & 12.4 & 1.75 & 1 & 163.66 & 76.1 & 1 & & G6Ia/Iab & 22 & SkKM 132\
J00550944-7230556 & 0 55 09.44 & -72 30 55.7 & 10.52 & 0.85 & 13.7 & 1.42 & 1 & 148.74 & 90.9 & 1 & & G7.5Iab & 12 &\
J00551077-7227095 & 0 55 10.76 & -72 27 09.6 & 11.39 & 0.27 & 12.56 & 0.43 & 1 & 166.99 & 29.9 & 1 & F0I & & &\
J00551256-7226111 a & 0 55 12.56 & -72 26 11.1 & 11.6 & 0.24 & 12.58 & 0.3 & 1 & 169.42 & 26.1 & 1 & F0I & & &\
J00551277-7230424 & 0 55 12.77 & -72 30 42.4 & 10.46 & 0.88 & 13.67 & 1.35 & 1 & 153.19 & 98.8 & 1 & & & & SkKM 134\
J00551819-7314205 & 0 55 18.21 & -73 14 20.8 & 12.25 & 0.02 & 12.41 & 0.03 & 1 & 153.26 & 55.1 & 1 & A0I & A0I & 13 & AzV 166; Sk 60\
J00552236-7216385 & 0 55 22.37 & -72 16 38.5 & 10.19 & 0.89 & 13.61 & 1.51 & 1 & 152.93 & 73.3 & 1 & & G7.5Iab & 12 & SkKM 135/6\
J00552265-7205218 & 0 55 22.65 & -72 05 21.8 & 11.31 & 0.29 & 12.54 & 0.4 & 1 & 182.26 & 34.9 & 1 & & & &\
J00552776-7243454 a & 0 55 27.77 & -72 43 45.6 & 11.74 & 0.17 & 12.54 & 0.21 & 1 & 117.32 & 35.2 & 1 & A5I & & &\
J00553015-7228496 a & 0 55 30.14 & -72 28 49.7 & 10.53 & 0.89 & 13.78 & 1.34 & 1 & 154.42 & 83 & 1 & & G8Ia/Iab & 22 &\
J00553056-7334291 & 0 55 30.57 & -73 34 29.3 & 9.17 & 0.32 & 10.51 & & 3 & -16.54 & 81.6 & 3 & & F9IV & 12 & HD 5549\
J00553664-7239157 & 0 55 36.64 & -72 39 15.9 & 10.7 & 0.29 & 12.07 & 0.57 & 1 & 31.69 & 129.5 & 3 & G0III & & &\
J00554163-7209429 & 0 55 41.65 & -72 09 42.8 & 11.42 & 0.3 & 12.73 & 0.53 & 1 & 17.96 & 66.7 & 3 & & & &\
J00554756-7416386 & 0 55 47.56 & -74 16 38.7 & 11.53 & 0.28 & 12.87 & & 3 & -23.15 & 68.1 & 3 & & & &\
J00555119-7346230 & 0 55 51.20 & -73 46 23.2 & 9.79 & 0.81 & 12.8 & 1.26 & 1 & 61.39 & 118.9 & 3 & K0III & K2III & 12 &\
J00555508-7240305 & 0 55 55.07 & -72 40 30.6 & 9.34 & 0.65 & 12.02 & 1.32 & 1 & 160.49 & 77.3 & 1 & & G3.5Ia & 12 &\
J00560334-7229081 & 0 56 03.35 & -72 29 08.0 & 10.96 & 0.33 & 12.32 & 0.56 & 1 & -2.28 & 106.9 & 3 & & F8V & 17 & Cl\* NGC 330 Arp i\
J00561519-7152311 & 0 56 15.19 & -71 52 31.2 & 10.31 & 0.92 & 13.81 & 1.62 & 1 & 157.36 & 93.2 & 1 & & K0Iab & 12 & SkKM 147\
J00562256-7228359 & 0 56 22.58 & -72 28 35.8 & 11.9 & 0.17 & 12.31 & 0.11 & 1 & 132.64 & 43.9 & 1 & & A5I & 17 & Cl\* NGC 330 Arp 9\
J00562354-7252534 & 0 56 23.55 & -72 52 53.5 & 10.5 & 0.37 & 12.02 & 0.66 & 1 & 49.84 & 109.7 & 3 & & G1V & 12 &\
J00562643-7328232 & 0 56 26.43 & -73 28 23.3 & 9.62 & 0.88 & 12.74 & 1.66 & 1 & 164.03 & 59.7 & 1 & & K4Ia/Iab & 22 & SkKM 152\
J00562700-7409243 & 0 56 27.00 & -74 09 24.3 & 9.09 & 0.68 & 11.65 & & 3 & 28.65 & 116 & 3 & & & &\
J00562776-7330270 & 0 56 27.75 & -73 30 27.2 & 10.43 & 0.86 & 13.63 & 1.5 & 1 & 164.23 & 39.9 & 1 & & G8Iab & 12 &\
J00563534-7336391 & 0 56 35.34 & -73 36 39.2 & 8.91 & 0.73 & 11.49 & 1.11 & 1 & 9.89 & 82.1 & 3 & K2III & & &\
J00563633-7226464 a & 0 56 36.34 & -72 26 46.5 & 9.86 & 0.92 & 13.54 & 1.78 & 1 & 144.32 & 84.1 & 1 & & G8Ia/Iab & 22 & SkKM 155\
J00563923-7300523 & 0 56 39.22 & -73 00 52.4 & 10.85 & 0.67 & 13.33 & 1.06 & 1 & 74.44 & 33.3 & 2 & & K1.5III & &\
J00564288-7158579 & 0 56 42.89 & -71 58 57.6 & 10.42 & 0.91 & 13.79 & 1.54 & 1 & 160.06 & 78.8 & 1 & & G8Ib & 12 & SkKM 156\
J00565219-7348448 & 0 56 52.18 & -73 48 44.9 & 9.44 & 0.79 & 12.3 & 1.18 & 1 & 11.59 & 84.1 & 3 & K2III & & &\
J00565449-7258094 & 0 56 54.49 & -72 58 09.6 & 9.01 & 0.64 & 11.37 & 0.99 & 1 & -31.06 & 128.2 & 3 & & & &\
J00570028-7424356 & 0 57 00.26 & -74 24 35.8 & 11.83 & 0.28 & 13.23 & 0.5 & 4 & 22.82 & 82.1 & 3 & & & &\
J00570053-7415257 & 0 57 00.53 & -74 15 25.7 & 10.16 & 0.71 & 12.92 & 1.21 & 5 & 111.32 & 93.4 & 2 & & & &\
J00570451-7229402 & 0 57 04.52 & -72 29 40.2 & 10.74 & 0.82 & 13.84 & 1.28 & 1 & 151.64 & 101.3 & 1 & & G6.5Ib & 12 &\
J00570920-7201181 & 0 57 09.19 & -72 01 17.8 & 11.61 & 0.3 & 12.83 & 0.51 & 1 & 4.96 & 66.1 & 3 & & & &\
J00571085-7332097 & 0 57 10.86 & -73 32 10.1 & 7.95 & 0.84 & 10.82 & & 3 & 54.93 & 124 & 3 & G8Ve & & &\
J00571147-7228399 & 0 57 11.48 & -72 28 39.9 & 12.89 & 0.03 & 13.14 & 0.06 & 1 & 158.74 & 35.1 & 1 & & B9(Iab) & 3 & AzV 66F\
J00571148-7129397 & 0 57 11.50 & -71 29 39.7 & 9.26 & 0.57 & 11.48 & & 3 & 29.5 & 158.7 & 3 & & & &\
J00571353-7323288 & 0 57 13.54 & -73 23 28.9 & 11.04 & 0.37 & 12.5 & 0.55 & 1 & 3.63 & 50.8 & 3 & G0V & & &\
J00571770-7242068 & 0 57 17.68 & -72 42 07.0 & 11.5 & 0.36 & 13.03 & 0.64 & 1 & -80.56 & 67.7 & 3 & & & &\
J00572370-7239586 & 0 57 23.67 & -72 39 58.7 & 11.79 & 0.23 & 12.81 & 0.42 & 1 & 25.34 & 54.3 & 3 & & & &\
J00572384-7256156 & 0 57 23.83 & -72 56 15.8 & 10.21 & 0.59 & 12.42 & 0.92 & 1 & 63.34 & 122.3 & 3 & & & &\
J00572982-7421032 & 0 57 29.82 & -74 21 03.4 & 11.51 & 0.26 & 12.73 & & 3 & 4.12 & 58.7 & 3 & & & &\
J00573683-7151251 & 0 57 36.83 & -71 51 25.2 & 9.75 & 0.69 & 12.29 & 1 & 1 & 9.26 & 88.9 & 3 & & K1IV & 12 &\
J00574139-7237523 & 0 57 41.38 & -72 37 52.2 & 10.57 & 0.91 & 13.89 & 1.44 & 1 & 142.54 & 76.3 & 1 & & G7Ib & 12 &\
J00574140-7404505 & 0 57 41.40 & -74 04 50.5 & 11.02 & 0.62 & 13.39 & 1.05 & 2 & 3.22 & 74.3 & 3 & & & &\
J00574622-7242207 & 0 57 46.20 & -72 42 20.8 & 12.31 & 0.18 & 12.89 & 0.13 & 1 & 143.24 & 18.6 & 1 & & & &\
J00575242-7200254 & 0 57 52.42 & -72 00 25.3 & 10.22 & 0.71 & 12.85 & 1.13 & 1 & 3.36 & 97.3 & 3 & & & &\
J00575495-7229451 & 0 57 54.95 & -72 29 45.2 & 10.43 & 0.36 & 11.93 & 0.65 & 1 & 55.49 & 86.3 & 3 & G0III & & &\
J00575574-7345341 & 0 57 55.73 & -73 45 34.1 & 8.39 & 0.62 & 10.77 & & 3 & -11.41 & 120.6 & 3 & K0III & G7III/IV & 22 & Flo 358; HD 5829\
J00575689-7333436 & 0 57 56.87 & -73 33 43.8 & 9.84 & 0.7 & 12.76 & 1.42 & 1 & 180.13 & 62.6 & 1 & & G1Ia/Iab & 22 &\
J00580226-7217566 & 0 58 02.28 & -72 17 56.6 & 10.58 & 0.29 & 11.86 & 0.45 & 2 & 204.99 & 31.8 & 1 & F0I & F2Ia & 13 & AzV 197\
J00580232-7206149 & 0 58 02.32 & -72 06 14.7 & 10.6 & 0.92 & 14.02 & 1.66 & 1 & 143.59 & 81.8 & 1 & & K0Ib & 12 &\
J00580363-7225261 & 0 58 03.63 & -72 25 26.0 & 10.36 & 0.66 & 12.79 & 1.08 & 1 & 8.91 & 82.8 & 3 & & & &\
J00580791-7238305 & 0 58 07.90 & -72 38 30.5 & 11.85 & 0.06 & 12.14 & 0.07 & 2 & 177.34 & 45.9 & 1 & & B8Ia & 7 & AzV 200; Sk 69\
J00581449-7301127 & 0 58 14.49 & -73 01 12.8 & 10.45 & 0.3 & 11.72 & 0.51 & 1 & 16.04 & 84.9 & 3 & & F5 & 6 & Flo 364\
J00581584-7229109 a & 0 58 15.84 & -72 29 10.9 & 11.46 & 0.23 & 12.55 & 0.47 & 1 & 7.21 & 64.9 & 3 & F2V & & &\
J00581662-7204150 & 0 58 16.63 & -72 04 14.9 & 10.04 & 0.46 & 11.83 & 0.73 & 1 & -9.11 & 85.9 & 3 & G5I & G4.5V & 12 &\
J00582319-7221352 & 0 58 23.21 & -72 21 35.1 & 11.79 & 0.12 & 12.27 & 0.12 & 1 & 148.49 & 37.5 & 1 & A3I & A0I & 13 & AzV 205; Sk 71\
J00582953-7211345 & 0 58 29.54 & -72 11 34.6 & 11.47 & 0.31 & 12.63 & 0.43 & 1 & 2.29 & 116.8 & 3 & F8V & & &\
J00584121-7226154 a & 0 58 41.23 & -72 26 15.5 & 11.08 & 0.12 & 11.51 & 0.1 & 2 & 172.41 & 75.6 & 1 & B8I & A0Ia & 10 & AzV 211; Sk 74\
J00584187-7335251 & 0 58 41.88 & -73 35 25.3 & 10.97 & 0.61 & 13.36 & 0.99 & 1 & 27.33 & 54.4 & 3 & & K2III/IV & 22 &\
J00584242-7209434 & 0 58 42.44 & -72 09 43.4 & 10.71 & 0.39 & 12.16 & 0.59 & 1 & -7.71 & 95 & 3 & G0V & & &\
J00584479-7300574 & 0 58 44.78 & -73 00 57.6 & 9.97 & 0.3 & 11.26 & & 3 & -3.46 & 79.3 & 3 & & F5 & 6 & HD 5913\
J00585073-7201179 & 0 58 50.74 & -72 01 17.8 & 10.59 & 0.25 & 11.8 & & 3 & 0.19 & 96.5 & 3 & F8I & & &\
J00585391-7212048 & 0 58 53.93 & -72 12 04.8 & 9.93 & 0.89 & 13.34 & 1.62 & 1 & 148.19 & 91.2 & 1 & K0:III: & & &\
J00585470-7250596 & 0 58 54.68 & -72 50 59.8 & 10.63 & 0.86 & 13.81 & 1.48 & 1 & 190.44 & 78 & 1 & & G7Iab & 12 & SkKM 183\
J00590884-7401188 & 0 59 08.87 & -74 01 18.9 & 11.31 & 0.35 & 12.84 & & 3 & 4.92 & 79.1 & 3 & & & &\
J00591980-7220368 & 0 59 19.80 & -72 20 36.8 & 11.23 & 0.28 & 12.44 & 0.52 & 1 & 14.89 & 81.4 & 3 & F8I & & &\
J00592249-7121142 & 0 59 22.49 & -71 21 14.3 & 11.61 & 0.34 & 13.02 & 0.53 & 4 & -0.3 & 118.7 & 3 & & & &\
J00593268-7322538 & 0 59 32.68 & -73 22 53.9 & 9.87 & 0.41 & 11.51 & & 3 & 9.33 & 72.1 & 3 & G5 & G2.5V & 12 & Flo 387\
J00593889-7216324 & 0 59 38.91 & -72 16 32.4 & 10.45 & 0.62 & 12.81 & 0.97 & 1 & 9.59 & 90.9 & 3 & K0III & & &\
J00594007-7213368 & 0 59 40.09 & -72 13 36.6 & 12 & 0.18 & 12.71 & 0.18 & 1 & 126.39 & 36.8 & 1 & A5V & & &\
J00595280-7219036 & 0 59 52.81 & -72 19 03.6 & 12.32 & 0.03 & 12.47 & 0.08 & 1 & 169.99 & 64 & 1 & B9I & B9(Ia) & 3 & Sk 83; RMC 16\
J00595623-7336330 & 0 59 56.24 & -73 36 33.1 & 10.91 & 0.63 & 13.32 & 0.85 & 4 & 61.79 & 38.3 & 3 & K0III & & &\
J00595831-7234188 a & 0 59 58.34 & -72 34 18.8 & 10.46 & 0.9 & 13.88 & 1.5 & 1 & 156.61 & 81.3 & 1 & & K0Ia/Iab & 22 &\
J01000957-7233592 & 1 0 09.60 & -72 33 59.3 & 10.19 & 0.9 & 13.6 & 1.57 & 1 & 146.29 & 88.4 & 1 & & & & SkKM 199\
J01001889-7233161 & 1 0 18.92 & -72 33 16.1 & 11.47 & 0.27 & 12.56 & 0.37 & 1 & 26.09 & 55.1 & 3 & F2:I: & F1 & 11 & AzV 75F\
J01002525-7149058 & 1 0 25.25 & -71 49 05.9 & 9.67 & 0.39 & 11.26 & & 3 & -10.4 & 81.9 & 3 & & G1IV & 12 & Flo 399\
J01003301-7211289 & 1 0 33.04 & -72 11 28.9 & 10.52 & 0.91 & 13.77 & 1.43 & 1 & 153.39 & 98.9 & 1 & & K0Ib & 12 &\
J01004306-7321576 & 1 0 43.08 & -73 21 57.8 & 10.73 & 0.3 & 12.01 & 0.53 & 1 & 10.03 & 36.1 & 3 & G5 & & &\
J01004445-7223551 & 1 0 44.45 & -72 23 55.1 & 12.85 & 0.01 & 12.73 & -0.05 & 1 & 172.49 & 38.8 & 1 & B2I & B2.5Iab & 7 & AzV 257; Sk 91\
J01004635-7341475 & 1 0 46.35 & -73 41 47.6 & 11.27 & 0.39 & 12.82 & 0.52 & 4 & 42.09 & 34.6 & 3 & G0V & & &\
J01005063-7228521 & 1 0 50.64 & -72 28 52.1 & 12.62 & 0.09 & 13.1 & 0.14 & 1 & 163.74 & 38.5 & 1 & & & &\
J01005338-7229023 & 1 0 53.39 & -72 29 02.4 & 10.07 & 0.91 & 13.41 & 1.55 & 1 & 162.29 & 83.1 & 1 & & G8Iab & 12 & SkKM 206\
J01005415-7251367 & 1 0 54.13 & -72 51 36.7 & 8.68 & 0.84 & 12.2 & 1.66 & 1 & 183.64 & 68.8 & 1 & & G5.5Ia & 12 & SkKM 207\
J01005444-7416521 a & 1 0 54.43 & -74 16 52.1 & 11.31 & 0.41 & 12.97 & 0.5 & 5 & 6.22 & 73.3 & 3 & & & &\
J01010075-7350069 a & 1 1 00.76 & -73 50 07.0 & 11.29 & 0.29 & 12.69 & & 3 & 16.89 & 34.1 & 3 & G0V & & &\
J01010873-7433309 & 1 1 08.74 & -74 33 31.1 & 9.58 & 0.35 & 11 & & 3 & -18.98 & 83.1 & 3 & & F7V & 6 & Flo 421\
J01011042-7132198 & 1 1 10.42 & -71 32 19.9 & 8.76 & 0.26 & 9.76 & 0.35 & 2 & 58.4 & 33.3 & 3 & & A9V & 9 & RMC 10F; HD 6193\
J01011237-7137191 & 1 1 12.37 & -71 37 19.1 & 10.5 & 0.66 & 12.84 & 0.97 & 1 & 21.2 & 111.7 & 3 & & & &\
J01011249-7216069 & 1 1 12.49 & -72 16 06.8 & 12.2 & 0.14 & 12.67 & 0.09 & 1 & 170.79 & 40.4 & 1 & A3V & A2 & 11 & AzV 80F\
J01011983-7217042 & 1 1 19.84 & -72 17 04.2 & 10.6 & 0.82 & 13.76 & 1.51 & 1 & 161.09 & 89.5 & 1 & & G7Ia/Iab & 22 &\
J01012102-7130084 & 1 1 21.04 & -71 30 08.5 & 10.41 & 0.9 & 13.89 & 1.33 & 5 & 135.3 & 82.2 & 1 & & & & SkKM 215\
J01012123-7314445 & 1 1 21.23 & -73 14 44.7 & 10.3 & 0.24 & 11.48 & & 3 & 20.49 & 48.8 & 3 & & F7V & 6 & Flo 422\
J01012215-7202033 & 1 1 22.15 & -72 02 03.2 & 9.03 & 0.86 & 11.93 & 1.24 & 1 & -32.39 & 103.2 & 3 & K0III & K2III & 12 &\
J01012272-7235404 & 1 1 22.73 & -72 35 40.4 & 11.42 & 0.3 & 12.77 & 0.56 & 1 & 7.29 & 70 & 3 & F8 & & &\
J01012743-7207062 a & 1 1 27.43 & -72 07 06.1 & 11.78 & 0.07 & 12.17 & 0.06 & 2 & 108.45 & 55.2 & 1 & A0I & A0 & 13 & AzV 273; Sk 99\
J01014157-7315244 & 1 1 41.57 & -73 15 24.5 & 11.72 & 0.35 & 13.12 & 0.6 & 1 & 6.69 & 35.5 & 3 & & & &\
J01014382-7218025 & 1 1 43.83 & -72 18 02.6 & 11.36 & 0.37 & 12.81 & 0.61 & 1 & 44.99 & 85.3 & 3 & G2III & & &\
J01014559-7223223 & 1 1 45.59 & -72 23 22.3 & 10.95 & 0.57 & 13.21 & 0.96 & 1 & 13.09 & 84.4 & 3 & & & &\
J01014948-7205454 a & 1 1 49.49 & -72 05 45.3 & 10.79 & 0.63 & 13.15 & 0.96 & 1 & 177.85 & 82.6 & 1 & G2 & G1Iab & 12 &\
J01015742-7204172 a & 1 1 57.43 & -72 04 17.1 & 11.58 & 0.17 & 12.25 & 0.15 & 1 & 123.65 & 42 & 1 & A5I & A3I & 13 & Sk 102; AzV 286; RMC 22\
J01020100-7122208 & 1 2 01.01 & -71 22 20.8 & 10.81 & 0.78 & 13.75 & 1.15 & 4 & 296.1 & 61.3 & 1 & & & &\
J01020466-7431060 & 1 2 04.68 & -74 31 06.1 & 9.36 & 0.84 & 12.33 & & 3 & 91.35 & 122.4 & 2 & & & &\
J01020591-7358015 & 1 2 05.91 & -73 58 01.7 & 10.97 & 0.59 & 13.31 & 0.85 & 5 & 84.1 & 97.7 & 3 & & & &\
J01020977-7155354 & 1 2 09.78 & -71 55 35.5 & 9.85 & 0.75 & 12.64 & 1.14 & 1 & 117.41 & 88.3 & 2 & G5I & & &\
J01020979-7200232 a & 1 2 09.80 & -72 00 23.2 & 12.21 & 0 & 12.1 & -0.03 & 1 & 169.45 & 56.2 & 1 & B5I & B8Ia & 7 & AzV 297; RMC 23\
J01021027-7348571 & 1 2 10.27 & -73 48 57.3 & 10.53 & 0.42 & 12.25 & & 3 & 4.59 & 60.2 & 3 & G5V & & &\
J01021230-7202516 a & 1 2 12.32 & -72 02 51.5 & 12.33 & 0.02 & 12.47 & 0.03 & 1 & 154.85 & 49.9 & 1 & A0I & A0I & 14 & AzV 298; RMC 24\
J01021987-7145570 & 1 2 19.87 & -71 45 57.0 & 11.28 & 0.42 & 12.93 & 0.6 & 1 & 12.51 & 59.5 & 3 & G2V & & &\
J01022044-7415374 a & 1 2 20.43 & -74 15 37.6 & 9.46 & 0.7 & 12.05 & 1.02 & 5 & 19.82 & 103.7 & 3 & & & &\
J01022196-7404492 & 1 2 21.97 & -74 04 49.3 & 10.66 & 0.35 & 11.92 & & 3 & -13.05 & 80.4 & 3 & & F2: & 6 & Flo 441\
J01022209-7146368 a & 1 2 22.09 & -71 46 36.9 & 11.59 & 0.29 & 12.74 & 0.43 & 1 & -13.8 & 64.7 & 3 & F2II & & &\
J01022649-7202284 & 1 2 26.50 & -72 02 28.4 & 9.84 & 0.36 & 11.29 & & 3 & 16.39 & 83.1 & 3 & F8V & G0V & 12 & Flo 438\
J01022712-7216454 & 1 2 27.13 & -72 16 45.5 & 11.88 & 0.22 & 12.74 & 0.24 & 1 & 131.59 & 36.7 & 1 & A3I & F5I & 8 & Flo 439; AzV 305\
J01023088-7223578 & 1 2 30.90 & -72 23 57.8 & 8.71 & 0.64 & 11.07 & 1.02 & 1 & 2.89 & 131.1 & 3 & K0III & G8II & 12 & Flo 443\
J01023575-7149423 & 1 2 35.77 & -71 49 42.4 & 9.98 & 0.91 & 13.52 & 1.57 & 1 & 186.91 & 88.5 & 1 & K2I & & & SkKM 233\
J01023730-7216250 & 1 2 37.31 & -72 16 25.0 & 8.97 & 0.9 & 12.55 & 1.74 & 1 & 158.29 & 82.2 & 1 & & G6Ia/Iab & 22 & SkKM 235\
J01023810-7208066 & 1 2 38.12 & -72 08 06.6 & 12.4 & 0.1 & 12.72 & 0.06 & 1 & 135.39 & 45.6 & 1 & A3I & & &\
J01023991-7211273 & 1 2 39.92 & -72 11 27.3 & 10.15 & 0.81 & 13.23 & 1.25 & 1 & 127.55 & 78.5 & 1 & G8III & G7Iab/Ib & 22 & SkKM 236\
J01024303-7207260 & 1 2 43.04 & -72 07 26.0 & 10.58 & 0.26 & 11.64 & 0.3 & 2 & 138.09 & 35.4 & 1 & F0I & F0Ia & 10 & Sk 105; AzV 310; RMC 26\
J01024960-7210145 & 1 2 49.60 & -72 10 14.5 & 10.55 & 0.11 & 10.9 & 0.08 & 2 & 145.01 & 53.5 & 1 & A0I & A0Ia & 7 & CPD-72 77; Sk 106; RMC 27\
J01025349-7213145 a & 1 2 53.49 & -72 13 14.5 & 8.9 & 0.65 & 11.25 & 0.97 & 1 & 24.95 & 126.6 & 3 & K0III & G7.5III & &\
J01025533-7206488 & 1 2 55.35 & -72 06 48.8 & 11.4 & 0.28 & 12.64 & 0.51 & 1 & 13.89 & 75.2 & 3 & F8V & & &\
J01025992-7232455 & 1 2 59.92 & -72 32 45.5 & 7.61 & 0.69 & 10.22 & & 3 & 1.89 & 118.5 & 3 & K0III & K1IV & 12 & Flo 454; HD 6407\
J01030213-7205170 & 1 3 02.12 & -72 05 17.0 & 8.41 & 0.29 & 9.63 & 0.46 & 2 & 36.51 & 92.8 & 3 & F5III & F9III & 12 & RMC 11F; HD 6406\
J01030464-7306289 & 1 3 04.66 & -73 06 29.1 & 7.28 & 0.91 & 10.6 & & 3 & -57.11 & 105.8 & 3 & & K2V & 6 & Flo 455\
J01030765-7212362 & 1 3 07.67 & -72 12 36.3 & 12.45 & 0.09 & 12.95 & 0.13 & 1 & 156.19 & 41.5 & 1 & F0Ib & A3 & 11 & AzV 90F\
J01030892-7155507 & 1 3 08.92 & -71 55 50.8 & 9.55 & 0.88 & 13.12 & 1.75 & 1 & 141.31 & 75.8 & 1 & G5: & K0Iab & 12 & SkKM 242\
J01031418-7221588 & 1 3 14.16 & -72 21 58.9 & 7.35 & 0.63 & 9.74 & 1.03 & 2 & 3.99 & 117.8 & 3 & K0III & G7IV & 12 & Flo 457; HD 6431\
J01032108-7204478 & 1 3 21.08 & -72 04 47.9 & 11.31 & 0.26 & 12.46 & 0.44 & 1 & 5.89 & 86.4 & 3 & F8III & & &\
J01032294-7236126 & 1 3 22.94 & -72 36 12.7 & 10.23 & 0.89 & 13.67 & 1.58 & 1 & 165.09 & 61.5 & 1 & G2? & K0Iab/Ib & 22 & SkKM 251\
J01033277-7224565 & 1 3 32.78 & -72 24 56.7 & 10.6 & 0.78 & 13.68 & 1.44 & 1 & 153.29 & 80.9 & 1 & & G5.5Iab & 12 & SkKM 254\
J01034304-7215299 a & 1 3 43.04 & -72 15 30.0 & 12.43 & 0.04 & 12.54 & 0.03 & 1 & 158.85 & 50.1 & 1 & A0I & A0I & 14 & AzV 338; Sk 110\
J01034677-7209106 & 1 3 46.78 & -72 09 10.8 & 10.88 & 0.41 & 12.42 & 0.64 & 1 & -13.09 & 103.1 & 3 & G5III & & &\
J01034794-7338348 & 1 3 47.96 & -73 38 35.1 & 9.69 & 0.7 & 12.31 & & 3 & 35.45 & 78.2 & 3 & & K0III & 12 &\
J01034909-7342442 & 1 3 49.09 & -73 42 44.6 & 10.68 & 0.78 & 13.61 & 1.47 & 4 & 194.85 & 49.1 & 1 & & G4.5Iab & 12 &\
J01034942-7207296 & 1 3 49.45 & -72 07 29.7 & 12.82 & 0 & 12.71 & -0.06 & 1 & 181.21 & 24.9 & 1 & B2I & B1Ia & 7 & AzV 340; RMC 33\
J01035385-7423020 & 1 3 53.83 & -74 23 02.1 & 11.58 & 0.37 & 13.01 & 0.72 & 5 & -7.35 & 81.8 & 3 & & K0.5III & 12 &\
J01035882-7327447 a & 1 3 58.83 & -73 27 44.8 & 5.51 & 0.91 & 8.7 & & 3 & 4.27 & 98.1 & 3 & & K2 & 9 & Flo 469; HD 6509\
J01035883-7223090 & 1 3 58.84 & -72 23 09.0 & 10.59 & 0.7 & 13.35 & 1.26 & 1 & 139.59 & 60.6 & 1 & & & & SkKM 263\
J01040165-7208254 & 1 4 01.67 & -72 08 25.3 & 7.83 & 0.87 & 11.14 & 1.42 & 1 & -0.21 & 105.9 & 3 & K5III & K4III/IV & 22 &\
J01040239-7412520 & 1 4 02.39 & -74 12 51.9 & 10.45 & 0.59 & 12.72 & 0.88 & 5 & 106.75 & 79.2 & 2 & & & &\
J01040596-7349058 & 1 4 05.96 & -73 49 05.8 & 12.02 & 0.21 & 13.08 & & 3 & 218.55 & 29.1 & 1 & & F0 & 3 & AzV 99F\
J01041045-7154025 & 1 4 10.46 & -71 54 02.5 & 11.31 & 0.4 & 12.82 & 0.69 & 1 & 61.31 & 120.7 & 3 & G2 & & &\
J01041108-7308494 & 1 4 11.08 & -73 08 49.4 & 10.73 & 0.35 & 12.21 & 0.66 & 1 & 37.69 & 105.2 & 3 & & & &\
J01041148-7143516 a & 1 4 11.48 & -71 43 51.6 & 10.28 & 0.58 & 12.61 & 1 & 1 & 11.3 & 76.4 & 3 & G8III & & &\
J01041227-7152032 & 1 4 12.28 & -71 52 03.2 & 11.9 & 0.02 & 12.1 & -0.01 & 2 & 146.71 & 47.8 & 1 & A0I & A0I & 13 & AzV 347; Sk 113\
J01041450-7311470 & 1 4 14.50 & -73 11 47.1 & 11.24 & 0.55 & 13.38 & 0.81 & 1 & 4.89 & 48.1 & 3 & & & &\
J01041658-7224012 & 1 4 16.58 & -72 24 01.2 & 9.36 & 0.28 & 10.49 & 0.44 & 2 & 7.39 & 83.5 & 3 & F2I & F2 & 6 & Flo 474; HD 6537\
J01041885-7203431 & 1 4 18.86 & -72 03 43.1 & 11.43 & 0.46 & 13.21 & 0.67 & 1 & 8.95 & 70.3 & 3 & G5V & & &\
J01042100-7215096 a & 1 4 20.98 & -72 15 09.7 & 6.1 & 0.6 & 8.42 & 1 & 2 & 1.25 & 129.6 & 3 & G8 & G8III & 12 & RMC 16F; HD 6536\
J01042391-7311565 & 1 4 23.90 & -73 11 56.6 & 8.92 & 0.27 & 9.92 & & 3 & -10.81 & 106.4 & 3 & & F9IV/V & 12 & Flo 481; HD 6561\
J01042527-7231188 & 1 4 25.27 & -72 31 18.8 & 10.1 & 0.65 & 12.57 & 1 & 1 & 42.19 & 80.9 & 3 & G5V & K0.5III & 12 &\
J01042815-7408183 & 1 4 28.14 & -74 08 18.3 & 11.34 & 0.42 & 13.07 & & 3 & 26.45 & 99 & 3 & & & &\
J01043056-7130447 & 1 4 30.56 & -71 30 44.8 & 9.48 & 0.63 & 12.03 & & 3 & -23 & 83.2 & 3 & & & &\
J01043573-7204155 & 1 4 35.74 & -72 04 15.5 & 10.54 & 0.8 & 13.65 & 1.47 & 1 & 154.89 & 97.8 & 1 & & & &\
J01044935-7206218 a & 1 4 49.35 & -72 06 21.8 & 11.34 & 0.03 & 11.36 & -0.02 & 2 & 189.85 & 50.3 & 1 & B1II & B3Ia & 7 & Sk 114; AzV 362; RMC 36\
J01045290-7208368 & 1 4 52.91 & -72 08 36.8 & 10.88 & 0.13 & 11.23 & 0.07 & 2 & 194.01 & 67.8 & 1 & B9I & B9Ia+ & 7 & Sk 117; AzV 367; RMC 37\
J01045356-7419475 & 1 4 53.54 & -74 19 47.6 & 10.72 & 0.33 & 12.13 & 0.39 & 4 & -47.15 & 79.9 & 3 & & F6V & 6 & Flo 489\
J01045902-7208503 & 1 4 59.04 & -72 08 50.2 & 8.09 & 0.72 & 11 & 1.04 & 4 & 2.59 & 99.2 & 3 & K2III & & & K1IIIe\
J01045990-7126108 & 1 4 59.90 & -71 26 10.9 & 11.63 & 0.34 & 13.18 & 0.67 & 4 & 68.5 & 89.4 & 3 & & & &\
J01051187-7143346 & 1 5 11.87 & -71 43 34.6 & 10.63 & 0.84 & 13.73 & 1.5 & 1 & 144.61 & 82.7 & 1 & & & & SkKM 283\
J01051190-7417474 & 1 5 11.89 & -74 17 47.4 & 10.46 & 0.61 & 12.86 & 1.03 & 4 & -33.25 & 129.5 & 3 & & & &\
J01051279-7159337 & 1 5 12.81 & -71 59 33.8 & 10.43 & 0.91 & 13.8 & 1.31 & 1 & 159.41 & 88.3 & 1 & K5? & & &\
J01052460-7255293 & 1 5 24.59 & -72 55 29.4 & 10.08 & 0.87 & 13.25 & 1.4 & 1 & 161.39 & 53.5 & 1 & G5?I? & & & SkKM 285; G8Iab\
J01052811-7127504 & 1 5 28.13 & -71 27 50.5 & 10.27 & 0.41 & 12.01 & 0.74 & 4 & -0.3 & 125.9 & 3 & & & &\
J01054102-7221246 & 1 5 41.04 & -72 21 24.6 & 10.43 & 0.91 & 13.71 & 1.53 & 1 & 191.59 & 70.2 & 1 & G2:III: & & & SkKM 290; G8Ia/Iab\
J01055631-7219448 a & 1 5 56.32 & -72 19 44.8 & 11.03 & 0.15 & 11.47 & -0.03 & 2 & 137.85 & 39 & 1 & B2I(e) & B2Ia & 16 & Sk 124; AzV 393; RMC 39\
J01060182-7217252 & 1 6 01.85 & -72 17 25.2 & 9.19 & 0.57 & 11.35 & 0.89 & 1 & -12.49 & 120.2 & 3 & G8III & & & G8IV\
J01065126-7255533 & 1 6 51.28 & -72 55 53.4 & 9.06 & 0.69 & 11.55 & 1.08 & 1 & 38.09 & 80.9 & 3 & & & & K0III\
J01065315-7357550 a & 1 6 53.16 & -73 57 55.2 & 11.03 & 0.28 & 12.34 & 0.33 & 4 & 23.88 & 63.4 & 3 & & & &\
J01070175-7310374 & 1 7 01.76 & -73 10 37.5 & 10.79 & 0.76 & 13.75 & 1.38 & 1 & 170.09 & 65.4 & 1 & & & & SkKM 300\
J01070284-7148269 & 1 7 02.84 & -71 48 27.1 & 8.41 & 0.47 & 10.39 & 0.84 & 2 & -9.39 & 129.5 & 3 & K0III & G8V & 6 & HD 6827\
J01070665-7346504 & 1 7 06.66 & -73 46 50.5 & 9.79 & 0.6 & 12.1 & & 3 & -31.75 & 109.8 & 3 & & & &\
J01070716-7336249 & 1 7 07.17 & -73 36 24.9 & 9.56 & 0.61 & 11.92 & & 3 & 53.65 & 84.6 & 3 & & & & K1V\
J01071373-7400250 & 1 7 13.74 & -74 00 25.0 & 8.51 & 0.65 & 11 & & 3 & 37.62 & 83.1 & 3 & & & &\
J01071439-7401353 & 1 7 14.39 & -74 01 35.3 & 7.71 & 0.84 & 10.86 & & 3 & -35.38 & 94.8 & 3 & & & &\
J01071905-7351075 a & 1 7 19.06 & -73 51 07.6 & 10.15 & 0.64 & 12.64 & 0.98 & 4 & 12.78 & 78.8 & 3 & & & &\
J01073340-7345474 & 1 7 33.40 & -73 45 47.6 & 10.15 & 0.36 & 11.72 & & 3 & -3.48 & 148.5 & 3 & & & &\
J01073493-7324304 & 1 7 34.92 & -73 24 30.4 & 6.99 & 0.76 & 9.82 & & 3 & 15.09 & 89.7 & 3 & & K2 & 6 & Flo 536; HD 6939; K2III\
J01073584-7213426 & 1 7 35.85 & -72 13 42.6 & 7.15 & 0.88 & 10.47 & & 3 & -0.49 & 96.9 & 3 & K3III & K5 & 6 & Flo 531; K4III/IV\
J01073986-7205208 a & 1 7 39.86 & -72 05 20.9 & 11.44 & 0.33 & 12.93 & 0.57 & 1 & 8.78 & 71 & 3 & F8V & & &\
J01074309-7212153 & 1 7 43.10 & -72 12 15.3 & 8.72 & 0.83 & 11.69 & 1.21 & 1 & -60.84 & 109.2 & 3 & & K5V & 2 & SMC 66510; K2III\
J01074659-7203564 & 1 7 46.60 & -72 03 56.5 & 10.66 & 0.19 & 11.62 & & 3 & 18.11 & 57.9 & 3 & F2III & & &\
J01080317-7149496 a & 1 8 03.18 & -71 49 49.7 & 10.35 & 0.57 & 12.6 & 0.94 & 1 & 45.08 & 77.5 & 3 & & & &\
J01080549-7208426 & 1 8 05.50 & -72 08 42.8 & 12.41 & 0.11 & 12.95 & 0.12 & 1 & 160.81 & 30.4 & 1 & A2I & A2I & 8 & AzV 431; Sk 134\
J01080736-7148502 a & 1 8 07.36 & -71 48 50.3 & 10.39 & 0.84 & 13.63 & 1.55 & 1 & 159.78 & 73.6 & 1 & & & & SkKM 307\
J01081243-7141278 & 1 8 12.44 & -71 41 27.9 & 11.44 & 0.31 & 12.64 & 0.38 & 1 & 176.86 & 30.5 & 1 & & F5I & 8 & AzV 434\
J01083857-7150069 a & 1 8 38.58 & -71 50 07.0 & 6.43 & 0.55 & 8.57 & & 3 & -8.82 & 124.2 & 3 & G0I & hG5gG5mF6 & 19 & HD 7031\
J01084312-7207344 & 1 8 43.13 & -72 07 34.4 & 9.68 & 0.52 & 11.72 & 0.87 & 1 & 12.11 & 87.7 & 3 & K0III & K1 & 6 & Flo 552; K3V\
J01084882-7316027 & 1 8 48.83 & -73 16 02.8 & 10.87 & 0.65 & 13.36 & 1.07 & 1 & -31.11 & 94.5 & 3 & & & &\
J01085451-7208214 a & 1 8 54.53 & -72 08 21.4 & 10.16 & 0.91 & 13.58 & 1.55 & 1 & 128.08 & 82.3 & 1 & & & & SkKM 313\
J01085685-7302344 & 1 8 56.85 & -73 02 34.4 & 10.78 & 0.14 & 11.35 & 0.13 & 2 & 178.79 & 50.8 & 1 & & A3Ia & 10 & Sk 136; AzV 442; RMC 43\
J01091296-7401274 & 1 9 12.93 & -74 01 27.4 & 9.59 & 0.68 & 12.17 & 0.87 & 4 & 19.92 & 85 & 3 & & & &\
J01091733-7151168 a & 1 9 17.33 & -71 51 16.8 & 8.7 & 0.65 & 11.17 & & 3 & 14.88 & 95.3 & 3 & K1III & G8 & 6 & Flo 563\
J01093700-7318030 & 1 9 37.01 & -73 18 03.1 & 12.77 & 0.04 & 13.08 & 0.08 & 1 & 159.99 & 34.2 & 1 & & A2II & 3 & 2dFS 2579; AzV 115F\
J01100842-7349103 & 1 10 08.40 & -73 49 10.5 & 11.16 & 0.29 & 12.4 & 0.45 & 4 & -13.68 & 39.3 & 3 & & & &\
J01101248-7237310 & 1 10 12.48 & -72 37 31.0 & 10.11 & 0.87 & 13.56 & 1.58 & 1 & 196.01 & 67.7 & 1 & & & & SkKM 320; G8.5Ia/Iab\
J01101910-7233515 & 1 10 19.10 & -72 33 51.5 & 11.47 & 0.41 & 13.19 & 0.8 & 1 & 10.81 & 29.4 & 3 & G8V & & &\
J01102381-7157353 & 1 10 23.79 & -71 57 35.3 & 10.51 & 0.91 & 13.85 & 1.57 & 1 & 192.86 & 64.9 & 1 & & & & SkKM 321\
J01104014-7228304 & 1 10 40.14 & -72 28 30.4 & 11.18 & 0.48 & 13.05 & 0.77 & 1 & 4.61 & 62.5 & 3 & G5 & & &\
J01104331-7148491 & 1 10 43.30 & -71 48 49.1 & 9.21 & 0.59 & 11.45 & & 3 & 8.76 & 95 & 3 & & G8 & 6 & Flo 588\
J01105850-7248346 & 1 10 58.48 & -72 48 34.6 & 9.78 & 0.82 & 12.8 & 1.34 & 1 & 33.71 & 37 & 3 & K3:IIIe & & & K3.5III\
J01110839-7143116 & 1 11 08.39 & -71 43 11.5 & 10.72 & 0.54 & 12.7 & 0.81 & 1 & -24.24 & 56.8 & 3 & & & &\
J01110988-7306296 & 1 11 09.87 & -73 06 29.7 & 11.21 & 0.56 & 13.45 & 1 & 1 & 1.34 & 57.1 & 3 & K0III & & &\
J01112045-7222169 & 1 11 20.44 & -72 22 16.9 & 10.06 & 0.26 & 11.25 & & 3 & 14.41 & 58.7 & 3 & & F2 & 6 & Flo 598\
J01112540-7317042 & 1 11 25.39 & -73 17 04.3 & 10.29 & 0.88 & 13.61 & 1.4 & 1 & 162.84 & 81.6 & 1 & G2I & & &\
J01112782-7258501 & 1 11 27.81 & -72 58 50.2 & 10.57 & 0.4 & 12.13 & 0.65 & 1 & -13.36 & 74.3 & 3 & G2V & & &\
J01114315-7207275 & 1 11 43.17 & -72 07 27.5 & 11.64 & 0.13 & 12.1 & 0.09 & 1 & 151.36 & 23.1 & 1 & & A2I & 13 & AzV 463; Sk 146\
J01115007-7202309 & 1 11 50.08 & -72 02 31.0 & 11.19 & 0.5 & 13.27 & 0.86 & 1 & 32.66 & 48.3 & 3 & & & &\
J01115008-7216215 a & 1 11 50.07 & -72 16 21.6 & 8.17 & 0.62 & 10.57 & & 3 & 0.83 & 124.9 & 3 & & G5III? & 6 & Flo 608; CD-72 55\
J01115339-7157036 & 1 11 53.39 & -71 57 03.6 & 11.52 & 0.28 & 12.79 & 0.47 & 1 & -6.74 & 45 & 3 & & & &\
J01115681-7202186 & 1 11 56.84 & -72 02 18.7 & 11.08 & 0.61 & 13.51 & 0.99 & 1 & 37.86 & 52.3 & 3 & & & &\
J01121070-7337312 & 1 12 10.69 & -73 37 31.2 & 8.29 & 0.74 & 11.06 & 1.24 & 1 & 52.52 & 92.7 & 3 & & K6: & 6 & Flo 616\
J01123076-7259262 & 1 12 30.76 & -72 59 26.4 & 10.15 & 0.78 & 13.01 & 1.12 & 1 & -10.36 & 107.2 & 3 & K0III & & &\
J01123517-7309353 & 1 12 35.17 & -73 09 35.5 & 10.6 & 0.85 & 13.85 & 1.47 & 1 & 173.74 & 78 & 1 & & & & SkKM 327\
J01124028-7352208 & 1 12 40.29 & -73 52 21.0 & 8.86 & 0.79 & 11.65 & 1.1 & 1 & 106.52 & 85.3 & 2 & & & &\
J01130112-7358126 & 1 13 01.14 & -73 58 12.8 & 10.2 & 0.36 & 11.53 & & 3 & 28.12 & 86.7 & 3 & & F7 & 6 & Flo 630\
J01130227-7241420 & 1 13 02.27 & -72 41 42.1 & 10.91 & 0.49 & 12.82 & 0.83 & 1 & -3.89 & 39.9 & 3 & & & &\
J01130838-7320236 & 1 13 08.40 & -73 20 23.8 & 11.2 & 0.42 & 12.81 & 0.61 & 1 & 188.94 & 46.6 & 1 & F8I & F5I? & 8 & AzV 473\
J01131654-7208516 & 1 13 16.56 & -72 08 51.6 & 10.49 & 0.4 & 12.2 & 0.72 & 1 & 17.86 & 80.3 & 3 & & & &\
J01132351-7328318 & 1 13 23.52 & -73 28 31.9 & 9.3 & 0.38 & 10.87 & & 3 & 1.24 & 121.9 & 3 & G2V & G1V & 6 & Flo 639\
J01132650-7400500 & 1 13 26.51 & -74 00 50.1 & 9.78 & 0.59 & 11.97 & 0.87 & 1 & 38.12 & 74.3 & 3 & & & &\
J01133207-7239069 & 1 13 32.08 & -72 39 06.9 & 10.99 & 0.28 & 12.21 & 0.52 & 1 & 26.21 & 78.8 & 3 & & & &\
J01133660-7255139 & 1 13 36.63 & -72 55 14.0 & 10.34 & 0.86 & 13.39 & 1.44 & 1 & 173.33 & 81.7 & 1 & G2V & & & SkKM 329\
J01135159-7325262 & 1 13 51.60 & -73 25 26.4 & 8.53 & 0.64 & 10.9 & & 3 & -42.36 & 114.2 & 3 & K1V & G9 & 6 & Flo 650\
J01135200-7327054 & 1 13 52.01 & -73 27 05.6 & 12.45 & 0.12 & 13.16 & 0.29 & 1 & 18.04 & 33.9 & 3 & A5-F2 & & &\
J01135403-7323500 & 1 13 54.03 & -73 23 50.2 & 11.74 & 0.34 & 13.09 & 0.46 & 1 & 192.94 & 32.3 & 1 & F0I & & &\
J01135405-7251396 & 1 13 54.06 & -72 51 39.6 & 9.84 & 0.89 & 13.18 & 1.54 & 1 & 132.84 & 84 & 1 & G0I: & & & SkKM 330\
J01135944-7402377 & 1 13 59.43 & -74 02 37.8 & 10.28 & 0.62 & 12.61 & 0.93 & 1 & -1.18 & 60.4 & 3 & & & &\
J01141827-7342084 & 1 14 18.28 & -73 42 08.5 & 10.8 & 0.29 & 12.04 & 0.48 & 1 & 27.42 & 89.2 & 3 & & F7 & 6 & Flo 657\
J01142725-7242297 & 1 14 27.27 & -72 42 29.8 & 11.16 & 0.36 & 12.64 & 0.69 & 1 & 25.91 & 37.1 & 3 & & & &\
J01142805-7239535 & 1 14 28.06 & -72 39 53.5 & 11.22 & 0.55 & 13.04 & 0.98 & 1 & 173.91 & 48.7 & 1 & & F5Ib & 15 & HV 2195; RMC 46\
J01143413-7302354 & 1 14 34.14 & -73 02 35.5 & 11.05 & 0.62 & 13.46 & 1.01 & 1 & 12.84 & 77.6 & 3 & K0III & & &\
J01144168-7219101 & 1 14 41.69 & -72 19 10.1 & 11.45 & 0.41 & 13.04 & 0.73 & 1 & 23.41 & 28 & 3 & & & &\
J01144335-7309572 & 1 14 43.36 & -73 09 57.3 & 10.79 & 0.33 & 12.16 & 0.54 & 1 & 11.74 & 73.9 & 3 & F8V & & &\
J01144672-7234513 & 1 14 46.72 & -72 34 51.3 & 11.37 & 0.3 & 12.67 & 0.55 & 1 & 1.81 & 40 & 3 & & & &\
J01144979-7319179 & 1 14 49.80 & -73 19 18.0 & 11.49 & 0.38 & 12.99 & 0.61 & 1 & 3.44 & 90.1 & 3 & F8V & & &\
J01145522-7309521 & 1 14 55.23 & -73 09 52.3 & 10.71 & 0.84 & 13.64 & 1.19 & 1 & 40.54 & 70.9 & 3 & & & & SkKM 332\
J01151147-7308341 & 1 15 11.48 & -73 08 34.2 & 10.31 & 0.87 & 13.69 & 1.58 & 1 & 171.84 & 73.9 & 1 & & & & SkKM 333\
J01151174-7355297 & 1 15 11.72 & -73 55 29.8 & 10.22 & 0.37 & 11.79 & 0.61 & 1 & -14.48 & 77.2 & 3 & & & &\
J01152452-7330443 & 1 15 24.53 & -73 30 44.4 & 10.48 & 0.43 & 12.08 & 0.63 & 1 & 181.24 & 52.5 & 1 & F2I & F2 & 6 & Flo 675\
J01152857-7315370 & 1 15 28.57 & -73 15 37.1 & 12.25 & 0.15 & 12.82 & 0.11 & 1 & 161.74 & 35.5 & 1 & A3I & A3II & 3 & 2dFS 3094\
J01153986-7221562 & 1 15 39.88 & -72 21 56.3 & 10.14 & 0.64 & 12.48 & 0.94 & 1 & 73.71 & 36.4 & 2 & & & &\
J01155317-7330500 & 1 15 53.18 & -73 30 50.0 & 9.62 & 0.31 & 10.84 & & 3 & -97.46 & 62.1 & 3 & hF5gF2mF0 & F2 & 16 & Flo 682\
J01161289-7231363 & 1 16 12.89 & -72 31 36.3 & 9.82 & 0.45 & 11.56 & & 3 & 30.71 & 99.9 & 3 & & G0 & 6 & Flo 685\
J01161931-7252274 & 1 16 19.30 & -72 52 27.5 & 11.18 & 0.58 & 13.56 & 1.1 & 1 & 12.04 & 73.9 & 3 & K2III & & &\
J01165808-7311321 & 1 16 58.10 & -73 11 32.1 & 10.55 & 0.62 & 12.86 & 0.98 & 1 & 6.84 & 106.7 & 3 & K0III & & & SkKM 336\
J01171108-7312214 & 1 17 11.09 & -73 12 21.5 & 11.42 & 0.41 & 13.21 & 0.8 & 1 & 7.94 & 92.3 & 3 & G5III & & &\
J01171303-7317398 & 1 17 13.03 & -73 17 39.8 & 10.36 & 0.62 & 12.71 & 0.86 & 1 & 188.74 & 56.2 & 1 & G5I & & &\
J01171843-7228088 & 1 17 18.42 & -72 28 08.8 & 11.09 & 0.2 & 12.01 & 0.4 & 1 & 12.31 & 37.1 & 3 & & B: & 6 & Flo 697\
J01171937-7306068 & 1 17 19.40 & -73 06 06.8 & 11.18 & 0.31 & 12.45 & 0.5 & 1 & 2.34 & 108.2 & 3 & F5III: & & &\
J01173565-7327031 & 1 17 35.65 & -73 27 03.2 & 9.39 & 0.64 & 11.8 & 0.95 & 1 & 172.64 & 82.3 & 1 & G9III & & &\
J01175070-7315067 & 1 17 50.73 & -73 15 06.7 & 11.11 & 0.39 & 12.72 & 0.71 & 1 & 17.34 & 99.6 & 3 & G0IV & & &\
J01175132-7316470 & 1 17 51.34 & -73 16 47.0 & 9.42 & 0.07 & 9.7 & & 3 & -69.56 & 13.1 & 3 & kA2hA5mF0 & A2V & 19 & HD 8096\
J01175354-7321127 & 1 17 53.56 & -73 21 12.8 & 10.62 & 0.34 & 12 & 0.54 & 1 & 34.24 & 103 & 3 & F8V & & &\
J01180479-7324302 & 1 18 04.79 & -73 24 30.3 & 9.04 & 0.71 & 11.74 & 1.22 & 1 & 48.74 & 89.4 & 3 & K0III & & &\
J01182311-7324445 & 1 18 23.12 & -73 24 44.5 & 9.52 & 0.65 & 11.92 & 1.04 & 1 & -12.46 & 113.3 & 3 & K0III & & &\
J01182380-7319158 & 1 18 23.81 & -73 19 15.8 & 9.73 & 0.72 & 12.35 & 1.16 & 1 & 23.94 & 114.1 & 3 & K2III & & &\
J01232375-7313342 & 1 23 23.77 & -73 13 34.2 & 9.67 & 0.67 & 12.16 & 1.14 & 1 & 17.98 & 88.8 & 3 & & & &\
J01234438-7332025 & 1 23 44.37 & -73 32 02.6 & 9.77 & 0.72 & 12.55 & 1.1 & 1 & -57.12 & 35.1 & 3 & & & &\
J01240629-7314454 & 1 24 06.29 & -73 14 45.4 & 9.16 & 0.54 & 11.19 & & 3 & 165.88 & 67.2 & 1 & & F7I & 6 & Flo 739\
J01243530-7313439 & 1 24 35.30 & -73 13 43.9 & 10.8 & 0.47 & 12.56 & 0.67 & 1 & 25.28 & 88.3 & 3 & & & &\
J01255005-7323415 & 1 25 50.06 & -73 23 41.6 & 10.64 & 0.41 & 12.2 & 0.62 & 1 & 5.48 & 82.4 & 3 & & & &\
J01260996-7323151 & 1 26 09.98 & -73 23 15.1 & 9.17 & 0.9 & 12.71 & 1.71 & 1 & 167.48 & 59.7 & 1 & & & & SkKM 351\
J01261633-7306310 & 1 26 16.33 & -73 06 30.9 & 11.67 & 0.26 & 12.71 & 0.45 & 1 & 27.38 & 46.8 & 3 & & & &\
J01262548-7330334 & 1 26 25.50 & -73 30 33.4 & 10.67 & 0.65 & 13.04 & 0.98 & 1 & 50.68 & 36.4 & 3 & & & &\
J01272108-7314431 & 1 27 21.10 & -73 14 43.2 & 11.18 & 0.34 & 12.72 & 0.68 & 1 & 38.68 & 57.4 & 3 & & & &\
J01283670-7314496 & 1 28 36.69 & -73 14 49.6 & 11.56 & 0.31 & 12.81 & 0.51 & 1 & -2.12 & 37.7 & 3 & & & &\
J01290330-7334131 & 1 29 03.30 & -73 34 13.1 & 10.93 & 0.27 & 12.13 & 0.51 & 1 & 20.48 & 29.9 & 3 & & & &\
J01303980-7322011 & 1 30 39.81 & -73 22 01.0 & 12.15 & 0.06 & 12.5 & 0.05 & 1 & 162.78 & 38.3 & 1 & & A2I: & 4 & Sk 185\
[l c c c c c c c c c]{} U-B & 0.78 & 0.75 & 4906 & & 0.72 & 4943 & & 0.68 & 4993\
B-V & 1.15 & 1.11 & 4768 & & 1.06 & 4857 & & 1.01 & 4945\
V-K & 2.94 & 2.82 & 4392 & & 2.67 & 4499 & & 2.53 & 4508\
J-K & 0.78 & 0.76 & 4394 & & 0.73 & 4461 & & 0.71 & 4508\
Avg. & & & 4615 && & 4690 && & 4766\
[l l]{} Mass & 9 M$_{\odot}$\
Age & 30 Myr\
T$_{\rm eff}$ & 4700 K\
$\log$ g \[cgs\] & 0.8\
M$_{\rm V}$ & -5.4\
$\log \frac{{\rm L}}{{\rm L}_{\odot}}$ & 4.2\
Radius & 190 R$_{\odot}$\
[^1]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
[^2]: <http://www.user.oats.inaf.it/castelli/colors/BCP/BCP_m05k2odfnew.dat>
[^3]: To compute the temperature from the models we applied simple linear interpolation between the colors and temperatures. Using the models required us to adopt a value for the surface gravity, $g$. We had to perform this process iteratively, as the surface gravity depends upon the adopted luminosity to compute the stellar radius, and luminosity depends upon the temperature for the bolometric correction. We assumed a mass of $9M_\odot$ based upon the stellar evolution tracks described in the next section. This process converged quickly, with a $\log g$ of 0.8 \[cgs\].
[^4]: Another star, J04482407-7104012 (CPD-71$^\circ$285, HD 268700) has a radial velocity of 401 km s$^{-1}$, but Gaia’s result results give a parallax measurement consistent with this star being a foreground halo giant at a distance of 350 pc.
|
-15mm
[**V.A.Buslov**]{}
[**Hierarchy Structure of Graphs and Weighted Condensations** ]{}
Abstract
Buslov V.A.
Introduction words
==================
The very first essential theorem on graphs (except Euler’s (1736) solution of Kënigsberg bridges problem) was formulated by Kirchhoff [@K] (1847) who considered graphs as networks of conducting wires. In this theorem Kirchhoff computed the number of connected subgraphs containing all vertices without circuits (spanning trees) for the aims of analyzing electric chains.
The next step in investigating tree-like structure of graph was the following. One can attach to every edge (unordered pair of vertices) some quantity (weight) and ask a question how to find the spanning tree of such graph (graph with weighted adjacencies) with minimal weight (having the minimal sum of edge weights). To clarify the idea of weights and the problem formulated one can use Kirchhoff’s approach. Let us assume that some number of points (vertices of graph) are connected by wires (edges of graph) in an arbitrary manner. The resistance (or length) of wire connecting two points is the weight of corresponding edge. The question is which wires to remove and which to reserve in order to get such network where all points are still connected by wires (may be through other points) but the summarized resistance (or length) of remaining wires be minimal. Of course such network must be a spanning tree. This significant problem is in graph theory one of few ones having a number of effective algorithms. According to one of them at the first step one takes an edge of minimal weight as a first intermediate graph (if it is not a single minimal weight edge one takes any of them), then the procedure is recurrent one. If $k$-th intermediate graph is constructed one have to add to it the edge of minimal weight among all others such as resulting graph does not contain circuits. At the $(N-1)$-th step, where $N$ is a number of all vertices, one gets spanning tree of minimal weight. So at every step one gets a forest (graph without circuits), which connected components are trees, and every step means that two of them are joined into single tree by adding some edge connecting them. But for directed graph (digraph), where edges replaced by arcs (ordered pairs of vertices), the equivalent procedure was absent to our knowledge, and Theorem 1 of present paper shows us the way of such constructing.
To make clear the idea of weights and the problem to solve in the situation of digraphs we need to replace the classical picture of graph as an electric network by another one. Let us consider the picture of potential relief on a smooth manifold $M$, defined by some real smooth function $V(x)$, $x\in M$. The points of local minimum of the potential we associate as vertices of digraph and the potential bar necessary to overtake in order to leave point $x_i$ of local minimum and leave the corresponding fundamental region $\Omega_i$ (the point $x_i$ attraction region of dynamic system $\dot x=-\nabla V(x)$) and fall into another fundamental region $\Omega_j$, which has a common boundary with $\Omega_i$, we assume to be weight $V_{ij}$ of arc $(i,j)$ of digraph.
The main idea of constructing spanning trees of minimal weight is similar to the method above. At $k$-th step we construct forest of minimal weight among forests having $k$ arcs by connecting by arc two trees of the previous step into a single tree but with serious addition. One of these two trees must be reconstructed into new one before, and only then we connect them.
Let us explain partially here the reason of this reconstructing. There are two different orientation types of directed trees in the situation of digraphs. In one of them moving from any vertex along arcs according to orientation one falls into a single vertex called a root (just this type of directed trees we consider at this parer). By changing the direction of all arcs we get another type of directed trees. The complexity of directed case can be demonstrated by the following fact: it is easy to construct directed graph with weighted adjacencies such that any of its spanning trees (independently of the orientation type) of minimal weight does not contain the minimal weight arc!
Returning to our example of potential field on manifold the problem of constructing spanning tree having minimal weight can be reformulated as the next one. The problem is to chose some point of local minima (the root of the tree – it would be the point of global minimum) and to bind points of local minima by one-side ways so that moving from any point along these ways one gets into a the chosen point and the sum of potential bars corresponding to these ways must be minimal.
But the main aim of present paper is not to construct directed spanning trees (it is only a collateral result), although tree remains the elementary brick of our construction. In fact we consider a sequence of ordered grainings of given graph nested into each other. Every of these grainings except the last one is represented by simpler graph (the “descendant”) still possessing essential properties of its “ancestor”.
To illustrate the idea of grainings let us return to our example of manifold. One can imagine the particle, moving on this manifold. If the particle is at the moment at some point of local minima and possesses some fixed additional energy and this energy is quite enough to overtake some potential bars, the particle does not notice these bars and does not distinguish the fundamental regions they (bars) connect. So our manifold divides into more rough regions than fundamental ones. Increasing the level of this additional energy one gets more rough subdivision etc.
So coming back to graphs it is naturally to ask such a question. Can we look on graphs not in detail, but rather roughly without noticing unessential connections and uniting the vertices themselves into some grainings, among which to establish new connections? Continuing this logic we could look another time on such graining graph and to enlarge it one more time and so on. We suggest such kind of hierarchy approach here. Inside the situation when arcs (for directed) or edges (for non-directed) graphs are of no weights, the proposition about graph hierarchy structure is rather poor. For usual (non-directed) graphs we can speak in such sense only about connected components, for directed – about graph of condensations, which elements are the sets of mutually accessible vertices. For directed (and even for non-directed) graphs with weighted adjacencies the situation is far richer. One could try to introduce by force some decompositions on clusters of vertices and then has a problem how to determine adjacencies connecting these clusters, but our approach is natural one. It implies that graph itself contains the whole information about the number of hierarchically nested decompositions and about clusters inside every decomposition and this is determined mainly not by the number of vertices and not by the number of arcs (edges) but mostly by forces of these adjacencies (by weights). In this connection it comes to light that one can neglect the part of adjacencies without any waste and such a neglect does not affect on clusters inside decomposition and the number of decompositions either.
On the structure of work. At section 2 we give the necessary definitions and designations. At section 3 we introduce some new definitions required to formulate the general method using in proofs. Next section is devoted to investigating the properties of directed forests, which are the factors of the initial graph and have the minimal weight among all forests of $k$ trees under different $k$. It turns out that such extreme forests allow to construct at section 5 nested system of algebras of subsets of the set of vertices (they determine the hierarchy) and to investigate their properties. At section 6 using results of the previous section we construct some kind of enlarged graphs, which we call by weighted condensations.
Main definitions
=================
In graph theory the unification of designations and even terminology proper have not complete yet. So let us give first necessary definitions and designations.
Let $G$ be graph (non-directed). By $\V G$ and $\E G$ we denote the set of its vertices and edges (unordered pairs of vertices) respectively.
Let $\X$ be non-empty set and $\X ^2$ – its Cartesian square and let $\U\subseteq\X^2$. Pair $G=(\X ,\U)$ is called [*directed graph (digraph)*]{}. The elements of the set $\X$ are called [*vertices*]{} and elements of the set $\U$ are called [*arcs*]{}. We use $\V G$ and $\A G$ to denote the set of vertices and arcs of $G$ respectively.
Let $a=(i,j)$ be an arc, vertices $i$ and $j$ are called [*an origin*]{} and [*a terminus*]{} of $a$ respectively. The arc $(i,i)$ with coinciding origin and terminus is called [*a loop*]{}. The number of arcs coming out of (into) the vertex $i$ is called [*outdegree*]{} $d^+(i)$ ([*indegree*]{} $d^-(i)$) of the vertex $i$ .
Digraph having several arcs with common origin and common terminus is called [*multidigraph*]{} and such arcs are called [*parallel*]{}. If every edge (arc) of (di)graph possesses some value (weight), such (di)graph is called [*(di)graph with weighted adjacencies*]{}.
We use sometimes the term “graph” in wide sense designating by it digraphs and multidigraphs with weighted adjacencies also if it is not lead to misunderstanding.
Graph $H$ is called [*subgraph*]{} of graph $G$ if $\V H\subseteq\V G$, $
\A H\subseteq\A G$. Subgraph $H$ is called [*spanning*]{} subgraph (or [*factor*]{}) if $\V H=\V G$. Subgraph $H$ is called [*induced*]{} (or more completely – subgraph induced by the set $\U\subset\V G$) if $\V H
=\U$ and $(i,j)\in\A H$ means that $(i,j)\in\A G$ and $\{i,j\}\subset
\U$. We designate subgraph of $G$ induced by the set $\U$ as $G|_\U$.
[*Directed circuit of length M*]{} is digraph with set of vertices $\{ x_1$,$x_2$,$\cdots $,$ x_M\}$ and with arcs $(x_j,x_{j+1}) ,
j=1,2,\cdots M-1$ and $(x_M,x_1)$.
[*Walk (noncyclical) of length $M-1$*]{} is digraph with set of vertices $\{ i_1$,$2$, $\cdots $,$M\} $ and with arcs $(i_j,i_{j+1})$, $i=1$, $\cdots $,$M-1$ . Such walk we designate $i_1\cdot i_M$-walk. [*Semiwalk of length $M-1$*]{} is digraph with set of vertices $\{
i_1,i_2,\cdots ,M\}$ and its arcs are either $(i_j,i_{j+1})$ or $(i_{j+1},i_j)$ , $i=1,2,\cdots ,M-1$. The vertex $j$ is said to be [*accessible*]{} ([*attainable*]{}) from the vertex $i$ in graph $G$ if there is $i\cdot j$-walk in $G$. Digraph is called [*strong*]{} (or [*strong connected*]{}) if all its vertices are mutually attainable. Digraph $G$ is called [*weak*]{} if for every pair of vertices there is a semiwalk connecting them in $G$.
Any maximal with respect to including weak subgraph of graph $G$ is called its connected component (or simply – component). [*Strong component*]{} of $G$ is any maximal with respect to including its strong subgraph.
Other definitions we will cite as it is necessary.
Other definitions, designations and predetermined operations
============================================================
Graph (non-directed) possessing no cycles is a [*forest*]{}. Connected component of a forest is a [*tree*]{}. For trees with $\S $ as a set of vertices we use the notation $T(\S )$.
There are two kinds of directed forests at the situation of digraphs. Here we call by [*forest*]{} the digraph without circuits, in which outdegree of every vertex is equal to zero or to one ($d^+(i)=0,1$). Arcwise connected components of forest are called [*trees*]{}. The only vertex $i$ of tree which outdegree is equal to zero ($d^+(i)=0$) is called [*root*]{} of a tree. The set of roots of the forest $F$ we designate by $\W ^F
$. The tree of $F$ with root $i$ we designate $T^F_i$.
Let $V$ be graph (directed or non-directed). We use notation $\F^k(V)$ for the set of spanning forests having $k$ trees and being subgraph of $V$.
We call the vertex $i$ [*rear*]{} to the vertex $j$ in graph $G$, and correspondingly $j$ [*front*]{} to $i$ if there is $i\cdot j$-walk in $G$. [*Front*]{} ([*rear*]{}) [*enclosing*]{} of the vertex $i$ in graph $G$ is the set of terminuses of arcs outcoming from (origins of arcs coming into) $i$. For such set we use notation $\N ^+_G(i)$ ($\N ^-_G(i)$).
[**Remark.**]{} If digraph is a forest, the front-rear relation is a relation of partial ordering. Such definition implicates that vertices can be connected by the rear-front relation only if they are in the same tree of the forest. The root of the forest is the front vertex to all vertices of the tree.
We say that in graph $G$ an arc comes out of the set $\U$ and comes into the set $\V$, if there is at list one arc which origin belongs to $\U$ and the terminus belongs to $\V$ in graph $G$. We also say that arc comes out of the set $\U$, if there is at least one arc which origin belongs to $\U$ but terminus does not.
If $\D$ is some subset of the set of all vertices of digraph $G$ we call the set of terminuses (origins) of arcs outcoming from (coming into) the set $\D$ as front (rear) enclosing of $\D$, which is naturally to designate as $\N^+_G(\D)$ ($\N ^-_G(\D)$).
In the following we will prove the existence of forests having some special properties. The general method of such proofs consists of sequent steps. It is necessary to take two concrete graphs with the same set of vertices and to select some subset $\D$ of vertices. Next, we exchange between each other arcs outcoming from the vertices of $\D$ in this graphs and then we investigate properties of the new graphs resulting in such exchange. Thereby is naturally to introduce the following definition.
Let $F$ and $G$ be two graphs with the same set of vertices and the set $\D$ is some subset of the set of vertices. We will say that the graph $H$ is $\D$-[*exchange*]{} of $F$ by $G$, if $H$ is a result of exchanging in graph $F$ arcs, outcoming from the vertices of the set $\D$, onto arcs that outcome from these vertices in graph $G$. Our interest relates to the situation, where $F$ and $G$ are forests and in addition $\D$-exchange of $F$ by $G$ and moreover at the same time $\D$-exchange of $G$ by $F$ are forests too. So at first let us formulate the criterion of $\D$-exchange to be a forest.
[**Criterion.**]{} [*Let $F$ and $G$ be two an arbitrary forests with the set of vertices $\N =\{ 1,2,\cdots ,N\}$, $\D $ – some subset of $\N
$. Let $H$ is $\D$-exchange $F$ by $G$. Then graph $H$ is a forest then and only then, if every vertex $i\in\N^+_G\D$ is not rear in $F$ with respect to those vertices from $\D$ which are rear to $i$ in $G$.*]{}
[**Proof.**]{} As any way starting from the vertex $i\in\N^+_G(\D)$ in graph $F$ (and as a sequence in $H$ too) by the condition can not include those vertices of the set $\D$, which are rear to $i$ in $G$ (and as a sequence in $H$), so $H$ does not contain circuits. Further, not more than one arc comes out from any vertex in $H$, so $H$ is a forest.
[**Sequence 1.**]{} [*Let $F$ – be a forest and $\D$ be some subset of the set of vertices, i) if there are not any arcs coming into $\D$ in $F$, so $\D$-exchange of $F$ by any forest $G$ is a forest; ii) if there are not any arcs coming out of $\D$ in $F$, so $\D$-exchange of any forest $G$ by $F$ is a forest.*]{}
[**Sequence 2.**]{} [*Let $T^F$ be a tree of the forest $F$ and $T^G$ – tree a of the forest $G$, and let $\D =\V T^F$ ($=\V T^F\cap\V T^G$, $=\V T^F\setminus\V T^G$) then $\D$-exchange of $F$ by $G$ and $
\D$-exchange of $G$ by $F$ are forests.*]{}
[**Sequence 3.**]{} [*Let $T^F$ be a tree of the forest $F$ and $T^G$ be a tree of the forest $G$, $\C =\V T^F\cap\V T^G$, and $\D\subseteq\C$, such, that there are not arcs coming into $\D$ in $F$ and terminuses of arcs coming from $\D$ do not belong to $\C$ , then $\D$-exchange of $F$ by $G$ and $\D$-exchange of $G$ by $F$ are forests.*]{}
[**Sequence 4.**]{} [*Let $T^F$ be a tree of the forest $F$ and $T^G$ be a tree of $G$, $\C =\V T^G\setminus \V T^F$, and $\D$ – be the set of all vertices from $\C$, such as the walk starting from any of them in the forest $G$ passes through the set $\V T^F$, then $\D$-exchange of $F$ by $G$ and $\D$-exchange of $G$ by $F$ are forests.*]{}
Related forests
===============
Let $V$ be digraph with real weighted adjacencies $v_{ij}$ on the set of vertices $\N =\{ 1,2,\cdots ,N\}$. We will consider factors of $F$ being forests and containing of $k=1,2,\ldots ,N$ trees (the set of such forests we designate $\F ^k(V)$). Under [*weight*]{} $\Sigma^F$ of $F$ we understand the following quantity:
$$\Sigma^F=\sum_{(i,j)\in \A F}v_{ij} \ .$$ The minimum of weight over all forests $F\in\F^k(V) $ consisting exactly of $k$ trees we designate as $\varphi_k$:
$$\varphi_k=\min_{F\in\F^k(V)}\Sigma^F \ .$$ If $\F^k(V)=\emptyset$ we suppose $\varphi_k=\infty$. In the following we write $\F^k$ instead of $\F^k(V)$ in cases when it is clear subgraphs of which graph $V$ are under consideration.
Let us pick out the subset $\tfk $ from the set of forests $\F^k$, consisting of forests with the minimum weight: $F\in\tfk \Leftrightarrow
\F\in\F^k$ and $ \ \Sigma^F=\varphi_k $. Such forests we call extreme.
Let us study extreme (giving minimum) forests from $\tfk (V)$ under different $k$. It turns to be that they have some kind of “genetic” link. In particular it is valid the following
[**Proposition 1.**]{} [*Let under some $k=1,2,\cdots ,N-1$, the set $\F^k$ is not empty, then for any forest $F\in\tf^{k+1}$ there is at least one $G\in\tfk$ (and for any forest $G\in\tfk$ there is $F\in\tf^
{k+1}$) such that the set of vertices of any tree of the forest $F$ is contained in the set of vertices of some tree of the forest $G$*]{}.
[**Remark.**]{} Just the formulation of this proposition means that as the forest $G\in\tfk$ contains one tree less than “relative” to it forest $F\in\tf^{k+1}$, so the sets of vertices of $k-1$ trees of the forest $G$ coincide with the sets of vertices of corresponding trees of the forest $F$, the set of vertices of the last tree of the forest $G$ is conjunction of sets of vertices of last two trees of the forest $F$.
In actual we will prove more powerful fact. Preliminary we give one definition.
Let us agree upon to call the forest $F\in\F^{k+1}$ with roots (exactness to the numeration) $ \ 1,2,\cdots ,k+1 \ $ as an [*ancestor*]{} of the forest $G\in\F^k$ with roots $1,2,\cdots ,k$, and correspondingly the forest $G\in\F^k$ to call as a [*descendant*]{} of the forest $F\in\F
^{k+1}$ if $T^F_i=T^G_i, \ \ i=1,2,\cdots ,k-1$, $T^F_k\subset T^G_k$, and subgraph $G|_{\V T^F_{k+1}}$ of the forest $G$ (or, which is the same, subgraph of the tree $T^G_k$) induced by the set $\V T^F_{k+1}$ is a tree (under this it may coincide with the tree $T^F_{k+1}$ or not).
The following theorem tells us on the minimum changes one must to provide to get a forest belonging to the set $\tfk$ from a forest belonging to the set $\tf^{k+1}$ and vice versa.
[**Theorem 1 (on “relatives”).**]{} [*Let under some $k=1,2,\cdots ,N-1$ the set $\F^k$ is not empty, then any forest $F\in\tf^{k+1}$ has a descendant in the set $\tfk$ and any forest $G\in\tfk$ has an ancestor in the set $\tf^{k+1}$.*]{}
[**Proof.**]{} Let us prove that any forest $F\in\tf^{k+1}$ has a descendant in the set $\tfk$. Let $F$ and $H$ be arbitrary forests from the sets $\tf^{k+1}$ and $\tfk$ respectively. As the power of the set of roots $\W^F$ of the forest $F$ is one unit more than the power $|\W^H|=k$, and as in any forest not more than one arc goes out of any vertex, so there is at least one vertex (let it be the vertex $j$) in the set $\W^F\setminus
\W^H$, which is not attainable in the forest $F$ from the set $\W^H
\setminus\W^F$, and hence the tree of the forest $F$, having the vertex $j$ as a root, has not intersection with the set $\W^H\setminus\W^F$. This way, all the vertices of the tree $T^F_j$ except the root $j$ itself, belong to the set $(\N\setminus\W^F) \cap (\N\setminus\W^H)$, so arc goes out of every vertex from the set $\V T^F_j\setminus\{ j\}$ in the forest $H$ (and in $F$ naturally).
Let us construct preliminary forest $E\in\tfk $, which is necessary to the final constructing of the descendant $G\in\tf^{k+1}$ of the forest $F$. We take $\V T^F_j$-exchange of $F$ by $H$ as this auxiliary graph $E$, and we designate $\V T^F_j$-exchange of $H$ by $F$ as $Q$. By force of [Sequence 2]{} from [Criterion]{} the graphs $E$ and $Q$ are forests. The forest $E$ contains one arc more than $F$, as there are no arc coming from the vertex $j$ in $F$, but there is one in $H$ ($j\in\W^F\setminus\W^H$). So $E\in\F^k$ and, analogically, $Q\in\F^{k+1}$ and
\_k \^E , \_[k+1]{}\^Q . If we designate by $\Delta$ the quantity $\Sigma^E-\Sigma^F=\Sigma^E-
\varphi_{k+1}$, then, obviously,
\^Q=\^H- . Using (\[1\]) and (\[2\]) we get $\Sigma^E=\varphi_{k+1}+\Delta\le
\Sigma^H -\Delta+\Delta=\varphi_k \ ,$ and hence $\Sigma^E=\varphi_k$, what means that $E\in\tfk$.
Let the vertex $j$ in the forest $E$ belong to the tree $T^E_m$ with vertex $m$ as a root. Consider the maximal walk being a subgraph of the tree $T^E_m$ and starting from the vertex $j$, all vertices of which belong to the set of vertices of the tree $T^F_j$. Let $n$ be final vertex of this way. Designate as $T$ maximal subtree of the tree $T^E_m$ with vertex $n$ as a root, all vertices of which belong to the set of vertices of the tree $T^F_j$. Notice, that all trees of the forest $F$ with the exception of the tree $T^F_j$ are subtrees of the trees of the forest $E$ with the same roots, but the vertices of the set $\V T^F_j$ are “divided” among the trees of the forest $E$. So, we can confirm that there are no arcs coming into the set $\V T$ in the forest $E$ and by force of [Sequence 3]{} from [Criterion]{} graph $G$ being $\V T$-exchange of $F$ by $E$ is a tree, and obviously it belongs to the set $\F^k$. If we consider $\V T$-exchange of $E$ by $F$, which by force of the same [Sequence 3]{} from [Criterion]{} is a thee, analogically to the previous we are convinced that really $G\in\tfk$, but by the construction it is a descendant of the forest $F$. To the other side the affirmation of the theorem is proved analogically.
The theorem on “relatives” lets us easy prove known system of convexity inequalities [@VF; @V]. Exactly, it is valid
[**Proposition 2.**]{} [*The quantities $\varphi_k$ satisfy to the following chain of convexity inequalities*]{}
\_[k-1]{}-\_k\_k- \_[k+1]{} .
[**Proof.**]{} By the theorem on “relatives” any forest $H\in\tf^{k-1}$ can be constructed using redirection of arcs coming from the vertices of the only tree of some forest $G\in\tfk$, which, in its turn, can be constructed by redirection of arcs coming from the vertices of the only tree of some forest $F\in\tf^{k+1}$. Let $F$, $G$, $H$ be just such “relative” forests. Then there is at least one tree of the forest $F$, from every vertex of which an arc goes out in the forest $H$. Let the vertex $i$ be root of this tree and let us designate by $f$ the sum of weights of arcs coming in forest $F$ from the vertices of the set $\V
T^F_i$, and by $h$ – the sum of weights of arcs outgoing from the vertices of the same set in $H$.
Let $P$ be $\V T^F_i$-exchange of $F$ by $H$, and $Q$ be $\V T^
F_i$-exchange of $H$ by $F$. By force of the [Sequence 2]{} from the [Criterion]{} both these graphs are forests and belong to the set $\F^k$ (because there is not an arc coming from the vertex $i$ in the forest $F$, but there is one coming from this vertex in forest $H$) and, hence
$$\Sigma^P=\Sigma^F+h-f=\varphi_{k+1}+h-f
\ge\varphi_k \ ,$$
$$\Sigma^Q=\Sigma^H-h+f=\varphi_{k-1}-h+f
\ge\varphi_k \ .$$ The Proposition is a direct sequence of the last two inequalities.
Note, that the following inequalities
\_[n-i]{}-\_n\_[m+i]{}-\_m , mn , (N-m,n)i0 , are the sequences from the system of convexity inequalities (\[convex\]).
Let us prove the following auxiliary
[**Proposition 3.**]{}
*Let $F\in\tf^n$ and $G\in\tf^m \ , \ \ m\ge
n$, and let $\D $ be subset of the set of vertices $\N $, such that graphs $P$ and $Q$, being $\D$-exchange of $F$ by $G$ and $\D$-exchange of $G$ by $F$ correspondingly, are forests. Then if*
a\) $\D$ contains $l\ge 0$ roots of the forest $F$ more than roots of the forest $G$, then $P\in\tf^{n-l}$ and $Q\in\tf^ {m+l}$;
b\) $\D $ contains $l\ge m-n$ roots of the forest $G$ more than roots of the forest $F$, then $P\in\tf^{n+l}$ and $Q\in\tf^{m-l}$.
[**Proof.**]{} We prove point b) (point ) can be proved analogically). Designate as $\Delta$ the following quantity $\Delta=\Sigma^P-\Sigma^F=
\Sigma^G-\Sigma^Q \ .$ It is followed from the condition, that $P\in
\F^{n+l}$ and $Q\in\F^{m-l}$, so
$$\Sigma^P=\varphi_n+\Delta\ge\varphi_{n+l} \ ,\ \ \Sigma^Q=\varphi_m-
\Delta\ge\varphi_{m-l} \ .$$ Combining these two inequalities one gets $\varphi_{m}-\varphi_{m-l}\le
\varphi_{n+l}-\varphi_n \ . $ However from (\[convex2\]) under $m\le
n+l$ it is followed reverse inequality and hence $\Sigma^P=\varphi_{n+l}$ and $\Sigma^Q= \varphi_{m-l}$ and this proves the proposition directly.
Algebras of subsets
===================
At the present paragraph we will construct the system of embedded algebras $\alk , \ k=1,2,\cdots N,$ of subsets of the set of all vertices $\N $ and investigate the properties of the elementary sets of these algebras.
Let us consider all connected components $T$ (trees) of the forests $F\in\tfk$. The sets of vertices of the trees $T$ are the base of the algebra $\alk$ (i.e. algebra $\alk$ is generated by the sets of vertices $\V T$ of the trees of the forests $F\in\tfk$).
[**Theorem 2.**]{}
*The sequence of algebras $\alk$ is an increasing one:*
$$\{\N ,\emptyset\}=\al_1 \subseteq \al_2
\subseteq \cdots \subseteq \al_{N-1} \subseteq \al_N =2^\N \
,$$ where $2^\N $ is the set of all subsets of the set $\N $.
[**Proof.**]{} Direct sequence of [Theorem 1]{}.
Let us give a [definition]{}. We call the vertex $j$ as [*marked point*]{} ([*vertex*]{}) [*of the level*]{} $k$, if there exists at least one forest $F\in\tfk$, where $j$ is a root (i.e. there exists connected component $T^F_j$).
Elementary sets of algebras $\alk$ can as contain as not contain marked vertices. Elementary set can contain few marked vertices at once. Those elementary sets, that contain marked vertices we will call [*marked sets*]{}.
Let $\S$ be some subset of the set of vertices $\N$. As $\tfk |_{\cal S}$ we will designate the set of subgraphs of the set of forests $\tfk$ induced by the set $\S$.
Let us see what the properties of extreme forests are in case, if under some $k$ there is equality in the system of convexity inequalities (\[convex\]):
\_[k-1]{}-\_k =\_k- \_[k+1]{} .
[**Theorem 3.**]{}
*Let (\[equal\]) be fulfilled, then*
1\) $\alk=\al_{k+1}$,
2\) $\tf ^{k-1}|_\E \subseteq \tfk |_\E
\supseteq\tf ^{k+1}|_\E$, where $\E$ is an arbitrary elementary set of the algebra $\alk$.
[**Proof.**]{} According to the [Theorem “on relatives”]{} every forest $H\in\tf^{k-1}$ possesses at least one ancestor $F\in\tfk$, which in its own, possesses at least one ancestor $G\in\tf^{k+1}$. Let $H$, $F$ and $G$ be such relative forests. There are 2 possible scenarios of getting granddescendant $H$ from grandancestor $G$. It is easy to see, that by one of them 4 trees of the forest $G$ participate in the construction of the forest $H$, and by another one – only 3. Let us see on the first possible scenario.
So, let $T^G_i$, $T^G_j$, $T^G_l$ and $T^G_m$ be trees of the forest $G$ with the roots $i$, $j$, $l$ and $m$ correspondingly. Let the forest $F$ be constructed from the forest $G$ by uniting trees $T^G_i$ and $T^G_j$ with may be redirecting of arcs coming from vertices of, for example, the tree $T^G_j$, i.e. $T^F_i|_{\V T^G_i}=T^G_i \ , \ \ T^F_i|_{\V T^G_j}$ is a tree and $\V T^F_i=\V T^G_j\cup\V T^G_i$, other trees of the forests $F$ and $G$ coincide between each other correspondingly. The forest $H$ in its turn is received from the forest $F$ by uniting trees $T^F_l$ and $T^F_m$ with may be redirecting arcs outgoing from the vertices of, for example, the tree $T^G_m$, i.e. $T^H_l|_{\V T^F_i}=T^F_l \ , \ \ T^H_l|_{\V
T^F_m}$ is a tree and $\V T^H_l =\V T^F_m\cup\V T^F_l$, other trees of the forests $F$ and $G$ coincide between each other correspondingly (note, that also $T^G_l= T^F_l$ and $T^F_m=T^G_m$). Designate as $F' \ \ $ $\V
T^G_j$-exchange of $H$ by $G$. It is obvious (by [Sequence 2]{} from [Criterion]{} and [Proposition 3]{}), that $F'\in\tilde \F^k$. By this every tree of the forest $G$ and every tree of the forest $H$ is either a tree of the forest $F$ or a tree of the forest $F'$, that confirms both points of the theorem. Another variant of the scenario is considered analogically.
We say, that the vertex $j$ is attainable from the vertex $i$ at the level $k$ or simply $j$ is $k$-attainable from $i$, if there is at least one forest $F\in\tfk $, such as there is $i\cdot j$-walk in $F$.
Let us see what are the properties of extreme forests in case if under some $k$ there is strong inequality in the system of convexity inequalities:
\_[k-1]{}-\_k >\_k- \_[k+1]{} .
[**Proposition 4.**]{} [*Let (\[nonequal\]) be taken place and $i$ and $j$ be level $k$ marked vertices. Let also the vertex $i$ be attainable from the vertex $j$ on the level $k$, then the vertex $j$ is attainable from the vertex $i$ on the level $k$ and, moreover, the vertices $j$ and $i$ belong to the same marked set of this level.*]{}
[**Proof.**]{} Under condition there is such forest $F\in\tfk $, where the vertex $i$ is rear comparative to the vertex $j$. Without loss of generality one can consider that, the vertex $j$ is a root in the forest $F$ (otherwise, if some marked vertex $m$ is a root of the tree containing the vertices $i$ and $j$ at this forest, the following discussions one can lead for any pair of vertices $i$ and $m$ or $j$ and $m$). Suppose, that there is such forest $G\in\tfk $, in which the vertex $j$ is a root, and the vertex $i$ does not belong to the tree having $j$ as a root. Let $\D =\V T^F_i \cap \V T^G_j$, and $P$ and $Q$ are $\D$-exchanges of $F$ by $G$ and of $G$ by $F$ correspondingly. Then by [Proposition 3]{} $ \ \ P\in\tf^{k+1}$ and $Q\in\tf^{k-1}$. Let us denote by $f$ and $g$ the sums of weights of arcs coming from the vertices of the set $\D $ at forests $F$ and $G$ correspondingly, then
$$\varphi_{k+1}=\Sigma^P=\Sigma^F-f+g=\varphi_k-f+g \ ,$$
$$\varphi_{k-1}=\Sigma^Q=\Sigma^G+f-g=\varphi_k+f-g \ ,$$ whence it follows that $\varphi_{k-1}-\varphi_k =\varphi_k- \varphi_{k+1}
\ , $ which contradicts (\[nonequal\]). So, in any forest $G\in\tfk$, in which the vertex $j$ is a root, the vertex $i$ belongs to the set of vertices of the tree $T^G_j$. From here it easy follows, that there is not such a forest in the set $\tfk$, in which the vertices $i$ and $j$ belong to different trees, which means validity of the proving proposition.
Note, that this proposition means in particular that if (\[nonequal\]) is fulfilled, so every marked set of algebra $\alk$ contains exactly one root of an arbitrary forest $F\in\tfk$, and it is valid the following.
[**Theorem 4.**]{} [*Let (\[nonequal\]) be fulfilled, then the algebra $\alk$ contains exactly $k$ marked elementary sets.*]{}
[**Proof.**]{} Any forest $F\in\tfk$ consists of $k$ trees and hence, there are not less than $k$ marked elementary sets in $\alk$. These $k$ marked sets are those elementary sets that contain the roots of the trees of $F$. Any root of an arbitrary forest $G\in\tfk$ naturally belongs to one of the trees of the forest $F$ and, hence, some root of the forest $F$ is accessible from it (root of $G$), and it means by Proposition 4 that this root belongs to one of mentioned elementary sets. Thus, there are exactly $k$ marked sets in $\alk$.
Let us call as [$k$-attraction domain]{} of marked vertex $i$ such set of vertices, which consists of such vertices $j$ that $i$ is accessible from $j$ in at least one forest $\F\in\tfk$.
[**Proposition 5.**]{} [*Let (\[nonequal\]) be fulfilled for some $k$, then for every marked vertex $i$ there is such forest $\F\in\tfk$, in which the vertex $i$ is a root and the set of vertices of the tree $T^F_i$ coincides with ${k}$-attraction domain of the vertex $i$, and also the sets of ${k}$-attraction domains of mutually ${k}$-attainable vertices coincide with each other.*]{}
[**Proof.**]{} Let $F$ and $G$ be forests belonging to the set $\tfk$, in which mutually $k$-attainable vertices $i$ and $j$ (in particular they can coincide) are roots of the trees $T^F_i$ and $T^G_j$ correspondingly. It is sufficient to show, that there is such a forest $H\in\tfk$, where the vertex $i$ is a root and $\V T^H_i \supseteq \V T^F_i \cup \V T^G_j$. Let $\D = \V T^G_j\setminus\V T^F_i$, then by Proposition 3 $\D$-exchange $F$ by $G$ is required forest $H$.
[**Proposition 6.**]{} [*Let (\[nonequal\]) be fulfilled, $F\in\tfk$ and $\E$ is elementary set belonging to algebra $\alk$, then there is such forest $G\in\tfk$, where all arcs coming out from the vertices of the set $\E$, coincide with ones coming out from them in the forest $F$, and also there are no arcs coming into the set $\E $ from the outside in $G$.*]{}
[**Proof.**]{} Let there be an arc coming into the set $\E $ from some elementary set $\E _1$ in the forest $F$. Since the sets $\E$ and $\E _1$ are elementary, so there is such forest $H\in\tfk$, where both these sets belong to different trees. Let $\E$ belong to the tree with $i$ as a root in $F$, and $\E _1$ belong to the tree with $j$ as a root in the forest $H$. Let $\D$ be the set $\V T^F_i\cap\V T^H_j$. Let $G$ be $\D$-exchange $F$ by $H$. By Proposition 4 the vertices $i$ and $j$ simultaneously belong or do not belong to the set $\D$. So $G\in\tfk$ and there are not any arcs coming into the set $\E$ from the set $\E _1$ in this forest, and also there are not more additional arcs coming into the set $\E$ in $G$, in comparison to ones coming into $\E$ in the forest $F$. If there are some arcs coming into the set $\E$ in $G$, one can repeat the procedure above now concerning the forest $G$ and get the forest, where no one arc comes into the set $\E$, but all arcs coming from it coincides with those coming from vertices of $\E$ in the forest $F$.
Next proposition being direct consequence of Proposition 6 is in some sense inverse to Proposition 5. If Proposition 5 tells how big tree of extreme forest can be, but in the following one we explain how small it can be.
[**Proposition 7.**]{} [*Let (\[nonequal\]) be fulfilled for some $k$, then for every marked elementary set $\M$ of algebra $\alk$ there is such forest $F\in\tfk$, where $\M$ is a set of vertices of one of trees of $F$, and also there is not such a forest belonging to $\tfk $, where arcs come out of the set $\M$.* ]{}
[**Proof.**]{} Let us suppose inverse. Let $F\in\tfk$ be a forest, where at least one arc comes out of $\M$ with, let us say, the vertex $m$ as an origin. By Proposition 6 without loss of generality one can suppose that there are not any arcs coming into $\M$ from outside. In addition, according to Proposition 4, the set $\M$ contains exactly one root of $F$. But then the tree of $F$ having this root does not contain the vertex $m$ and is contained in $\M$, which is in contradiction with elementary character of $\M$.
Proposition 7 means in particular, that any subgraph of an arbitrary forest $F\in\tfk$, induced by marked elementary set of algebra $\alk$, is a tree if (\[nonequal\]) is fulfilled. It is prove to be that indicated property is valid for unmarked elementary sets too.
[**Theorem 5.**]{} [*Let (\[nonequal\]) be fulfilled, then induced by any elementary set $\E$ of algebra $\alk$ subgraph of any forest $F\in\tfk$ is a tree.*]{}
[**Proof.**]{} It is necessary to show, that not more than one arc can come out of an arbitrary elementary set $\E $. Let $F\in\tfk$. According to Proposition 6 one can suppose, that there are not any arcs coming into $\E$ from outside in $F$. Let us verify firstly, that not more than one arc can come out of the set $\E$ into any other elementary set. On the contrary, we assume that there are, for example, two arcs at the forest $F\in\tfk$ coming out of the set $\E$ into some elementary set $\E _1$ of algebra $\alk$. Let also the arcs coming out of the set $\E$ into $\E_1$ have their origin at the vertices $a$ and $b$ and let the sets $\A$ and $\B$ be sets of rear vertices with respect to vertices $a$ and $b$ correspondingly (including vertices $a$ and $b$ themselves). The sets $\A$ and $\B$ do not intersect with each other and $\A\cup\B =\E$. As the sets $\E$ and $\E _1$ are elementary, so there is such forest $G\in\tfk$, where these sets belong to different trees, let us say, to the trees $T^G_j$ and $T^G_m$ correspondingly. Let $H$ be $\A$-exchange of $G$ by $F$. Obviously, that $H\in\tfk$. In addition, since $\E$ is elementary and, hence, its vertices at any forest from the set $\tfk$ must belong to the same tree, among them at $H$ too. It is possible only if the vertices of the set $\B $ are rear with respect to the vertex $a$ at the forest $G$ (only in this case elementary set $\E $ belongs entirely to single tree at $H$, namely to the tree $T^H_m$). Analogously, if $Q$ is $\B$-exchange of $G$ by $F$, so $Q\in\tfk$ and the vertices of the set $\A$ must be rear with respect to the vertex $b$ at the forest $G$. So the vertices $a$ and $b$ are rear with respect to each other at $G$, which is impossible because $G$ is a forest.
Other cases, where arcs could come out of $\E$ into several elementary sets one can examine analogously.
[**Theorem 6.**]{}
*Let (\[nonequal\]) be fulfilled, then*
i\) induced by any elementary set $\E$ belonging to the algebra $\alk$ subgraph of an arbitrary forest $F\in\tf^{k-1}$ is a tree,
ii\) if $\U$ is unmarked elementary set belonging to the algebra $\alk$, then $\tf^{k-1}|_\U = \tfk |_\U$.
[**Proof.**]{} Let $F$ and $G$ be relative forests belonging correspondingly to $\tfk$ and $\tf^{k-1}$, and let also one can construct the forest $G$ from $F$ by adding an arc coming out of the root $i$ of some tree $T^F_i$, and, may be, by redirecting of arcs that come out of other vertices of this tree. According to Proposition 6, without loss of generality, one can consider that $\M =\V T^F_i$ is marked elementary set, and by theorem on “relatives” the graph $G|_\M$ is a tree. In this case, if $\U$ is unmarked elementary set of the algebra $\alk$, so $G|_\U =F|_\U $.
Theorems 5 and 6 are very important for the consequent constructions, since based on Theorem 5 one can construct enlarged graphs and to determine adjacencies (and their weights) connecting enlarged vertices (elements of decomposition of the set of all vertices). Theorem 6 allows based on one level of enlargement to construct the following one.
Weighted condensations
======================
Proved above properties of extreme forests allow us to look on them and at all on directed graphs with weighted adjacencies in “an enlarged way”, without interest on details of their arc connections inside elementary sets, but paying attention only on connections among elementary sets, understanding elementary sets themselves as a vertices of some enlarged graph. Let us convert what has been said above into precise definition. Beforehand we remind existing definition of condensation for non-weighted directed graph, which just allows understand graphs in an enlarged way. Here is the corresponding definition.
Let $\{ {\cal S}_1,{\cal S}_2,\cdots ,{\cal S}_M\} $ be strong components (strong component is the set of inter-attainable vertices) of digraph $G$. [*Condensation*]{} of digraph $G$ is digraph $\hat G$ with the set of vertices $\{ s_1,s_2,\cdots ,s_M\} $, where the pair $(s_i,s_j)$ is an arc in $\hat G$ if and only if there is an arc in $G$ with origin belonging to ${\cal S}_i$, and terminus belonging to ${\cal S}_j$.
Mentioned definition is rather poor, since, for example, for strong digraphs (where all vertices are inter-attainable) condensation is trivial and consists of only one vertex, and hence, there are not any arcs in it. So we essentially modify the concept of condensation for weighted digraphs.
Let us firstly consider the case of non-directed graphs. In some sense the following simple theorem [@E] is more strong reformulation of Theorem on relatives but for non-directed graphs.
[**Theorem 7.**]{} [*Let the edge $e$ of non-directed graph $P$ possesses the minimal weight among all edges, in which exactly one endpoint belongs to the tree $T$ which is subgraph of $P$. Then there is at least one spanning tree containing $T\cup e$ and having minimal weight among all spanning trees of $P$ containing $T$.*]{}
According to this theorem all examinations drawn are valid but essentially simplify. For example the division on marked and unmarked sets vanishes (every set is marked) and also there is no necessity to replace edges under joining trees as it was in case of directed graphs (one only need add an edge to connect two trees). Of course for non-directed graph $P$ with weighted adjacencies inequalities of convexity are fulfilled and if (\[nonequal\]) is valid then algebra $\alk$ contains exactly $k$ elementary sets. The main property resulting from this theorem, that is useful for us, we point out as following.
[**Property 1.**]{} [*Subgraph of any forest $F\in\tf^n$ induced by elementary set $\E$ of algebra $\alk$, $n\le k$, is a tree.*]{}
[**Property 2.**]{} [*For every forest $F\in\tf^n$ there exists such forest $G\in\tfk$, $n\le k$ (and, into opposite side, for any $G\in\tfk$ there exists such $F\in\tf^n$ ) that $F|_\E =G|_\E$, where $\E$ is an arbitrary elementary set of algebra $alk$.*]{}
[**Definition.**]{} Let $P$ be non-directed graph with weighted adjacencies $p_{ij}$, and let (\[nonequal\]) be fulfilled, $\alk$ – algebra of subsets of the set of all vertices generated by the sets of vertices of trees belonging to $\tfk(P)$. We call non-directed graph $P^k$ with $k$ vertices as [*weighted condensation of the level*]{} $k$ (simply – [*k-weighted condensation)*]{} of $P$ if weights of it adjacencies are equal to the following numbers
p\^k\_[xy]{}=\_[ij]{} p\_[ij]{} , where $\X$ and $\Y$ are elementary sets of algebra $\alk$. If there is not any edge in $P$, such that one of its ends belongs to the elementary set $\X$, and another to the elementary set $\Y$, so we suppose that there is not corresponding edge $(x,y)$ in $P^k$.
It seems natural to consider that in graph of weighted condensation not only arcs possess weights but vertices too, which are actually elementary sets of corresponding algebra. We determine weight of vertex $s$, or which is the same, weight of elementary set $S$ corresponding to vertex $s$, as minimum of weight of spanning tree of graph $P|_{\cal S}$, i.e. as the quantity
$$\mathop{\min}\limits_{\sc T\subset P \atop
\sc \V T={\cal S}} \sum\limits_{(i,j)\in T}p_{ij} \ .$$ As weighted condensations, represent themselves usual graphs with weighted adjacencies, so all previous properties are valid for them (introduction of weights of vertices is not change anymore because we consider only spanning subgraphs, which include all vertices by definition). In particular, one can consider factor-forests of $P^k$ and determine the sets $\F^n(P^k)$ and also their subsets $\tf^m(P^k)$ possessing minimal weight. The weight itself of the forest $F\in\tf^n(P^k)$ we determine as stated above in the following way
\^F=\_[(x,y)F]{}p\^k\_[xy]{}+ \_ \_[TV T=]{} \_[(i,j)T]{}p\_[ij]{} , where $\E $ is elementary set of algebra $\alk$. For example, any forest $F\in\F^k(P^k)$ is empty graph ($k$ vertices (however possessing their own weights) and no edges), any $F$ belonging to $\F^1(P^k)$ is a spanning tree of $P^k$.
Under definition (\[SFN\]) it is obvious that if we introduce the numbers $\varphi^k_n$, $n\le k$, by the rule
\^k\_n =\_[F\^n(P\^k)]{}\^F , then by force of Property 1 \_n = \_n\^k , nk , and, of course, inequalities of convexity are valid:
\_[n-1]{}\^k-\_n\^k\_n\^k-\_[n+1]{}\^k , n=2,3,,k-1 . Equalities (\[varphi=\]) mean exactly, that minimum weights of spanning trees, consisting of equal number of trees $n\le k$, of weighted condensation $P^k$ and graph $P$ proper coincide with each other.
Now we consider analogical examination for directed graphs. Let $V$ be digraph with weighted adjacencies $v_{ij}$, and let (\[nonequal\]) be fulfilled, $\alk$ – algebra of subsets of the set of vertices of $V$, generated by the sets of vertices of trees of forests belonging to $\tfk(V)$. Algebra $\alk$ contains at least $k$ elementary sets, to be precisely, it contains $k+l$ elementary sets, where $l$ is the number of unmarked sets (this number can be equal to zero).
[**Definition.**]{} Let us call digraph $V^k$ with $k+l$ vertices as [*weighted condensation of the level*]{} $k$ (simply – [*k-weighted condensation)*]{} if weights of its adjacencies are equal to the following numbers
v\^k\_[xy]{}=\_[ij]{} (\_[T\_i()]{}\^[T\_i([X]{})]{}+v\_[ij]{}) , where $\X $ and $\Y $ are elementary sets of algebra $\alk$, $T_i(\E)$ is a tree with $\E$ as a set of vertices and $i$ as a root. If there is not any arc in $V$, such as its origin belongs to the elementary set $\X$, and the terminus to the elementary set $\Y$, and under this $i$ is a root of at least one spanning tree of digraph $V|_\X$ we suppose that there is not arc $(x,y)$ in $V^k$.
The necessity of weights determination in a different way than it was in non-directed situation is caused by the fact that one must be sure that the set $\Y$ is attainable from every vertex of $\X$ and in this case only it is justified to introduce an arc $(x,y)$ into graph $V^k$. Note, that weight minimum of tree $T_i(\X)$ depends on vertex $i$, so generally speaking in the situation of directed graphs it is not possible to introduce the weight of elementary set and one needs add “it” (look at (\[vad\])) to corresponding arc going out of this set. Nevertheless, if there are not arcs going out of some set $\X$ in digraph, it is possible to determine weight of $\X$ as minimum by all $i\in\X$ of weights of trees $T_i(\X)$.
As graph $V^k$ has at least $k$ ($k+l$ to be precisely) vertices one can consider, in particular, spanning forests of it and to determine the sets $\F^m(V^k)$ and also their subsets $ \tf^m(V^k)$ possessing minimal weight. However the weight itself of the forest $F\in\tf^m(V^k)$ we must determine in other way than in non-directed situation, because arc weights (\[vad\]) are determined not analogous to edge ones (\[vadn\]). Namely:
\^F=\_[(x,y)F]{}v\^k\_[xy]{}+ \_[d\^+(e)=0]{} \_[ TV T= ]{} \_[(i,j)T]{}v\_[ij]{} , where $F\in\F^m(V^k)$, $e$ is a root of $F$ corresponding to the elementary set $\E\in\alk$. So weight of $F\in\F ^m(V^k)$ is determined as sum of all arc weights $v^k_{xy}$ plus “weights” of those elementary sets of algebra $\alk$, corresponding to which vertices in $F$ are roots.
Now one can introduce the quantities $\varphi^k_n \ , \ n=1,2,\cdots ,k$ by the rule analogous to (\[defvarphikn\])
$$\varphi_n^k=\min\limits_{F\in\tf^n(V^k)}\Sigma^F \ .$$ and, of course, for these quantities the inequalities of convexity (\[nonequalkn\]) continue to be fulfilled, but (\[varphi=\]) is not true now and one can assert only that
$$\varphi_n\le \varphi_n^k \ ,$$ as the minima $\varphi_m$ are calculated using graph $V$ itself, but the numbers $\varphi_m^k$ – only using its weighted condensation. However, by force of definition of weighted condensations and its adjacencies (\[vad\]) $\varphi_k^k=\varphi_k$. Moreover, since by Theorem 6 subgraph of any graph belonging to $\tf ^{k-1}$ induced by an arbitrary elementary set of algebra $\alk$ is a tree, so $\varphi_{k-1}^k=
\varphi_{k-1}$. Note, that (\[varphi=\]) is a sequence of Property 1, which is not valid here generally speaking.
Point here that one can use the definition of weighted condensations in case of non-fulfillment of (\[nonequal\]) also. Namely, let under some $k$ and $n\le k-1$
\_[k-n-1]{}-\_[k-n]{}>\_[k-n]{}-\_[k-n+1]{}== \_[k-1]{}-\_k >\_k-\_[k+1]{} , then, as it follows from Theorem 3, algebras $\al_{k-n+1}$, $\al_{
k-n+2},\ldots ,\alk$ coincide with each other and so the definition of weighted condensations, initially introduced for index equal to $k$, one can spread to indices $k-1$, $k-2$, $\ldots $, $k-n+1$. Under that it is obvious that all this condensations are the same, so the number of different condensations equal to the number of sign $'>'$ at the system of convexity inequalities (\[nonequal\]) plus one. Theorems 3 and 6 mean also that under (\[kn\]) subgraph of any forest belonging to one of sets $\tf^{k-l}$, $l=1,2,\ldots ,n$, induced by arbitrary elementary set of algebra $\alk$, is a forest and hence
$$\varphi_{k-l}=\varphi_{k-l}^k \ , \ \ l=1,2,\cdots ,n \ .$$ One could think that (\[varphi=\]) is valid for digraphs, however it is not so, because under (\[kn\]) one has not any reason to expect that subgraph of $F\in\tf^{k-n-1}$, induced by elementary set of $\alk$, is a forest (and really it is not so, one can easy construct such example). Nevertheless (\[varphi=\]) takes place if the adjacencies $v_{ij}$ of digraph $V$ can be written in the form
v\_[ij]{}=p\_[ij]{}-p\_[ii]{} , where the numbers $p_{ij}\in R^1$ are weights of edges of some non-directed graph $P$ ($p_{ij}=p_{ji}$). This property we will call as [*potentiality* ]{} of weights of digraph $V$. Such definition is bound up with the fact, that under fulfillment of (\[pot\]) the weights $v_{ij}$ can be realized as potential bars necessary to overtake in order to get into point $j$ from point$i$ (the number $p_{ij}$ is transition potential from $i$ to $j$, $p_{ii}$ – potential of point $i$). Equalities (\[varphi=\]) succeed from the following
[**Theorem 8.**]{} [*Let digraph $V$ possess potential weights and its adjacencies satisfy (\[pot\]), then Property 1 is valid for $V$ and (\[varphi=\]) takes place.*]{}
[**Proof.**]{} From the definition of potentiality it is followed that if there is an arc $(i,j)$ in digraph $V$, so there is an opposite arc $(j,i)$ there. Further, for potential graph it is not difficult to see that if some $i\cdot j$-way possesses minimum weight (minimum sum of arc weights (potential bars)) among all ways from $i$ to $j$, then if one changes all these arcs to opposite ones in this $i\cdot j$-way, one gets $j\cdot i$-way with minimum weight among all ways from $j$ to $i$ in $V$. Now let us turn to Theorem 1 (on “relatives”). According to it any forest $G\in\tf^{k+1}$ one can construct from some forest $F\in\tfk$ by adding an arc connecting two trees and may be by redirecting of arcs in that tree, from which this additional arc would go out. Let $F$ and $G$ be such relative forests, and let $G$ one can get from $F$ by adding arc $(i,m)$, where $i$ belongs to the set of vertices of tree $T^F_j$ with $j$ as a root, and ,it is clear, if $i$ does not coincide with $j$, by reconfiguration of arcs of this tree in such a way as to get on the set $\V T^F_j$ a new tree, but with $i$ as a root. In this connection this tree $G|_{\V T^F_j}$ must possess minimum weight among all trees on the set $\V T^F_j$ with $i$ as a root. Let us construct new tree $G'\in\tfk$ from $F$ by adding the same arc $(i,m)$, but reconfiguration of arcs of $T^F_j$ will be done in the following manner. Consider $i\cdot j$-way belonging to tree $T^F_j$. It is, of course, the only in this tree and it possesses minimum weight among all $i\cdot j$-ways in induced subgraph $V|_{\V T^F_j}$. Now change in $F$ arcs of this $i\cdot j$-way into opposite ones (one gets under this a tree on the set $\V T^F_j$ with minimum weight among all trees on this set with $i$ as a root) and add arc $(i,m)$. This forest let call $G'$. It is extreme, of course, because it was constructed under really minimum changes of forest $F$. Note, that this forest $G'$ possesses one important property. If one takes away the orientation from $F$ and $G'$, then these graphs coincide with each other, except adjacency $(i,m)$ proper. It appears from the above the validity of Property 1 for potential digraphs.
Theorem 8 shows, that the analysis of potential digraphs is not more difficult than the same of non-directed graphs, and for them instead of Property 2 it is valid
[**Property 2’.**]{} [*Let $V$ be potential digraph, then for any forest $F\in\tf^n(V)$ there exist such forest $G \in \tfk(V)$, $n\le k$ (and for any $G \in \tfk (V)$ there is such $F \in\tf^n(V)$), that induced by any elementary set $\E \in\alk$ subgraphs of $F$ and $G$ coincide with each other to within the orientation.*]{}
So, our considerations above mean the following. Let us suppose that we constructed $k$-weighted condensation of some directed graph $V$ and try to build up condensation $V^n$, $n<k$, of some next level $n$. The question appears: Can one do it using the information on already constructed condensation only? It turns out that can not, generally speaking. More exactly, one can construct the corresponding algebra $\al_n$, but new adjacencies – can not. One have to use information on arcs of the initial digraph $V$. Nevertheless, if weights of $V$ are potential (or, moreover, graph is non-directed), it is not necessary to use any additional information and to realize the transition to next hierarchy level one can forget “prehistory” of graph and use the adjacencies of $V^k$ only. This reduces considerably the number of calculations required.
Instead of discussion
=====================
The method suggested can have a lot of applications in different brunches of science such as economy and finances, biology and neuron-nets, probability theory and random processes, mathematical and theoretical physics. This is forced just by the necessity to determine the structure and the hierarchy of complicated objects and using this information to give a conclusion which processes are essential on each level and which are not. For example, at exponentially large times in dynamic systems under small random perturbations some sublimit distributions appear [@VF]. They correspond in fact to distributions concentrated at marked elementary sets of some algebra $\alk$, the number of nontrivial possible time scales is equal to the number of different algebras. Under this, the generators of Fokker-Plank type equations (being singular perturbed ones [@L]), which govern distribution functions of stochastic differential equations, possess very special spectrum. Its low-frequency spectrum and corresponding eigenfunctions are determined by weighted condensations of some special digraph [@V1; @V2], which analysis connects with the opportunity of representing of characteristic polynomial in terms of tree-like structure of corresponding digraph [@CDZ; @FS; @V3].
[This work was supported RFBR, grants N-99-01-00696 and N-98-01-01063.]{} .5cm
[20]{}
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abstract: 'Distributed systems often face transient errors and localized component degradation and failure. Verifying that the overall system remains healthy in the face of such failures is challenging. At Netflix, we have built a platform for automatically generating and executing chaos experiments, which check how well the production system can handle component failures and slowdowns. This paper describes the platform and our experiences operating it.'
author:
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bibliography:
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title: Automating chaos experiments in production
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chaos engineering, fault injection, distributed systems, experimentation
Introduction
============
Consider a service delivered to users over the Internet. All such services are implemented as distributed systems. The smallest such service would involve two machines (a single client and server), while the larger ones, such as Netflix, are composed of thousands of servers.
Large-scale distributed systems contain many failure modes, as there are many opportunities for individual component failures and unexpected interactions among components[@Gunawi2014]. To ensure that a system remains available to users, engineers build resiliency into the system through strategies such as timeouts, retries, and fallbacks [@release-it].
Ideally, a distributed system will degrade gracefully if an individual component runs into trouble. In general, it is difficult to know whether engineering resiliency mechanisms will actually work to keep the overall system healthy if some part of the system goes bad.
*Chaos engineering* [@chaoseng; @chaosengbook] is an emerging approach in industry to evaluate the resiliency of a distributed system by running experiments against the system, in production. These experiments can identify weaknesses that could lead to outages if left unchecked.
At Netflix, we have developed a platform to automate the generation and execution of chaos experiments. These experiments run directly against our production environment. The authors are all members of the Resilience Engineering team, which is a centralized team that develops chaos engineering tools. In this paper, we discuss the design of our system and our experiences operating it. This platform was built in about three years by a team whose size varied between one and four software engineers.
Context: Netflix
================
Overview
--------
Netflix is a service that enables customers to stream television shows and movies over the Internet, on a variety of different types of devices, such as smart TVs, set-top boxes, smartphones, tablets, laptops, and game consoles.
One of the most important key performance indicators for Netflix is *availability*. In our context, we consider the system to be available if users are successfully able to stream video on their devices. We usually express service availability as a percentage of users who are successfully able to stream video, over some interval of time. For example: 99.99% availability (“four nines”) means, roughly, that 99.99% of the time, when a user tried to start watching a video, they were successful.
While Netflix does not have the same availability requirements as, say, a telecommunications company, availability is still important for the business: if customers experience service interruptions, they are more likely to cancel their subscriptions.
Microservice architecture
-------------------------
From the perspective of the end-user, Netflix is a single service. Internally, Netflix is implemented using a microservice architecture [@microservices]: a collection of services[^1] that communicate with each other via remote procedure call (RPC). Fig. \[fig-architecture\] shows a Vizceral[@vizceral] visualization of a microservice architecture, where each node represents a cluster of servers that make up a single service, and requests, referred to collectively as *traffic*, flow through the system from left to right.
{width="\textwidth"}
Many features that Netflix exposes to users are localized to specific services. For example, Netflix allows users to search for a specific title: this functionality is implemented by the *search* service. Fig. \[fig-ui\], which shows a screen shot of part of the Netflix user interface, demonstrates how the user interface is assembled from the output of different microservices. There is one service that is responsible for presenting the match score, another one for showing metadata about the video stream (HD, 5.1), and another one that will process the results if a user clicks the “Rate this title” option and give the title a thumbs up or thumbs down.
![Screen shot of the top-left corner Netflix UI. Note that information such as the match score (99% Match), and the badges which show video and audio metadata (HD, 5.1) are retrieved from different microservices.[]{data-label="fig-ui"}](ui-single-show-double-cropped.png){width="\columnwidth"}
Netflix uses a microservice architecture to improve velocity: services are owned by different teams, and each team can deploy a service independently, without needing to coordinate with other teams. A microservice architecture can also increase availability by reducing the size of fault domains: if one service gets into a bad state, it doesn’t necessarily put the overall system into a bad state. For example, if the service that processes user rating fails, the overall system should handle that failure gracefully and the user should not even notice that this service has failed.
Resilience through timeouts, retries, and fallbacks
---------------------------------------------------
The Netflix control plane operates in an unforgiving environment: one of constant change within a public cloud. In this type of environment, there are many potential sources of failure, stemming from the infrastructure itself (e.g., degraded hardware, transient networking problem) or, more often, because of some change deployed by Netflix engineers that did not have the intended effect.
Three of the strategies that Netflix employs to achieve resilience are *timeouts*, *retries*, and *fallbacks*. All RPCs are configured with timeouts. An RPC might time out for a number of reasons, including problems with the particular server being called (e.g., server is overloaded) or problems with the networking infrastructure.
Many failures are either transient or isolated to a specific server, which means that a retry to a different server can often resolve the situation. However, not all problems are transient: if a bad code push results in a downstream service being in a bad state (e.g., returning errors for all requests), then a retry will not resolve the situation. In these cases, we rely on *fallbacks*, which is a sensible default response. For example, if we cannot suggest a row of viewables that are personalized for an individual user, we can fall back by serving a row of viewables that are not personalized (e.g., what is currently popular on Netflix). Fallbacks enable the system to degrade gracefully when encountering localized failures.
Many Netflix services use the Java-based Hystrix[@hystrix] library to implement fallbacks and the circuit breaker pattern[@release-it]. The Hystrix library has the notion of *commands*. A Hystrix command is a wrapper around logic that can potentially fail. A common Hystrix use case is to wrap a call for an RPC client. The Hystrix command can be configured to support timeouts, and fallbacks to handle errors. Hystrix commands are multithreaded: by default, a Hystrix command is associated with a threadpool with ten threads.
One of the challenges with timeouts and fallbacks is that these behaviors are not exercised as frequently as the *happy path*, which means we have less confidence they will work as expected. This is one of the key motivations for the development of the Chaos Automation Platform.
ChAP
====
Overview
--------
We have developed a system called the Chaos Automation Platform (ChAP)[@issre] for running chaos engineering experiments within the Netflix microservice architecture.
Most ChAP experiments focus on evaluating whether the overall system would remain healthy if one of the services degraded. Two failure modes we focus on are a service becoming slower (increase in response latency) or a service failing outright (returning errors). The service-level view of failure is useful because many different types of faults can be modeled as a service slowing down or returning errors. In particular, many bad code pushes (deployment of code with defects) can be modeled as a service that returns errors, and many forms of resource exhaustion (e.g., CPU, threads, memory, network bandwidth) can be modeled as a service slowing down.
ChAP is effectively an orchestration system that interacts with a number of internal Netflix services in order to carry out experiments. ChAP leverages a fault[^2] injection system developed inside of Netflix called FIT[@fit], which does fault injection at the application level. FIT takes advantage of the fact that applications deployed inside of the Netflix control plane use a common set of Java libraries. These libraries have hooks in them that enable us to inject faults at runtime.
Fault injection is typically done by annotating incoming requests with metadata that indicate that a call should be failed. This metadata is passed along as requests propagate through the system.
FIT supports two types of fault injection:
- failure - throw an exception instead of executing the call
- latency - add latency before executing the call
FIT supports injection into a number of different libraries, including RPC clients (both REST and gRPC), Hystrix[@hystrix], EVcache[@evcache], and Cassandra database clients.
Motivating example: failing the bookmarks service
-------------------------------------------------
To understand what ChAP does, we’ll walk through an example experiment. There is a service, which we’ll call [*bookmarks*]{}, which is responsible for keeping track of where a user when they previously watched a video. For example, if you previously watched 45 minutes of the romantic comedy “Set it up”, quit the Netflix app, and then returned to watch, the bookmarks service is responsible for identifying the location in the video to continue watching from.
The bookmarks service adds value for users, but it is not essential to the functioning of the overall system. If the bookmarks service failed, then we would still expect that users would be able to use the service and watch videos: they simply wouldn’t be able to continue from the last place where they watched.
Our hypothesis is that users should still be able to stream video when the bookmarks service fails. We can test this hypothesis by running an experiment where we intentionally cause the bookmarks service to fail for a sample of the user population, and verify that this sample of users are still able to stream videos successfully.
Although ChAP supports both server-side and client-side RPC fault injection, in our experiments we generally inject faults into the RPC clients: the service that is the subject of the experiment is the caller upstream of the service that we intend to fail. In this example, the API service[@api] is the upstream that calls the bookmarks service.
For this example, we’ll assume that the API cluster contains 180 servers. We also assume that the experiment will impact 1% of users. This means that we will select randomly select 1% of active users to be in the experiment (treatment) group. We will also randomly select 1% of users to be in the control group. Fig. \[fig-system-diagram\] depicts some of the internal Netflix services that ChAP must orchestrate to run an experiment. A ChAP experiment in progress is depicted in Fig. \[fig-bookmarks\].
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Define the experiment
---------------------
The user, a Netflix engineer, creates an experiment using the ChAP UI . In this case, the experiment is: *fail RPCs to the bookmark service and verify the system in general and the API service in particular remain healthy.*
The user specifies:
- Failure scenario: fail calls to the bookmark service
- Service to observe: API
Create the baseline and canary clusters
---------------------------------------
ChAP first calls out to Spinnaker[@spinnaker] . Spinnaker is Netflix’s continuous delivery system. Spinnaker serves as an interface for two important services: Amazon EC2 and Netflix’s internal dynamic properties system.
Netflix engineers use Spinnaker’s web-based user interface (UI) to define deployment pipelines for deploying their services onto Amazon EC2. Netflix engineers also use Spinnaker’s UI as an interface into Netflix’s internal dynamic properties system. Netflix services support dynamic configuration: configuration properties can be changed at runtime through Spinnaker.
ChAP uses Spinnaker to make requests against Amazon EC2 to provision two smaller copies of the API cluster inside of the Netflix control plane . These clusters are referred to as the “baseline” and “canary” clusters. Traffic from users in the control group will be routed to the baseline cluster, and traffic from users in the experiment group will be routed to the canary cluster. We borrow the baseline/canary terminology from canary deployments[@continuous-delivery]. Spinnaker is also used to copy any dynamic configuration settings from the original cluster to the baseline and canary . The dynamic configuration settings are copied before creating the new clusters so that the settings are in place before the new instances boot.
We size the baseline and canary clusters to be 1% of the size of the original cluster, in this example that corresponds to two servers per cluster.
Netflix services use routing identifiers called VIPs (virtual IPs)[@release-it]. VIPs are strings which behave similarly to DNS hostnames. Servers advertise VIPs to Eureka[@eureka], the service discovery service. RPC clients query Eureka to translate a VIP into a list of IP addresses. The baseline and canary clusters are each configured to advertise a VIP that is different from the original cluster. For example, if the servers in the original API cluster advertise the VIP [`api`]{}, then the baseline cluster would be assigned the VIP [`api-chap-baseline`]{} and the canary cluster would be assigned the VIP [`api-chap-canary`]{}. Because the RPC clients upstream of API are only configured to call out to the [`api`]{} VIP, the baseline and canary clusters do not receive any traffic when they first come up.
Start low-latency monitoring job
--------------------------------
As will be discussed in more detail in Section \[sec:dashboard\], ChAP relies on the Atlas[@atlas] telemetry system as a source for many of the dashboard graphs as well as the final analysis of the experimental results.
However, the most recent data we can reliably query from Atlas is typically five minutes old. If an experiment has revealed a significant vulnerability and customers are being severely impacted, we wish to detect this as soon as possible so we can abort the experiment.
In order to obtain lower latency telemetry data on critical business metrics that indicate whether customers are being impacted, we rely on an internal stream processing system called Mantis[@mantis].
Mantis is a platform that allows Netflix engineers to define jobs that consume events that are generated by different microservices. ChAP starts a Mantis[@mantis] job which keeps a count of the number of successful and failed video start play and download events, for users in both the baseline and canary groups. The Mantis job sends this data back to ChAP once a second. The latency of this data is on the order of seconds rather than minutes.
Sample from users, configure routing and fault injection
--------------------------------------------------------
ChAP publishes an event through an internal data publishing and subscription (pub/sub) service to indicate that the experiment should begin.
The event contains the following information:
------------------------- ---------------------------------------
Experiment size 1% of users
Failure to inject fail calls to the *bookmarks* service
VIP of original cluster [`api`]{}
VIP of baseline cluster [`api-baseline`]{}
VIP of canary cluster [`api-canary`]{}
------------------------- ---------------------------------------
Zuul[@zuul] is the *front door* to the Netflix control plane. It is a reverse-proxy service that receives all inbound requests from Netflix devices and routes the traffic into the control plane. Zuul also supports the notion of *filters*, which are functions that can be added to Zuul in order to perform processing on incoming requests. The Resilience Engineering team owns a Zuul filter which provides the functionality required for doing ChAP experiments, as described below:
The ChAP Zuul filter receives the event published by ChAP and randomly assigns 1% of users to the baseline group, and 1% of users to the canary group.
If an end-user is selected to be part of the baseline group, then all requests associated with that user will be annotated with a header with the following routing rule: [`api`]{} $\rightarrow$ [`api-baseline`]{}.
Any RPC client that is configured to make calls to the [`api`]{} VIP will instead make the call to `api-baseline` VIP. This will ensure that the users in the baseline group have their traffic routed to the api-baseline cluster.
If an end-user is selected to be part of the canary group, then all requests associated with that user will be annotated with a header with the following routing rule: [`api`]{} $\rightarrow$ [`api-canary`]{}
In addition, all requests will be annotated with a header with the following fault injection rule: *fail calls to the bookmark service*. Any RPC client configured to call the bookmarks service will instead return an error.
Zuul emits a Mantis event for each request. When a request involves an end-user that has been selected to be a member of the baseline or canary group, this event is consumed by the ChAP Mantis job , which keeps track of which end-users are in the baseline and canary groups.
Other services in the Netflix control plane emit events that are associated with “start play” events (successes, errors). These events are also consumed by the ChAP Mantis job: if a relevant event is associated with a user in a baseline or canary group, then the relevant metric counter is incremented.
Display dashboard {#sec:dashboard}
-----------------
Once the experiment starts, the user is presented with a dashboard which plots relevant metrics . Netflix relies on two internal system for generating operational dashboards: Atlas and Lumen.
Atlas[@atlas] is a telemetry platform for collecting and graphing time-series data in order to provide operational insight. Netflix engineers often interact with Atlas using a user interface called AtlasUI that allows users to generate graphs using either a graphical user interface or through a custom query language called Atlas Stack Language.
AtlasUI is useful for ad hoc generation of individual graphs, but engineers typically want to view a collection of previously specified graphs on a single page. Lumen[@lumen] is a platform that allows Netflix engineers to generate dashboards based on graphs generated by the Atlas back-end.
ChAP uses Lumen dashboards for displaying metrics data. This data is used to determine whether or not the injected faults had a negative impact on system health.
There are two main classes of metrics we use to determine whether the injected fault has had a negative impact on system health:
- key performance indicators (KPIs) for the users in the canary group versus the baseline group
- health metrics for the canary cluster versus the baseline cluster
The key performance indicators we are concerned with are around the ability of users to be able to stream video. In particular, we track a count of stream-starts per second (SPS), which is a count of the number of successful video stream starts. We collect SPS counts from both the server side and the device side, and we also track SPS errors, as shown in Fig. \[fig-kpis\]. Note that because error counts are generally much lower than success counts, it is much more likely that the error counts differ between the two groups due to random variation.
![Example graph from the dashboard generated by ChAP. This one shows one of the KPIs for the users in the baseline (control) and experiment (canary) group. The specific KPI shown here is the cumulative percentage of startplay errors reported by the client device. The area shaded in green indicates the time when fault injection is active. The y axis is intentionally obscured here to hide proprietary information.[]{data-label="fig-kpis"}](startplay-errors-cumulative.png){width="\columnwidth"}
We also compare health metrics between the baseline and canary deployments. We examine metrics such as request rate, latency, error rate, and CPU utilization. One example, CPU utilization, is shown in in Fig. \[fig-server-stats\].
![CPU utilization (%) is one of health metrics tracked in experiments.[]{data-label="fig-server-stats"}](cpu-utilization.png){width="\columnwidth"}
Monitor ongoing experiment
--------------------------
ChAP monitors the experiment as it runs to verify that there is no significant negative impact to customers. See Section \[sec-safety\] for more details.
Cleanup
-------
Once the experiment has completed, ChAP unpublishes the experiment event to stop Zuul from re-routing traffic and injecting faults. ChAP makes calls against Spinnaker to tear down the baseline and canary clusters.
Analyze the results
-------------------
Finally, ChAP calls out to Kayenta[@kayenta] , Netflix’s automated canary analysis system. Netflix engineers often use canary deployments to verify that new code being pushed to production has not introduced a regression. Kayenta performs a statistical analysis of metrics collected from a canary cluster and compares it to a baseline cluster, in order to determine whether there has been a statistically significant impact on any metrics of interest.
ChAP leverages Kayenta to perform an automated analysis of the relevant metrics for a chaos engineering experiment. While Kayenta was originally designed for evaluating the health of canary deployments, we found it to be an excellent match for analyzing the results of ChAP experiments.
{width="\textwidth"}
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Safety mechanisms {#sec-safety}
=================
Because ChAP experiments involve fault injection, every experiment carries risk that it could lead to an incident. To mitigate this risk, we have built in a number of safety mechanisms that limit the blast radius of an experiment.
**Business hours.** ChAP experiments only run during business hours: weekdays from 9AM to 5PM. This ensures that if something does go wrong, then engineers are likely to be at work and can respond more quickly than if there was an issue after hours.
**Automatic stop.** If ChAP detects excessive customer impact during an experiment, then it will stop the experiment early. This customer impact test is a cruder test than the non-parametric statistical tests used by Kayenta, but it limits the harm in the case where an experiment might have significant impact on affected users.
**Total traffic.** The set of all concurrently executing ChAP experiments cannot impact more than 5% of the total traffic, in any one of the three geographical regions that Netflix servers run in.
**Failover.** The Netflix control plane is deployed in three different Amazon geographical regions. If there is a problem with one of the regions, Netflix engineers can evacuate traffic from the troubled region and redirect it to the other two regions, a process we refer to as failover[@activeactive]. ChAP does not permit experiments to run during a failover as a safety mechanism, since a failover can violate assumptions that ChAP makes about the number of requests that are flowing through a region.
Monocle
=======
In designing ChAP, we considered two models of usage:
- a *self-serve* model where engineers define and run their own experiments.
- a *fully automated* model where a centralized team defines and runs experiments.
We initially adopted a self-serve model, where users were responsible for defining their own experiments. ChAP integrated with the Spinnaker deployment system so that users could add ChAP experiments to their deployment pipelines, but users were responsible for setting this up themselves. Later, we transitioned to a hybrid approach where we would automatically generate and run experiments, as well as support users in running self-serve experiments.
We developed an additional service called *Monocle* that has two functions, introspecting services and generating experiments.
Service introspection
=====================
Monocle introspects Netflix services to collect information about its dependencies. Here, a *dependency* refers to either a configured RPC client or a Hystrix command.
Monocle integrates data from multiple sources: the telemetry system (Atlas[@atlas]), the tracing system (based on Dapper[@dapper]) and by querying running servers directly for configuration information such as timeout values.
Monocle provides a UI which summarizes information about dependencies. For each Hystrix command, Monocle displays:
- The percentage of inbound requests that trigger an invocation of the Hystrix command.
- Whether it is believed to be safe to fail
- Whether it is configured with a fallback
- Configured timeouts
- Observed latencies over the past two weeks (mean, P90, P99, P99.5)[^3]
- Thread pool size
- Observed number of active threads over the past two weeks
- Which RPC clients it wraps, if any
- Any known impacts associated with the Hystrix command
The Monocle UI provides a tabular view from Monocle of a list of Hystrix commands We use heuristics such as the presence of a configured fallback and telemetry data that shows that the fallback has succeeded when executed. When accessing this view, users have the ability to toggle back-and-forth between viewing all of the Hystrix commands for a cluster or viewing all the RPC dependencies for a cluster.
Yellow and red attention icons show tooltips when moused over, and provide additional details around warnings and vulnerabilities. For example, mousing over red tooltip reveals the following message:
> Warning: Hystrix Command Timeout is Misaligned with RPC client.
>
> Timeout (1000 ms) is less than the max computed timeout of the wrapped RPC client (4000 ms). This means that Hystrix will give up waiting on RPC. This may be OK for non-critical calls, but you should review the config settings to confirm the desired behavior.
For each RPC Client, Monocle shows:
- Timeout and retry configurations
- The percentage of inbound requests that trigger an RPC call
- Maximum observed invocation rate (requests per second) over the past two weeks
- Which Hystrix commands wrap it, if any, and if they are safe to fail
- Any known impacts associated with the RPC client or the Hystrix commands that wrap it
Fig. \[fig-niws\] shows a tabular view from Monocle of a list of RPC dependencies (note: NIWS Client reference in the figure is Netflix Internal Web Service Framework, referred to as RPC Client throughout the paper \[20\]). Note how the commands that are not configured with fallbacks are shown as not being safe to fail. We use heuristics such as the presence of a configured fallback and telemetry data that shows that the fallback has succeeded when executed. When accessing this view, users have the ability to toggle back-and-forth between viewing all of the Hystrix commands for a cluster or viewing all the RPC dependencies for a cluster.
Experiment generation
=====================
Monocle is also responsible for generating experiments automatically. Today, Monocle generates fault injection experiments for dependencies (RPC clients and Hystrix commands). It generates three types:
- Failure
- Latency just below configured timeout (highest timeout - P99 latency over the past 7 days + 5% buffer)
- Latency causing failure (highest configured timeout + 5% buffer)
Monocle uses heuristics to try to identify the experiments with the highest likelihood of finding a vulnerability.
Criticality score
-----------------
Monocle assigns a *criticality score* to each dependency, which is used as input for prioritizing experiments. We are interested in prioritizing experiments on the more critical dependencies sooner and more frequently because these are the ones that we believe will cause the most harm if they behave incorrectly.
Beyond direct experimentation; we have also discussed using scores to generate a list of critical dependencies. That list could be used to identify and prioritize reliability work for our most critical components.
The criticality score is the product of the following four values:
1. dependency priority (RPC client $\rightarrow 1$, Hystrix command $\rightarrow 100$)
2. maximum percentage of inbound requests that trigger the dependency over the past 7 days as compared to all of the dependencies in the cluster ($<0.1\% \rightarrow
0, <1\% \rightarrow 10, <10\% \rightarrow 100, \leq100\% \rightarrow 1000$)
3. retry factor (1 + configured number of retries)
4. number of interactions associated with the dependency (if the dependency is an RPC Client, this would be the number of Hystrix commands, if it is a Hystrix command, this would be the number of RPC Clients)
Hystrix commands are prioritized over RPC clients because Hystrix commands wrap RPC client calls, and we want to verify that the Hystrix fallbacks are working correctly first.
Prioritization score
--------------------
Monocle calculates a score for each experiment as a the product of:
1. criticality score (see above)
2. safety score (safe $\rightarrow +1$, unsafe $\rightarrow -1$)
3. experiment weight[^4] (failure $\rightarrow 3$, latency $\rightarrow 2$, latency causing failure $\rightarrow 1$)
Monocle assigns a safety score of -1 to any experiment that it deems as unsafe. This includes situations such as:
- Dependency being experimented on has been blacklisted (no experiment types should be created for this dependency)
- Dependency data associated with the experiment has been not been recently collected, and does not have up-to-date information
- Dependency being experimented on contains a critical vulnerability and no experiment types should be created. For example, the dependency is an unwrapped RPC Client (no Hystrix command)
- Dependency has a known impact associated with its failure (added by a user and directly related to a key performance indicator: SPS, downloads per second, login, or signup)
- Experiment being created is a latency experiment, and the dependency is both missing fallbacks and requires timeout tuning
- Experiment being created is a failure experiment and the dependency is not safe to fail due to a missing fallback
Monocle computes scores for all possible experiments and the resulting score represents whether a test is safe to run (i.e., scores $>$ 0 are safe). For the current use cases, there is not much need for scores of unsafe experiments except for debugging. This is an implementation detail, and another option would be to pre-process the experiments and identify safety before scoring them.
Monocle then runs the experiments in priority order. Only experiments which have a positive score will be run. When going through the prioritized list, Monocle checks if the experiment should run by ensuring
- The experiment is not already running
- The experiment is not in a failed state (it has previously failed and has not been looked at by a user)
- The experiment has not been run in the last (configured number) of days
Results
=======
To date, experiments generated by Monocle have revealed several cases where timeouts were set incorrectly and fallbacks revealed the service to be more business critical than the owner had intended. Fig. \[fig-hystrix\] shows an example of an interaction problem that can arise between a Hystrix command’s timeout configuration and its threadpool size. In this experiment, about 900 ms of latency is injected into a Hystrix command to mimic the scenario of a downstream dependency going latent.
![A Hystrix metric graph that shows elevated threadpool rejections for the canary group[]{data-label="fig-hystrix"}](hystrix-threadpool-rejected.png){width="\columnwidth"}
The increase in the *countThreadPoolRejected* metric indicates that the work could not be scheduled to a thread because all of the threads in the threadpool are blocked, resulting in the Hystrix command serving a fallback. This leads to a short-circuiting behavior where the Hystrix command unconditionally serves fallbacks for a period of time.
The revealed problem is that the timeout is too high relative to the size of the threadpool. In this particular case, the configured timeout was much higher than the 99^th^ percentile latency reported by our telemetry system, and so the solution was to decrease the timeout accordingly.
Challenges and lessons learned
==============================
Failure modeling
----------------
The existing tooling limits the type of faults that we can inject into the system. In particular, FIT can only cause one type of error per injection point, but there are often multiple ways that a service can fail. For example, we had a situation where we ran an experiment by injecting failure into a particular service. The injected fault behaved as if the service returned an error. The experiment showed that the system could handle this failure gracefully. However, we had an incident where none of the servers associated with that service were registered in our service discovery mechanism.
Application-based fault injection
---------------------------------
Netflix implements fault injection using defined injection points in Java-based platform libraries. These libraries are common to all Netflix applications. Deploying new libraries can take many months, as we have to wait for all of the services to pick them up.
Historically, applications in the Netflix control plane have been Java-based. However, the trend inside of Netflix is to move towards more support for polyglot, most notably JavaScript running on Node.js. This makes an application-based approach challenging because we need to implement new libraries for each supported language.
There are alternative approaches which do fault injection out-of-process, obviating the need for language-specific bindings. Istio[@istio] is one example of a service mesh based approach to fault injection.
Even if you build it, they might not come
-----------------------------------------
When ChAP was first made available for internal users, we only had a few teams making regular use of the service. While we strove to make the interface as simple as possible for new users, running an experiment on production traffic is itself a complex task, and there is a limit as to how much we can possibly simplify the interface to support this. We employed a consulting model with internal teams who would make good candidates for potential users, but we did not see widespread usage.
Challenge of automation
-----------------------
Our alternate approach to the self-serve model, automatically generating the experiments ourselves, presented its own set of challenges. We needed to develop heuristics to determine which types of faults could be injected which were not known to cause customer impact. This was necessarily a smaller space of faults then if we have domain knowledge about the individual services.
In addition, we had to ensure that we had a low false positive rate, otherwise the service teams would lose confidence in the results of the experiments. This meant that if an automatically generated ChAP experiment revealed a potential vulnerability, a member of the Resilience Engineering team had to spend time analyzing the results of the experiment to verify that it did, in fact, reveal a genuine vulnerability. This was generally a time-consuming and tedious process. It is possible to build supplementary tooling to reduce the effort involved in doing the analysis, but that itself requires an additional investment.
Small sub-populations
---------------------
Small sub-populations are a challenge to deal with. Netflix software runs a variety of different types of devices, and these devices behave differently. A problem might only manifest on, say, a particular brand of Smart TV. If only a tiny fraction of our userbase watches Netflix on this type of television, then even if these users can’t stream video at all, it’s unlikely to be detected by looking at total SPS success counts. We could potentially oversample from device types that are less common, but this means that we are increasing the blast radius for that device type, which we have been reluctant to do.
Error counts
------------
Although intuitively error rates seem like a reliable signal for identifying problems with the system, we found them to have some undesirable properties for use as an experimental measure.
Error counts can help with the small sub-population issue, because error counts are generally quite low, so even a smaller sub-population may end up contributing significantly to the error rate. However, this creates the opposite problem: a single device in the baseline or canary group can cause a significant increase in the error count if the device keeps erroring over and over. This means that if an error-prone device happens by chance to be assigned to a baseline or error group, it can generate a spurious error signal.
In particular, for our automatic stop safety mechanism, we had to substantially increase the threshold which we trigger a stop on the error metrics because of how noisy they were.
Value of visualization
----------------------
To be able to automatically generate experiments, we needed to build tooling to obtain visibility into the configuration and observed behavior of RPC clients and Hystrix commands. While this information has always been available within Netflix through telemetry or configuration endpoints, it had not previously been aggregated and displayed, which proved useful for surfacing vulnerabilities and interactions even without using this data for automatic experiment generation. For example, by integrating this information into a single view, we were able to identify some cases of inconsistently configured Hystrix and RPC client timeouts without even needing to run a ChAP experiment.
Conclusion
==========
Our work demonstrates that it is possible to automatically and safely generate and run chaos experiments. These experiments have identified vulnerabilities that could lead to outages if left untreated.
While the original goal for ChAP was to run fault injection experiments, we have discovered that the platform itself can be used for other types of experiments. We are currently extending ChAP to support load testing experiments, similar in spirit to RedLiner[@redliner]. In addition, some self-serve users have begun experimenting with using ChAP for canary deployments[@release-it], because of the additional analysis that ChAP provides.
[^1]: For the remainder of this paper, we use the term *service* to refer to a microservice.
[^2]: In this paper, we sometimes use *fault* and *failure* interchangeably, which is consistent with the Chaos Engineering industry community usage.
[^3]: P90 stands for “90^th^ percentile”
[^4]: This is a simplification, we actually flip the weights if the score is negative in order to maintain the ordering of the experiments.
|
---
abstract: 'We use a three-dimensional molecular dynamics simulation to study the single particle distribution function of a dilute granular gas driven by a vertically oscillating plate at high accelerations ($15g - 90g$). We find that the density and the temperature fields are essentially time-invariant above a height of about 35 particle diameters, where typically 20% of the grains are contained. These grains form the nonequilibrium steady state granular gas with a Knudsen number unity or greater. In the steady state region, the distribution function of horizontal velocities (scaled by the local horizontal temperature) is found to be nearly independent of height, even though the hydrodynamic fields vary with height. The high energy tails of the distribution functions are described by a stretched exponential $\sim \exp(-{\cal B}c_x^{\alpha})$, where $\alpha$ depends on the normal coefficient of restitution $e$ ($1.2 < \alpha < 1.6$), but $\alpha$ does not vary for a wide range of the friction parameter. We find that the distribution function of a [*frictionless*]{} inelastic hard sphere model can be made similar to that of a frictional model by adjusting $e$. However, there is no single value of $e$ that mimics the frictional model over a range of heights.'
author:
- Sung Joon Moon
- 'J. B. Swift'
- 'Harry L. Swinney'
title: Steady state velocity distributions of an oscillated granular gas
---
introduction
============
A dilute gas in thermal equilibrium is sufficiently characterized by the pressure and temperature and is described by a simple relation, the equation of state. However, when a gas is far from equilibrium, there is no general, finite set of variables specifying the state. The single particle distribution function $f({\bf r},{\bf v},t)$ is often sufficient to characterize the statistical properties of a dilute nonequilibrium gas when correlations are negligible. Given this function, other quantities, such as moments of the distribution and transport coefficients, can be evaluated. Dilute granular materials subject to an external forcing exhibit gaseous behaviors that share many analogies with a molecular gas, and they are often called granular gases. Such a granular gas is always far from equilibrium due to the dissipative collisions, and the deviation of $f({\bf r},{\bf v},t)$ from the Maxwell-Boltzmann (MB) distribution has been of great interest in recent years [@warr95; @olafsen98; @losert99; @kudrolli00; @rouyer00].
The velocity distributions of a vibro-fluidized granular gas were first measured by Warr [*et al.*]{} [@warr95]; they studied the distribution functions of grains confined between two transparent plates and concluded that the distribution was consistent with the MB distribution function. Recently, the same system has been studied by Rouyer [*et al.*]{} [@rouyer00], who found a [*universal*]{} distribution function of the form $\sim \exp(-B|{\bf v}|^{1.5})$ for the [*entire*]{} range of velocities studied, where $B$ was a parameter. The authors reported that this functional form fit their measurements for a wide range of oscillation parameters for various materials; thus the granular temperature, the second moment of the distribution, was the only parameter of the distribution function.
There have been numerical studies of vibrated inelastic hard disks, subject to a saw-tooth type oscillation, in the presence of gravity [@brey03] and in the absence of gravity [@barrat02]. Such forcing is often used in theoretical studies as a simplification of the sinusoidal oscillation used in experiments, assuming that the asymptotic behavior of the hydrodynamic fields far from the plate is the same; however, it is not known [*a priori*]{} how far from the oscillating plate one must be in order for this assumption to be valid.
In this paper, we perform a simulation that is as close as possible to three-dimensional experiments on vertically oscillated granular gases. We use a previously validated molecular dynamics (MD) simulation [@bizon98; @moon02a]. The hydrodynamic fields are oscillatory near the plate, and their oscillatory behavior decays with height. Above some height, the fields are not correlated with the oscillation of the plate and are essentially time-invariant. We study the distribution functions in this nonequilibrium steady state region. To focus on the distributions due to the intrinsic dynamics of the granular gas, we do not impose sidewalls or include air. We also study how the distribution changes with the friction. In many theoretical or numerical studies of granular fluids, granular materials are modeled as frictionless inelastic hard disks or spheres; however, no granular materials are frictionless, in the same way that none of them are elastic. We check if the role of friction can be incorporated into the inelasticity by adjusting the value of the normal coefficient of restitution. In this paper we discuss the distributions only in the steady state region; those in the oscillatory region near the plate will be discussed in a separate paper [@moon03vdf2].
The rest of the paper is organized as follows. In Section II, the system under consideration, the data analysis method, and the collision model are described. Results are presented in Section III and discussed in Section IV.
method
======
System and data analysis
------------------------
We use both a frictional and frictionless, inelastic hard sphere MD simulation. We consider 133 328 monodisperse spheres of unit mass and of diameter $\sigma = 165~\mu$m in a container with square bottom of area 200$\sigma\times 200\sigma$ (the average depth of the layer at rest is approximately 3$\sigma$), where periodic boundary conditions are imposed in both horizontal directions. We choose the same particle size as in Ref. [@bizon98], as the patterns were quantitatively reproduced for a wide range of oscillation parameters with this particle size; however, as long as the collision model is valid, all the length scales can be normalized by $\sigma$. We assume the bottom plate of the container is made of the same material as grains; we use the same material coefficients for the inelasticity and the friction as grains. The bottom plate is subject to a vertical sinusoidal oscillation with an amplitude $A$ and a frequency $f$. We vary the oscillation parameters in the range of $3\sigma < A < 10\sigma$ and $40~{\rm Hz} < f < 170~{\rm Hz}$, which approximately corresponds to $0.35~{\rm m/s} < V_{max}~(= 2\pi fA) < 0.75~{\rm m/s}$ and $15g < a_{max}~[= A(2\pi f)^2] < 90g$, where $g$ is the acceleration due to gravity. We check that with our parameters no mean flow develops and that grains rarely reach to the top, which is fixed at 300$\sigma$.
Hydrodynamic fields and the distribution functions are analyzed by binning the box into horizontal slabs of height $\sigma$, as the system is invariant under the translation in both horizontal directions, in the absence of any mean flow. We use the granular volume fraction $\nu$ for the density, which is the ratio of the volume occupied by grains to the volume of each horizontal slab. We consider the following three granular temperatures separately:
$$\begin{aligned}
T_x &=& {1 \over 2} \left<\left(v_x-\left<v_x\right>\right)^2+\left(v_y-\left<v_y\right>\right)^2\right>,\\
T_z &=& \left<(v_z-\left<v_z\right>)^2\right>,\\
T &=& {1 \over 3} \left<|{\bf v}-\left<{\bf v}\right>|^2\right>\\
&=& {1 \over 3} \left(2T_x+T_z\right),\end{aligned}$$
where $x$ and $y$ are horizontal directions that are indistinguishable, $z$ is the vertical direction, [**v**]{} is a velocity vector for each grain, and the ensemble average $\left<~\right>$ is taken over the particles in the same bin at the same phase angle during 40 cycles, after initial transients have decayed. We define the scaled horizontal velocity to be $$\label{scale}
c_x = (v_x - \left< v_x \right>)/\sqrt{2T_x}.$$
collision model
---------------
We implement the collision model that was originally proposed by Maw [*et al.*]{} [@maw], simplified by Walton [@walton], and then experimentally tested by Foerster [*et al.*]{} [@foerster]. This model updates the velocity after a collision according to the three parameters, the normal coefficient of restitution $e$ ($\in [0,1]$), the coefficient of friction $\mu$, which relates the tangential force to the normal force at collision using Coulomb’s law and then determines the tangential coefficient of restitution $\beta$ ($\in [-1,1]$), and the maximum tangential coefficient of restitution $\beta_0$, which represents the tangential restitution of the surface velocity when the colliding particles slide discontinuously at the contact point.
At collision, it is convenient to decompose the relative colliding velocities into the components normal (${\bf v}_n$) and tangential (${\bf v}_t$) to the normalized relative displacement vector $\hat{\bf r}_{12} \equiv ({\bf r}_1 - {\bf r}_2)/|{\bf r}_1 - {\bf r}_2|$, where ${\bf r}_1$ and ${\bf r}_2$ are displacement vectors of grains 1 and 2, and the same notation is used for ${\bf v}$: $$\begin{aligned}
{\bf v}_n &=& ({\bf v}_{12}\cdot \hat{\bf r}_{12})\hat{\bf r}_{12}
\equiv v_n\hat{\bf r}_{12},\\
{\bf v}_t &=& \hat{\bf r}_{12}\times ({\bf v}_{12}\times \hat{\bf r}_{12}) = {\bf v}_{12} - {\bf v}_n.\end{aligned}$$ The relative surface velocity at collision, ${\bf v}_s$, for monodisperse spheres of diameter $\sigma$ is $${\bf v}_s = {\bf v}_t + {\sigma \over 2}\hat{\bf r}_{12} \times
({\bf w}_1+{\bf w}_2) \equiv v_s\hat{\bf v}_s,$$ where ${\bf w}_1$ and ${\bf w}_2$ are the angular velocities of grains 1 and 2, respectively.
For monodisperse spheres of diameter $\sigma$ and unit mass, the linear and angular momenta conservations and the definitions of the normal coefficient of restitution $e \equiv -v_n^*/v_n$ and the tangential coefficient of restitution $\beta \equiv -v_s^*/v_s$ (post-collisional velocities are indicated by superscript $*$, and pre-collisional values have no superscript) give the changes in the velocities at the collision: $$\begin{aligned}
\Delta {\bf v}_{1n} &=& -\Delta {\bf v}_{2n} = {1\over 2}\left(1+e\right){\bf v}_n,\\
\Delta {\bf v}_{1t} &=& -\Delta {\bf v}_{2t} = {K\left(1+\beta\right)\over 2\left(K+1\right)}{\bf v}_s,\\
\Delta {\bf w}_1 &=& -\Delta {\bf w}_2 = {\left(1+\beta\right)\over \sigma\left(K+1\right)}\hat{\bf r}_{12}\times{\bf v}_s,\end{aligned}$$ where $K = 4I/\sigma^2$ is a geometrical factor relating the momentum transfer from the translational degrees of freedom to rotational degrees of freedom, and $I$ is the moment of inertia about the center of the grain. For a uniform density sphere, $K$ is $2/5$.
We use a velocity-dependent normal coefficient of restitution as in Ref. [@bizon98], to account for the viscoelasticity of the real grains: $$e = {\rm max}\left[e_0,1 - (1-e_0)\left({v_n\over \sqrt{g\sigma}}\right)^{3/4}\right],$$ where $e_0$ is a positive constant less than unity. Since we impose high forcing ($V_{max} > 0.35$ m/s while $\sqrt{g\sigma} = 0.04$ m/s) the collision probability for relative colliding velocities $v_n$ less than $\sqrt{g\sigma}$ is small, and using a velocity-independent $e$ does not result in any noticeable difference, compared to using a velocity-independent one $e = e_0$; the same was true in Ref. [@bougie02]. We use the symbol $e$ for $e_0$ hereafter.
In collisions of real granular materials, not only is the relative surface velocity reduced, but also the stored tangential strain energy in the contact region can often reverse the direction of the relative surface velocity. To account for this effect, the tangential coefficient of restitution $\beta$ could be positive, leading to the range of $\beta$ as $[-1,1]$. There are two kinds of frictional interaction at collisions, sliding and rolling friction, which are accounted for by the following formula for $\beta$: $$\beta = {\rm min}\left[\beta_0,-1 + \mu\left(1+e\right)\left(1+{1\over K}\right){v_n\over v_s}\right],$$ where $\beta_0$ is the maximum tangential coefficient of restitution. For sliding friction, the tangential impulse is assumed to be the normal impulse multiplied by $\mu$. When $\beta$ is identically negative unity (or simply $\mu = 0$), this model reduces to the frictionless interaction. For the special case $v_s = 0$, the collision is treated as frictionless. This friction model is still a simplification of the real frictional interaction; there is no clear-cut distinction between the two types of frictions for real grains, and even a transfer of energy from the rotational to translational degrees of freedom, which results in $e$ larger than unity, has been observed [@louge]. However, this collision model is accurate enough to reproduce many phenomena, including standing wave pattern formation in vertically oscillated granular layers, when the parameters are properly chosen. In Refs. [@bizon98] and [@moon02a], $e = 0.7,~\beta_0 = 0.35$, and $\mu = 0.5$ were used.
[![ \[FF\] The volume fraction $\nu$ (thick solid line) and the granular temperature $T$ (thick dashed line) as a function of height at different times during a cycle, where $e = 0.9,~\beta_0 = 0.35,~\mu = 0.5$, $V_{max} = 0.55$ m/s ($a_{max} = 60g$, $A = 3\sigma$, and $f = 169$ Hz), and $ft$ is set to zero when the plate is at the equilibrium position moving upward. Above some height ($z/\sigma \approx 35$), the hydrodynamic fields do not vary much in time. The horizontal temperature $T_x$ (thin dashed line) is smaller than the vertical temperature $T_z$ (thin solid line). The container bottom is indicated by the vertical gray line. ](./fig1.eps "fig:"){width="0.805\columnwidth"}]{}
[![ \[T-INV\] This superposition of the volume fraction and temperature fields at five different times in a cycle (see Fig. \[FF\]) illustrates that the fields are nearly time-independent above $z/\sigma \approx 35$, which we call the steady state region. ](./fig2.eps "fig:"){width="0.8\columnwidth"}]{}
[![ \[CPP\] The number of grain-grain collisions per grain during a cycle $N_{coll}$ (solid line), and time-averaged volume fraction $\nu_{avg}$ (dot-dashed line, multiplied by 100) and time-averaged granular temperature $T_{avg}$ (dashed line, multiplied by 100), over sixty different equally spaced times during a cycle. $N_{coll}$ is less than 1 in the steady state region ($z/\sigma > 35$). Inset : The same quantities (without multiplications) on a logarithmic scale. ](./fig3.eps "fig:"){width="0.8\columnwidth"}]{}
results
=======
Hydrodynamic fields and steady state
------------------------------------
Due to the oscillatory boundary forcing, the hydrodynamic fields, the volume fraction $\nu$ and the granular temperatures ($T,~T_x$, and $T_z$), depend on height $z$ and time $t$ (Fig. \[FF\]) near the oscillating plate; the temperatures exhibit stronger oscillatory behaviors than the density. Since the energy is injected mainly through the vertical velocities, the granular temperature is anisotropic, as illustrated in Fig. \[FF\]; $T_z$ is larger than its horizontal counterpart $T_x$, and the former is significantly larger near the bottom plate, where the hydrodynamic fields are oscillatory. The vertical temperature increases almost linearly with height for $z/\sigma > 35$; however, the temperature $T$ increases slower than linearly, as the slope of $T_x$ decreases with height and $T_x$ levels off for $z/\sigma > 120$ (Fig. \[FF\]). A similar increase of the temperature with height was observed in an open system of frictionless inelastic hard disks or spheres subject to a thermal bottom heating [@soto99] and a saw-tooth type vibration [@brey01]. We characterize the oscillation parameters only by $V_{max}$, as we observe for the parameters in our study that the hydrodynamic fields in the steady state are nearly the same for the same $V_{max}$, even for different combinations of $a_{max}$ and $f$; such scaling behavior was also observed in Ref. [@lee95].
During each cycle, a normal shock forms at the impact from the bottom plate and propagates upward [@bougie02]. As the shock propagates, it decays and becomes undetectable above some height ($z/\sigma \approx 35$), rather than propagating up through the entire granular media (which was the case in Ref. [@bougie02]). Above this height, the hydrodynamic fields are invariant in time, and the granular gas forms a nonequilibrium steady state (Fig. \[T-INV\]), where about 20% of the grains are contained in this case; this fraction depends on the oscillation parameters.
With the parameters used in this paper the granular temperatures are nonzero throughout the cycle, as grains do not solidify after the shock passes through, in contrast to the case in Ref. [@bougie02]. As a result, when the bottom plate moves down, the granular gas expands, and an expansion wave propagates downward (see the temperature peaks near the plate for $ft > 0.4$ in Fig. \[FF\]).
We count the number of grain-grain collisions per grain during a cycle ($N_{coll}$ in Fig. \[CPP\]), and find that $N_{coll}$ is less than unity in the steady state region; the granular gas in the steady state is nearly collisionless. We estimate the mean free path using the formula for a gas of hard spheres $\lambda(z)/\sigma = (2\sqrt{2\pi}n\sigma^3)^{-1} =
\sqrt{\pi}/(12\sqrt{2}\nu)$ (where $n$ is the number density) [@huang], which ranges between 5 and 280 for $40 < z/\sigma < 100$ (Fig. \[mfpKn\]). When we estimate the mean free path using the measured collision frequency and the thermal speed, we get a similar result. We fit the density with piecewise exponential functions \[$\sim \exp(-(z-z_0)/l_{\nu})$\] in the steady state region, and obtain a hydrodynamic length scale $l_{\nu}/\sigma$ between 12 and 15 for $40 < z/\sigma < 100$. We obtain a similar length scale from piecewise linear fitting of the temperature $T$ in the same region. We calculate the Knudsen number $Kn$, defined as the ratio of the mean free path to the length scale of the macroscopic gradients [@kogan69], using $\lambda(z)$ and $l_{\nu}$; $Kn$ ranges from 0.5 to 20 in the region $40 < z/\sigma < 100$ (Fig. \[mfpKn\]).
[![ \[mfpKn\] The mean free path $\lambda$, estimated from a formula for a gas of hard spheres, and the Knudsen number $Kn$, estimated by using $\lambda$ and the length scale of the density $l_{\nu}$ (see text), in the steady state region.](./lambdaKn.eps "fig:"){width="0.75\columnwidth"}]{}
Height-independence of the distribution
---------------------------------------
The distribution of scaled horizontal velocities should be symmetric as a consequence of the symmetry of the system. We calculate the skewness of the distribution, $\gamma_3 = {\cal M}_3/{\cal M}_2^{3/2}$, where ${\cal M}_n$ is the $n^{th}$ moment of the distribution
[![ \[TK\] Above some height ($z/\sigma \approx 42$, indicated by a vertical dashed line), the kurtosis $\gamma_4$ is nearly time-invariant. The horizontal granular temperature $T_x$ (dashed lines) and $\gamma_4$ (solid lines) of the horizontal velocity distribution function are shown at five different times during a cycle ([*cf.*]{} Fig. \[FF\]). ](./fig4.eps "fig:"){width="0.8\columnwidth"}]{}
[![ \[M4\] The horizontal velocity distribution functions at four different heights (compared with $f_{MB}$, the solid line) obtained at $ft = -0.2$. There is no noticeable difference among the distributions in the steady state region, $40 < z/\sigma < 80$. ](./fig5.eps "fig:"){width="0.8\columnwidth"}]{}
$$\label{EQmoment}
{\cal M}_n = \int c_x^nf(c_x) dc_x,$$
and check that $|\gamma_3|$ is less than 0.01 for all the distributions we study. The lowest order deviation from the MB distribution is characterized by the flatness of the distribution, which is called the fourth cumulant or the kurtosis. It was used to quantify the deviation from the MB distribution of the homogeneously cooling state [@vannoije98; @brilliantov00; @nie02], the homogeneously heated state [@vannoije98; @moon01], and granular gases subject to a boundary forcing [@brey03; @blair01]. The kurtosis $\gamma_4$ ($\equiv {\cal M}_4/{\cal M}_2^2 -3$) is defined so that it vanishes for the MB distribution, and we find that it also does not change in time above some height (Fig. \[TK\]). Further, in the steady state region, $\gamma_4$ is nearly independent of the height, even though both the density and the temperature change; the distributions at different heights in the steady state region are hardly distinguishable (Fig. \[M4\]). Also, the kurtosis in the steady state region does not vary much for a wide range of the oscillation parameters: for $0.35~{\rm m/s} < V_{max} < 0.75~{\rm m/s}$ (and other parameters fixed), $\gamma_4$ changes less than 10%. A similar absence of height dependence of the velocity distribution function was found in a recent experiment on a vertically oscillated quasi-2d granular gas [@vanzon].
[![ \[kurtosis\_e\] Kurtosis for three values of $e$, as a function of height at $ft = -0.2$, where $\mu$ and $\beta_0$ are set to 0.5 and 0.35. Different forcings are applied for each case to achieve similar profiles of the hydrodynamic fields: $V_{max}~(a_{max}) = 0.4$ m/s ($43g$), 0.55 m/s ($60g$), and 0.66 m/s ($72g$) for $e = 0.95$, $e = 0.9$, and $e = 0.85$, respectively. ](./fig6.eps "fig:"){width="0.8\columnwidth"}]{}
[![ \[VDF\_e\] The distribution functions of scaled horizontal velocities $f(c_x)$ for the cases in Fig. \[kurtosis\_e\], on linear (top panel) and logarithmic (bottom panel) scales. Although the kurtosis slightly decreases with $e$, the difference between the distributions is hardly distinguishable on both scales. The ranges for the averaging in height were between $30\sigma$ and $45\sigma$ for $e=0.95$, $40\sigma$ and $55\sigma$ for $e = 0.9$, and $45\sigma$ and $60\sigma$ for $e = 0.85$; the averaging was done over relatively similar heights in the steady state regions (see Fig. \[kurtosis\_e\]). The solid line is the MB distribution. ](./fig7a.eps "fig:"){width="0.8\columnwidth"}]{} [![ \[VDF\_e\] The distribution functions of scaled horizontal velocities $f(c_x)$ for the cases in Fig. \[kurtosis\_e\], on linear (top panel) and logarithmic (bottom panel) scales. Although the kurtosis slightly decreases with $e$, the difference between the distributions is hardly distinguishable on both scales. The ranges for the averaging in height were between $30\sigma$ and $45\sigma$ for $e=0.95$, $40\sigma$ and $55\sigma$ for $e = 0.9$, and $45\sigma$ and $60\sigma$ for $e = 0.85$; the averaging was done over relatively similar heights in the steady state regions (see Fig. \[kurtosis\_e\]). The solid line is the MB distribution. ](./fig7b.eps "fig:"){width="0.8\columnwidth"}]{}
[![ \[lnlnYF\] Double logarithm of the distributions of scaled horizontal velocities for the cases in Fig. \[kurtosis\_e\] as a function of the logarithm of $c_x$. The slope corresponds to the negative exponent, $-{\alpha}$, of a stretched exponential function $\exp(-c_x^{\alpha})$. ${\alpha}$ is the same for small velocities for three different $e$’s, however, it depends on $e$ in high energy tails. Dashed lines correspond to ${\alpha}$ = 1.9, and the solid lines (from the top) correspond to ${\alpha}$ = 1.2, 1.4, and 1.6, respectively (indicated by the numbers). ](./fig8.eps "fig:"){width="0.8\columnwidth"}]{}
Velocity distributions
----------------------
We first examine the dependence of the distribution on $e$; we measure $\gamma_4$ for three different values of $e$, while $\beta_0$ and $\mu$ are kept at 0.35 and 0.5, respectively. We find that $\gamma_4$ significantly decreases with increasing $e$ in the oscillatory state, however, it decreases only slightly in the steady state region (Fig. \[kurtosis\_e\]).
Now we compare our results with the MB distribution of variance $1/\sqrt{2}$,
$$f_{MB}(c_x) = {1 \over \sqrt{\pi}}\exp(-c_x^2).$$
The steady state distributions obtained for the parameters in Fig. \[kurtosis\_e\] are overpopulated in the high energy tails and underpopulated at small velocities (Fig. \[VDF\_e\]), compared to $f_{MB}$. The differences of the distributions for various $e$’s are hardly noticeable both on linear and logarithmic scales (Fig. \[VDF\_e\]), but on a double logarithmic scale plot the tails of the distributions are described by different functions (Fig. \[lnlnYF\]). We investigate the functional form of the distributions by fitting them (after the normalization by the value at $c_x = 0$) with a stretched exponential function $\exp(-c_x^{\alpha})$. We find that the exponent ${\alpha}$ changes from 1.9 (indicated by dashed lines) to some smaller value (solid lines), depending on $e$, as the velocity increases. We have not investigated lower values of $e$ to avoid issues such as cluster formation.
We now keep $e$ and $\beta_0$ at 0.9 and 0.35, respectively, and change the value of $\mu$. The profile of $\gamma_4$ in the oscillatory region changes significantly with $\mu$, however, it is nearly unchanged in the steady state region (Fig. \[kurtosis\_mu\]). The velocity distributions in this region for three different values of $\mu$ in Fig. \[kurtosis\_mu\] are also hardly distinguishable. We observe that the distribution function depends also on the density, as in Refs. [@moon01; @barrat02; @brey03]; however, we do not investigate this dependency systematically.
[![ \[kurtosis\_mu\] Kurtosis of the distributions of scaled horizontal velocities as a function of height at $ft = -0.2$, for three values of $\mu$ ($\beta_0$ and $e$ are kept at 0.35 and 0.9, respectively). In the oscillatory region, $\gamma_4$ increases with $\mu$; however, $\gamma_4$ is nearly the same within the uncertainty in the steady state region. The same forcing ($V_{max} = 0.55$ m/s) is applied to the three cases. ](./fig9.eps "fig:"){width="0.8\columnwidth"}]{}
Frictionless inelastic hard sphere model
----------------------------------------
In theoretical and numerical studies, granular materials are often modeled as smooth (frictionless) inelastic hard disks or spheres, assuming that the friction is a secondary effect that can be neglected or that both the inelasticity and the friction can be incorporated together into a so-called effective coefficient of restitution. In this Section, we discuss how the velocity distribution changes when the friction is not included, and we show that the frictionless model exhibits qualitative differences from the frictional model.
[![ \[kurtosis\_NF\] Kurtosis of the distributions of scaled horizontal velocities of frictionless spheres, as a function of height for three values of $e$ obtained at $ft = -0.2$. Different forcings (the same as in Fig. \[kurtosis\_e\]) are applied for each case: $V_{max}~(a_{max}) = 0.4$ m/s ($43g$), 0.55 m/s ($60g$), and 0.66 m/s ($72g$) for $e = 0.95$, $e = 0.9$, and $e = 0.85$, respectively. ](./fig10.eps "fig:"){width="0.8\columnwidth"}]{}
The rotational kinetic energy is two orders of magnitude smaller than its translational counterpart for the cases studied in this paper. However, the presence of the friction reduces the expansion of the granular gas significantly, because the friction reduces the mobility of the grains and increases the collision frequency [@moon03fric]. The mean height of frictional inelastic hard spheres exhibits a different scaling behavior with the plate velocity from that of frictionless spheres. Only the frictional sphere model reproduces the experimental observations [@moon03fric; @luding95].
[![ \[lnlnNF\] Double logarithm of the rescaled horizontal velocity distribution functions (normalized by its value at zero) for the cases in Fig. \[kurtosis\_NF\], as a function of the logarithm of $c_x$. Dashed lines correspond to ${\alpha}$ = 2.0, and the solid lines (from the top) correspond to ${\alpha}$ = 1.65, 1.85, and 1.9, respectively. ](./fig11.eps "fig:"){width="0.8\columnwidth"}]{}
The $\gamma_4$’s obtained from the simulations of frictionless particles for the same forcings as in Fig. \[kurtosis\_e\] are illustrated in Fig. \[kurtosis\_NF\]. In the absence of friction, the layer expands much more, and the steady state occurs at greater height, $z/\sigma > 100$. In both the oscillatory and the steady state regions, values of $\gamma_4$ are smaller than those of frictional spheres (compare Fig. \[kurtosis\_NF\] with Figs. \[kurtosis\_e\] and \[kurtosis\_mu\]); the distribution deviates from $f_{MB}$ only slightly. The kurtosis decreases with increasing $e$, and the difference among the distributions for the three $e$’s are small. These distributions have four crossovers with $f_{MB}$; they are overpopulated both at very small and high velocities and are underpopulated in between, compared to $f_{MB}$. We fit them with a stretched exponential function, and find that ${\alpha}$ is 2.0 for small velocities, and that it depends on $e$ for the high energy tails (Fig. \[lnlnNF\]), as in frictional hard spheres.
[![ \[EffectiveE\] The volume fraction $\nu$ (gray scale and contour lines) at $ft = 0.25$ as a function of height and $v_z$ \[(a) and (c)\], and of height and $v_x$ \[(b) and (d)\], obtained from simulations of frictional hard spheres \[(a) and (b); $e = 0.9,~\mu = 0.5$\], and of frictionless hard spheres \[(c) and (d); $e = 0.7,~\mu = 0$\] at $V_{max} = 0.55$ m/s; $e$ is adjusted in the frictionless case to obtain comparable overall dissipation and similar velocity distribution functions in the steady state region. The frictionless spheres spread more smoothly in height, and they do not yield the sharp gradient in the density around $z/\sigma \approx 13$ as in the frictional case, no matter what the value of $e$ is; the density profile of the frictionless spheres is qualitatively different from that of frictional spheres. Contour lines correspond to (0.03,0.3,0.6,0.9,1.2,1.5)$\times 10^{-3}$ from outside. ](./fig12a.eps "fig:"){width="0.93\columnwidth"}]{} [![ \[EffectiveE\] The volume fraction $\nu$ (gray scale and contour lines) at $ft = 0.25$ as a function of height and $v_z$ \[(a) and (c)\], and of height and $v_x$ \[(b) and (d)\], obtained from simulations of frictional hard spheres \[(a) and (b); $e = 0.9,~\mu = 0.5$\], and of frictionless hard spheres \[(c) and (d); $e = 0.7,~\mu = 0$\] at $V_{max} = 0.55$ m/s; $e$ is adjusted in the frictionless case to obtain comparable overall dissipation and similar velocity distribution functions in the steady state region. The frictionless spheres spread more smoothly in height, and they do not yield the sharp gradient in the density around $z/\sigma \approx 13$ as in the frictional case, no matter what the value of $e$ is; the density profile of the frictionless spheres is qualitatively different from that of frictional spheres. Contour lines correspond to (0.03,0.3,0.6,0.9,1.2,1.5)$\times 10^{-3}$ from outside. ](./fig12b.eps "fig:"){width="0.93\columnwidth"}]{}
Since the functional form of the distribution depends on $e$, we can get a similar distribution function for the steady state by adjusting $e$. For instance, for $\mu = 0$, $e = 0.7$, and $V_{max} = 0.55$ m/s (the same forcing as in Fig. \[kurtosis\_e\]), we obtain $\gamma_4 \approx 0.5$ for the steady state region; the steady state distribution is similar to the one in Fig. \[kurtosis\_e\] for $e = 0.9$ and $\mu = 0.5$. However, we find that no single value of $e$ mimics the hydrodynamic fields or the distribution function of frictional hard spheres, both in the oscillatory and steady state regions; the effect of friction cannot be taken over by an adjusted normal coefficient of restitution. The difference between the results of the two models is illustrated in Fig. \[EffectiveE\], where velocities are not rescaled for better comparison. The outermost contour lines in both models become similar when $e$ in the frictionless model is adjusted as a free parameter \[compare Figs. \[EffectiveE\](a) and (c), or (b) and (d)\]; however, the overall shape of the density contours cannot be matched by adjusting only $e$. Note that the density changes rapidly with height and $\nu > 1.5 \times 10^{-3}$ at $z/\sigma \approx 10$ near $v_z \approx v_x \approx 0$ in the frictional model, whereas in the frictionless model, the particles spread more smoothly in height, and there is no region for $\nu >1.2 \times 10^{-3}$.
conclusions
===========
We have studied the horizontal velocity distribution function of vertically oscillated dilute granular gas, using a molecular dynamics simulation of frictional, inelastic hard spheres. The hydrodynamic fields are oscillatory in time near the oscillating bottom plate due to a shock wave and an expansion wave. However, the fields are nearly stationary above some height, thus constituting a granular gas in a nonequilibrium steady state. The steady state region forms a granular analog of a nearly collisionless Knudsen gas (Figs. \[CPP\] and \[mfpKn\]). We find that the dependence of the distribution functions in this granular Knudsen gas regime on the forcing and material parameters is very weak, even though the distributions in the collisional bulk at lower heights depend strongly on the forcing and material parameters (Figs. \[kurtosis\_e\] and \[kurtosis\_mu\]). The behavior of an ordinary Knudsen gas is determined by boundary conditions [@kogan69]. Although we do not know whether boundary conditions or collisions are dominant in determining the behavior of our granular Knudsen gas, we note that this gas does not depend much on the properties of its only boundary, which is the oscillatory region close to the plate.
The functional form of the horizontal velocity distribution in the steady state region is nearly independent of height, when velocities are scaled by horizontal temperature (Fig. \[M4\]), even though the hydrodynamic fields continue to change. The distribution function is broader than the MB distribution, being underpopulated at small velocities and overpopulated in the high energy tails (Fig. \[VDF\_e\]). We do not observe a universal functional form for the distribution function (Fig. \[lnlnYF\]). The functional form of the high energy tail changes with the dissipation parameters ($e$ and $\mu$) and the oscillation parameter ($V_{max}$). The dependence on $\mu$ in the steady state region is very weak (Fig. \[kurtosis\_mu\]).
Our conclusions regarding the absence of a universal distribution function differ from that of Ref. [@rouyer00], because: (1) We studied the local distribution function, while in Ref. [@rouyer00] the authors obtained the distribution by averaging over space and time; they assumed that the spatial and temporal variation was negligible near the center of the oscillating box, based on their observation of a weak dependence of the density. We find that the time dependence of the density is weak, but that of the temperature is strong in the oscillatory state region (Figs. \[FF\] and \[T-INV\]). Note that if a distribution is averaged over different temperatures, even the MB distribution function leads to a different resultant distribution function. (2) Our system is different from that in Ref. [@rouyer00]: we do not have either air or sidewalls, and our container is much taller, so that the bottom plate is the only energy source in our case. How air and sidewalls affect the dynamics of a granular gas is yet to be clarified.
We also studied the velocity distributions of frictionless inelastic hard spheres, and examined the possibility of including frictional effects using an effective normal coefficient of restitution. We found that no single effective restitution coefficient could describe the frictionless gas at different heights.
Velocities of a granular gas, even in the dilute limit, are strongly correlated, and the correlations depend on the density and the coefficient of restitution [@moon01]. The dependence of the distribution on the density implies that the single particle distribution of a dilute granular gas cannot play a role equivalent to that in a dilute ordinary gas; it is not sufficient to specify statistical properties of the gas. However, the knowledge of the single particle distribution of this complex nonequilibrium gas is still of great importance for the purpose of the first approximation.
The authors thank J. Bougie, D. I. Goldman, E. Rericha, and J. van Zon for helpful discussions. This work was supported by DOE Grant DE-FG-0393ER14312 and Texas Advanced Research Program Grant ARP-055-2001.
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abstract: 'The Einstein-Gordon equations for Friedmann-Robertson-Walker (FRW) geometries in feedback reaction with the quartically self-interacting physical field, arisen from the \`\`inner parity” spontaneous breaking, are explicitly formulated. The Hamiltonian density non-positive extrema would classically forbid both spatially closed and flat homogeneous and isotropic worlds if these were to allow the physical field to (repeatedly) go through and to (finally) settle down in a ground state. In this respect, the [*fixed point*]{} exact solutions of the spontaneous $Z_2-$symmetry breaking Einstein-Gordon equations (mandatory) describe $(k=-1)-$FRW manifolds which actually are either Milne or anti-de Sitter Universes. Setting the $Z_2-$invariance breaking scale at the one of the electroweak symmetry, we speculate on the cosmological implications of the Higgs-anti-de Sitter bubbles and derive a set of particular closed-form solutions to the $S^2-$cobordism with a spatially-flat FRW Universe.'
author:
- |
Ciprian Dariescu[^1]\
[*Institute of Theoretical Science*]{}\
[*5203 University of Oregon*]{}\
[*Eugene OR 97403-5203*]{}\
email: ciprian@physics.uoregon.edu
title: 'Evolving Maximally Symmetric Spacetime Bubbles from Spontaneous $Z_2-$Violation at Electroweak Symmetry Breaking Scale'
---
[*Keywords*]{}:
- Friedmann-Robertson-Walker Cosmology,
- 3-dimensional spacelike manifold of negative curvature,
- spontaneously broken $Z_{2}-$symmetry,
- \`\`field reflection” non-invariant Einstein-Gordon equations,
- Milne and anti-de Sitter spacetimes,
- electroweak symmetry breaking scale, Higgs boson,
- CMBR, galaxies and quasars, Inflation,
- Higgs$-$anti-de Sitter bubbles,
- $S^{2}-$cobordism.
PACS numbers: 11.27.+d, 04.40.Nr., 04.20.Jb\
1.5em
Introduction
============
The resourceful M Theory \[1-10\] and the celebrated AdS/CFT correspondence \[11,12\] have released a high tide of (new) investigations on the anti-de Sitter (AdS) spacetime, its extended versions to more than four dimensions and on the features of the (quantum) matter-fields evolving therein \[13-22\]. The applications to High Energy Physics, General Relativity and Cosmology \[14,23-32\], covering topics like 4D gauge field theories [*via*]{} M Theory, \`\`conformal Higgs”, mass hierarchy (problem), dimensional compactification, either small or large extra dimensions, new unified theories, etc., together with the late observational data on the Universe (cosmological) acceleration \[33-37\], determinedly go far beyond the (so-called) Standard Model(s). Along with the growing interest in Pre-Big-Bang and other new types of Inflation \[38-49\] $-$which could also solve the [*cosmological constant problem*]{}$-$ and in the models with various \`\`dark energy” contents \[48,50\], there is also a need for a better understanding of the geometrodynamical link between the spacetime structure and the nature of spontaneously symmetry breaking vacua. A comprehensive account on this matter, emphasizing the False Vacuum Physics \`\`subtleties” and quoting most of the previous papers in the field, has been given by T. Banks \[51\]; somewhat similarly, for an external continuous symmetry, J. W. Moffat \[52\] has explicitly worked out the intriguing, particularly astrophysical, consequences of the $SO(3,1)-$invariance spontaneous breaking. Last but not least, with respect to the subject we would like to speculate on, there is also the recent paper of A. Dev [*et al.*]{} \[53\], where the conclusion is drawn, based on a careful analysis of the late(st) [*gravity lensing*]{} and [*high-z supernova*]{} data \[54\], that the possibility of a (quasi)Milne stage, i.e. $(k=-1)-$FRW spacetime with linearly expanding scale function (so that the Universe could undergo a uniform expansion), has actually not been completely ruled out.
Formally, what we are dealing with, in the present paper, is a geometrodynamical analysis of the \`\`extremal” spacetime structures derived as exact solutions of the Einstein’s field equations, for (initially) a generic FRW background with a quartically self-interacting scalar field as matter-content, in the system fixed points, i.e. at the three $Z_2-$symmetric extrema of the (fourth-degree polynomial) Hamiltonian density. The \`\`catch” is that, while the central fixed point, the local maximum, is \`\`gravitationally” inconsistent [*only*]{} with the [*spatially-closed*]{} FRW geometries, [*each*]{} of the [*other*]{} two [*fixed points*]{}, the absolute minima representing the matter-source degenerate vacua, is (geometrodynamically) consistent (in the sense of an $R-$valued solution to the corresponding Einstein equations) [*only*]{} with the $(k=-1)-$[*family*]{} of FRW manifolds. As a matter of fact, once the vacuum has been set in one of the (two possible) ground states, the spontaneous$Z_2-$symmetry breaking does \`\`instantly” create an anti-de Sitter Universe; when slightly perturbed, it gets filled with massive particles representing the physical field (quantum) excitations around the settled ground state. Considering therefore \`\`$k=-1$” as a [*compulsory*]{} condition, the previously mentioned [*central fixed point*]{} corresponds to a [*Milne phase*]{}, which, being unstable against the coherent field fluctuations, does primarily turn into an [*anti-de Sitter one*]{} of (very) small curvature (in absolute value).
Informally from the rigour perspective, we set the $Z_2-$invariance breaking scale at the one of the electroweak symmetry,\`\`taking” the Higgs-boson mass somewhere inbetween $115$ and $300 \, GeV$, and analyzing the respective values for the \`\`gravitationally sustained” proper-pulsation, energy and power of the Higgs$-$anti-de Sitter bubbles, we speculate on some of their cosmological implications, such as a stronger CBR anisotropy on the frequencies ranging from (about) $190 \, MHz$ upto (some) $1.4 \, GHz$, more prominent towards the Giant Void(s), seizable deviations from the \`\`whole sky”-averaged intensity-level of the $21 \, cm(s)$ Hydrogen line, inner parity violating seeds of galaxy formation and Higgs-vacuum-based anti-de Sitter power-sources feeding the quasars’ cores.
A particular set of closed-form solutions to the $S^2-$cobordism between an anti-de Sitter bubble and a spatially-flat Universe is readily worked out in the final part of the paper. It (generically) points out that, as seen [*from*]{} the $(k=0)-$FRW spacetime, the coordinate-radius of the small anti-de Sitter bubbles, as well as of the ones (not necessarily small) that might have existed in the course of Inflation, does asymptotically vanish at some high exponential rate. However, analytically, similar conclusions on the [*large*]{} bubbles evolution, in a [*subexponentially*]{} expanding conformally-flat Universe, cannot be drawn so easily, because of the highly nonlinear character of the respective $S^2-$cobordering equation(s).
Spontaneously Broken ${\mbox{\boldmath $Z_2-$}}$Symmetry
========================================================
Let us consider the inner parity (i.e. the field reflection $\Phi
\rightarrow - \, \Phi$) invariant Lagrangian density $${\cal L}[ \Phi ]\, = \, - \, \frac{1}{2} \, \eta^{ab} \, \Phi_{|a}
\Phi_{|b} \, + \, \frac{1}{2} \, \mu^2 \, \Phi^2 \, - \,
\frac{\lambda}{24} \, \Phi^4$$ of a quartically self-interacting real scalar field $\Phi$, where, ${\mbox{\boldmath $\eta$}}\, = \, diag[1,1,1,-1]$ is the fundamental metric tensor for a pseudo-orthonormal tetrad $\lbrace
e_a = e_{a}^{i} \, \partial_i \rbrace_{a=\overline{1,4}}^{i=
\overline{1,4}}$ whose dual $\lbrace \omega^a = \omega^{a}_{i} \,
dx^i \,\, | \,\, \langle \omega^a \, , \, e_b \rangle =
\delta^{a}_{b} \, \rbrace^{i=\overline{1,4}}_{a,b=\overline{1,4}}$ generates the spacetime metric $$ds^2 \, = \, \eta_{ab} \, \omega^a \, \omega^b \,\, ,$$ the $( \, \cdot \, )_{|a}$ notation stands for the tetradic derivative with respect to $e_a$, i.e. $( \, \cdot \, )_{|a} \, =
\, e_{a} ( \, \cdot \, ) \, =\, e_{a}^{i} \, \partial_{i}( \,
\cdot \, )$, and $\mu^2$ $-$ with ${\rm mass}^2$ dimension $-$ and $\lambda$ (dimensionless) are the two positive parameters that \`\`accommodate” the spontaneously (discrete) symmetry breaking mechanism. Working out, from the functional expression $${\mbox{\boldmath $T$}} \, = \, - \, \frac{2}{\sqrt{- \, g}} \,
\frac{\delta}{\delta {\mbox{\boldmath $g$}}} \left[ \sqrt{- \, g}
\, {\cal L} \right] \, \, ,$$ the covariant components (with respect to the \`\`rigid” tensor basis\
$\lbrace \omega^a \otimes \omega^b
\rbrace_{a,b=\overline{1,4}}\,$) of the conservative stress-energy-momentum tensor, for the considered scalar field, it results in full $$T_{ab} \, = \, \Phi_{|a} \, \Phi_{|b} \, - \, \frac{1}{2} \,
\eta_{ab} \, \left[ \Phi^{|c} \, \Phi_{|c} \, - \, \mu^2 \, \Phi^2
\, + \, \frac{\lambda}{12} \, \Phi^4 \, \right] \, \, ,$$ so that, the extrema of the Hamiltonian density $${\cal H} \, = \, T_{44} \, = \, \frac{1}{2} \, \delta^{ab} \,
\Phi_{|a} \, \Phi_{|b} \, - \, \frac{\mu^2}{2} \, \Phi^2 \, + \,
\frac{\lambda}{24} \, \Phi^4$$ are given by the equation $$\frac{\partial {\cal H}}{\partial \Phi} \left(
\, \Phi_0 \, \right) \, = \,
\Phi_0 \,\left( \frac{\lambda}{6} \, \Phi_0^2 \,
- \, \mu^2 \right) \, = \,0 \,\, ,$$ i.e. $\lbrace \Phi_{0}^{\alpha} \rbrace_{\alpha=-,0,+} = \left
\lbrace - \mu \sqrt{\frac{6}{\lambda}}, \, 0 \, , \, \mu
\sqrt{\frac{6}{\lambda}} \right \rbrace$. Inspecting the sign of the second derivative $$\frac{\partial^{2}{\cal H}}{\partial \Phi^2} \left( \, \Phi_0 \,
\right)\, = \,
\frac{\lambda}{2}\, \Phi_0^2 \, - \, \mu^2$$ for each of the three roots $\lbrace \Phi_{0}^{\alpha}
\rbrace_{\alpha=-,0,+}$, it instantly results that $\Phi_{0}^{0}$, where $\frac{\partial^{2}{\cal H}}{\partial \Phi^2}= \,- \mu^2$, is an [*unstable*]{} fixed point, while $\Phi_{0}^{\pm}$, where $\frac{\partial^{2} {\cal H}}{\partial \Phi^2}= \,2 \mu^2$, are the \`\`real” [*minima*]{} which correspond to the two possible ground states of the initially fictitious (i.e. apparently deprived of direct particle interpretation) scalar field $\Phi$.
Choosing $v = \Phi_{0}^{+} = \mu \sqrt{\frac{6}{\lambda}}$ as the vacuum expectation value of $\Phi$, in its ground state, and accordingly shifting the field $$\Phi \, = \, v \, + \, \chi \, , \; \; {\rm where} \; \; \chi :
M_4 \rightarrow {\rm{\bf R}} \, ,$$ such that $\chi = 0$ represents the [*true*]{} vacuum of the theory, one ends up with the spontaneously ${\mbox{\boldmath
$Z$}}_2$ broken Lagrangian density $${\cal L}[ \chi ] \, = \, - \, \frac{1}{2} \, \chi^{|c} \, \chi_{|c} -
\,\frac{1}{2}(2 \mu^2) \chi^2 \, -\, \mu
\sqrt{\frac{\lambda}{6}}\, \chi^3 \, - \, \frac{\lambda}{24} \,
\chi^4 \,+ \, \frac{3 \, \mu^4}{2 \, \lambda}$$ of the physically observable massive, $m_{\chi} \,= \, \sqrt{2} \,
\mu$, (real) scalar field $\chi$, subsequently obeying the inner parity violating ternary nonlinear (generalized) Gordon equation $$\Box \chi \, - \, \left( 2 \, \mu^2 \right) \, \chi \, = \,
3 \, \mu \sqrt{\frac{\lambda}{6}} \, \chi^2 \, + \,
\frac{\lambda}{6} \, \chi^3 \, \, ,$$ where $$\Box \chi \, = \, \frac{1}{\sqrt{- \, g}} \,
\frac{\partial}{\partial x^i} \left[\sqrt{- \, g} \,
g^{ik} \frac{\partial \chi}{\partial x^k} \right]$$ is the d’Alembert operator on $M_4$, in terms of some local coordinates $\lbrace x^i \rbrace_{i=\overline{1,4}}$.
Therefore, either from (9) and (3), or straightly from (4) with the shift (8), it yields for the components of the energy-momentum tensor ${\rm{\bf T}}$ (of the physical field $\chi$, and with respect to the tensor basis $\lbrace \omega^a \otimes \omega^b
\rbrace_{a,b=\overline{1,4}}$) the actual expression $$T_{ab} \, = \, \chi_{|a} \, \chi_{|b} \, - \,
\frac{1}{2}\, \eta_{ab} \, \left[ \chi^{|c} \, \chi_{|c} \, + \,
2 \, V(\chi) \, - \, \frac{3 \, \mu^4}{\lambda} \right] \, \, ,$$ where the total, semi-classical (i.e. without quantum corrections) potential $$V(\chi) \, = \, \mu^2 \, \chi^2 \,+ \, \mu
\sqrt{\frac{\lambda}{6}} \, \chi^3 \, + \, \frac{\lambda}{24} \,
\chi^4$$ is clearly no longer invariant under the discrete transformation $\chi \, \rightarrow - \, \chi$.
In the Minkowski spacetime, which keeps on being flat whatever the matter energy-momentum is, the time-translation isometry, and the respective action-functional invariance, accounting for energy conservation, allows gauging the energy scale by any constant amount. Hence, any constant, field-independent, contribution to the (44)-component of the conservative energy-momentum tensor, such as $-3 \mu^4/(2 \lambda)$ in (12), does actually leave no observable signature in the field dynamics and so, it can just simply be thrown away. However, in general and physically more realistic situations, where gravity cannot be neglected, the matter stress-energy-momentum tensor does clearly affect the metric of the Lorentzian base manifold, so that any of its additional terms, even a constant one, cannot be omitted any longer unless there are some serious, both mathematical and physical, reasons.
Inner Parity Non-Invariant Einstein-Gordon Equations in FRW Cosmologies
=======================================================================
We have come to the point where we can address the question of what type of cosmic-time evolving homogeneous and isotropic 3-geometry \`\`fits” the massive scalar source-field $\chi$ in such a way to produce an exact solution to the Einstein-(generalized) Gordon equations. Since both homogeneity and isotropy require a [*maximal*]{} $G_6-$group of motion on the 3-dimensional (sub)manifold $N_3$ (of $M_4$), this must possess [*constant*]{} curvature, i.e. $k = \lbrace 1,0, \, -1 \rbrace$, and thus, can only be the sphere $S^3$, the Euclidian $\textbf{R}^3$, or the disconnected wings of the hyperboloid $H^3$ defined, in the flat $\textbf{R}^4$ of Cartesian coordinates $\left( X^{\alpha}, \, T \right)_{\alpha=\overline{1,3}}$ and metric signature 2, by the \`\`typical” equation $T^2 \, - \,
\delta_{\alpha \beta}\, X^{\alpha} \, X^{\beta} \, = \, 1$. Hence, in terms of dimensionless Euler-like coordinates $$(\alpha, \beta, \theta) \, = \left \lbrace
\begin{array} {l}
(0,2 \, \pi) \times (0,2 \, \pi) \times (0, \frac{\pi}{2}
\,{\rm or} \, \pi) \; \; {\rm on} \; \; S^3 \, , k=1 \\
{\rm{\bf R}}^3 \, , k=0 \\
{\rm{\bf R}} \times (0,2 \, \pi) \times {\rm{\bf R}} \; \;
{\rm on} \; \; H^3
, \, k=-1 \\
\end{array} \right.$$ the metric on $N_3$ does respectively read $$dl_{N_3}^2 \, = \left \lbrace
\begin{array} {l}
\cos^2 \theta \, (d \alpha)^2 \, + \, \sin^2 \theta \,
(d \beta)^2 \, + \, (d \theta)^2 \, , \, k=1 \\
(d \alpha)^2 \, + \, (d \beta)^2 \, + \, (d \theta)^2 \, , \,
k=0 \\
\cosh^2 \theta \, (d \alpha)^2 \, + \, \sinh^2 \theta \,
(d \beta)^2 \, + \, (d \theta)^2 \, , k=-1 \\
\end{array} \right.$$ and therefore, considering the spacetime $M_4 \, = \, N_3 \,
\times \, {\rm{\bf R}}$ as a continuous \`\`tower” of $\lbrace
t = cst \, | \, \forall \, cst \in R \rbrace -$ cosmic-time orthogonal foliations $ {\cal N}_3$ homothetic to $N_3$, the metric on $M_4$ gets the well-known Friedmann-Robertson-Walker (FRW) form $$ds^2 \, = \, a^2 \, e^{2 \, f} \, dl_{N_3}^2 \, - \, (dt)^2 \,\, ,$$ where $a$ is a scale parameter with dimension of length and the modified metric function $f\, : \, {\rm{\bf R}} \rightarrow
{\rm{\bf R}}$ does actually express the primitive $$f(t) \, = \, \int^t H(t') \, dt'$$ of the celebrated Hubble function $$H(t) \, \stackrel{\Delta}{=} \, \frac{e^{-f}}{a} \, \frac{d}{dt}
\left( a \, e^f \right) \, \equiv \frac{df}{dt}$$
Consequently, with respect to (16) and (2), the dually related pseudo-orthonormal bases $\lbrace {\mbox{\boldmath $\omega$}} ;
{\mbox{\boldmath $e$}} \rbrace$ are respectively given by the concrete expressions $$\begin{aligned}
& & \omega^1 = a e^f \cos(\theta) d \alpha \, , \, \omega^2 = a
e^f \sin(\theta) d \beta \, , \, \omega^3 = a e^f \, d \theta \, ,
\, \omega^4 = dt \nonumber \\* & (a) & \omega^1 = a e^f d \alpha
\, , \, \omega^2 = a e^f d \beta \, , \, \omega^3 = a e^f d \theta
\, , \, \omega^4 = dt \nonumber \\* & & \omega^1 = a e^f
\cosh(\theta) d \alpha \, , \, \omega^2 = a e^f \sinh(\theta) d
\beta \, , \, \omega^3 = a e^f d \theta \, , \, \omega^4 = dt
\nonumber \\*
& & e_1 =
\frac{e^{-f}}{a} {\rm sec} (\theta) \partial_{\alpha} \, , \, e_2 =
\frac{e^{-f}}{a} {\rm cosec} (\theta) \partial_{\beta } \, , \, e_3
= \frac{e^{-f}}{a} \partial_{\theta} \, , \, e_4 =
\partial_4
\nonumber \\*
& (b) & e_1 =
\frac{e^{-f}}{a}
\partial_{\alpha} \, , \, e_2 = \frac{e^{-f}}{a}
\partial_{\beta} \, , \, e_3 = \frac{e^{-f}}{a}
\partial_{\theta} \, , \, e_4 = \partial_t
\nonumber \\*
& & e_1 = \frac{e^{-f}}{a} {\rm sech} (\theta)
\partial_{\alpha} \, , \, e_2 = \frac{e^{-f}}{a} {\rm cosech}
(\theta) \partial_{\beta} \, , \, e_3 = \frac{e^{-f}}{a}
\partial_{\theta} \, , \, e_4 = \partial_t\end{aligned}$$ which move the exterior-forms formalism, through the Cartan’s Equations $$\begin{aligned}
&(a)& d \omega^a = \Gamma^{a}_{\; bc} \, \omega^b \wedge \omega^c
\, , \; {\rm without \; torsion} \, , \nonumber \\* &(b)& {\rm{\bf
R}}_{ab} = d \Gamma_{ab} \, + \, \Gamma_{ac} \wedge \Gamma^{c}_{\;
b} \, \, ,\end{aligned}$$ where $$\Gamma_{ab} \, = \, \eta_{ad} \, \Gamma^{d}_{\; b} \, = \,
\Gamma_{abc} \, \omega^c \, \, ,$$ all the way down to the essential components $R_{abcd}$ of the curvature 2-forms, $${\rm{\bf R}}_{ab} \, = \, \frac{1}{2} \,R_{abcd} \, \omega^c
\wedge \omega^d \, \, ,$$ namely, $$\begin{aligned}
&(a)& R_{1212} \, = \, R_{1313} \, = \, R_{2323} \, = \, \left(
f_{|4} \right)^2 \, + \, \frac{k}{a^2} e^{-2 \, f} \, , \,
\nonumber \\* &(b)& R_{1414} \, = \, R_{2424} \, = \, R_{3434} \,
= \, - \, \left[ f_{|44} \, + \, \left( f_{|4} \right)^2 \right]
\, .\end{aligned}$$ Thus, the Ricci tensor gets no off-diagonal components and therefore it reads $$\begin{aligned}
&(a)& R_{\alpha \beta} \, = \, \left[ f_{|44} \, + \, 3 \left(
f_{|4} \right)^2 \, + \, \frac{2 \, k}{a^2} \, e^{-2 \, f} \right]
\, \delta_{\alpha \beta} \, \, , \nonumber \\* &(b)& R_{44} \, =
\, -3 \, \left[ f_{|44} \, + \, \left( f_{|4} \right)^2 \right] \,
\, ,\end{aligned}$$ where $\alpha, \beta = \overline{1,3}$, leading to the scalar curvature $$R \, = \, 6 \left[ f_{|44} \,+ \, 2 \left( f_{|4} \right)^2 \, +
\, \frac{k}{a^2} \, e^{-2 \, f} \right]$$ and altogether to the algebraically essential components of the Einstein tensor $$\begin{aligned}
&(a)& G_{\alpha \beta} \, = \, - \left[ 2 \, f_{|44} \, + \, 3
\left( f_{|4} \right)^2 \, + \, \frac{k}{a^2}\, e^{-2 \, f}
\right] \, \delta_{\alpha \beta} \nonumber \\* &(b)& G_{44} \, =
\, 3 \left[ \left( f_{|4} \right)^2 \, + \, \frac{k}{a^2} \, e^{-2
\, f} \right]\end{aligned}$$
As it can be noticed, for both $k \, = \, \lbrace 0,1 \rbrace$ the $G_{44}$-component does always stand non-negative and that is an important restriction, through the Einstein equation $G_{44} =
\kappa_0 \, T_{44}$, on the type of matter-sources that can be fit (in the sense of an exact solution) into such geometries: excepting the Minkowskian ($k=0$)-vacuum case, all the other matter-sources $-$ if combined $-$ should have a positive total energy density, $w \, = \, T_{44} > 0$, i.e. they should behave on the whole, for $k=0$ or $1$, as [*conventional*]{} matter, fulfilling the Hawking’s [*weak*]{} energy condition, $T_{ab} \,
u^a u^b \geq 0$ (for any non-spacelike 4-vector ${\mbox{\boldmath
$u$}}$). On the contrary, and completely nontrivial from a geometrodynamical perspective, in the hyperbolic case, $k=-1$, the sign of $G_{44}$ and (together with it) the one of the (resulting) energy density gets undefined, unless $f_{|4}=0$ when the Einstein equations demand a deeply exotic kind of matter, of state-equation $$P \, = \, - \, \frac{1}{3} \, w \, , \; {\rm where} \; w \, = \,
- \, \frac{3/ \kappa_{0}}{a^2} \, ,$$ i.e. $$P \, = \, \frac{1}{3} \, |w| \, \, ,$$ which can be comprehended as a sort of [*ghost*]{}-black-body radiation. Anyway, in the general situation, of evolving (time-orthogonal) $H^3-$foliations, the geometrodyamics gets much more involved, since it can accommodate some reasonably mixed matter-sources, made both of \`\`ordinary” and \`\`exotic” matter. Thus, letting apart for the moment the well-known conventional sources, such as the thermalized electromagnetic (or other massless-field) radiation and the baryonic dust, we deal with the case where the $(k = -1)-$FRW geometry is driven by the spontaneously inner parity breaking massive scalar field, $\chi$, alone. As all of the essential Einstein tensor components are on-diagonal and only time-dependent, the source field $\chi$ can only be [*coherent*]{} and therefore, its conservative stress-energy-momentum tensor does also become diagonal, exhibiting the components $-$ subsequently derived from (12) $-$ $$\begin{aligned}
&(a)& T_{\alpha \beta} \, = \, - \, \frac{1}{2} \left[ - \left(
\chi_{|4} \right)^2 \, + \, 2 \, V( \chi) \, - \, \frac{3
\mu^4}{\lambda} \right] \, \delta_{\alpha \beta} \nonumber \\*
&(b)& T_{44} \, = \, \frac{1}{2} \left[ \left( \chi_{|4} \right)^2
\, + \, 2 \, V( \chi) \, - \, \frac{3 \mu^4}{\lambda} \right]\end{aligned}$$ Hence, the whole set of \`\`quartically generalized” Einstein-Gordon equations $$G_{ab}[f] \, = \, \kappa_0 \, T_{ab}[ \chi]$$ immediately goes down to the following functionally 2-dimensional nonlinear differential system $$\begin{aligned}
&(a)& 2 \, f_{|44} \, + \, 3 \left( f_{|4} \right)^2 \, - \,
\frac{e^{-2 \, f}}{a^2} \, = \, \frac{\kappa_0}{2} \left[ - \,
\left( \chi_{|4} \right)^2 \, + \, 2 \, V( \chi) \, - \, \frac{3
\mu^4}{\lambda} \right] \, \, , \nonumber \\* &(b)& 3 \left[
\left( f_{|4} \right)^2 \, - \, \frac{e^{-2 \, f}}{a^2} \right] \,
= \, \frac{\kappa_0}{2} \left[ \left( \chi_{|4} \right)^2 \, + \, 2 \,
V( \chi) \, - \, \frac{3 \mu^4}{\lambda} \right] \, \, ,\end{aligned}$$ where the semi-classical potential $V$ is given by (13). We have not included in (26) the generalized Gordon equation (10), explicitly worked out for the considered $(k = -1)-$FRW spacetime dynamically sustained by the coherent massive scalar $\chi$, since all three of them, i.e. (26.a, b) and (10), taken together, are not functionally independent because of the twice contracted Second Bianchi Identity $$0 \, \equiv \, \left( R^{ab} \, - \, \frac{1}{2} \, g^{ab} \,R
\right)_{ ; \, b} \, = \, G^{ab}_{; \, b} \, = \,\kappa_0 \, T^{ab}_{;
\, b}
\, \Rightarrow \, T^{ab}_{; \, b} \, = \,0 \, ,$$ i.e. if the energy-momentum tensor ${\mbox{\boldmath $T$}}$ is correctly derived from the field Lagrangian density ${\cal L}$ then its 4-divergenceless property (the one of being [*conservative*]{}) does accurately account for the field dynamics prescribed by the Euler-Lagrange equations $$\frac{\delta {\cal L}}{\delta \chi} \, = \,0 \, \, .$$ To put it shortly, the source-field equation (10), particularized to the form $$\chi_{|44} \, + \, 3 \, f_{|4} \, \chi_{|4} \, + \, 2 \, \mu^2 \,
\chi \, = \,- \, 3 \mu \sqrt{\frac{\lambda}{6}}\, \chi^2 \,- \,
\frac{\lambda}{6} \, \chi^3 \; ,$$ must spring out from (26) just by taking first-order derivatives and subsequently doing algebraic manipulations. Indeed, taking the time-derivative of(26.b) it yields $$2 \left[ f_{|44} \, + \, \frac{e^{-2 \,f}}{a^2} \right] \, = \,
\frac{\kappa_0}{3} \, \frac{ \chi_{|4}}{f_{|4}} \left[ \chi_{|44}
\, + \, \frac{dV}{d \chi} \right]$$ and, by insertion in (26.a), written as $$2 \left[ f_{|44} \, + \, \frac{e^{-2 \,f}}{a^2} \right] \, +
\, 3 \left[ \left( f_{|4} \right)^2 \, - \, \frac{e^{-2
\,f}}{a^2} \right] \, = \, \frac{\kappa_0}{2} \left( \chi_{|4}
\right)^2 \, + \, \kappa_0 \, V( \chi) \, - \, \kappa_0 \frac{3
\mu^4}{2 \lambda} \, \, ,$$ i.e., using (26.b), $$\begin{aligned}
2 \left[ f_{|44} \, + \, \frac{e^{-2 \,f}}{a^2} \right] & + &
\frac{\kappa_0}{2} \left( \chi_{|4} \right)^2 \, + \, \kappa_0 \,
V( \chi) \, - \, \kappa_0 \frac{3 \mu^4}{2 \lambda} \nonumber \\*
= & - & \frac{\kappa_0}{2} \left( \chi_{|4} \right)^2 \, + \,
\kappa_0 \, V( \chi) \, - \, \kappa_0 \frac{3 \mu^4}{2 \lambda} \;
, \nonumber\end{aligned}$$ it gives $$\frac{1}{3} \, \frac{\chi_{|4}}{f_{|4}} \left[ \chi_{|44} \, + \,
\frac{dV}{d \chi} \right] \, + \, \left( \chi_{|4} \right)^2 \, =
\, 0 \, \, ,$$ i.e. $$\chi_{|44} \, + \, 3 f_{|4} \, \chi_{|4} \, = \, - \,
\frac{dV}{d \chi} \, , \; \; {\rm with} \; \; \chi_{|4} \neq 0 \neq
f_{|4} \, \, ,$$ where the last equation does exactly come to the Gordon one (28) just by plugging in the $\chi$-derivative of the fourth-degree polynomial potential (13).
Nevertheless, is good to know that we can play all three equations, (26.a, b) and (28), since, in some concrete calculations, the result might be got easier in some particular combination of them, instead of working only with (26) as they stand.
In the above given proof of the compatibility of Euler-Lagrange equation (28) with the Einstein’s ones (26), we have asked for $f_{|4} \neq 0$ and $\chi \neq 0$. If $f_{|4} = 0$ then $f$ can be scaled to [*zero*]{} and the system (26) becomes $$\begin{aligned}
\frac{1}{a^2} & = & \frac{\kappa_0}{2} \left( \chi_{|4} \right)^2
\, - \, \kappa_0 \, V(\chi ) \, + \, \frac{3 \kappa_0 \mu^4}{2
\lambda} \nonumber
\\*
\frac{3}{a^2} & = & - \, \frac{\kappa_0}{2} \left( \chi_{|4}
\right)^2 \, - \, \kappa_0 \, V(\chi ) \, + \, \frac{3 \kappa_0
\mu^4}{2 \lambda} \nonumber\end{aligned}$$ Subtracting the first equation from the second one, it yields $$\left( \chi_{|4} \right)^2 = \, - \, \frac{2/\kappa_0}{a^2} \; \;
\Rightarrow \; \; \chi = \chi_0 \pm i \, \sqrt{\frac{2}{\kappa_0}}
\, \frac{t}{a} \, ,$$ so that, the massive scalar $\chi$ would be actually a [*ghost*]{}; in addition, since $\chi_{|44} \equiv 0$ (in this case), the nonlinear Gordon equation (28) just turns into the algebraic equation $$\chi^2 \, + \, 3 \mu \sqrt{\frac{\lambda}{6}} \, \chi \, + \,
\frac{12 \mu^2}{\lambda} \, = \, 0 \, , \; {\rm with} \; \; \chi
\sim \frac{t}{a} \, , \; (\forall ) \, t \in {\rm{\bf R}} \, ,$$ which obviously cannot be satisfied as $\chi$ is clearly time-dependent. Hence, the $(f = const)-$particular case is definitely forbidden for the considered matter-source.
Maximally Symmetric Fixed Points
================================
The other particular case, $\chi_{|4} =0$, is by far of much interest for it reveals the simplest $(k=-1)-$FRW spacetime dynamics in the fixed points of the nonlinear Gordon equation for the physical field $\chi$ left-over by the spontaneous breaking of the discrete inner-symmetry $\phi \rightarrow - \phi$. The shortest path to the solution(s) is paved by the observation that for $\chi_{|4} =0$ the right-hand-side of the two Einstein equations (26) gets the same and therefore, subtracting them, it yields the modified metric function essential equation $$f_{|44} \, + \, \frac{e^{-2f}}{a^2} \, = \, 0$$ which can be inserted back into (26.a), with $\chi_{|4} =0$, getting at once the same equation (26.b) (with $\chi_{|4} =0$). Hence, among the three Einstein-(generalized) Gordon equations we have just to solve, in this particular case, the very simple system $$\begin{aligned}
& (a) & \left( f_{|4} \right)^2 - \, \frac{e^{-2f}}{a^2} \, = \,
\frac{\kappa_0 }{3} \, V(\chi ) \, - \, \frac{\kappa_0 \mu^4}{2
\lambda} \nonumber \\* & (b) & \chi \left[ 2 \mu^2 \, + \, 3 \mu
\, \sqrt{\frac{\lambda}{6}} \, \chi \, + \, \frac{\lambda}{6} \,
\chi^2 \right] = \, 0\end{aligned}$$ where $V(\chi )$ is given by (13), i.e. $$V(\chi ) \, = \, \frac{\lambda}{24} \, \chi^2 \left[ \chi^2 \, +
\, 4 \mu \, \sqrt{\frac{6}{\lambda}} \, \chi \, + \, \frac{24
\mu^2}{\lambda} \right]$$ Ordered by their magnitudes, the roots of (30.b) $-$ meaning the matter-field fixed-point values $-$ do respectively read $$\chi_L \, = \, - \, 2 \mu \, \sqrt{\frac{6}{\lambda}} \, , \;
\chi_M \, = \, - \, \mu \, \sqrt{\frac{6}{\lambda}} \, , \; \chi_R
\, = \, 0 \, ,$$ where the indices $L, \, M, \, R$ come from \`\`left, Milne, right”, respectively. By their \`\`turn by turn” insertion in (31), one immediately gets the corresponding values of the semi-classical (4-nominal self-interaction) potential, namely $$V_L \, = \, 0 \, , \; V_M \, = \, \frac{3 \mu^4}{2 \lambda} \, ,
\; V_R \, = \, 0 \, ,$$ so that, the nonlinear first-order differential equation (30.a) of the metric function $f$ does only take the following two particular forms $$\begin{aligned}
& (a) & \left( f_{|4} \right)^2 - \, \frac{e^{-2f}}{a^2} \, = \, -
\, \frac{\kappa_0 \mu^4 }{2 \lambda} \, , \; {\rm for } \; \; V_L
= V_R =0 \, , \nonumber \\* & (b) & \left( f_{|4} \right)^2 - \,
\frac{e^{-2f}}{a^2} \, = \, 0 \, , \; {\rm for } \; \; V_M = \,
\frac{3 \mu^4}{2 \lambda} \, ,\end{aligned}$$ which will be correspondingly leading to the only two types of $k=-1$ homogeneous and isotropic universes that can accommodate and are also physically supported by the massive, $m_{\chi} =
\sqrt{2} \, \mu$, real scalar field $\chi$ in any of its three 4-dimensionally constant main states. It is worth noticing that without integrating the equation (34) we can already say, by the help of equation (29), what the two generated spacetimes are. Indeed, in the simpler case (34.b), because of (29), it also results that $$f_{|44} \, + \, \left( f_{|4} \right)^2 \, = \, 0$$ and thus, having a look at the components (21) of the curvature tensor, we instantly realize that all of them vanish. Hence, the spacetime corresponding to (34.b), supported by the static physical field $\chi_M = - \, \mu \,
\sqrt{6/ \lambda}$, is [*flat*]{}, being basically (geometrically) the Minkowski spacetime. However, especially from a cosmological perspective, the difference is that this spacetime is patched in a different coordinate-system, namely the Milne’s one, which sharply presents the evolution of the $H^3-$spacelike-foliation, instead of the static picture of the Cartesian ${\rm{\bf R}}^3$-foliations (of constant Minkowskian time, $x^4 =t$). Concerning the other $(k=-1)-$FRW model, the one related to the equation (34.a), we get, using again the essential equation (29), that $$f_{|44} \, + \, \left( f_{|4} \right)^2 \, = \, - \,
\frac{\kappa_0 \mu^4}{2 \lambda}$$ and so, inserting it, together with (34.a), in the expressions (21) of the Riemann tensor components and in the expression (23), with $k=-1$, of the scalar curvature, it yields $$\begin{aligned}
& (a) & R_{\alpha \beta \alpha \beta} \, = \, - \, R_{\alpha 4
\alpha 4 } \, = - \, \frac{\kappa_0 \mu^4}{2 \lambda} \, , \;
\alpha = \overline{1,3} \, , \; \beta \neq \alpha \, , \; \beta =
\overline{1,3} \, , \nonumber \\* & (b) & R \, = \, - \, 12 \left(
\frac{\kappa_0 \mu^4}{2 \lambda} \right) ,\end{aligned}$$ which clearly points out that the solution to the \`\`basic” equation (34.a) does surely sustain a $(k=-1)-$FRW Universe of constant negative (4D) curvature and that can only be the anti-de Sitter spacetime. Although investigating it, and some of its possibly observable cosmological consequences, mostly related to the spontaneous breaking of the field-reflection inner symmetry, is the main goal of the present paper, we would like (first) to make some remarks on the model described by the solution of (34.b), better known as the Milne Universe, and to comment a bit its linear stability within the context of the coherent linear perturbations of the massive source-field $\chi$ around its fixed-point value $\chi_M$.
Milne Spacetime Coherent Perturbations
======================================
As it is almost obvious, because of its very simple form, the equation (34.b) can be immediately written as $$\left( \frac{dS}{dt} \right)^2 \, = \, 1 \, ,$$ where $$S \, = \, a \, e^f$$ is the \`\`standard” cosmological scale function $-$ casting the FRW-metric into the \`\`history making” form $$ds^2 \, = \, S^2(t) \, dl_{N_3}^2 \, - \, (dt)^2 \, ,$$ (where $N_3$ is one of the $S^3$, ${\rm{\bf R}}^3$, $H^3$ manifolds) $-$ and so, independently of both the sign we choose for the solution and the corresponding integration constant, the scale function is basically reading $$S(t) \, = \, |t| \, ,$$ so that, in $(k=-1)-$spherical (physically dimensionless) coordinates $\lbrace r, \theta , \varphi \rbrace$ on the upper wing (let’s say) of $H^3$, the [*generic*]{} metric (40) does explicitly turn into the one of the Milne Universe $$ds_{Mln}^2 \, = \, t^2 \left[ \frac{(dr)^2}{1+r^2} \, + \, r^2 \,
d \Omega^2 \right] \, - \, (dt)^2 \, ,$$ where $$d \Omega^2 \, = \, (d \theta)^2 \, + \, \sin^2 \theta \, (d \varphi)^2$$ is the well-known metric on the unit sphere $S^2$. Since, as we have already shown, this spacetime is properly [*flat*]{}, there must be a globally defined local coordinates transformation that takes the Minkowskian metric, $$d s^2_{Mnk} \, = \, \left( d R \right)^2 \, + \, R^2\, d \Omega^2 \,
- \, \left( d T \right)^2 \, \, ,$$ into the Milne’s one, (42), and vice versa. In this respect, as the angular $( \theta, \varphi)-$coordinates are the same (in the two metrics and on the two manifolds), it is quite obvious that the usual radial coordinate $R$ is given in terms of the Milne coordinates \`\`$r,t$” by the simple relation $$R(r,t) \, = \, r \, |t| \, , \; {\rm with} \; r \geq \, 0 \, .$$ Then, the Minkowskian time $T$ must also be a function of the two coordinates $(r,t)$, i.e. $$T \, = \, T(r , t ) \, ,$$ such that, plugging in (44) the square of its differential and the one of (45), it should equate the two (1+1)-metrics, i.e. $$\left( d R \right)^2 \, - \, \left( d T \right)^2 \, = \,
\frac{t^2}{1 \, + \, r^2} \, (d r)^2 \, - \, (d t)^2 \, \, ,$$ which concretely comes to the self-embedding condition $$\left( |t| \, dr \, + \, r \, \frac{d |t|}{dt} \, dt \right)^2
- \left( \frac{\partial T}{\partial r} \, dr \, + \,
\frac{\partial T}{\partial t} \, dt \right)^2 = \,
\frac{t^2}{1 + r^2} \, (dr)^2 \, - \, (dt)^2 \, ,$$ leading therefore to the following set of equations that (46) must satisfy: $$\begin{aligned}
&(a)& \left( \frac{\partial T}{\partial r} \right)^2 \, = \,
\frac{t^2 \, r^2}{1 + r^2} \nonumber \\* &(b)& \frac{\partial
T}{\partial r} \, \frac{\partial T}{\partial t} = \, r \, t
\nonumber \\* &(c)& \left( \frac{\partial T}{\partial t} \right)^2
\, = \, 1 + r^2\end{aligned}$$ Hence, setting to zero the integration constants, the two branches that spring out from the equations (49.a, c), compatible with (49.b), are respectively defined by $$\begin{aligned}
&(a)& \frac{\partial T}{\partial t} = - \, \sqrt{1 + r^2} \,
\Rightarrow \, \frac{\partial T}{\partial r} = - \frac{r
t}{\sqrt{1 + r^2}} \, \Rightarrow \, T = \, - \, t \, \sqrt{1 +
r^2} \, \, , \nonumber \\* &(b)& \frac{\partial T}{\partial t} =
\, \sqrt{1 + r^2} \, \Rightarrow \, \frac{\partial T}{\partial r}
= \, \frac{r t}{\sqrt{1 + r^2}} \, \Rightarrow \, T = \, t \,
\sqrt{1 + r^2} \, \, ,\end{aligned}$$ where only (50.b) preserves the same orientation on the considered manifold, i.e. $$\left|
\begin{array}{l}
\frac{\partial R}{\partial r} \; \; \; \; \frac{\partial R}{\partial t}
\\
\frac{\partial T}{\partial r} \; \; \; \; \frac{\partial T}{\partial t}
\end{array}
\right| \, = \, \frac{|t|}{\sqrt{1 + r^2}} > 0 \, , \; \forall \;
t \in {\rm{\bf R}} - \lbrace 0 \rbrace \, .$$ Thus, collecting the two results (45) and (50.b), we have actually got the celebrated [*orientation-preserving*]{} Milne transformation, $$\begin{aligned}
&(a)& R \, = \, r \, |t| \, \, , \nonumber \\* &(b)& T \, = \, t
\, \sqrt{1 + r^2} \, ,\end{aligned}$$ where $r \geq 0$ and $t \in {\rm{\bf R}}$, which casts (44) into the form (42) and, with a bit of care (again, about orientation), we can immediately write down, from (52), its (proper) [*inverse*]{} $$\begin{aligned}
&(a)& r \, = \, \frac{r}{\sqrt{T^2 - R^2}} \nonumber \\* &(b)& t
\, = \, sgn(T) \, \sqrt{T^2 - R^2} \, \, ,\end{aligned}$$ with $T^2 - R^2 \geq \, 0 \, , \, \, R \geq \, 0 \, , \, \, T \in
\textbf{R}$, which actually defines the Milne coordinates $(r,t)$ in terms of the Minkowskian ones $(R,T)$ and accordingly turns (42) into (44). Although the bulk of the Milne’s Universe properties and structure is very well known since quite long ago, its connection to the spacetime geometry erected by the \`\`remnant” massive scalar field $\chi$, \`\`after” the $\textbf{Z}_{2}-$invariance got spontaneously broken, has not been investigated to a large extent in the literature. It might be shedding a brand new light on the real (physical) significance of the Milne Universe $-$ previously considered just as a \`\`toy model” $-$ and on a seemingly not yet extensively explored bunch of cosmological implications of the decaying remnant-field blown up Milne bubbles in the observable Universe. That is why is worth studying their behaviour at least with respect to the linear perturbations of the source-field $\chi$ around its static value $\chi_M$.
Hence, let us consider the coherently evolving small field-perturbation $\psi$, such that $$\chi = \chi_M \, + \, \psi \, , \; {\rm where} \; \; | \psi | << |
\chi_M | = \sqrt{\frac{6}{\lambda}} \, \, ,$$ whose dynamics is effectively controlled by (28), which concretely becomes $$\frac{d^2 \, \psi}{d t^2} \, + \, \frac{3}{t} \, \frac{d \psi}{d t} \,
- \, \mu^2 \, \psi \, = \, 0 \, \, ,$$ once the expression $$f \, = \, \ln \left( \frac{|t|}{a} \right) \, \, ,$$ derived from (39) and(41), and the identity $$\frac{\lambda}{6} \, \chi_M^2 \, + \, 3 \mu \sqrt{\frac{\lambda}{6}} \,
\chi_M
\, + \, 2 \mu^2 \, \equiv \, 0 \, \, ,$$ stated by (30.b), are taken into account. Because of the matter-field fluctuations encoded in (55), the initial Milne background starts exhibiting metric perturbations which no longer fulfill the unperturbed equations (34.b) and (35); instead, their evolution is controlled by the two Einstein equations (26.a, b). Fortunately, since the last two terms on the right hand side of (26) are exactly the same, one gets a single essential equation for the perturbed modified metric function, $f$, namely $$f_{|44} \, + \, \frac{e^{-2 \, f}}{a^2} \, = \, - \,
\frac{\kappa_0}{2}\, \left( \chi_{|4} \right)^2 \, ,$$ just by subtracting (26.b) from (26.a). As it can be noticed, although one can reduce the order of differentiability in (58), by the well-known substitutions $$p = \frac{df}{dt} \, , \, dt = \frac{df}{p} \; \Rightarrow \;
\frac{d}{dt} = p \, \frac{d}{df} \, , \; q = p^2 \, \, ,$$ casting it into the form of the linear, non homogeneous, first-order differential equation $$\frac{d q}{d f} \, + \, \kappa_0 \, \left( \frac{d \psi}{d f} \right)^2
q \, = \, - \, \frac{2}{a^2}\, e^{-2 \, f}$$ $- \; \; \kappa_0 \left( \chi_{|4} \right)^2 = \kappa_0 \, p^2
\left( \frac{d \chi}{d f} \right)^2 = \kappa_0 \left( \frac{d
\psi}{d f} \right)^2 q$ because of (54) and (59) $-$ whose solution reads $$q = e^{- \, \kappa_0 \, \int^f \left( \frac{d \psi}{d \xi}
\right)^2 \, d
\xi} \, \left[ {\cal C} \, - \, \frac{2}{a^2} \int^f \, e^{-2 \left[
\xi - \frac{\kappa_0}{2} \int^{\xi} \left( \frac{d \psi}{d \eta}
\right)^2 \, d
\eta \right]} \, d \xi \right] \, ,$$ where ${\cal C} = const. \, \geq \, 0 \,$, so that $$\frac{d f}{d t} \, = \pm \, e^{- \frac{\kappa_0}{2} \int^f \left(
\frac{d \psi}{d \xi} \right)^2 \, d \xi} \, \left[{\cal C} \, - \,
\frac{2}{a^2} \int^f \, e^{-2 \left[ \xi - \frac{\kappa_0}{2}
\int^{\xi} \left( \frac{d \psi}{d \eta} \right)^2 \, d \eta
\right]} \, d \xi \right]^{1/2}$$ and therefore $$t \, = \, t_0 \, \pm \, \int \frac{e^{\frac{\kappa_0}{2} \int^f
\left( \frac{d \psi}{d \xi} \right)^2 \, d \xi}\, \, df} {\left[
{\cal C} \, - \, \frac{2}{a^2} \int^f \, e^{-2 \left[ \xi -
\frac{\kappa_0}{2} \int^{\xi} \left( \frac{d \psi}{d \eta}
\right)^2 \, d \eta \right]} \, d \xi \right]^{1/2}} \, , \, t_0
\in {\rm{\bf R}} \, ,$$ one cannot actually get an explicit closed-form solution quite because of the matter-source contribution $\frac{\kappa_0}{2}
\left( \frac{d \psi}{d f} \right)^2$ which, in principle, should be worked out from the awkward (although linear) second-order differential equation of the linear field-fluctuation around $\chi_M$ $$\frac{d^2 \, \psi}{d f^2} \, + \, \left[ 3 \, + \, \frac{d}{df}
\left( \ln \sqrt{q} \, \right) \right] \frac{d \psi}{d f} \, - \,
\frac{\mu^2}{q} \, \psi \, = \, 0 \, \, ,$$ representing the image of (55), softly generalized as $$\frac{d^2 \, \psi}{d t^2} \, + \, 3 \, \frac{d f}{d t} \, \frac{d
\psi}{d t} \, - \, \mu^2 \, \psi \, = \, 0 \, \, ,$$ under the whole set of substitutions (59). Hence, even for small perturbations around the Milne value of the physical field $\chi$, it is clearly unlikely to get an exact solution $\lbrace \psi\, ,
\, q \rbrace (f) \, : \textbf{R} \rightarrow \lbrace \textbf{R} \,
, \, \textbf{R}_{+} \rbrace$, i.e., by (59), $\lbrace \psi \, , \,
f \rbrace(t) \, : \textbf{R} \rightarrow \textbf{R}$, to the highly nonlinearly coupled differential equations (60) and (61) and therefore, analytically, the best one can do, at least for tackling the subject, is to consider the linear fluctuation $h$ in the modified metric function $f$, around its Milne form (56), i.e. $$f \, = \, \ln \left| \frac{t}{a} \right| \, + \, h \, \, ,
{\rm with} \; e^h \, \cong \, 1 + h$$ and to subsequently linearize the essential (inhomogeneous) Einstein equation (58), getting the far much simpler form $$\frac{d^2 h}{d t^2} \, - \, \frac{2}{t^2} \, h \, = \, - \,
\frac{\kappa_0}{2}\, \left( \frac{d \psi}{d t} \right)^2 \, \, ,$$ where the source-term on the right hand side is going to be evaluated, in the first-order approximation, by integrating the fluctuating remnant field equation (55). In this respect, setting $$\tau = \mu \, t \; \; {\rm and} \; \; \psi \, = \, \tau^{\nu} \,
U(\tau) \, , \, \, \nu \in {\rm{\bf R}} \, \, ,$$ it yields the differential equation $$\frac{d^2 U}{d \tau^2} \, + \, \frac{2 \nu + 3}{\tau} \,
\frac{d U} {d \tau} \, - \, \left(1 \, - \,
\frac{\nu (\nu + 2)}{\tau^2} \right) \, U \, = \, 0 \, \, ,$$ for the new function $U$, so that, for $$\nu \, = \, -1 \, \, ,$$ it concretely gets the form of the \`\`[*modified*]{} Bessel functions” equation, $$\frac{d^2 U}{d \tau^2} \, + \, \frac{1}{\tau} \,
\frac{d U} {d \tau} \, - \left(1 \, + \,
\frac{1}{\tau^2} \right) U \, = \, 0 \, \, ,$$ and therefore, referring to the perturbation field $\psi$ in (64) with (66), one instantly gets the two (functionally) linearly independent modes $$\psi_{+}(t) \, = \, \frac{{\cal N}_{+}}{\mu t} \, I_{1}(\mu \, t)
\, \, , \, \, \psi_{-}(t) \, = \, \frac{{\cal N}_{-}}{\mu t} \,
K_{1}(\mu \, t) \, \, ,$$ where the real constants ${\cal N}_{\pm} \, \, -$ of renormalization dimension $D = 1 \, \, -$ set the amplitude of the physical field fluctuation at some reference-moment \`\`$t_0$”. In spite of the way it heads into the future (timelike) infinity, decaying extremely fast, the mode $\psi_{-}$ is strongly singular at the Milne-time origin, $t=0$, and takes very large values around it, so that, at such early epochs, it hardly can be considered as a [*small*]{} perturbation that should fulfill the requirement $ | \psi_{-}(0 < \mu t << 1)| << \mu \sqrt{6/ \lambda}
= |\chi_{M}|$. Therefore, only the $(t=0)-$nonsingular growing mode $\psi_{+}$ contributes to the source term in (63) and drives the evolution of the modified metric function perturbation, $h$. With respect to the dimensionless time-like parameter $\tau$ (defined in (64)) the equation (63) does simply read $$\frac{d^2 h}{d \tau^2} \, - \, \frac{2}{\tau^2} \, h \, = \,
- \, \frac{\kappa_0}{2} \left( \frac{d \psi_{+}}{d \tau} \right)^2$$ and we work out its general solution(s) by starting with the function substitution $$h \, = \, \tau^{\gamma} \, F(\tau) \, \, ,$$ which turns (69) into the inhomogeneous Euler equation, for the function $F$, $$\frac{d^2 F}{d \tau^2} \, + \, \frac{2 \gamma}{\tau} \, \frac{d
F}{d \tau} \, + \, \frac{\gamma (\gamma - 1) - 2}{\tau^2} \, F \,
= \, - \, \frac{\kappa_0}{2} \, \left( \frac{d \psi_{+}}{d \tau}
\right)^2 \tau^{- \gamma} \, .$$ As it can be noticed regarding the term in F, this can be nontrivially vanished by taking $\gamma$ as each of the two roots of the very simple 2-nd degree equation $$\gamma^2 \, - \, \gamma \, - \, 2 \, = \, 0 \, \Rightarrow \,
\gamma = \lbrace - \, 1 \, , \, 2 \rbrace$$ and therefore we get $-$ in principle $-$ two branches of solutions, respectively stated by the mathematical relations $$\begin{aligned}
&(a)& h_{-} \, = \, \frac{1}{\tau} \, F_{-}( \tau) \, \, , \, \,
\frac{d F_{-}}{d \tau} \, = \, G_{-}( \tau) \, \, , \nonumber
\\* &(b)& \frac{d G_{-}}{d \tau} \, - \, \frac{2}{\tau} \,
G_{-} \, = \, - \, \frac{\kappa_0}{2} \, \left( \frac{d
\psi_{+}}{d \tau} \right)^2 \, \tau \, , \, \, {\rm for} \, \gamma
\, = \, - 1 \, \, ; \nonumber \\* &(c)& h_{+} \, = \, \tau^2 \,
F_{+}( \tau) \, \, , \, \, \frac{d F_{+}}{d \tau} \, = \, G_{+}(
\tau) \, \, , \nonumber \\* &(d)& \frac{d G_{+}}{d \tau} \, + \,
\frac{4}{\tau} \, G_{+} \, = \, - \, \frac{\kappa_0}{2} \,
\frac{1}{\tau^2} \left( \frac{d \psi_{+}}{d \tau} \right)^2\, , \,
\, {\rm for} \, \gamma \, = \, 2 \, \, .\end{aligned}$$ Thus, by (73.a, c), we have reduced the second-order differential equation(69) to the inhomogeneous first-order differential equations (73.b, d) whose solutions can be easily derived as $$\begin{aligned}
&(a)& G_{-}( \tau) \, = \, G_{-}^{0} \tau^2 \, - \,
\frac{\kappa_0}{2} \, \tau^2 \, \int^{\tau} \frac{d s}{s} \left(
\frac{d \psi_{+}}{d s} \right)^2 \, , \, G_{-}^{0} = cst \in
{\rm{\bf R}} \, , \nonumber \\* &(b)& G_{+}( \tau) \, = \,
\frac{G_{+}^{0}}{\tau^4} \, - \, \frac{\kappa_0}{2} \,
\frac{1}{\tau^4} \, \int^{\tau} s^2 \left( \frac{d \psi_{+}}{d s}
\right)^2 d s \, , \, G_{+}^{0} = cst \in {\rm{\bf R}}\, ,\end{aligned}$$ where $\psi_{+}$ is given by (68), with $ \mu t$ replaced by $s$, i.e. $$\psi_{+}(s) = \frac{{\cal N}_{+}}{s} \, I_{1}(s) \, \Rightarrow \,
\left \lbrace
\begin{array}{l}
\frac{d \psi_{+}}{d s} = \frac{{\cal N}_{+}}{s} \,
\left[ \frac{d I_{1}}{d s} \, - \, \frac{I_{1}}{s} \right] \, \, , \\
\\ \frac{d I_{1}}{d s} = \frac{1}{2}
\left[ I_{0}(s) \, + \, I_{2}(s) \right] \, \, . \\
\end{array} \right.$$
Making (a respective) use of (73.a, c), once (74.a, b) have been gotten, we come to the concrete expression(s) of the $-$ \`\`formally” [*two*]{} $-$ (most) general solutions, $$\begin{aligned}
&(a)& h_{-}( \tau) = F_{-}^{0} \tau^{-1} + \frac{1}{3} G_{-}^{0}
\tau^2 - \frac{\kappa_0}{2} \, \tau^{-1} \int \tau^2 \left(
\int^{\tau} \frac{d s}{s} \left( \frac{d \psi_{+}}{d s} \right)^2
\right) d \tau \, , \nonumber \\* &(b)& h_{+}( \tau) = -
\frac{G_{+}^{0}}{3} \tau^{-1} + F_{+}^{0} \tau^2 -
\frac{\kappa_0}{2} \, \tau^2 \, \int \frac{d \tau}{\tau^4} \left(
\int^{\tau} s^2 \left( \frac{d \psi_{+}}{d s} \right)^2 \, d s
\right) ,\end{aligned}$$ with $F_{\pm}^{0} = cst \in {\rm{\bf R}}$, of the linearized Einstein-Gordon equation (69) $\Leftrightarrow$ (63), which describes, envisaging(62), the coherent Milne’s Universe metric fluctuations induced by an initially small perturbation $-$ $$| \psi_{+}(0) | \, = \, \frac{1}{2} | {\cal N}_{+} | \, << \, \mu
\sqrt{\frac{6}{\lambda}} \, = \, | \chi_{M} | \, \, ,$$ right on the singular, but [*free*]{} of \`\`real” geometrical singularities, $\lbrace t = 0 \rbrace-$foliation $-$ in the physical field $\chi$, leftover around $\chi_M$ by the spontaneous breaking of the inner parity invariance. Nevertheless, in a straight computational manner, it is not quite trivial to realize that the [*two*]{} branches [*do*]{} actually [*coincide*]{}.
The difference in the way they look is only apparent and comes from an additional part-by-part integral that has been subtly performed $-$ \`\`by itself”, actually $-$ in the switch from (73.a, b) to (73.c,d). Indeed, one can notice first that $$\int^{\tau} \frac{d s}{s} \left( \frac{d \psi_{+}}{d s} \right)^2
\, = \, \int^{\tau} \frac{d s}{s^3} \, s^2 \left( \frac{d
\psi_{+}}{d s} \right)^2 = \int^{\tau} \frac{1}{s^3} \left[ s^2
\left( \frac{d \psi_{+}}{d s} \right)^2 \right] d s$$ and setting $$\begin{aligned}
& & u \, = \, \frac{1}{s^3} \; \; , \; \; d v \, = \, s^2 \left(
\frac{d \psi_{+}}{d s} \right)^2 d s \, \, , \nonumber \\* & & d u
\, = \, - \, \frac{3 d s}{s^4} \; \; , \; \; v \, = \, \int^{s}
\xi^2 \left( \frac{d \psi_{+}}{d \xi} \right)^2 d \xi \, \, ,\end{aligned}$$ it yields (integrating \`\`by parts”) $$\int \frac{d \tau}{\tau} \left( \frac{d \psi_{+}}{d \tau}
\right)^2 \, = \, \frac{1}{\tau^3} \, \int s^2 \left( \frac{d
\psi_{+}}{d s} \right)^2 d s \, + \, 3 \, \int \frac{d
\tau}{\tau^4} \left( \int^{\tau} s^2 \left( \frac{d \psi_{+}}{d s}
\right)^2 d s \right) \, \, .$$ That is to be used in the part-by-part integral $$\int \tau^2 \left( \int^{\tau} \frac{d s}{s}
\left( \frac{d \psi_{+}}{d s} \right)^2 \right) d \tau =
\frac{\tau^3}{3} \int^{\tau} \frac{d s}{s} \left(
\frac{d \psi_{+}}{d s} \right)^2 \, - \,
\frac{1}{3} \int \tau^2
\left( \frac{\psi_{+}}{d \tau} \right)^2 d \tau \, \, ,$$ i.e. $$\begin{aligned}
\int \tau^2 \left( \int^{\tau} \frac{d s}{s} \left( \frac{d
\psi_{+}}{d s} \right)^2 \right) d \tau \, &=& \frac{1}{3}
\int^{\tau} s^2 \left( \frac{d \psi_{+}}{d s} \right)^2 d s \, +
\nonumber \\* & & + \, \tau^3 \, \int \frac{d \tau}{\tau^4} \left(
\int^{\tau} s^2 \left( \frac{d \psi_{+}}{d s} \right)^2 d s
\right) \, - \nonumber
\\* & & - \, \frac{1}{3} \int^{\tau} s^2 \left( \frac{d
\psi_{+}}{d s} \right)^2 d s \, = \nonumber \\* &=& \tau^3 \int
\frac{d \tau}{\tau^4} \left( \int^{\tau} s^2 \left( \frac{d
\psi_{+}}{d s} \right)^2 d s \right) ,\end{aligned}$$ explicitly stating that $$\frac{1}{\tau}\, \int \tau^2
\left( \int^{\tau} \frac{d s}{s}
\left( \frac{d \psi{+}}{d s} \right)^2 \right) d \tau \, = \,
\tau^2 \, \int \frac{d \tau}{\tau^4}
\left( \int^{\tau} s^2
\left( \frac{d \psi_{+}}{d s} \right)^2 d s \right)$$ and therefore, with $$\begin{aligned}
& & - \, \frac{G_{+}^0}{3} \, = \, F_{-}^0 \, = \, F_0 \, \, ,
\nonumber \\* & & F_{+}^0 \, = \, \frac{G_{-}^0}{3} \, = \, - \,
\frac{1}{6} \, G_0 \, \, , \nonumber\end{aligned}$$ it has been entirely proven that the two seemingly different branches (76) are actually the same, being subsequently described by the modified metric function perturbation $$h \, = \, \frac{F_0}{\tau} \, - \, \frac{G_0}{6} \, \tau^2 \, - \,
\frac{\kappa_0}{2} \, \tau^{-1} \int \tau^2 \left( \int^{\tau}
\frac{d s}{s} \left( \frac{d \psi_{+}}{d s} \right)^2 \right) d
\tau \, \, .$$ With that and (62), the [*proper*]{} scale function (39) becomes $$S( \tau) \, = \, \frac{F_0}{\mu} \, + \, \frac{\tau}{\mu}
\left[ 1 \, - \, \frac{G_0}{6} \, \tau^2 \, - \,
\frac{\kappa_0}{2} \, \tau^{-1} \, \int \tau^2 \left( \int^{\tau}
\frac{d s}{s} \left( \frac{d \psi_{+}}{d s} \right)^2 \right) d
\tau \right] ,$$ so that, the seemingly divergent term $F_0 / \tau$, in (82), brings nothing more than a [*constant*]{} (universal) shift in the Milne’s cosmic-time, making no contribution at all to the curvature disturbances produced by the source-field (linear) fluctuations $\psi_{+}$. Hence, it simply can be dropped away just by setting $F_0 = 0$.
Quite on the contrary, the [*seemingly arbitrary*]{} constant $G_0$ makes an important contribution to the curvature of the coherently $\psi_{+}-$ perturbed Milne Universe, mostly with respect to its [*stability*]{}.
To give the details of this matter, let us first work out the curvature perturbations straight from the relations (82), (62) and (21, with $k = - \, 1$). It primarily results $$\begin{aligned}
R_{\alpha \beta \alpha \beta } & = & \frac{2 \mu^2}{\tau^2} \left[
\tau \, \frac{dh}{d \tau} + h \right] = \mu^2 \, \frac{2}{\tau^2}
\, \frac{d \;}{d \tau} ( \tau h) \, , \; \; \alpha , \beta =
\overline{1,3} \, , \nonumber \\* R_{\alpha 4 \alpha 4} & = & - \,
\mu^2 \left[ \frac{d^2 h}{d \tau^2} + \frac{2}{\tau} \frac{dh}{d
\tau} \right] = - \, \frac{\mu^2}{\tau^2} \, \frac{d \;}{d \tau}
\left( \tau^2 \frac{d h}{d \tau} \right) \, , \; \alpha =
\overline{1,3} \; , \nonumber\end{aligned}$$ with no summation on the repeated indices, and plugging the (82) in, it yields $$\begin{aligned}
R_{\alpha \beta \alpha \beta} & = &
- \, \mu^2 \left[ G_0 + \kappa_0 \int \frac{d \tau}{\tau } \left(
\frac{d \psi_+}{d \tau} \right)^2 \right] \nonumber \\* R_{\alpha
4 \alpha 4} & = & - \, R_{\alpha \beta \alpha \beta } \, + \,
\frac{\kappa_0}{2} \, \mu^2 \, \left( \frac{d \psi_+}{d \tau}
\right)^2\end{aligned}$$ On the other hand, completely independent of the form of $h$, the same curvature components can be derived from the exact form (26) of the Einstein-(generalized) Gordon equations, where the potential (13) reads, within the linear approximation assumption, $$\left. \left. V( \chi ) = V( \chi_M + \psi ) = V( \chi_M ) +
\frac{dV}{d \chi } \right|_{\chi_M} \psi + \frac{1}{2}
\frac{d^2V}{d \chi^2} \right|_{\chi_M} \psi^2 + {\cal O}\left(
\psi^{n > 2} \right)$$ i.e., since $\chi_M$ is an extremum of $V$, $$V( \psi ) \, = \, \frac{3 \mu^4}{2 \lambda} \, - \,
\frac{\mu^2}{2} \, \psi^2$$ Thus, one gets the relations $$\begin{aligned}
\left( f_{|4} \right)^2 - \, \frac{e^{-2f}}{a^2} & = &
\frac{\kappa_0}{6} \left[ \left( \psi_{|4} \right)^2 \, - \, \mu^2
\, \psi^2 \right] \nonumber \\* 2 \left[ f_{|44} + \left( f_{|4}
\right)^2 \right] & = & - \, \frac{\kappa_0}{2} \left[ \left(
\psi_{|4} \right)^2 \, + \, \mu^2 \, \psi^2 \right] - \left[
\left( f_{|4} \right)^2 - \, \frac{e^{-2f}}{a^2} \right]\end{aligned}$$ which straightforwardly lead, because of (21, with $k=-1$), to the entirely $\psi_+ - $dependent expressions of the essential curvature components, $$\begin{aligned}
R_{\alpha \beta \alpha \beta} & = & \frac{\kappa_0}{6} \left[
\left( \frac{d \psi_+}{dt} \right)^2 - \, \mu^2 \, \psi_+^2
\right] \nonumber \\* R_{\alpha 4 \alpha 4 } & = &
\frac{\kappa_0}{6} \left[ 2 \left( \frac{d \psi_+}{dt} \right)^2 +
\, \mu^2 \, \psi_+^2 \right] \nonumber\end{aligned}$$ (no summation, $ \alpha \neq \beta \in \lbrace 1,2,3 \rbrace$), which can obviously be written, in terms of the dimensionless variable $\tau$, as $$\begin{aligned}
& (a) & R_{\alpha \beta \alpha \beta} \; = \; \frac{\kappa_0}{6}
\, \mu^2 \left[ \left( \frac{d \psi_+}{d \tau } \right)^2 - \,
\psi_+^2 \right] \nonumber \\* & (b) & R_{\alpha 4 \alpha 4 } \; =
\; \frac{\kappa_0}{6} \, \mu^2 \left[ 2 \left( \frac{d \psi_+}{d
\tau } \right)^2 + \, \psi_+^2 \right]\end{aligned}$$ Adding the two expressions, we instantly get the second result (83) and that is a very good cheek-out since, basically, the formulae (86) and (83) have been independently derived, speaking of the concretely employed methods. Hence, it is quite sufficient to refer the calculations just to the spatial sectional curvature components and, if we worked well, the two expressions (86.a) and (the first of) (83) should produce the same result. As, because of (75), we would be dealing with the three Bessel functions, $\left
\lbrace I_n (\tau ) \right \rbrace_{n= \overline{0,2}}$, the analytical closed-form estimation of the integral involved in (83) would certainly be out of (the normal) reach. So that, we are going to consider only the first two terms, i.e. $$I_1 ( \tau ) \cong \frac{\tau}{2} \, + \, \frac{\tau^3}{16} \, ,$$ from the power-series expression of the modified Bessel function $I_1$. Consequently, $$\psi_+ \, = \, \frac{{\cal N}_+}{2} \left[ 1 + \frac{\tau^2}{8}
\right] , \; \frac{d \psi_+}{d \tau} \, = \, \frac{{\cal N}_+}{8}
\, \tau$$ and therefore, the first of (83) becomes $$R_{\alpha \beta \alpha \beta } \, = \, - \, \mu^2 \, G_0 \, - \,
\frac{\kappa_0 \mu^2}{128} \, {\cal N}_+^2 \, \tau^2 \, ,$$ while (86.a) does concretely read $$R_{\alpha \beta \alpha \beta } \, = \, - \, \frac{\kappa_0
\mu^2}{24} \, {\cal N}_+^2 \, - \, \frac{\kappa_0 \mu^2}{128} \,
{\cal N}_+^2 \, \tau^2$$ Hence, as we have said, the integration constant $G_0$, in (82), is just seemingly arbitrary for it must actually equate the static contribution $\frac{\kappa_0}{24}{\cal N}_+^2$ of the perturbation field $\psi_+$. Subsequently, either from (83) or straightly from (86.b), the mixed components of the perturbed curvature, close to the singular epoch $t=0$, are given by $$R_{\alpha 4 \alpha 4} \, = \, \frac{\kappa_0 \mu^2}{24} \, {\cal
N}_+^2 \, + \, \frac{\kappa_0 \mu^2}{64} \, {\cal N}_+^2 \, \tau^2$$ and, just for the sake of completeness, the metric perturbation function reads $$h \, = \, - \, \frac{\kappa_0 {\cal N}_+^2}{144} \left[ 1 \, + \,
\frac{9 \tau^2}{80} \right] \, \tau^2$$ It can be concluded so far, inspecting the \`\`early” coherently perturbed components (88), (89) of the curvature tensor, that the spontaneously inner-parity breaking generated Milne phase is clearly unstable, no matter how small the source-field perturbation is, and it primarily runs into an anti-de Sitter phase of scalar curvature $R[0] = - \,
\frac{\kappa_0}{2} \left( \mu {\cal N}_+ \right)^2$.
Higgs$-$anti-de Sitter Spacetime Bubbles
========================================
That is nicely closing the circle, for it brings us back to the only Einstein equation (34.a) characterizing the spacetime \`\`supported” by the other two fixed point values of the field $\chi$. Written with respect to the cosmological scale function (39) and introducing the notation $$\omega_0^2 \, = \, \frac{\kappa_0 \mu^4}{2 \lambda} \; \;
\Leftrightarrow \; \; \omega_0 = \mu^2 \sqrt{ \frac{ \kappa_0}{2
\lambda}} \, ,$$ the above equation, (34.a), becomes extremely simple $$\left( \frac{dS}{dt} \right)^2 = \, 1 - \, \omega^2_0 \,
S^2 \;
\; \Rightarrow \; \; \frac{dS}{dt} \, = \, \pm \,
\sqrt{1-(
\omega_0 S)^2 } \, ,$$ so that, its general solution reads, \`\`by the book”, $$S(t) \, = \, \omega_0^{-1} \, \sin ( \omega_0 t + \gamma_0 ) \, ,$$ where the constant phase-factor $\gamma_0$ accounts for both the sign-choices ($\pm$). Actually, considering a positive scale factor \`\`$a$” $-$ with physical dimension of length $-$ and because $f : {\rm{\bf R}} \to {\rm{\bf R}}$, the scale function defined by (39) must be non-negative, reading therefore $$S(t) \, = \, \frac{1}{\omega_0} | \sin (\omega_0 t + \gamma_0 ) |
, \; \; {\rm such \; that} \; \; f = \ln | \sin (\omega_0 t +
\gamma_0 ) |$$ Hence, for the other two fixed-point values $\chi_{L,R}$ given by (32), double roots of the potential (31), we have been through quite fast with the non-linear Einstein-Gordon system (30), once we had (34.a) integrated, getting its exact solution(s) as a pair of anti-de Sitter Universes, whose metric does explicitly read (in terms of $(k=-1)-$spherical coordinates $\lbrace r, \theta) ,
\varphi \rbrace$ $$ds^2 \, = \, \frac{1}{\omega_0^2} \, \sin^2 ( \omega_0 t +
\gamma_0 ) \left[ \frac{(dr)^2}{1+r^2} + r^2 \, d \Omega^2
\right] - \, (dt)^2 \, ,$$ actually representing two harmonically oscillating $(k=-1)-$bubbles, that go through an eternal sequence of cosmic Bangs and Crunches, one of them filled up with the remnant field $\chi_L = - 2 \mu \sqrt{6/ \lambda }$ and the other seemingly empty as the massive source-field $\chi$ vanishes everywhere inside, but not before it left an enormous amount of exotic vacuum-energy. In this respect, let us see what the numbers would be if one took $\lambda \approx 6$ and considered the smallest symmetry breaking scale, namely the one involved in the Higgs sector of the Standard Model, where (probably, for now, as the Higgs has not been experimentally detected yet) its mass, $m_H =
\sqrt{2} \mu$, lies somewhere inbetween 115 and 300 $GeV$, i.e. (in Kilos) $$2 \cdot 10^{-25} \; (kg) \leq m_H = \sqrt{2} \, \mu < 5.(3) \cdot
10^{-25} \; (kg)$$ First, with the fundamental (universal) constants $c$ and $\hbar$ plugged in, the vacuum-energy density $${\cal H}_0 \, = \, T_{44} [0] = \, - \, \frac{3 \mu^4}{2 \lambda}
\, ,$$ sustaining the anti-de Sitter \`\`bubble” where the Higgs cools down to its undynamized ground state $\chi = 0$, does explicitly read $${\cal H}_0 \, = \, - \, \frac{3 m_H^4 c^5}{8 \lambda \hbar^3} \,
= \, - \, \frac{1.246 \div 63}{\lambda} \cdot 10^{45} \;
({J/m^3}) ,$$ yielding in modulus, for $\lambda \approx 6$, the impressive values $$| {\cal H}_0 | \cong 2 \cdot 10^{44} \div 10^{46} \; (J/m^3)$$ which, nevertheless, have been frequently encountered in the Domain Walls Theory. Similarly, the proper pulsation (91), measured in $s^{-1}$, is given by the formula $$\omega_0 \, = \, \frac{c \, m_H^2}{\hbar} \sqrt{\frac{\pi
G c}{\hbar \lambda}} \, \cong \, \frac{2.94 \div
20.82}{\sqrt{\lambda}} \cdot 10^9 \; (s^{-1})$$ that subsequently leads to the proper frequency $$\nu_0 \, = \, \frac{c \, m_H^2}{2 \pi \hbar} \sqrt{\frac{\pi G c}{
\hbar \lambda}} \cong \frac{0.468 \div 3.314}{\sqrt{\lambda }}
\; \; (GHz)$$ and to the geometrical period of the Bang-Crunch cycles in these Higgs-anti-de Sitter spacetime bubbles, $$T \, = \, \frac{\pi}{\omega_0} \, \cong \, ( 0.151 \div 1.071 )
\sqrt{\lambda} \; (ns) \; .$$ For the considered $\lambda$, their respective values are $$\begin{aligned}
& (a) & \omega_0 \cong (1.2 \div 8.5 ) \cdot 10^9 \; (s^{-1}) \, ,
\nonumber \\* & (b) & \nu_0 \cong 0.19 \div 1.35 \; (GHz) \, ,
\nonumber \\* & (c) & T \cong 0.35 \div 2.626 \; (ns) \, .\end{aligned}$$
Concerning the cosmological length-scale parameter $\omega_0^{-1}$, which is nothing else but the amplitude of the anti-de Sitter scale function oscillation, it reads (in international units) $$\omega_0^{-1} \, = \, \frac{\hbar}{m_H^2} \sqrt{\frac{\hbar
\lambda}{\pi Gc}} \cong \frac{1.44 \div 10.2}{\sqrt{\lambda }} \;
\; (cm)$$ and is getting, as $\omega_0$ did in (103.a), the numerical values $$\omega_0^{-1} \cong 3.53 \div 25 \; (cm)$$ Based on these data, we can speculate a bit, in a sort of [*what if...*]{}-manner, on the possible cosmological consequences of the existence, in some regions of our Universe, of some (2+1)-dimensional windows towards the bulk-space extra-dimensions where such Higgs–anti-de Sitter (harmonically oscillating) bubbles might be living. For instance, the upper limit of the proper frequency $\nu_0$ is pretty close to the famous 21 $cm(s)$ Hydrogen-line so that, inspecting the whole sky, if the Higgs-boson mass were around 300 $GeV$, there would (presumably) be some conventionally unexplained deviations from the averaged level of the electromagnetic radiation received from the known and ordinary excited baryonic astrophysical objects. Similarly, referring to the rest of the $\nu_0-$values, as the present thermalized-photons temperature is too small to significantly dynamize the Higgs-like field $\chi$ around its ground state, $\chi =0$, one can presume that, watching for instance the Giant Voids, which are pretty much deprived of the other forms of conventional matter, there might be detected some disturbances in (or, eventually fluctuating, anisotropy of) the $\lbrace n \,
\nu_0 \rbrace_{n= \overline{1,3}}$ channels of the Cosmic Microwave Background Radiation, coming from the junction with (such) an electroweak spontaneously broken$-$anti-de Sitter \`\`small” scale Universe. In some respect, the situation is very much alike the one in Chaotic Inflation $-$ where inflating Baby Universes pop up (chaotically) from the spacetime foam $-$ except that now we deal with harmonically oscillating $(k=-1)-$regions, geometrodynamically exactly accommodating the initial self-interacting field $\Phi$ in one of the absolute minima of its quartic potential, that pop up in an already inflated bubble, which is our own Universe. The reason why we have included the third harmonic of $\nu_0$ among the frequencies on which there might be some deviations from the black body radiation law of the Cosmic Microwave Background, lies in the manner the total energy of an anti-de Sitter three-dimensional ball depends upon time. Indeed, considering the well-known formula (for the energy of a $\lbrace t = cst. \rbrace-$compact filled in by the matter-density ${\cal H}_0$ ) $$E = \int_{N_3 ( t=cst.)} \sqrt{-g} \; {\cal H}_0 \; d^3 x \, ,$$ where $d^3 x = dr \, d \theta \, d \varphi$, $\sqrt{-g} = \frac{|
\sin^3 (\omega_0 t)|}{\omega_0^3} \, \frac{r^2 \sin
\theta}{\sqrt{1+r^2}}$ (derived from (95), discarding $\gamma_0$) and ${\cal H}_0$ being given by (98), it yields for the instantaneous (total) energy of the $\lbrace t=cst.
\rbrace -H^3-$ball, of dimensionless radius $r_0$, the expression $$E(t) \, = \, - \, {\cal E} F(r_0) \, | \sin (\omega_0 t )|^3 \, ,$$ where the radial volume-function $F(r_0)$ and the (physically dimensional) energy amplitude ${\cal E}$ are respectively given by $$\begin{aligned}
& (a) & F(r_0) \, = \, = \, \frac{1}{2} \left[ r_0 \sqrt{1+r_0^2}
\, - \, \ln \left( r_0 + \sqrt{1+r_0^2} \right) \right] ,
\nonumber
\\* & (b) & {\cal E} \, = \, \frac{3 \hbar c^4}{2 G m_H^2}
\sqrt{\frac{\hbar \lambda}{ \pi G c}} \, , \; [{\cal E}]= Joule\end{aligned}$$ Since $\sin^3 x \equiv \frac{3}{4} \sin x - \frac{1}{4} \sin
(3x)$, it clearly results a $25 \% $ energy-distribution on the $3
\, \nu_0-$channel. With (108) and (107), the mean-energy during an anti-de Sitter cycle (102), $$\langle E \rangle_T \, = \, \frac{1}{T} \int_0^T E(t) \, dt \, =
\, - \, \frac{{\cal E}F(r_0)}{\pi} \int_0^{\pi} \sin^3 \gamma \, d
\gamma \, ,$$ i.e. $-$ formally $-$ $$\langle E \rangle_T = \, - \, \frac{4 {\cal E}}{3 \pi} \, F(r_0)
\, ,$$ does actually read $$\langle E \rangle_T = \, - \, \frac{2 \hbar c^4}{\pi G m_H^2} \,
\sqrt{\frac{\hbar \lambda}{\pi G c}} \, F(r_0) \, ,$$ and, in terms of absolute values, it already gives an idea about the effectively involved power $${\cal P}_{ef} \, = \, \frac{| \langle E \rangle_T|}{T} \, ,$$ namely, in $watts$, $${\cal P}_{ef} \, = \, \frac{2c^5}{\pi^2 G} \, F(r_0)$$ Of course, in a rigorous manner, the instantaneous power ${\cal
P}(t)$ comes being expressed from (107) as $${\cal P} \, = \, \frac{dE}{dt} \, = \, - \, 3 \, \omega_0 \,
{\cal E}
F(r_0) \sin^2 ( \omega_0 t) \cos ( \omega_0 t) \, , \; ( \forall )
\; t \in \left[ 0, \, \frac{\pi}{\omega_0} \right] ,$$ i.e. $-$ inserting (100) and (108.b) $-$ $${\cal P} \, = \, - \, \frac{3c^5}{2G} \, \sin^2 (\omega_0 t) \cos
(\omega_0 t) \, F(r_0) \, ,$$ so that it takes symmetric values during an anti-de Sitter cycle, being negative in its first half, when the bubble blows to its maximum size $\omega_0^{-1} r_0$, and respectively positive, on the second half, while the deflating bubble goes into the $T$-crunch. Hence, although the sooth averaged power is zero, yet one can meaningfully define the [*anti-de Sitter semi cycle mean-power*]{}, (in absolute value), $$\langle {\cal P} \rangle_{1/2} \, = \,
\frac{3 c^5}{2 G} \left[ \frac{2 \omega_0}{\pi}
\int_{0}^{\frac{\pi}{2 \omega_0}} \sin^2 (\omega_0 t) \cos( \omega_0 t)
d t \right] \, F(r_0) \, \, ,$$ i.e. $$\langle {\cal P} \rangle_{1/2} \, = \,
\frac{c^5}{\pi G} \, F(r_0) \; (W) \, ,$$ which is released for instance in the crunch-directed decaying phase; compared to ${\cal P}_{ef}$, it is $\frac{\pi}{2}$-times larger but, nevertheless, of the same order of magnitude.
At the electroweak symmetry breaking scale, that has been considered here, the [*energy amplitude*]{} \`\`alone” already achieves [*intriguing*]{} numerical values, $${\cal E} \, \cong \, (0.7 \div 4.8) \cdot 10^{43} \; (J) \, ,$$ which are $-$ \`\`astrophysically speaking” $-$ of the same order (of magnitude) with the ones of a [*medium size*]{} galaxy. Hence, speculating again, envisaging the modulus of the mean-energy (110), $$| \langle E \rangle_T | \, \cong \, (0.3 \div 2) F(r_0) \cdot
10^{43} \; (J) \, ,$$ it might turn out that decaying Higgs-vacuum$-$anti-de Sitter bubbles, no larger than few tens of $\omega_{0}^{-1}$ (given by (105)), could (in principle) provide enough energy to act as [*seeds*]{} in the galaxy formation process. In what it concerns the power (114), $$\langle {\cal P} \rangle_{1/2} \, \cong \, F(r_0) \cdot 10^{52} \; (W)
\, ,$$ which would be released if the bubble stopped growing again after it crunched, that might account for the one emitted by quasars, if some understandable anti-de Sitter-Higgs$-$electromagnetic conversion (mechanism), acting in the core of the quasar, could be figured out. Nevertheless, it should exist, since the derived power expressions (111-114) are completely independent not only of the electroweak breaking scale parameters, but also of the universal Planck constant, being therefore entirely of generally relativistic gravitational origin.
${\mbox{\boldmath $S^2$-}}$Cobordism and \`\`Wick Companions”
=============================================================
Finally, there are two more features (of the topic we are discussing) that we would like to address in the remaining part of the paper.
The first concerns the $\lbrace r = cst. \rbrace -(2 \, + \,
1)$-dimensional cobordism of the anti-de Sitter sphere of coordinate-radius $r_0$ to a spatially flat FRW-Universe of scale function $a(T)$. It comes about by equating the corresponding metrics, $$\frac{r_{0}^{2}}{\omega_{0}^{2}} \, \sin^2 (\omega_0 t) \,
d \Omega^2 \, - \, (d t)^2 \, = \, a^2 (T) (d R)^2 \, +
\, a^2 (T) R^2 d \Omega^2 \, - \, (d T)^2 \, \, ,$$ such that the [*first cobordering equation*]{} reads $$a(T) R \, = \, \frac{r_0}{\omega_0} \, | \sin(\omega_0 t) | \, \, ,$$ leading therefore, to the [*second*]{} one $$(d T)^2 \, - \, a^2 (T) (d R)^2 \, = \, (d t)^2 \, \, ,$$ i.e. $$\left( \frac{d T}{d t} \right)^2 \, - \,
a^2 (T) \left( \frac{d R}{d t} \right)^2 \, = \, 1 \, \, .$$ Extracting $R$ from (119) and taking its derivative with respect to $t$, then plugging the result back into (120), the latter becomes $$\left[ 1 - \frac{r_{0}^{2} \sin^2 (\omega_0 t)}{\omega_{0}^{2} \,
a^2(T)}
\left( \frac{d a}{d t} \right)^2 \right]
\left( \frac{d T}{d t} \right)^2 +
\frac{r_{0}^{2} \sin(2 \omega_0 t)}{\omega_0 \, a(T)} \frac{d a}{d t}
\frac{d T}{d t} \, - \left[ 1 + r_{0}^{2}
\cos^2 (\omega_0 t) \right] = 0$$ and, as it can be noticed, although is just a first-order differential equation, it actually is a highly nonlinear one, especially when general forms of the $(k=0)-$scale function $a(T)$ are to be considered. Moreover, because of the trigonometric functions involved in each of the three terms, (the dimensionless coordinate-radius) $r_0$ [*alone*]{} is getting us in trouble, even for simple forms of $a(T)$, particularly when it achieves [*large*]{} values. Hence, a [*closed form exact solution*]{} to the second cobordering equation (121) does not come easy.
However, some particular $-$ but not trivial $-$ cases can be worked out completely, even if they might be looking a bit nasty, and we are talking here about the \`\`critical” case where $$\frac{1}{a} \frac{d a}{d t} \, = \,
\frac{\omega_0}{r_0} \, | \sin ( \omega_0 t) |^{-1} \, \, ,$$ such that it reduces the degree of (121), regarded as an algebraic equation in $\left( \frac{d T}{d t} \right)$, yielding the far much simpler equation $$\frac{d T}{d t} \, = \, \frac{1 + r_{0}^{2} \cos^2 (\omega_0 t)}{2
r_0 \cos (\omega_0 t)} \, \, ,$$ whose solution $$T \, = \, T_0 \, + \, \omega_{0}^{-1} \left[ \frac{1}{2 r_0}
\ln \left| \frac{1 + {\rm tg} \left(
\frac{\omega_0 t}{2} \right)}{1 - {\rm tg}
\left( \frac{\omega_0 t}{2} \right)}
\right| \, + \, \frac{r_0}{2} \sin ( \omega_0 t) \right] \, , \, \,
T_0 = cst. \in {\rm{\bf R}} \, \, ,$$ gives the concrete dependence (in this case) of the $(k = 0)-$RW universal time on the one in the anti-de Sitter bubble. Because of (123), the $(k = 0)-$scale function equation (122) reads $$\frac{1}{a} \, d a \, = \, \left( \frac{1}{r_{0}^{2}} +
\frac{1}{2} \right) \frac{d( \omega_0 t)}{\sin (2 \omega_0 t)} +
\frac{1}{2} \, {\rm ctg} (2 \omega_0 t) \, d( \omega_0 t) \, \, ,$$ getting therefore the solution $$a(t) \, = \, a_0 [1 - \cos ( 2 \omega_0 t)]^{\frac{1}{4}} \,
| {\rm tg} ( \omega_0 t) |^{- \frac{1}{2 r_0^2}} \, , \, \,
a_0 = cst. \in {\rm{\bf R}}_{+} \, \, ,$$ which, together with (124), give the complete parametric representation of the scale function $a(T)$. Consequently, the first cobordering equation (119) does explicitly set the behaviour of the \`\`true” radius $R$ of the anti-de Sitter sphere as it is actually seen from (within) the corresponding spatially flat Universe (of scale function derived from) (125) and (124); that is $$R(t) \, = \, \frac{\omega_{0}^{-1} r_0}{2^{1/4} a_0}
| \sin (\omega_0 t) |^{\frac{1}{2}} \,
| {\rm tg} ( \omega_0 t) |^{- \frac{1}{2 r_0^2}} \, \, .$$ These are non-perturbative (exact) results. Considering now a small enough anti-de Sitter bubble, such that $r_0^2 << 1$, they got major simplification since (124) becomes $$2 r_0 \, \omega_0 (T - T_0) \cong
\ln \left| \frac{1 + {\rm tg} ( \frac{\omega_0 t}{2})}
{1 - {\rm tg} ( \frac{\omega_0 t}{2})} \right| \, ,$$ so that $${\rm tg} (\frac{\omega_0 t}{2}) \, = \, {\rm th}
[ r_0 \, \omega_0 (T - T_0)] \, ,$$ which leads, after a short calculation, to the scale function expression $$a(T) \, = \, 2^{1/4} a_0 \, | {\rm th}
[2 r_0 \omega_0 (T - T_0)] |^{\frac{1}{2}} \,
| {\rm sh} [2 r_0 \omega_0 (T - T_0)] |^{\frac{1}{2 r_0^2}}$$ and to the $(k = 0)-$radial coordinate evolution law (in RW-time) $$R(T) \, = \frac{\omega_0^{-1} r_0}{2^{1/4} a_0} \,
| {\rm th} [2 r_0 \, \omega_0 (T - T_0)] |^{\frac{1}{2}} \,
| {\rm sh} [2 r_0 \, \omega_0 (T - T_0)] |^{- \frac{1}{2 r_0^2}}$$ Late into the future, for $T - T_0 >> (2 \, r_0 \,
\omega_0)^{-1}$, each of them does respectively go as $$\begin{aligned}
&(a)& \, a(T) \, \cong \, b_0 \, \, e^{\frac{\omega_0}{r_0} T} \,
\, , \nonumber \\* &(b)& \, R(T) \, \cong \, \frac{\omega_{0}^{-1}
r_0}{b_0} \, \, e^{- \frac{\omega_0}{r_0} T} \, \, ,\end{aligned}$$ with $b_0 = a_0/2^{\frac{1}{2 r_0^2}}$, lighting up clearly a strongly decaying Higgs-vacuum (small scale) bubble, $S^{2}-$connected to an extremely fast inflating universe.
As a matter of fact, this beautiful picture can also be obtained as an \`\`uncritical” [*exact solution*]{} of the cobordering equation (121) in the case where the physical radius $\omega_0^{-1} r_0$ of the anti-de Sitter sphere does sharply equate the inverse, $H_0^{-1}$, of the Hubble constant of a de Sitter Steady-State Universe, $$a(T) \, = \, e^{H_0 T} \; , \, \, H_0 > 0 \, \, .$$ So, using (131) and the aforementioned coordinate-radius constraint, $$r_0 \, = \, \frac{\omega_0}{H_0} \, \, ,$$ the equation (121) gets the much more tractable form $$\left[ \cos \tau \, \frac{d {\cal T}}{d \tau} \, + \,
\frac{\omega_0}{H_0} \sin \tau \right]^2 - \left( 1 +
\frac{\omega_0^2}{H_0^2} \right) = 0 \, ; \, \, \tau = \omega_0 t
\, , \, \, {\cal T} = \omega_0 T \, ;$$ whose [*time-orientation preserving*]{} positive branch, $$\cos \tau \, \frac{d {\cal T}}{d \tau} \, + \,
\frac{\omega_0}{H_0} \sin \tau \, = \, \sqrt{1 + \left(
\frac{\omega_0}{H_0} \right)^2} \, \, ,$$ does immediately lead to the de Sitter$-$anti-de Sitter [*synchronization*]{} law $${\cal T} - {\cal T}_0 = \frac{\omega_0}{H_0}
\ln | \cos \tau | + \sqrt{1 + \frac{\omega_0^2}{H_0^2}}
\ln \left| \frac{1 + {\rm tg} (\tau/2)}{1 - {\rm tg} (\tau/2)} \right|
\, \, ,$$ i.e., in physically dimensional quantities, $$T - T_0 = H_0^{-1} \left[
\ln | \cos (\omega_0 t) | +
\sqrt{ \left( \frac{H_0}{\omega_0} \right)^2 + 1} \,
\ln \left| \frac{1 + {\rm tg} \left( \frac{\omega_0 t}{2} \right)}
{1 - {\rm tg} \left( \frac{\omega_0 t}{2} \right)} \right| \right]$$ and subsequently, through (131) and the first cobordering equation (119), to the variation law of the de Sitter-radial coordinate, $$R(t) \, = \, H_0^{-1} | {\rm tg} (\omega_0 t) | \cdot
\left[ \frac{| \cos (\omega_0 t) |}{1 + \sin (\omega_0 t)}
\right]^{\sqrt{1 + (H_0/ \omega_0)^2}} \, \, .$$ As it can be noticed, using (135) written as $$\frac{ | \cos ( \omega_0 t) |^{\sqrt{1 + (H_0/ \omega_0)^2} - 1}}
{\left[ 1 + \sin ( \omega_0 t) \right]^
{\sqrt{ 1 + (H_0/ \omega_0)^2}}} \, = \, e^{- H_0 T} \, \, ,$$ where we have set (for convenience) $T_0 = 0$, the expression (136) reads $$R \, = \, H_0^{-1} | \sin (\omega_0 t) | e^{- H_0 T}$$ and asymptotically achieves the (130.b)-like behaviour, $$R(T) \cong H_0^{-1} \, e^{- H_0 T} \, \, ,$$ at late events (into the future), where, as $T \to \infty$, $| \sin ( \omega_0 t) | \to 1$.
However, in the general case, where $r_0^2 << 1$ is clearly invalidated, we could not find other exact solutions, in [*closed-form*]{}, besides the ones given above; eventually, a numerical study of the cobordering equations (121), (119) for power-like scale functions, $a \sim T^{\nu}$, with $\nu > 0$, such as the ones in the radiation dominated era, $\nu = 1/2$, or in the one of \`\`dusty”-matter, $\nu = 2/3$, even supplied with an accelerating cosmological term, might be quite important for it could point out some sort of bifurcations in the $(k =
0)-$cosmological evolution of [*large*]{} Higgs$-$ anti-de Sitter spactime bubbles. What else could be done in this respect, would be to look for the proper [*simultaneous*]{} embeddings of the two $S^{2}-$connected universes, so that to get a clear and \`\`very pictural” understanding of the resulting spacetime [*global*]{} structure; that can further be used as the [*unperturbed*]{} background in similar $-$ but seriously improved $-$ models with conventional matter-sources and [*dynamical*]{} \`\`remnant” field.
The second matter we would have liked to refer to had regarded the [*instanton*]{} companion (gotten by a Wick-rotation) of the Higgs$-$anti-de Sitter spacetime, which is precisely the \`\`never-started $-$ never-ending” $(k = 1)-$de Sitter Universe, $$d s^2 = \frac{1}{\omega_0^2} \, {\rm ch}^{2}(\omega_0 t)\,
d l_{S^3}^{2} \, - \, (d t)^2 \,\, , \;
\omega_0 = \mu^2 \, \sqrt{\frac{\kappa_0}{2 \lambda}} \, ,$$ in the (static source-field) [*fixed point*]{} case, and does beautifully turn into the Linde’s [*Inflationary*]{} Universe with quadratic driven-source, $\mu^2 \, \phi^2$, where $\phi$ is a genuine [*inflaton*]{}, when the spontaneous $Z_2-$invariance breaking resulting field gets excited.
[**Acknowledgement**]{}
The warm hospitality, enjoyable atmosphere and fertile environment of the Institute of Theoretical Science, University of Oregon are deeply acknowledged. Special thanks go to J. Isenberg and X. Tata for useful discussions and fruitful suggestions.
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[^1]: on leave of absence from Dept. of Theoretical Physics, \`\`Al. I. Cuza” University, Iaşi, 6600, Romania
|
---
abstract: 'Let $S$ be a cyclic $n$-ary sequence. We say that $S$ is a [*universal cycle*]{} ($(n,k)$-[*Ucycle*]{}) for $k$-subsets of $[n]$ if every such subset appears exactly once contiguously in $S$, and is a [*Ucycle packing*]{} if every such subset appears at most once. Few examples of Ucycles are known to exist, so the relaxation to packings merits investigation. A family $\{S_n\}$ of $(n,k)$-Ucycle packings for fixed $k$ is a [*near-Ucycle*]{} if the length of $S_n$ is $(1-o(1))\binom{n}{k}$. In this paper we prove that near-$(n,k)$-Ucycles exist for all $k$.'
author:
- |
Dawn Curtis[^1], Taylor Hines[^2], Glenn Hurlbert[^3], Tatiana Moyer[^4]\
Department of Mathematics and Statistics\
Arizona State University, Tempe, AZ 85287-1804
title: 'Near-Universal Cycles for Subsets Exist'
---
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\def$[[)]{}]{} $${{\Biggl[}}
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Introduction {#Intro}
============
A [*universal cycle (Ucycle) for $k$-subsets of $[n]$*]{}, denoted $(n,k)$-UCS, is a cyclic sequence of integers from $[n]=\{1,2,\ldots,n\}$ with the property that each $k$-subset of $[n]$ appears exactly once consecutively in the sequence. For example, $1234524135$ is a universal cycle for pairs of $[5]$. Chung, et al. [@CDG], defined universal cycles for a general class of combinatorial structures, generalizing both deBruijn sequences and Gray codes (see [@Sav]). The books [@Knu; @Rus] contain a wealth of information about generating such things efficiently.
A necessary condition for the existence of an $(n,k)$-UCS is that $n\mid\binom{n}{k}$. This is because symmetry demands that each symbol appears equally often. The authors of [@CDG] conjectured that the necessary condition is sufficient for large enough $n$ in terms of $k$ (some evidence [@Jac2] suggests $n\ge k+3$) and offered \$100 for its proof. [^5]
\[CDG\][@CDG] For all $k\ge 2$ there exists $n_0(k)$ such that for $n\ge n_0(k)$ Ucycles for $k$-subsets of $[n]$ exist if and only if $k$ divides $\binom{n}{k}$.
Progress on the conjecture has been slow. The $k=2$ case is trivial, corresponding to the existence of eulerian circuits in $K_n$ if and only if $n$ is odd. Jackson [@Jac1] proved the conjecture for $k=3$, and constructed ucyles for $k=4$ and odd $n$, leaving the case $n\equiv 2\mod 8$ unresolved. In [@Hur] we find the following result.
\[OldRes\][@Hur] Let $n_0(3)=8$, $n_0(4)=9$, and $n_0(6)=17$. Then $(n,k)$-UCs exist for $k=$ 3, 4, and 6 with $n \geq n_0(k)$ and gcd$(n,k)=1$.
Note that $n_0(k)=3k$ suffices for $k\in \{3,4,6\}$. It would be nice to lower $3k$ as much as possible. To this end, Stevens et al. [@Ste] proved the following.
\[stevens\][@Ste] No $(k+2,k)$-UC exists for $k\ge 2$.
Combined with particular computer examples found by Jackson [@Jac2] (e.g. $(n,k)=(10,4)$), this suggests that $n_0(k)=k+3$ may suffice.
In cases where no Ucycles have been found, or none exist, it is natural to look for cycles with as many distinct subsets as possible; that is, a [*Ucycle packing*]{}, such as the sequence $$S = 1345682 \ 4678135 \ 7812468 \ 2345713 \ 5678246 \ 8123571 \ 3456824 \ 6781357$$ for $(n,k)=(8,4)$. Note that no $(8,4)$-UCS exists, and that $S$ accounts for 56 of the possible 70 subsets. One might notice that these blocks shift upward by $3 \mod{8}$ from one to the next. This is important in the techniques that follow.
In their paper, Stevens, et al., show that the longest possible packing of a $(k+2,k)$-UCS has length $k+2$ and a packing achieving this bound always exists. Compared to the potential $\binom{k+2}{k}$ length, this shows that for $n=k+2$, we cannot get close to to a full universal cycle. To establish this notion formally, we define a [*near-Ucycle*]{} packing as a sequence of Ucycle packings, one for each $n$, such that as $n \rightarrow \infty$, asymptotically few $k$-subsets are omitted from the $(n,k)$-UCS packing. That is, the length of cycle $S_n$ is $(1-o(1))\binom{n}{k}$. For example, if $n$ is even and $M$ is any perfect matching in $K_n$ then $K_n-M$ is eulerian. In particular, any eulerian circuit is a near-(n,2)-UCS of length $(1-\frac{1}{n-1})\binom{n}{2}$. The purpose of this paper is to prove that near-Ucycles exist for all $k$.
\[Near\] For all $k$, near-$(n,k)$-Ucycles exist.
We proceed in the proof of this theorem by analyzing the construction of universal cycles. As we will show, we can create a Ucycle packing by selecting only those subsets that avoid certain structure. We prove that there are asymptotically few such structured subsets.
General Technique {#GenT}
=================
The general technique used to construct Ucycles originated from [@Jac1]. It consists of classifying the component subsets by their structure and ordering them accordingly. We write the $k$-subset $S$ of $[n]$ as $S = \{s_1, \dots, s_k\}$, with $s_i < s_{i+1}$, and define the [*form*]{} of $S$ as $F = (f_1, \dots, f_k)$ by $f_i = s_{i+1}-s_i$, where indices are modulo $k$ and arithmetic is modulo $n$. That is, the form of a set is the ordered collection of distances between set elements. (see Figure \[135form\])
![Form visualization of $\{1,3,5\}$[]{data-label="135form"}](135_form.ps){height="1.75in"}
Consider the following example. For $(n,k)=(8,3)$, the set $\{1,3,5\}$ has form is $(2,2,4)$. We consider the cyclic permutations of a form to be equivalent, so $(2,4,2)$ and $(4,2,2)$ can both serve as the form of $\{1,3,5\}$. The choice of form has to do with how the form appears in the cycle. For example, the form $F=(4,2,2)$ makes the sets $\{1,3,5\},\{2,4,6\},\dots,$ and $\{8,4,2\}$ appear as $513, 624,\dots,$ and $482$, respectively. Note that the last 2 in $F$ is not used for these sets, and so we may represent $F$ as $(4,2;2)$, or more simply $(4,2)$. In our techniques below, we will restrict our attention to forms $(f_1,\dots,f_{k-1}; f_k)$ having unique $f_k$. In fact, we choose $f_k$ to be the largest unique entry.
It is important to note that the method used here to construct the forms is somewhat limited. As part of our definition, every form has entries whose sum is $n$. We call such forms [*simple*]{}. However, it is possible to relax this condition. For example, the set $\{5,3,1\}$ would have the form $(6,6,4)$. If we allow the sum of form entries to be a multiple of $n$, we would have more freedom in representing the subsets, and hence a better method of constructing Ucycles could be developed. These forms we define as [*crossing*]{}. In this paper, we will only use simple forms.
The purpose of the forms is to model what occurs in a Ucycle, namely, that if $s_0s_1 \dots s_k$ appears in a Ucycle then the forms $(f_1,\dots,f_{k-1})$ and $(f_2,\dots,f_k)$ of the sets $\{s_0,\dots,s_{k-1}\}$ and $\{s_1,\dots,s_k\}$ overlap on $(f_2,\dots,f_{k-1})$. This motivates the following definitions.
For a given form rep. $(f_1, \dots, f_{k-1})$, we define $(f_1, \dots, f_{k-2})$ to be its [*prefix*]{}, and $(f_{2}, \dots, f_{k-1})$ to be its [*suffix*]{}.
The [*transition graph*]{}, denoted $\cT_{n,k}$, is a graph whose vertices are the prefixes (and suffixes) of the form representations of $(n,k)$-UCS. The directed edges are the form representations, drawn from prefix to suffix. In order for this construction to produce a Ucycle, it is necessary that this transition graph be Eulerian.
Consider the cases of $(n,k)=(8,3)$ and $(n,k)=(10,4)$, and the forms of each:
We use these forms to construct $\mathcal{T}_{8,3}$ and $\mathcal{T}_{10,4}$, as shown in Figure 2.
$$\begin{array}{ccc}
\includegraphics[height=1.5in]{T_83.ps}\label{T_83} &
\qquad&
\includegraphics[height=2in]{finished_graph.ps}\label{T_104_bad} \\
&&\\
\cT_{8,3}&&\cT_{10,4}\\
\end{array}$$
The condition of evenness is highlighted by these two examples. We know that a Ucycle is possible for $(n,k)=(8,3)$ but not for $(n,k)=(10,4)$. As shown in the following result, this is directly connected to the fact that $\cT_{8,3}$ is Eulerian and $\cT_{10,4}$ is not.
\[OldLem1\][@Hur] If $\cT_{n,k}$ is Eulerian for some choice of form representations, then an $(n,k)$-UCS exists.
As we can see in $\cT_{10,4}$, the vertex $44$ has no out degree. This is not necessarily the case. If we write the form $(1,1,4;4)$ as $(4,4,1;1)$, then we would instead connect $44 \rightarrow 41$. Since this form has no unique entry, this is left ambiguous. We call such forms *bad*. We define all forms with at least one unique element, a clear representative, as [*good*]{}.
Consider the previous example. We can ignore all bad forms of $(n,k)=(10,4)$, and construct the transition graph, as shown in Figure 2.
![Good $\cT_{10,4}$[]{data-label="T104"}](finished_graph_good.ps){height="2in"}
In this case, ignoring the bad forms yields an even transition graph. As stated in the following result, this is true for any connected transition graph. Note, however, that it is not Eulerian since $((33))$ is not connected. This problem disappears if we also ignore all such isolated cycles, as stated in the following result. As we will show, the number of such isolated cycles is negligible as $n \rightarrow \infty$.
\[OldLem2\][@Hur] If $\cH_{n,k}$ is connected and there are no bad forms for $k$-subsets of $[n]$, then $\cT_{n,k}$ is Eulerian.
We can construct a Ucycle packing by ignoring such bad forms for any $(n,k)$ pair and restricting our attention to the largest component of $\cT_{n,k}$, the component containing $((1 \dots 1))$.
Main Result {#Main}
===========
In order to prove that a near Ucycle packing is possible for any $k$, we must show that an asymptotically large proportion of sets can be included in a Ucycle. As we will show, the [*good sets*]{}, those belonging to good forms, can easily be included in a Ucycle packing, while the remaining [*bad sets*]{} cannot. As we have indicated, we will only include good sets. Of course, not all good sets can be included. It therefore remains to show that the good sets which can be included asymptotically outnumber all other sets, and that a Ucycle packing that includes each of these sets can be created.
Counting Good Sets {#CountGS}
------------------
Consider the example of the $(8,4)$-UCS. We know that each form must have four entries, and, since each form is simple, the sum of these entries must be eight. The number of times each entry appears is also important, since the bad forms will have no entry appearing only once. Therefore, we do not need all of its entries in order to determine whether or not a form is bad. We need only to know how many times each entry appears. For example, the form $(1,2,2;3)$ has the unordered [*pattern*]{} $\la1,2,1\ra = \la2,1,1\ra$, essentially the list of multiplicities. [^6] Since this pattern contains a 1 as an entry, it has a unique element and therefore all forms with this [*good pattern*]{} are good. In this case, the pattern entries define a partition of $4$, since each form has $4$ entries. In general, every pattern of an $(n,k)$-UCS is a partition of $k$.
Consider a [*bad pattern*]{} $P = \la p_1, \dots, p_t \ra$. Clearly, $p_i \geq 2$ for all $1 \leq i \leq t$, so there exists a corresponding good pattern $P^{\pr} = \la p_1,\dots,p_t-1,1\ra$. We define the function $\phi: \G \rightarrow \b$ from the set of of good patterns to the set of bad patterns as $\p(P) = P^{\pr}$. Since this map applies to any bad pattern, we can see that $|\b| \leq |\G|$. [^7]
As we can see, many forms may belong to the same pattern. In fact, the forms belonging to a particular pattern satisfy the equation $\sum_{i=1}^t p_ix_i = n$, for some positive integers $x_1,\dots,x_t$, where $\la p_1,p_2,\dots,p_t\ra$ are the pattern entries. However, this equation imposes no order on the solution, and thus does not distinguish between two different forms that share the same entries. We say that two such forms belong to the same [*class*]{}.
For example, the forms $(1,2,2;3)$, $(2,1,2;3)$, and $(2,2,1;3)$ both share the class $[1,2,2;3]$. By convention, we order the entries of a class from smallest to largest entry.
As we know, the classes belonging to a pattern $P$ are all representations of $n$ as a positive distinct integer linear combination of $p_1, \dots, p_t$. We count the classes with the aid of Shur’s theorem. For $P=\la p_1,\dots,p_t\ra$, define the function $$\psi(P)=\left| \left\{ (x_1,\dots,x_t) \mid \sum_{j=1}^{t} p_j x_j = n,
x_j \geq 0 \right\} \right|\ .$$
\[Schur\][@Wilf] Suppose that $gcd(p_1,\dots, p_t) = 1$, and define $k=\sum_{j=1}^{t} p_j$ and $Q = \prod_{j=1}^{t} p_j$. Then $$\psi(P)\ \sim\ \frac{n^{t-1}}{(t-1)! Q}\ .$$
Counting the classes requires a slightly more general result. Namely, we want to extend Schur’s theorem as follows. We define $$\psi^\prime(P)=\left| \left\{ (x_1,\dots,x_t) \mid\ \sum_{j=1}^{t} p_j x_j = n,
\ x_j \geq 1, \ x_j \neq x_i \ \forall i \neq j \right\} \right|\ .$$
\[GenSchur\] Define $g=gcd(p_1,\dots, p_t)$, $k=\sum_{j=1}^{t} p_j$, and $Q =\prod_{j=1}^{t} p_j$. Then $$\psi^\prime(P)\
\sim\ \frac{(n-k)^{t-1}}{(t-1)! Qg^{t-1}} \ -
\ \frac{(n-k)^{t-2}}{(t-2)!Qg^{t-2}} \sum_{i < j}\frac{p_i p_j}{p_i+p_j}\ .$$
Since all form entries are positive integers, we count only integer solutions to $\sum_{j=1}^{t} p_j x_j = n$ such that $x_j \geq 1$. This is equivalent to finding all integer solutions to the equation $\sum_{j=1}^{t} p_j (x_j+1)$ $= \sum_{j=1}^{t} p_j x_j + k$ $= n$, without the positivity constraint.
We further modify the system to act for the condition $gcd(p_1,\dots, p_t) = g$. The equation $\sum_{j=1}^{t} \frac{p_j}{g} x_j$ $= \sum_{j=1}^{t} q_j x_j$ $ = \frac{n-k}{g}$ is the same as our original equation, and clearly $gcd(q_1,\dots, q_t) = 1$.
Finally, we need to account for the fact that for a pattern of size $t$, each corresponding class has exactly $t$ distinct entries. If, for example, $P = \la 2,1,1\ra$, then $C = [1,1,3,5]$ is one possible class. However, $[3,3,3,1]$, although it is a valid solution to $\sum_{i=1}^{3}p_ix_i = n$, does not fulfill the requirement that $x_i \neq x_j$ for all $i \neq j$. This is easily seen if we adopt the notation $C = [x_1^{p_1}, \dots, x_t^{p_t}]$. Written in this way, $[1,1,3,5] = [1^2 3^1 5^1]$ and $[3,3,3,1] = [3^2 3^1 1^1]$ In the latter case, $x_1 = x_2$, so the class $[3^2 3^1 1^1]$ is not valid. Therefore, we must account the number of such non-distinct solutions. If $x_i = x_k$, then the equation reduces to $(p_i+p_k)x_i + \sum_{j=1, j\neq i, j\neq k}^{t} p_j x_j = n$. We can then apply Schur’s theorem to this equation as we did before. For the new system, $Q^{\pr}$ $= (p_i+p_k)\prod_{j=1,j\neq i, j\neq k}^{t} p_j$ $= Q \frac{p_i + p_j}{p_i p_j}$, so the total number of non-distinct solutions to this equation is $\sim \frac{(n-k)^{t-2}}{(t-2)!Qg^{t-2}} \sum_{i < j}\frac{p_i p_j}{p_i+p_j}$
In order to count the classes, we now apply Lemma 8 to each pattern. We want to show that as $n$ gets large, a given bad pattern $P$ has far fewer classes than its good component $\phi(P)$. First, we must determine the size of $\sum_{i < j}\frac{p_i p_j}{p_i+p_j}$ for any pattern $P$.
For any $p_i$, we know that $\frac{p_i p_j}{p_i+p_j}$$= \frac{p_i(c-p_i)}{c}$ for some $c < k$, which is maximized when $p_i = c/2$. If $p_i = c/2$, then $\sum_{j=1}^t p_i$$= \frac{tc}{2}$$ = k $. Thus $c = \frac{2k}{t}$. Since each $p_i = c/2$, we maximize the value of this expression when $p_i = k/t$. Thus, $\sum_{i < j}\frac{p_i p_j}{p_i+p_j}$$\leq \binom{t}{2} \frac{(k/t)^2}{2k/t}$$\sim \frac{t^2}{2}\frac{k}{2t}$ $ = \frac{kt}{4} \leq k^2/4$.
Using this upper bound, we calculate that the number of classes belonging to a pattern $P$ is
$$c(P) \sim \displaystyle\frac{(n-k)^{t-1}}{(t-1)! Q g^{t-1}} - \displaystyle\frac{(n-k)^{t-2}}{(t-2)! Q g^{t-2}} \frac{k^2}{4} \sim \displaystyle\frac{(n-k)^{t-1}}{(t-1)! Q g^{t-1}} .$$
With this application of Schur’s theorem, we can count the number of classes belonging to good patterns compared to the number of classes belonging to bad patterns as follows. $$\frac{c(P)}{c(\phi(P))} \sim
\displaystyle\frac{(n-k)^{t-1}}{(t-1)! g^{t-1} \prod_{j=1}^{t} p_j}\displaystyle\frac{t! \prod_{j=1}^{t-1} p_j (p_t - 1)}{(n-k)^{t}}
\sim \frac{t(p_t-1)}{p_t n} \rightarrow 0$$ That is, the good classes asymptotically outnumber the bad classes.
It still remains to show that the number of forms of a bad class $C$ belonging to a bad pattern $P$ is no greater than the number of forms of a good class $C$ of $\phi(P)$. As we know, the forms of a class are all permutations of the class entries modulo cyclic rotation. Therefore, each class $C \in P$ has $\frac{(k-1)!}{\prod_{j=1}^{t}p_j!}$ forms, while classes $C \in \phi(P)$ has $ \frac{k!}{\prod_{j=1}^{t-1}p_j! (p_t-1)}$ forms. That is, the good forms outnumber the bad forms by a factor of $kp_t$.
Finally, it remains to count the sets. By definition, all good forms have at least one unique element. Thus the good sets have at least one unique difference between two elements. This implies that every good form contains exactly $n$ sets. [^8] Therefore, the number of sets per good form is at least the number of sets per bad form.
By counting the patterns, classes, forms and sets, one can see that almost every set is good. However, it remains to be shown that asymptotically many of these good sets can be arranged into a Ucycle packing. We will use the following lemmas to prove that this is true, and therefore a near Ucycle packing is always possible.
\[even\] If $\cT_{n,k}$ is restricted to the good classes then it is a union of cycles, and hence even.
For a given class $C$ and transition graph $\cT_{n,k}$, we define the graph $\cT_{n,k}(C)$ to be the restriction of $\cT_{n,k}$ to the edges belonging to the forms of $C$.
If the class $C$ is good, then $\cT_{n,k}(C)$ is a cycle or union of cycles. Each form of $C$ has a unique representative $c_k$, thus for any permutation $F$ of $\{c_1, \dots, c_{k-1}\}$, all cyclic permutations of $F$ are also forms of $C$. Since $\cT_{n,k}(C)$ is a union of cycles for each good $C$, it is clear that $\cT_{n,k}$ restricted to the good classes will be a union of Eulerian subgraphs, and is therefore even.
If the restriction of $\cT_{n,k}$ to good classes is connected, then it is Eulerian.
Since the restriction of $\cT_{n,k}$ to good classes is a union of cycles, it is easily seen that if $\cT_{n,k}$ is connected, it is surely Eulerian.
By the result in [@Hur], if $\cT_{n,k}$ is Eulerian, then an $(n,k)$-UCS exists. Therefore, if we can show that the restriction of $\cT_{n,k}$ to good classes is connected, then it follows that a Ucycle packing exists that includes all good sets. However, Fig. 2 demonstrates that this is not always the case. Instead, we prove that an asymptotically large component is connected. Since we have proven that $(1-o(1))\binom{n}{k}$ sets are good, a Ucycle packing that includes all good sets is a near-packing.
Finding a Large Component {#LargeComp}
-------------------------
In order to study the components of the restriction of $\cT_{n,k}$ to good classes, we define the [*class graph*]{}, denoted $\cH_{n,k}$, as the undirected graph whose vertices are all classes of $(n,k)$-UCS . An edge is drawn between the class representations that differ by only one entry. For example, $[1,2,2;5]$ and $[1,1,2;6]$ are connected in $\cH_{10,4}$.
If $C_1$ and $C_2$ are connected in $\cH_{n,k}$, then this means that $\cT_{n,k}(C_1)$ and $\cT_{n,k}(C_2)$ will also share vertices. For example, $[1,2,2;5]$ and $[1,1,2;6]$ are connected in $\cH_{10,4}$, and correspond to the cycles $$((12)) \ \rightarrow \ ((22)) \ \rightarrow \ ((21)) \ \rightarrow \ ((12))$$ and $$((11)) \ \rightarrow \ ((12)) \ \rightarrow \ ((21)) \ \rightarrow \ ((11)) \ .$$
Just because $C_1$ and $C_2$ are connected in $\cT{n,k}$, this does not guarantee that the union of $\cT_{n,k}(C_1)$ and $\cT_{n,k}(C_2)$ will be connected. It could happen, for example, that each $C_i$ has two components that connect, resulting in two components for $C_1 \cup C_2$. However, as proven in [@Hur], if $\cH_{n,k}$ is connected, then the union over all classes produces a connected $\cT_{n,k}$. We will clarify this with the map $\kappa$, defined as follows.
Let $C = [c_1^{p_1}, \dots, c_{t-1}^{p_{t-1}}; c_t]$ be a class, where the entry $c_i$ appears $p_i$ times, and $c_t$ is the largest singleton. To connect the class good classes, we define the map $\k:c(\Gamma)\rightarrow c(\Gamma)$ by $[c_1^{p_1}, \dots, c_{t-1}^{p_{t-1}}; c_t] \rightarrow [1,c_1^{p_1},\dots,c_{t-1}^{p_{t-1}-1};c_t+c_{t-1}-1]$ We then apply $\k$ again, each time adding another 1.
For example, $\k$ connects the class $[2,2,2;4]$ to the class $[1,1,1;7]$ of $\cH_{10,4}$ by the path $$[2,2,2;4] \ \rightarrow \ [1,2,2;5] \ \rightarrow \ [1,1,2;6] \ \rightarrow \ [1,1,1;7] \ .$$
In this way, we are able to connect the majority of the classes to the class $[1,\dots,1;k-t+1]$. However, using $\k$ does not work for every good class. For example, the class $[3,3,3;1]$ is surely good. However, $\k$ maps $[3,3,3;1]$ to $[1,3,3;3]$; that is, to itself. In general, if $c_t$ is the largest singleton of a class $C$, then $c_t+c_{t-1}-1$ will be the largest singleton of $\k(C)$ only if $c_t > 1$.
Counting Awesome Sets {#CountAS}
---------------------
In order to circumvent the problem introduced above, we restrict our attention to the [*awesome classes*]{}, the good classes whose largest singleton is greater than 1. Using Schur’s theorem, we can show that the number of classes that are not awesome is negligible, as shown below.
$$\left| \left\{ (x_1,\dots,x_t) \mid \sum_{j=1}^{t}p_jx_j=n, \ 1=x_1=p_1<p_2<\dots<p_t, \ x_j>1 \right\} \right|$$ $$\begin{aligned}
&=& \left| \left\{ (x_2,\dots,x_t) \mid \sum_{j=2}^{t}p_jx_j=n-1, \ 1<p_2<\dots<p_t, \ x_j>1 \right\} \right|\\
&=& \left| \left\{ (y_2,\dots,y_t) \mid \sum_{j=2}^{t}p_jy_j=n-2k-1, \ 1<p_2<\dots<p_t, \ y_j\geq0 \right\} \right|\\
&\sim& \frac{(n-2k-1)^{t-2}}{(t-2)!g^{t-2}Q}\\
&\ll& n^{t-1}\end{aligned}$$ for $n > 2k$.
Since the number of non-awesome classes is negligible compared to the number of total classes, we can disregard them and restrict our attention to the awesome classes, which still comprise an asymptotically large proportion of all subsets.
Proof of Theorem \[Near\] {#Proof}
-------------------------
Since almost all classes are awesome, we know that each awesome class is connected to $[1,1,\dots,1;n-k+1]$ in $\cH_{n,k}$. Since the awesome classes of $\cH_{n,k}$ are connected, the restriction of $\cT_{n,k}$ to the awesome classes is also connected. If the restriction of $\cT_{n,k}$ to awesome classes is connected, then by lemma \[even\], we know that all [*awesome sets*]{}, the sets belonging to awesome classes, can be connected to form a Ucycle packing. Since, as shown above, the awesome sets represent an asymptotically large proportion of the total sets, it follows that this Ucycle packing is a near Ucycle packing.
Remarks {#Remarks}
=======
Many of the techniques presented here can be extended to other forms of Ucycle approximations. For example, it is possible to include any set in a Ucycle packing by simply inserting the set elements anywhere in the cycle. We could therefore construct a [*Ucycle covering*]{} by simply appending all non-awesome sets onto a near Ucycle created using the techniques we have described. However, this is very inefficient because it increases the length of the cycle by $k$ for each added set instead of the desired 1. In order to find a more elegant method of constructing Ucycle coverings, more complicated analysis is required. As stated earlier, each bad form may only produce a factor of $n$ sets. Therefore, the method used to connect awesome sets into the Ucycle will not work, since traversing an Eulerian transition graph $n$ times is impossible for many bad forms.
One possible method of connecting bad sets in a Ucycle is to consider multiple form classes. Currently, for a form $F = (f_1,\dots,f_k)$, we require $\sum_{i=1}^{k}f_i = n$. However, it could be useful to consider $\sum_{i=1}^{k}f_i = \a n$ for $\a > 1$. This would allow much more freedom in representing sets, and therefore more ways of connecting sets.
Finally, due to our proof that near Ucycles exist, we believe that we deserve asymptotically much of the prize money, or $\$[1-o(1)](250.04)$. Since we do not know the speed of the $o(1)$ term, we have made a conservative estimate of $\$249.99$.
[99]{}
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B. Jackson, [*Universal cycles of $k$-subsets and $k$-permutations*]{}, Discrete Math. [**117**]{} (1993), 141–150.
B. Jackson, personal communication (2004).
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B. Stevens, P. Buskell, P. Ecimovic, C. Ivanescu, A. Malik, A. Savu, T. Vassilev, H. Verrall, B. Yang, and Z. Zhao, [*Solution of an outstanding conjecture: the nonexistence of universal cycles with $k=n-2$*]{}, Discrete Math. [**258**]{} (2002), 193–204.
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[^1]: dawn.curtis@asu.edu
[^2]: taylor.hines@asu.edu
[^3]: hurlbert@asu.edu
[^4]: tatiana.moyer@asu.edu
[^5]: At the 2004 Banff Workshop on Generalizations of de Bruijn Cycles and Gray Codes it was suggested by the second author that a modest inflationary rate should revalue the prize near \$250.04.
[^6]: We can also denote the form $(1,2,2,3)$ as $(3^1 2^2 1^1)$, a notation we will find useful later
[^7]: Note that the number of good partitions of $k$ is equal to the number of partitions of $k-1$, denoted $p(k-1)$. Since $p(k)\sim e^{\pi\sqrt{2k/3}}/4\sqrt{3}k\sim p(k-1)$, almost all patterns are good.
[^8]: The bad forms do not have a unique element, so this is not always the case for bad sets. Instead, we only know that the number of sets contained in a bad form is a factor of $n$. (Symmetry can reduce the number of sets; e.g. $\la 3 \ra$ has forms $(1,4,7)$, $(2,5,8)$, and $(3,6,9)$ when $n=9$.)
|
---
abstract: |
The cluster formation in Three Dimensional Wireless Sensor Networks (3D-WSN) give rise to overlapping of signals due to spherical sensing range which leads to information redundancy in the network. To address this problem, we develop a sensing algorithm for 3D-WSN based on dodecahedron topology which we call Three Dimensional Distributed Clustering (3D-DC) algorithm. Using 3D-DC algorithm in 3D-WSN, accurate information extraction appears to be a major challenge due to the environmental noise where a Cluster Head (CH) node gathers and estimates information in each dodecahedron cluster. Hence, to extract precise information in each dodecahedron cluster, we propose Three Dimensional Information Estimation (3D-IE) algorithm. Moreover, Node deployment strategy also plays an important factor to maximize information accuracy in 3D-WSN. In most cases, sensor nodes are deployed deterministically or randomly. But both the deployment scenario are not aware of where to exactly place the sensor nodes to extract more information in terms of *accuracy*. Therefore, placing nodes in its appropriate positions in 3D-WSN is a challenging task. We propose a Three Dimensional Node Placement (3D-NP) algorithm which can find the possible nodes and their deployment strategy to maximize information accuracy in the network. We perform simulations using MATLAB to validate the 3D-DC, 3D-IE and 3D-NP, algorithms respectively.\
Keywords: *Spatial correlation, information estimation, node placement, three dimensional sensor networks*
author:
- 'Jyotirmoy Karjee and H.S Jamadagni$\dag$ [^1] [^2] [^3]'
title: Information Estimation with Node Placement Strategy in 3D Wireless Sensor Networks
---
Introduction
============
Wireless Sensor Networks (WSN) [@jk1] have made sufficiently great attention in the field of signal processing and wireless communications. Sensor nodes are made up of MEMS [@jk2], [@jk3] and smaller in size having low processing speed, generally used for sensing and extracting data in a network. WSN are used in military applications, under water [@jk4], environmental monitoring [@jk5], health care applications and many more. Sensor nodes are capable to measure an event features (like temperature, pressure, humidity, etc.). In WSN, a group of sensor nodes perform similar sensing task of an event features (like measuring moisture content of agricultural field, reading temperature of an indoor room, measuring seismic event, detecting event target, detecting fire in a forest etc). An event is defined as a physical occurrence which is measured by groups of sensor nodes in a network. Sensor nodes continuously measure the physical phenomenon of an event and report measured information to the sink node.
Sensor nodes does collaborative sensing task in two dimensional [@jk6]-[@jk11] networks, where nodes are deployed in agricultural field, forest, etc. However, WSN can also be applied in three dimensional [@jk15]-[@jk17] space by measuring movement and behavior of birds or insects and fixing sensor nodes with probes in underwater [@jk4]. The information collected by sensor nodes deployed in three dimensional space are generally spatially correlated within a network. Sensor node uses correlated information to form clusters. In a 3D clusters based sensing model [@jk15], the sensing range of a cluster is considered as sphere. Since the sensing range of clusters are spherical, it creates tessellate (overlapping or gapping) of sensing coverage as shown in Figure 1. In this figure, three clusters are given namely $A$, $B$, $C$ respectively. Each cluster have its own neighboring nodes within its spherical sensing range. As shown in figure, cluster A and B respectively overlaps in its spherical range which leads to increase the bandwidth utilization in 3D network. Moreover, in between cluster’s $A$, $B$ and $C$ respectively, there is a gap (tessellate) among spherical sensing range which leads to poor sensing coverage in 3D networks.
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The assessment of information is a major issue in 3D-WSN as sensor nodes have a limitation of having low processing power and have sort communication range. Based upon the statistical knowledge of data [@jk14], sensor node extract data (with prior knowledge or without prior knowledge of data statistics) from the environment, process it and finally transmits to the Base Station (BS). While extracting data, in some cases sensor networks assume that they have prior knowledge of data statistics (like variance and covariance of sensed data) in the network. But most often, sensor nodes extract data in a regular interval of time without having prior knowledge of data statistics. Thus, sensor nodes extract statistical information from the environment under two scenarios. Firstly, sensor nodes extract information within a network under priori knowledge of statistical information (e.g variance and covariance) of the environment. For example, sensor networks have prior knowledge of variance and covariance of temperature of an indoor room. In WSN, extracting information without having prior knowledge of information statistics is a challenging task in three dimensional space. Hence, in this paper, we focus on extracting information in 3D space without having prior knowledge of signal statistics. Here, sensor nodes observe and sense information dynamically and transmit it to CH node or base station.
Real time sensor data is monitored continuously over temporal and spatial [@jk6] domains, respectively. The information extracted by sensor nodes are generally spatially correlated within a network. As the node density increases, the spatially proximal [@jk12], [@jk13] observation of correlated information among the nodes also increases in 3D-WSN. Since, spatial information received at the sink node are highly correlated, it increases information redundancy in the network. Hence, it is required to minimized information redundancy in WSNs. Moreover, in each cluster, cluster Head (CH) node calculates a distortion factor [@jk6]-[@jk11] (information accuracy) in 3D space. Distortion factor or information accuracy [@jk8] is defined as the degree of closeness of measured data to its actual sensed data in a network. Data accuracy [@jk6]-[@jk11] models are developed under spatial data correlation. These models calculates a minimum set of sensor nodes which are sufficient to give the desired data accuracy level as achieved by the whole network. Data accuracy can be model under online [@jk14] data extraction to select optimal sensor nodes in the network. Generally, information accuracy is maximized to get accurate information but leads to increase in energy consumption in the network. To balance these, a trade off [@jk10] between information accuracy and energy consumption is built to increase the lifetime of sensor networks. It also selects an optimal sensor nodes, thereby reducing the communication overhead in the network.
In Figure 2, a 3D space is considered having three coordinates where we are interested to measure the information profile of each sensor node. In this 3D space lets assuming ten sensor nodes are deployed randomly. Out of ten sensor nodes, lets say six sensor nodes have reached the desired maximum information level. Hence node number: $1$, $3$, $4$, $7$, $8$ and $10$ are considered in the network to maximize the information accuracy. The sensor nodes with bellow of that desired information level may have lower information profile. In reality, it may be possible that node id: $1$, $3$, $4$, $7$, $8$ and $10$ are directly put under the sunlight, due to which the signal variance goes above the desired information accuracy level, where as node id: $2$, $5$, $6$ and $9$ are placed under the shade of a tree. This may be the reason due to which the signal variance of the sensor nodes are bellow that desired information accuracy level of that specific location. Hence, it may be recommended that nodes with lower information profile in that specific location are added with extra deployed sensor nodes so that a desirable information accuracy can be extracted on that specific location.
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To extract accurate information, nodes deployment in 3D-space plays a crucial role in wireless sensor networks. Sensor nodes can be deployed in two ways: (a) It can be deterministically deployed by manually putting it in the sensing field by human effort. But these type of nodes deployment is inefficient in terms of time and cost. (b) It can be randomly deployed in the sensing field by aerial or human effort. But both the procedure, we are not aware of extracting maximum information in a specific location due to exact placement of sensor nodes in 3D wireless sensor networks. Therefore, placement of sensor nodes plays a vital role in terms of maximizing the information accuracy with node placement strategy in wireless sensor networks. In this paper, we consider the application of environmental monitoring to measure temperature data in 3D-WSN. The paper is organized as follows: In section II, we discuss the related works. In section III, we have addressed the problem definitions of our work. In section IV, we describe the mathematical models of 3D-DC, 3D-IE and 3D-NP algorithms respectively. Validations and simulations of the propose work are presented in Section V. Finally, we conclude our work in Section VI.
Related Work
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In [@jkA27], author’s proposed the confident information coverage model where sensor nodes are deployed such that they cover the entire geographical area in wireless sensor networks. In [@jkA28], author’s proposed an optimal node placement patterns based on confident information coverage (CIC) considering the collaboration of sensor information and spatial correlation of physical data. In [@jkA29], author’s investigate optimal node placement for long belt coverage in wireless sensor networks. Moreover, in [@jkA30]-[@jkA32], authors discussed the information coverage problem implemented in two dimensional wireless sensor networks, but none of the above works illustrate the information coverage problem in three dimensional networks, which is still a open challenge in communication networks. The above literatures, also haven’t addressed the information coverage problem, in three dimensional distributed scenario. However, many clustering algorithms [@jk10]-[@jk13] are proposed for WSN in two dimensional space. But none of the work addresses the formation of clusters using jointly sensing nodes in 3D space. In [@jk15], a 3D spherical based sensing model is proposed, but the model creates tessellate (overlapping or gapping) in between the spherical ranges which utilizes more bandwidth in the network. However, in [@jk16] authors addressed the problem to model the sensing coverage by tessellating polyhedra but lags the formation of cluster in 3D-WSN. Moreover, various data estimation models [@jk6]-[@jk11], [@jk15]-[@jk17], are developed with *a-prior* knowledge of data statistics in two dimensional space and three dimensional space, but none of the works focus on estimating information without having *a-priori* knowledge of information statistics in 3D-WSN. In [@jk6]-[@jk11], data accuracy models are developed under spatial data correlation. These models calculates a minimum set of sensor nodes which are sufficient to give the desired data accuracy level as achieved by the whole network. Similarly, in [@jk14], [@jk1411] data accuracy model is developed under online data extraction to select optimal sensor nodes in the network. In [@jk10], a trade off between data accuracy and energy consumption is developed to select an optimal number of sensor nodes, thereby reducing the communication overhead in the network. But none of the works have determine which sensor nodes are to be selected for achieving maximum information accuracy in the network. In the above literature, sensor nodes are either deterministically or randomly deployed in the sensing field to find an optimal set of sensor nodes maintaining a desired information accuracy level. Therefore, It also doesn’t explore which sensor nodes are to be selected. Moreover, it doesn’t clarifies where exactly is to be place sensor nodes in 3D-space to get maximum information accuracy.
Problem Formulations with Specific Use-case
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In this section, we emphasize to formulate our problem definitions as follows.
*1. Formation of distributed clusters using dodecahedran topology for better sensing coverage is a subject of interest in 3D-WSN*: In 3D-WSN, sensing coverage of clusters are generally modelled as spherical topology. But spherical range of clusters form tessellate (overlapping or gaping) space within clusters. This lead to poor sensing coverage (due to space among spherical range of cluster) or more bandwidth utilization (due to overlapping of spherical range of clusters) in 3D-WSN. Moreover deployment of more nodes in the sensing range creates unnecessary data redundancy in 3D-WSN due to spatially correlated data. To overcome this problem, we develop Three Dimensional Distributed Clustering (3D-DC) algorithm using dodecahedron topology as shown in Figure 3. In this figure, three clusters namely $D$, $E$ and $F$ are shown with dodecahedron topology. Each dodecahedron clusters has its own neighboring sensor nodes within its range. As shown in figure, there may not have tessellate between the clusters which reduce the information redundancy in 3D networks.
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*2. Information estimation without having priori knowledge of signal statistics (i.e online information estimation) is a challenging task in 3D-WSN:* Sensor nodes form distributed clusters in two dimensional space and three dimensional space using spatially correlated data among sensor nodes. Data correlation among sensor nodes have both priori and without priori information statistics (i.e known and unknown variance or covariance) in the network. Using the knowledge of information statistics, information estimation models are developed in two dimensional WSN. Moreover, information estimation with prior knowledge of signal statistics is implemented in 3D-WSN but non of the work emphasize on estimating information without priori knowledge of signal statistics in 3D-WSN. Hence, information estimation in 3D-WSN is a challenging task (e.g estimating accurate information in under-water).
*3. To extract maximum information in a sensor network, maximum nodes deployment is necessary for some specific sensing region where sensor signal penetration may be difficult*: As discussed in the previous section, sensor nodes can be deployed deterministically and randomly in a sensing field to extract information in a network. But both the deployment scenarios fails to extract accurate information in a network as we are not aware where to place sensor nodes to get more information. To clarify this, lets assume sensor nodes deployment scenario in a forest to sense temperature in a specific sensing region like 3D-space. Throughout the paper, we assume deployment of sensor nodes in a 3D-space to measure temperature as an example. As nodes are deployed randomly, it may be possible that at some sensing region (3D-space), placing more sensor nodes are required to maximize the information. Generally, sensor signal penetration is less in dense forest (tree dominated) region in 3D space. In such region, more number of sensor nodes can be deployed to extract more information. But it is unclear as what exactly the upper bound of sensor nodes to be deployed in such a region to extract adequate information. Here, optimal node placement plays an important role to maximize information accuracy where sensor signal penetration is difficult.
*4. Sensor nodes are placed closer to the target event to maximize the information accuracy* : We take another scenario, where we deploy sensor nodes closer to the event [@jk6] target to maximize information accuracy in 3D network. Placing sensor nodes closer to the target event may extract more information than placing sensor node far apart from the event. But it doesn’t give any clear idea at what minimum distance, nodes are to be deployed with respect to the target event and by which topology nodes can be deployed to extract more information in a network. It may be possible that placing sensor nodes closer to target event may cause more noise distortion [@jk8] due to signal interference. In this case, a deployment topology in 3D space can be developed by which sensor nodes can be placed with minimum distance with respect to the target event to extract information without noise distortion.
*5. We give emphasize on temporal data collection as there may be variation of data collected depending upon the node placement in 3D space*: Generally, data collected in a sensor networks have spatio-temporal correlation among them. Some times, it is more important to give emphasise on temporal information than spatial information for node deployment strategy in 3D space. Considering a network scenario in a forest, where a set of sensor nodes are deployed under the shade of dense trees and another set of nodes are placed directly under sunlight. Assuming that both the set of nodes are placed close to each other in the sensing region. Since both set of sensor nodes are closer and collects an event feature (i.e temperature), there is a difference in variation of temporal data collected by them as one set of nodes is placed under shade of tree and other placed directly under sunlight. In this case, we deploy sensor node such that more nodes are to be placed under shade of a tree due to week sensor signal where as less number of sensor nodes are placed directly under sunlight. Still, it is unclear that what is the maximum number (upper bound) of nodes to be placed under shade of a tree and minimum number (lower bound) of nodes to be deployed under sunlight to get information accuracy in a network. Thus optimal deployment of nodes is necessary for this scenario.
*6. Consideration may be given to spatial data correlation among sensor nodes to determine minimum number of sensor nodes required to maximize information accuracy*: We consider a network scenario, where a set of sensor nodes are close to each other and directly collects data from sunlight in a sensing region. In this case, sensor nodes have almost similar signal variance and therefore, consideration is given to spatial data correlation among sensor nodes in a network. We make use of spatial data correlation among sensor nodes to determine optimal sensor nodes in a network with maximum information accuracy.
Mathematical Model
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In this section, we develop the following mathematical basis of our models: firstly, we develop a three dimensional distributed clustering (3D-DC) algorithm in 3D space. Secondly, with in each cluster, we develop three dimensional information estimation (3D-IE) model at CH node of the cluster. 3D-DC and 3D-IE algorithms are the extension of work presented in [@jk15]. Finally, three dimentional node placement (3D-NP) algorithm is developed for the network which maintains a desired information accuracy.
Cluster Formation in Three Dimensional Space
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Considering a static source event which propagates its signal spherically in 3D space [@jk15]. We ignore the attenuation or propagation delay of signal from the source event. Assuming that the nodes which are within the spherical signal range can only sense the source event. The information extracted by sensor nodes for the source event can be modelled as spatial data correlation using exponential model [@jk18], [@jk19] given by $C_{exp}(d)= exp^{\left(-\frac{|d|^{\alpha}}{\theta} \right)} $ where $\alpha =1$, $\alpha$ is the smoothness parameter which gives the geometric attributes of the propagated signals, $\theta$ is the range parameter which determines the delay of signal w.r.t to distance and $d$ is the distance among the source event to the sensor nodes in 3D space. Using exponential model, we develop the 3D-DC algorithm as follows: Data correlation between source event to number of sensor nodes in 3D space; Data correlation model among sensor nodes; Formation of distributed clusters in 3D space. The 3D-DC is explained in Algorithm 1.
**Require:** A source event $S$ occurred and sensor nodes $i$ and $j$ are randomly deployed in a three dimensional space.
**Returns:** Formation of clusters in three dimensional space.
Start
Compute information correlation among source event $\emph{s}$ and sensor node *i*
Define a threshold $\tau_{e}$ to verify whether the information is strongly correlated among $s$ and $s_i$ or weakly correlated
If the information correlation is strong, then calculate euclidean distance $d_{s,i}$ among event $s$ source to $i$th sensor nodes
Compare euclidian distance $d_{s,i}$, with the radius of the spherical range of event source $r_e$
Compute volume of spherical range of event source $ V_e$ from step 4
Assuming the sensing range of sensor nodes *i* and *j* as dodecahedron which lie within $\emph{V}_{e}$.
Calculate radius of circumscribed sphere of a node as $ r_n$
Calculate the volume of dodecahedron sensing range of a node as $V_n$.
Find spatial information correlation among $s_i$ and $s_j$.
Define a threshold value $\tau_n$ such that information is strongly correlated.
Calculate circumscribed spherical radius of a node as $r_n $ where vertices’s of dodecahedron touches the spherical range.
Update $V_n$ using step 11, where $V_n \subseteq V_e$
To form clusters in 3D space, assuming set of sensor node $\textit{A}$ with dodecahedron sensing range lie within $V_e$.
Within sensing range $V_e$, $\forall i \in \textit{A}, \hspace{0.1in} \textup{let} \hspace{0.1in} \mathcal{E}(i)=\{j \in \textit{A} : d(i,j) \leq r_n, i\neq j \} $ where *d(i,j*) is the euclidian distance between *i* and *j* having dodecahedron sensing range.
$\mathcal{G} = \{j \in \mathcal{L} : \mathcal{E}(j)=\max\mathcal{E}(i)\}, i \in \mathcal{L} $ , we define $d_{max}(i)=\displaystyle \max_{j\in \mathcal{E}(i)}d(i,j)$, where *d(i,j)* is the Euclidian distance between *i* and *j* within $v_n$
Consider $\mathcal{P}=\displaystyle \arg \min_{i\in \mathcal{G}}d_{max}(i) $ and $C=C\cup \{(\mathcal{P},\mathcal{E}(\mathcal{P}) )\}$ where $C$ is the set of cluster of dodecahedron range
For each [$\emph{c}$=($\emph{x}_c$ , $\emph{y}_c$)]{},verifies the Euclidian distance between *x* to $s$ to select $C=\displaystyle \min_{i\in \mathcal{G}} d_{x_s,S}$ with $\emph{c} \in C$
$\textit{A}=\textit{A}-\{\mathcal{P}\}-\mathcal{E}\{\mathcal{P}\} $.
If $\textit{A} \neq \{ \phi \} $, go to step 16.
End
*(a). Data correlation between source event to number of sensor nodes in 3D space*: We are assuming that a source event $s$ occur in 3D space. The event $s$ propagates its signal spherically in 3D space. The sensor node $i$ which are situated within the effect of spherical range $s$, can form information correlation among $s$ and node $i$ such that $\emph{s}$ and *i*: $\rho[\emph{s},\emph{s}_i]=$ $C_{exp}(d_{s,i})= exp^{\left(-\frac{|d_{s,s_i}|^{\alpha}}{\theta} \right)}$ for $\alpha=1$. We define a threshold value $\tau_e$ [@jk9; @jk10] to determine the strength of information correlation. If $\rho[\emph{s},\emph{s}_i] \geq \tau_e $, information shows strong correlation among $s$ and $s_i$ otherwise weakly correlated i.e $C_{exp}(d_{s,i})= exp^{\left(-\frac{|d|^{\alpha}}{\theta} \right)} \geq \tau_e $. If it shows strong information correlation, then the euclidian distance among event source to $i$th sensor nodes is given as $d_{s,i} \leq \left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_e} \right) \right)^{1/\alpha}$ in 3D space. We compare euclidian distance $d_{s,i}\leq \left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_e} \right) \right)^{1/\alpha}$, with spherical source event radius $r_e$ is given by $\left(\frac{3V_e } {4\pi} \right)^{1/3}$ to get $\left(\frac{3V_e } {4\pi} \right)^{1/3}=\left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_e} \right) \right)^{1/\alpha}$. Finally, we compute volume of spherical range of event source as $ V_e=\left( \frac{4 \pi } {3 }\right)\left(\theta \hspace{0.04in} log \left(\frac{1}{\tau_e} \right) \right)^{3/\alpha} $. The parameters $\tau_e$, $\theta$ and $\alpha$ are dependent on $v_e$. If $\tau_e$ increases, $V_e$ decreases exponentially [@jk15] with fixed value of $\theta$ and $\alpha$.
*(b). Data correlation model among sensor nodes:* In second phase of 3D-DC algorithm, we illustrate spatial information correlation among sensor nodes which belongs to the spherical range of source event $s$. Thus, sensing nodes which occurs with in volume $V_e$ can only form jointly sensing nodes to explore spatial correlation between them. We assume that sensing range of nodes are dodecahedron [@jk16] in shape which are within the range $V_e$. The sensing range of a node with dodecahedron shape is circumscribed in a spherical range where the vertices’s of dodecahedron touches the spherical range. The radius of circumscribed sphere of a node is given as $ r_n=\frac{\nu \sqrt{3}}{4}(1+ \sqrt{5})$ where $\nu$ is the edge range of dodecahedron. The volume of dodecahedron sensing range of a node is given as $
V_n=\frac{\nu}{4}(15 + 7 \sqrt{5})$. Considering the value of $r_n$ in $V_n$, we the volume of dodecahedron sensing range $V_n=\frac{4}{\sqrt{3}} \frac{(15 + 7 \sqrt{5})}{(1+ \sqrt{5})}r_n $. We are interested to find spatial information correlation among dodecahedron sensing range of sensor nodes $s_i$ and $s_j$ which is given as $\rho[\emph{s}_i,\emph{s}_j]=C_{(.)}(d_{s_i,s_j})$. We define another threshold value $\tau_n$ to determine spatial data correlation among nodes $s_i$ and $s_j$ with dodecahedron sensing range. If $\rho[\emph{s}_i,\emph{s}_j] \geq \tau_n $, information are strongly correlated else weakly correlated within $V_e$. For strongly correlated information i.e $\rho[\emph{s}_i,\emph{s}_j] \geq \tau_n $, we get $d_{s_i,s_j} \leq \left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha} \leq d_{s,s_i}$. Comparing this with three dimensional euclidian distance between nodes $i$ and $j$, we get the circumscribed spherical radius of a node as $r_n \leq \left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha}$ where the vertices of dodecahedron touches the spherical range. This means $r_n=\frac{\nu \sqrt{3}}{4}(1+ \sqrt{5})=\left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha}$. Therefore $V_n=\frac{4}{ \sqrt{3}} \frac{(15 + 7 \sqrt{5})}{(1+ \sqrt{5})}\left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha} $ where $V_n \subseteq V_e$. The size of $V_n$ depends upon the value of $\tau_n$, $\theta$ and $\alpha$ and always $V_n \leq V_e$. For a fixed value of $\theta$ and $\alpha$, if we increase $\tau_n$, the size $V_n$ decreases exponentially thereby the average number of distributed clusters decreases exponentially within $V_e$ [@jk15].
*(c). Formation of distributed clusters in 3D space*: In the third phase of 3D-DC algorithm, the sensing nodes co-operate each other to form clusters. In [@jk15], authors proposed 3D-ESCC algorithm where jointly sensing nodes form clusters having spherical sensing range. The major drawback of 3D-ESCC algorithm is that the spherical range of clusters form tessellate (overlapping or gapping) among different clusters. Hence it is essential to develop a sensing coverage model which can fulfill the gap among two clusters and remove the overlapping of sensing coverage among clusters. To overcome these problems, we propose a sensing model in this paper where sensing coverage of clusters have dodecahedron [^4] topology which may overcome the tessellate space among the clusters. For the formation of clusters, each sensor node finds its dodecahedron sensing range according to Algorithm 1 within the source event volume $V_e$. The dodecahedron sensing range of a sensor node having maximum number of neighbouring nodes within its range can form the first cluster. Similarly, the formation of next clusters is considered to have decreasing number of sensor nodes within its dodecahedron sensing range and so on. For example, if a sensor node $i$ form the first cluster having five neighboring sensor nodes within its dodecahedron sensing range then the next sensor node $j$ can form cluster having less than five neighbouring sensor nodes within its dodecahedron sensing range without tessellate in 3D-space. Moreover, if sensor nodes $i$ and $j$ have same number of neighbouring nodes within its dodecahedron sensing ranges, then which form the first cluster is a subject of interest? The question is trivial. In this case, sensor nodes $i$ and $j$ finds the farthest position of neighboring nodes from the centre position of nodes $i$ and $j$ with its dodecahedron sensing range respectively, then it calculates which of the neighboring node is closer to its position $i$ and $j$ nodes respectively. The neighbouring node with minimum distance from the centre position of nodes $i$ and $j$, can form the first cluster, since closer the sensing node, the information correlation among sensing node increases, the detailed formation of clustering algorithm is explained in Algorithm 1 (step 14 to step 20).
Data Estimation in Three Dimensional Space
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Initially, we deploy sensor nodes in three dimensional (3D) space. In 3D space, sensor nodes form distributed cluster according to 3D-DC algorithm within $V_e$. Here, we define Three Dimensional Information Estimation (3D-IE) algorithm by adopting 3D-ESCC [@jk15] algorithm and extending the work when CH node of cluster extract information in dynamic condition (without having prior knowledge of information). In each cluster, each sensor node transmits its observed data to its respective CH node of the cluster. In 3D space, each cluster is responsible for estimating single source point event $s$. Let $u_c^i$ is the observation done by a cluster $C$ where $i$ denotes the number of cluster formed in 3D space. Hence, observation done by all the clusters is given by $\sum^{p}_{i=1} u_c^i$ where $p$ is the total number of clusters in 3D space.
We consider a single cluster $C$, where each sensor node $j$ can sensed the data $s_j(t)$ from the source point event $s$ at a time interval $t$ under additive white gaussian noise (AWGN) given by
$$\label{eq:jkx}
u_{c_j}(t)=s_j(t)e^{kwt}+\alpha_j(t) \hspace{0.2in} \textup{\emph{j=1,2,. . , m}} \hspace{0.2in}$$
where *k*=$\sqrt{-1}$, carrier frequency $\omega=2\pi f$ and $m=$ number of sensing nodes in a cluster $C$. We denote $\Delta t$ as the total time required for signal prorogation from point event source $s$ to each sensor node *j* of cluster $C$ with distance $\emph{d}$ given by
$$\label{eq:jkx}
\Delta t=\frac{d_{S,S_{c_j}} }{\Omega}=\frac{2\pi d_{S,S_{c_j}}}{\omega_c \lambda}$$
where $\Omega$ is the velocity of data prorogation in atmospheric medium and $\lambda$ is the wavelength of data signal [@jk15].
**Data Propagation:** For wireless sensor networks, we assume two channels by which sensor data can propagate: sensing channel and communication channel. Since sensor nodes are deployed in three dimensional space, each sensor node can extract data through sensing channel. Once the sensor node extract raw data, they transmits data among the networks through the communication channel. Thus, communication channel is any medium (e.g air, water, space) by which sensor nodes can transmit data among them in a cluster $C$.
If each sensor node $j$ observed data as $v_{c_j}(t)=s_j(t)e^{kwt} + \alpha_j (t)$ at time interval $t$, then the observation done by another node $l$ with delay varying time $\Delta t$ is given as $v_{c_l}(t)=s_l(t+\Delta t)e^{k\omega (t + \Delta t}+ \alpha_l(t)$ in any medium. According to slow varying [@jk15], [@jk20] data signal, we consider $s(t+\Delta t)\cong s(t)$. Hence, in a cluster $C$, each node observed data as
$$\label{eq:jkx}
v_{c_l}(t)=s_l(t)e^{k\omega t}e^{k \frac{2\pi d_{S,S_{c_l}}}{ \lambda} }$$
We conclude that data prorogation from node $j$ with observation $u_{c_j}$ is delayed with $\Delta t$ with observation $v_{c_l}$ within a cluster $C$. In a cluster $C$, the data received by each sensor nodes are converted to baseband. This makes the carrier signal $e^{-kwt}$ to be vanished. Considering the $qth$ count of observations done by sensor node at time $t$ is given as
$$u_{c_l}(t)=s_l(t)e^{k \frac{2 \pi q}{\lambda} d_{S,S_{c_l}} } + \alpha_j(t) \hspace{0.2in} \emph{q=0,1,2,. . .,m-1}$$
We describe a linear model using Gauss-Markov Theorem [@jk20], at each CH node of cluster given by
$$U(t)=L.s(t)+ \Psi(t)$$
where
$L=\left(1 \hspace{0.1in} e^{j \frac{2 \pi }{\lambda} d_{S,S_1} } \hspace{0.1in} e^{j \frac{4 \pi }{\lambda} d_{S,S_2 }} \hspace{0.1in} \cdot \hspace{0.1in} \cdot \hspace{0.1in} e^{j \frac{2 \pi (m-1)}{\lambda} d_{S,S_{m-1}} } \right)^T$,
$\Psi(t)=\left( \hspace{0.05in} \psi_0(t) \hspace{0.1in} \psi _1(t) \hspace{0.1in} \psi_2(t) \hspace{0.1in} \cdot \hspace{0.1in} \cdot \hspace{0.1in} \psi_{m-1}(t) \hspace{0.05in} \right)^T$
We define $L$ as a $m \times 1$ vector of sensor nodes locations, $s(t)$ is the actual source event to be estimated at time $t$, $\Psi(t)$ in a additive with gaussian noise at time $t$ with zero mean and covariance $\Lambda$. We are interested to find how much accurate information, we can extract at CH node of a cluster under noisy environment. Thus, we formulate an estimator using mean square error [@jk20] represented as
$$I(m)=E[s(t)-\hat{s}(t)]^2$$
We construct $\hat{s}(t)$ using best linear unbiased estimator (BLUE) as $\hat{s}(t)=\frac{L^T \Lambda^{-1}U(t)}{L^T\Lambda^{-1}L}$. We assume that sensor nodes extract reliable information under additive white gaussian noise which is uncorrelated with $s$ with zero mean and form covariance matrix $\Lambda=\sigma^2_{{n}_i}I$. Finally, $m$ sensor nodes in cluster are extracting accurate information at CH node under noisy correlation given as
$$\hat{s}(t)=\frac{1}{m } \sum^{m-1}_{n=0} \sum^{m}_{i=1} U_{n}(t)e^{-j \frac{2 \pi q}{\lambda} d_{S,S_i} }$$
To get the normalized information at the CH node of a cluster, we modify
$$I_{A}(m)=1-\frac{I(m)}{E[s^2(t)]}=\frac{1}{E[s^2(t)]}[2E[s(t)\hat{s}(t)]-E[\hat{s}^2(t) ]$$
To get the normalized information at CH node of a cluster, we modify [@jk4] as; $Cov[s,s_i]=\sigma^2_{s}Corr[s,s_i]=\sigma^2_{s}\rho(s,s_i)=\sigma^2_{s}C_{exp}(d_{s,s_i})$. Similarly, $Cov[s_i,s_j]=\sigma^2_{s}Corr[s_i,s_j]=\sigma^2_{s}\rho(s_i,s_j)=\sigma^2_{s}C_{exp}(d_{s_i,s_j})$. The $d_{s,s_i}$ is defined as $ d_{s,s_i}=\parallel s - s_j \parallel$ as euclidian distance among source $s$ to number of node $i$ in a cluster. Similarly, $d_{s_i,s_j}$ is given as $ d_{s_i,s_j}=\parallel s_i - s_j \parallel$ as euclidian distance among nodes $i$ and $j$ at time interval time $t$. $\sigma^2_{s_i}$ is the signal variance extended by sensor nodes $i$ and $j$ at time interval $t$. Finally, we calculate the information accuracy using 3D-IE in a CH node of cluster using known signal statistics given as
$I_{A}({m})={\frac{1}{m}}{\left(2{\sum_{i=1}^{m}\rho_{S,S_i}}\right)}-{\frac{1}{m^2}} \left(
{\sum_{i=1}^{m}}{\sum_{j\neq i}^{m}}{\rho_{S_i,S_j}} \right)$
$$- {\frac{1 }{ m^2} }
\left(\frac{m\sigma_S^2 + \sum_{i=1}^{m}\sigma_{n_i}^2} {\sigma_{S }^2} \right)$$
*Case study when nodes fails to operate in 3D networks:* In 3D space, it may be possible that any of the nodes in network fails to operate due to environmental conditions [@jk15]. In such situation, information accuracy may degrade. To overcome this problem, data prediction can be done for that specific node failure in the 3D space. Assuming there are *O* sensor nodes within sensing range $V_e$ which maximizes the information accuracy with locations at $L_{1}$, $L_{2}$, . . $L_{O}$. The sensor observation done by nodes can be expressed as $\emph{s}(\emph{L}_{1})$, $\emph{s}(\emph{L}_{2})$, . . $\emph{s}(\emph{L}_{m})$. Let say with *O* sensor nodes, *n* sensor nodes are dead and *m* are active for doing communication process such that *O=n+m*. Our goal is to predict observed data for *n* dead nodes. The unobserved value of dead nodes can be represented as ${s_d}(\emph{L}_{1})$, ${s_d}(\emph{L}_{2})$, . . ${s_d}(\emph{L}_{n})$ at locations $L_{1}$, $L_{2}$, . . $L_{n}$ where the coordinates of sensor positions are known. Hence, in a 3D space, the sensor observation considered to predict information $\int \emph{s}(\emph{L})\emph{dL}$ represented as
$$\label{eq:11}
s(L)=\chi +I(L) %\hspace{0.2in}$$
where $\chi$ is an unknown factor and *I*(*L*) is intrinsically stationary, IS [@jk22]. Taking into consideration (\[eq:11\]) for specific location $Z_{0}$ in 3D space, our aim is to find unobserved information values ${s_d}(\emph{Z}_{1}), {s_d}(\emph{Z}_{1}), ....., {s_d}(\emph{Z}_{n}) $. To evaluate the best predicted of unobserved information of ${s}_d(L_{1})$, ${s}_d(L_{2})$, . . ${S}_d(L_{n})$ from the following observed information ${s}(L_{n+1}), {s}(L_{n+2}), ......s(L_m)$ in 3D space given by $$\label{eq:jk2}
\hat{s}(L_d)=E[s(L_d)|s_m]$$
where $L_d=L_1, L_2,........L_n$ and $s_m=[s(L_{n+1}), s(L_{n+2}), . . \emph{s}(L_{m})]$. The prediction $\hat{\emph{s}}(L_d)$ is defined as the average of the observed information extracted by *O* sensor nodes in 3D space as
$$\label{eq:jk2}
\hat{s}(L_{d})=\frac{1}{O}\sum^{O}_{i=1}\emph{S}(\emph{L}_{i}) \hspace{0.2in} i=1,2,. . . O$$
where $n+m=O$. Using minimum minimum mean square, we design the predictor as
$\hspace{0.6in} P_d=E[s(L_{d})-\hat{s}(L_{d})]^2 \hspace{0.2in}$
To get better estimation of predicted information, we normalized the information as
$ P_{d(Nor)}=1-\frac{P_d}{E[s^2(L_{d})]}$
$$\hspace{0.7in} =\frac{2}{O}\sum^{O}_{i=1}\rho(d_{L_d,L_i})-\frac{1}{m^2} \sum^{O}_{i=1} \sum^{O}_{j\neq i}\rho(d_{L_i,L_j)}$$
$P_{d(Nor)}$ is the normalized form of predictor which predicts information for sensor node within the sensing range $V_e$.
Node Placement in 3D Networks
-----------------------------
In this subsection, we are interested to find which nodes are to be selected to maximize information estimation within the sensing range $V_e$. Hence, we propose three dimensional Node Placement (3D-NP) algorithm, which can find an optimal nodes using global optimization problem [@jk23]- [@jk26] in 3D space. Using 3D-NP algorithm each node position is search in the geographical location which actually maximizes information accuracy in the network. Hence, the goal is to maximize information in that specific location to be searched where a set of sensor nodes collaboratively does the operation. In some cases, sensor nodes placed in 3D network doesn’t have prior knowledge of location where the information is maximized and which sensor nodes are to be selected for maximizing the information. It may be possible that sensor nodes have prior knowledge about its locations where the information can be maximized. For example, sensor nodes placed directly under sunlight gains more information accuracy than the nodes placed under a dense forest area. Hence, the effective strategy is to find the sensor nodes which are placed in such a location where the information accuracy may be more (i.e nodes placed directly under sunlight). In this optimization problem, each sensor node have a searching space. Hence, in 3D space each node have a cost value which are calculated using a cost function which is to be optimized. For this optimization problem, sensor nodes does the searching operation by updating its iterations. In each iteration, each node signal variance is updated by the following two *best* signal variances. The first variance of cost value we called $\sigma^2_b$. Another signal variance is tracked using the optimization problem obtained by any sensor nodes within the whole 3D network. The best signal variance is called the global variance ($\sigma^2_{gb}$). The sensor nodes participated in the whole 3D network within its neighbouring nodes, we assume the best variance is the local signal variance ($\sigma^2_l$) in the network. After finding the two best signal variances, sensor node updates its signal variance to maximize information accuracy in its location [@jk23] as following.
$$\label{eq:jkk1}
I_A=I_A + \phi_1 (\sigma^2_b - \sigma^2_p) + \phi_2 (\sigma^2_{gb} - \sigma^2_p)$$
$$\label{eq:jkk2}
\sigma^2_p=\sigma^2_p + I_A$$
where $I_A$ is the information accuracy of sensor nodes, $\sigma^2_p$ is the present signal variance of sensor node and $\phi_1$, $\phi_2$ are the adaptation factor to learn signal variance w.r.t to time. 3D-NP algorithm is summarize as follows in Algorithm 2.
For each sensor node Initialize signal variance of sensor node in 3D space End For each sensor node Calculate cost function value which is better than the best signal variance ($\sigma^2_b$) Set present cost value as the new best signal variance ($\sigma^2_b$) End Select sensor node with the best cost function value (signal variance) among all sensor nodes in 3D network as the global variance ($\sigma^2_{gb}$) For each sensor node Calculate the maximum information accuracy using (\[eq:jkk1\]) Update the position of sensor nodes using (\[eq:jkk2\]) in 3D space End
Thus a set of sensor nodes can be selected with in a 3D network which maximizes information accuracy using 3D-NP algorithm.
Simulations and Validations
===========================
In this section, we validate 3D-DC, 3D-IE and 3D-NP algorithms respectively by taking sensor data from Intel Berkeley Research Lab [@jk21]. We consider 54 Mica2Dot sensor nodes which are deployed in three dimensional space to collect temperature data in the network. In our simulations, 54 sensor nodes collects 800 observations done on 28th February 2004.
In the first simulation setup, our goal is to form clusters using 3D-DC algorithm in 3D space. We assume that source event occurs in 3D-space such that it propagates its signal spherically where the volume of spherical event sensing range is $V_e$. The sensor nodes which lies within this range $V_e$ are able to extract the source event observations in 3D space. Within $V_e$, sensor nodes form clusters using 3D-DC algorithm having dodecahedron topology. The circumscribed spherical sensing range of radius of a node is given by $r_n \leq \left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha}$ where the vertices’s of dodecahedron touches the spherical sensing range such that $r_n=\frac{\nu \sqrt{3}}{4}(1+ \sqrt{5})\approx\left( \theta \hspace{0.04in} log \left(\frac{1}{\tau_n} \right) \right)^{1/\alpha}$ according to Algorithm 1. In our simulations, we consider $\tau_n=0.85$ where $0<\tau \leq 1$, $\theta=30$, $\alpha=1$ to calculate the radius of the dodecahedron clustered topology. The sensing coverage radius of each node is assumed to be 6 mts. We capture the 3D coordinates of each node in TABLE I. We assume all the sensor nodes falls within $V_e$ form a cluster. According to Algorithm 1, 3D-DC algorithm creates distributed clusters in 3D space. In TABLE II, seven dodecahedron clusters are formed using 3D-IE algorithm. Each dodecahedron cluster have a CH node with its associate neighboring sensor nodes. In each dodecahedron cluster, neighboring sensor nodes collaboratively transmits sensor data to its respective fusion node where fusion node estimates information accuracy for that cluster. From Table I, it is clear that the sensor nodes which are close to each other can form clusters as shown in TABLE II, based on their 3D coordinates. For example, in Table II, the cluster number III have fusion node 47 along-with associate nodes 5,6,12,28,32,38 who’s 3D coordinates are almost close to each other illustrated in TABLE I, Therefore, they form cluster. Similarly, cluster number VI, consists of only a single fusion node id-16 who’s coordinates are $x=9.612, y=9.591, z=9.373$. The coordinates of node id-16 is approximately far way from other nodes. Hence, no sensor nodes are associated with node id-16. In Table II, we compare our information accuracy done using 3D-IE algorithm with the existing models [@jk6], [@jk7], [@jk8], [@jk11]. Result shows that 3D-IE algorithm does better estimation compared to existing models at the CH node of each dodecahedron cluster. In cluster number VI and VII, the fusion node id- 16 & 11 shows information accuracy still they are not associated with other nodes due to the effect of sensing coverage from nearby clusters. Moreover, the table shows that the information accuracy achieved by cluster number VI is much lesser than other cluster as it has no associated nodes whereas cluster VII have somehow more information accuracy due to sensing effect of other clusters.
**Node id** **x** **y** **z** **Node id** **x** **y** **z**
------------- ------- ------- ------- ------------- ------- ------- -------
1 1.807 6.525 8.785 28 5.829 9.513 6.605
2 1.938 9.937 3.874 29 0.154 0.462 5.052
3 3.605 7.137 2.464 30 2.936 0.209 1.980
4 4.043 1.023 1.117 31 0.452 0.880 1.043
5 2.257 4.472 8.467 32 7.947 5.178 8.884
6 6.690 6.927 6.596 33 7.113 8.786 3.629
7 9.572 3.329 7.796 34 5.363 4.021 4.537
8 4.316 9.217 3.585 35 5.867 5.613 0.647
9 8.038 6.324 1.841 36 2.109 5.981 8.507
10 7.981 9.321 0.101 37 4.139 0.750 2.794
11 9.952 4.259 0.859 38 5.158 3.914 9.119
12 2.118 3.004 3.294 39 8.383 3.549 0.597
13 3.290 8.890 3.008 40 6.599 2.447 6.159
14 7.623 0.173 5.065 41 4.811 1.080 5.639
15 7.567 1.480 3.866 42 8.527 1.122 6.682
16 9.612 9.591 9.373 43 4.815 2.672 3.354
17 5.698 7.144 1.595 44 2.723 2.663 0.377
18 6.085 3.064 7.797 45 7.312 9.360 2.857
19 3.650 8.281 1.774 46 9.694 1.861 9.312
20 3.435 8.078 8.052 47 3.756 5.074 7.998
21 5.999 9.092 5.623 48 2.386 1.475 3.310
22 5.256 6.428 4.700 49 2.427 9.207 7.374
23 4.849 6.286 3.596 50 0.189 9.294 5.674
24 6.556 1.184 3.717 51 9.826 1.367 0.160
25 0.382 9.189 3.828 52 8.026 8.715 4.255
26 1.899 6.239 2.563 53 5.634 0.123 0.293
27 9.287 2.575 9.720 54 3.888 7.220 7.841
: Sensor position (i.e x-y-z coordinates) in 3D space[]{data-label="Tabjk3"}
[\*[17]{}[cc]{}]{}
Cluster & Fusion & Associated nodes & Accuracy & Accuracy & Accuracy & Accuracy &Accuracy\
\[1ex\] Number & Node & ID in cluster &Vuran at. all [@jk6] & Li at. all [@jk8] & Chai at. all[@jk7] & Karjee at. all[@jk11] & 3D-IE\
\[0.2ex\]
\
I & 25 & 1,2,3,8,13,19,20,21 &0.9412 &0.9412 &0.9300 &0.9310 &0.9424\
& &22,23,26,36,49,50,54 & & & & &\
\
II & 14 &4,7,15,18,24,27,30,34,37, & 0.9570 &0.9570 &0.9484 &0.9490 &0.9593\
& &39,40,41,42,43,46,48,51,53 & & & & &\
\
III & 47 &5,6,12,28,32,38 &0.9656 &0.9656 &0.9566 &0.9582 &0.9676\
\
IV & 33 &9,10,17,35,45,52 &0.9678 &0.9678 &0.9531 &0.9556 &0.9730\
\
V & 31 &29,44 &0.9604 &0.9604 &0.9495 &0.9539 &0.9641\
\
VI & 16 & - &0.9176 &0.9176 &0.9121 &0.9176 &0.9244\
\
VII & 11 & - &0.9427 &0.9427 &0.9376 &0.9427 &0.9460\
\
In the second simulation setup, we are interested to find the node placement strategy of sensor nodes using 3D-NP algorithm. We assume that all the 54 sensor nodes are with in the sensing range $V_e$. To achieve this goal, we need to perform the experiments under spatio-temporal effects of information accuracy. To validate the information correlation among sensor nodes in 3D space, we simulate to identify the sensor nodes which can maximize information accuracy in 3D space. This means, the sensor nodes that have more signal variance and covariance can be selected in the network. A cost function is calculated for node deployment scenario which perform the information accuracy level for the dedicated 3D networks. Moreover, we validate the temporal information correlation among sensor nodes in 3D space. 3D-NP algorithm searches for the nodes which maximizes information accuracy w.r.t to time using the cost function. As the time progresses, the cost function have temporal effect on the 3D network, as it reaches a saturation level after certain duration. In Figure 4, the sensor nodes having higher cost function are being selected from 54 nodes, which actually maximize the information accuracy in the network. The nodes with higher cost function have more signal variances and covariances, where as nodes having lower cost function have less variances and covariances. For example, sensor nodes deployed directly under sunlight have higher cost function w.r.t to nodes deployed under shade of trees, still they are placed under same geographical region. Here in the tree dominated sensing region, the cost function is low. To maximize information accuracy in such tree dominated sensing region, it is recommended to put more nodes to maximize the information accuracy. Therefore, the sensor node ids: 1 to 23 and node ids: 33 to 47 are selected, considering a benchmark of minimum cost function threshold value of 5. Above this benchmark, nodes have higher variances and covariances values and are selected for maximizing information accuracy for the network. Therefore in Figure 4, out of 54 sensor nodes, 37 sensor nodes are selected to maximize information accuracy. In Figure 5, the cost function achieves a desired information accuracy level as the number of sensor nodes increases. Cost function shows that approximately 38 sensor nodes are sufficient to give a desirable information accuracy as validated by Figure 4.
In Figure 6, cost function searches for the nodes having maximum information accuracy with respect to time (or number of rounds). We perform 300 rounds to select the nodes (as given in Figure 4), which have higher variances or covariances of signal above a desired benchmark threshold value. Similarly, in Figure 7, the cost function achieves a saturation level as the time progresses. This means, transmitting more number of data packet to the sink node doesn’t give extra advantage over cost function. Since, it has temporal effect on 3D networks, transmitting a subset of information is sufficient to achieve the desired information accuracy as number of round increases.
{width="47.00000%"}
{width="47.00000%"}
{width="50.00000%"}
{width="50.00000%"}
Conclusions
===========
A sensing model for 3D-WSN based dodecahedron topology called 3D-DC algorithm is developed to form cluster, where there is no tessellate space among them. In each dodecahedron cluster, Cluster Head (CH) node extracts accurate estimates of information using Three Dimensional Information Estimation (3D-IE) algorithm. Moreover, node deployment is an important factor to maximize information accuracy in 3D-WSN. We consider node deployment scenario using 3D-NP algorithm which can find an optimal way of placing sensor nodes in a sensing field to extract maximum information accuracy in the network. We validate 3D-NP algorithm using simulation results. 3D-DC, 3D-IE and 3D-NP algorithms may reduce communication overhead and energy consumption in sensor networks.
Acknowledgement
===============
This work was carried out by the authors at Indian Institute of Science, India. The work was submitted, while the corresponding author was associated with Manipal University Jaipur, India. The revised version of this paper is submitted while the corresponding author is associated with the above present address.
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[Jyotirmoy Karjee]{} received his PhD in Engineering from Indian Institute of Science, Bangalore, India. He did his Post-Doctoral research from Technische Universität München, Germany. Presently, he holds the position of Researcher at Embedded Systems and Robotics group, TCS Research and Innovation, Bangalore, India. He is a recipient of Heritage Erasmus Mundus scholarship (fellowship) to do postdoctoral research. He is interested in statistical signal processing, wireless sensor networks, wireless communications, robotic communications and embedded systems, machine learning, cloud computing and Internet of things.
[H.S Jamadagni]{} received his M.E and Ph.D degree in Electrical and Communication Engineering from Indian Institute of Science, Bangalore. Currently, He is the professor at Department of Electronic System Engineering, Indian Institute of Science. He is one of the main coordinators for the Intel higher education program, member of Telecommunications Regulatory Authority of India (TRAI) and key mentors for various intel workshops in india. His current research work includes in the areas of communication networks, embedded systems, VLSI for wireless networks, etc.
[^1]: Corresponding author
[^2]: Embedded Systems and Robotics Group, TCS Research and Innovation, Bangalore, INDIA; $\dag$Department of Electronic Systems Engineering, Indian Institute of Science, Bangalore, INDIA
[^3]: Email: jyotirmoy.karjee@tcs.com, hsjam@dese.iisc.ernet.in
[^4]: Theoretically sensor nodes with uniform dodecahedron sensing range can remove the tessellate space for cluster formation, where as in algorithm 1, we consider that there are tessellate among clusters since they are not uniform in 3D space.
|
---
title: 'A precise measurement of the $W$-boson mass with the Collider Detector at Fermilab'
---
abstract.tex
Introduction {#sec:introduction}
============
introduction.tex
Overview {#sec:overview}
========
overview.tex
The CDF II detector {#sec:detector}
===================
detector.tex
Tracking system {#sec:tracker}
---------------
tracker.tex
Calorimeter system {#sec:calo}
------------------
calorimeter.tex
Muon detectors
--------------
cmupcmx.tex
Trigger system {#sec:trigger}
--------------
triggers.tex
Detector simulation {#sec:model}
===================
model.tex
Charged-lepton scattering and ionization {#sec:ionization}
----------------------------------------
ionization.tex
Electron bremsstrahlung
-----------------------
bremsstrahlung.tex
Photon conversion and scattering
--------------------------------
conversion.tex
COT simulation and reconstruction {#sec:cotsim}
---------------------------------
cotsim.tex
Calorimeter response {#sec:ecal}
--------------------
elecal.tex
Production and decay models {#sec:production}
===========================
prodintro.tex
Parton distribution functions
-----------------------------
pdfs.tex
$W$ and $Z$ boson $p_T$
-----------------------
bosonpt.tex
Boson decay
-----------
wdecay.tex
QED radiation
-------------
qed.tex
$W$ and $Z$ boson event selection {#sec:wsample}
=================================
wdata.tex
Muon selection {#sec:muselection}
--------------
mu.tex
Electron selection {#sec:wesample}
------------------
ele.tex
Muon momentum measurement {#sec:muons}
=========================
muons.tex
COT alignment {#sec:alignment}
-------------
alignment.tex
$J/\psi\rightarrow \mu\mu$ calibration {#sec:jpsi}
--------------------------------------
jpsi.tex
### Data selection {#sec:jpsiselection}
jpsidata.tex
### Monte Carlo generation {#sec:jpsimc}
jpsimc.tex
### Momentum scale measurement {#sec:jpsicalib}
jpsimom.tex
### Systematic uncertainties {#sec:jpsisyst}
jpsisys.tex
$\Upsilon\to\mu\mu$ calibration {#sec:upsilon}
-------------------------------
upsilon.tex
Combination of $J/\psi$ and $\Upsilon$ calibrations {#sec:psiups}
---------------------------------------------------
psiupscomb.tex
$Z\to\mu\mu$ mass measurement and calibration {#sec:zmm}
---------------------------------------------
zmumu.tex
Electron Momentum Measurement {#sec:electrons}
=============================
electrons.tex
$E/p$ calibration {#sec:eop}
-----------------
eop.tex
$Z\to ee$ mass measurement and calibration {#sec:zee}
------------------------------------------
zee.tex
Recoil measurement {#sec:recoil}
==================
recoil.tex
Data corrections
----------------
recoildata.tex
Lepton tower removal
--------------------
leptonremoval.tex
Model parametrization
---------------------
recoilmodel.tex
### Recoil response
recoilscale.tex
### Recoil resolution {#sec:recoilresolution}
recoilresolution.tex
### Spectator and additional $p\bar{p}$ interactions {#sec:spectator}
spectator.tex
Model tests {#sec:recoilcheck}
-----------
recoilcheck.tex
Backgrounds {#sec:background}
===========
backgrounds.tex
$W$-boson-mass fits {#sec:fits}
===================
fits.tex
Summary {#sec:summary}
=======
summary.tex
acknowledgements.tex references.tex
|
---
abstract: 'Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Grüss type inequalities in inner product spaces. Some natural applications for integral inequalities are also pointed out.'
address: |
School of Computer Science and Mathematics\
Victoria University of Technology\
PO Box 14428, MCMC 8001\
Victoria, Australia.
author:
- 'S.S. Dragomir'
date: 'August 04, 2003.'
title: 'Reverses of Schwarz, Triangle and Bessel Inequalities in Inner Product Spaces'
---
Introduction\[s1\]
==================
Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product over the real or complex number field $\mathbb{K}$. The following inequality is known in the literature as *Schwarz’s inequality*:$$\left\vert \left\langle x,y\right\rangle \right\vert ^{2}\leq \left\Vert
x\right\Vert ^{2}\left\Vert y\right\Vert ^{2},\ \ \ \ x,y\in H; \label{1.1}$$where $\left\Vert z\right\Vert ^{2}=\left\langle z,z\right\rangle ,$ $z\in
H. $ The equality occurs in (\[1.1\]) if and only if $x$ and $y$ are linearly dependent.
In [@SSD1], the following *reverse* of Schwarz’s inequality has been obtained:$$0\leq \left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}-\left\vert
\left\langle x,y\right\rangle \right\vert ^{2}\leq \frac{1}{4}\left\vert
A-a\right\vert ^{2}\left\Vert y\right\Vert ^{4}, \label{1.2}$$provided $x,y\in H$ and $a,A\in \mathbb{K}$ are so that either$$\func{Re}\left\langle Ay-x,x-ay\right\rangle \geq 0, \label{1.3}$$or, equivalently,$$\left\Vert x-\frac{a+A}{2}\cdot y\right\Vert \leq \frac{1}{2}\left\vert
A-a\right\vert \left\Vert y\right\Vert , \label{1.4}$$holds. The constant $\frac{1}{4}$ is best possible in (\[1.2\]) in the sense that it cannot be replaced by a smaller constant.
If $x,y,A,a$ satisfy either (\[1.3\]) or (\[1.4\]), then the following reverse of Schwarz’s inequality also holds [@SSD2]$$\begin{aligned}
\left\Vert x\right\Vert \left\Vert y\right\Vert & \leq \frac{1}{2}\cdot
\frac{\func{Re}\left[ A\overline{\left\langle x,y\right\rangle }+\overline{a}%
\left\langle x,y\right\rangle \right] }{\left[ \func{Re}\left( \overline{a}%
A\right) \right] ^{\frac{1}{2}}} \label{1.5} \\
& \leq \frac{1}{2}\cdot \frac{\left\vert A\right\vert +\left\vert
a\right\vert }{\left[ \func{Re}\left( \overline{a}A\right) \right] ^{\frac{1%
}{2}}}\left\vert \left\langle x,y\right\rangle \right\vert , \notag\end{aligned}$$provided that, the complex numbers $a$ and $A$ satisfy the condition $\func{%
Re}\left( \overline{a}A\right) >0.$ In both inequalities in (\[1.5\]), the constant $\frac{1}{2}$ is best possible.
An additive version of (\[1.5\]) may be stated as well (see also [SSD3]{})$$0\leq \left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}-\left\vert
\left\langle x,y\right\rangle \right\vert ^{2}\leq \frac{1}{4}\cdot \frac{%
\left( \left\vert A\right\vert -\left\vert a\right\vert \right) ^{2}+4\left[
\left\vert Aa\right\vert -\func{Re}\left( \overline{a}A\right) \right] }{%
\func{Re}\left( \overline{a}A\right) }\left\vert \left\langle
x,y\right\rangle \right\vert ^{2}. \label{1.6}$$In this inequality, $\frac{1}{4}$ is the best possible constant.
It has been proven in [@SSD4], that$$0\leq \left\Vert x\right\Vert ^{2}-\left\vert \left\langle x,y\right\rangle
\right\vert ^{2}\leq \frac{1}{4}\left\vert \phi -\varphi \right\vert
^{2}-\left\vert \frac{\phi +\varphi }{2}-\left\langle x,e\right\rangle
\right\vert ^{2}; \label{1.7}$$provided, either $$\func{Re}\left\langle \phi e-x,x-\varphi e\right\rangle \geq 0, \label{1.8}$$or, equivalently,$$\left\Vert x-\frac{\phi +\varphi }{2}e\right\Vert \leq \frac{1}{2}\left\vert
\phi -\varphi \right\vert , \label{1.9}$$where $e=H,$ $\left\Vert e\right\Vert =1.$ The constant $\frac{1}{4}$ in [1.7]{} is also best possible.
If we choose $e=\frac{y}{\left\Vert y\right\Vert },$ $\phi =\Gamma
\left\Vert y\right\Vert ,$ $\varphi =\gamma \left\Vert y\right\Vert $ $%
\left( y\neq 0\right) ,$ $\Gamma ,\gamma \in \mathbb{K}$, then by (\[1.8\]), (\[1.9\]) we have,$$\func{Re}\left\langle \Gamma y-x,x-\gamma y\right\rangle \geq 0,
\label{1.10}$$or, equivalently,$$\left\Vert x-\frac{\Gamma +\gamma }{2}y\right\Vert \leq \frac{1}{2}%
\left\vert \Gamma -\gamma \right\vert \left\Vert y\right\Vert , \label{1.11}$$imply the following reverse of Schwarz’s inequality:$$0\leq \left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}-\left\vert
\left\langle x,y\right\rangle \right\vert ^{2}\leq \frac{1}{4}\left\vert
\Gamma -\gamma \right\vert ^{2}\left\Vert y\right\Vert ^{4}-\left\vert \frac{%
\Gamma +\gamma }{2}\left\Vert y\right\Vert ^{2}-\left\langle
x,y\right\rangle \right\vert ^{2}. \label{1.12}$$The constant $\frac{1}{4}$ in (\[1.12\]) is sharp.
Note that this inequality is an improvement of (\[1.2\]), but it might not be very convenient for applications.
Now, let $\left\{ e_{i}\right\} _{i\in I}$ be a finite or infinite family of orthornormal vectors in the inner product space $\left( H;\left\langle \cdot
,\cdot \right\rangle \right) ,$ i.e., we recall that $$\left\langle e_{i},e_{j}\right\rangle =\left\{
\begin{array}{ll}
0 & \text{if \ }i\neq j \\
& \\
1 & \text{if \ }i=j%
\end{array}%
\right. ,\ \ \ i,j\in I.$$In [@SSD5], we proved that if $\left\{ e_{i}\right\} _{i\in I}$ is as above, $F\subset I$ is a finite part of $I$ such that either$$\func{Re}\left\langle \sum_{i\in F}\phi _{i}e_{i}-x,x-\sum_{i\in F}\varphi
_{i}e_{i}\right\rangle \geq 0, \label{1.13}$$or, equivalently,$$\left\Vert x-\sum_{i\in F}\frac{\phi _{i}+\varphi _{i}}{2}e_{i}\right\Vert
\leq \frac{1}{2}\left( \sum_{i\in F}\left\vert \phi _{i}-\varphi
_{i}\right\vert ^{2}\right) ^{\frac{1}{2}}, \label{1.14}$$holds, where $\left( \phi _{i}\right) _{i\in I},$ $\left( \varphi
_{i}\right) _{i\in I}$ are real or complex numbers, then we have the following reverse of *Bessel’s inequality:*$$\begin{aligned}
0& \leq \left\Vert x\right\Vert ^{2}-\sum_{i\in F}\left\vert \left\langle
x,e_{i}\right\rangle \right\vert ^{2} \label{1.15} \\
& \leq \frac{1}{4}\cdot \sum_{i\in F}\left\vert \phi _{i}-\varphi
_{i}\right\vert ^{2}-\func{Re}\left\langle \sum_{i\in F}\phi
_{i}e_{i}-x,x-\sum_{i\in F}\varphi _{i}e_{i}\right\rangle \notag \\
& \leq \frac{1}{4}\cdot \sum_{i\in F}\left\vert \phi _{i}-\varphi
_{i}\right\vert ^{2}. \notag\end{aligned}$$The constant $\frac{1}{4}$ in both inequalities is sharp. This result improves an earlier result by N. Ujević obtained only for real spaces [@NU].
In [@SSD4], by the use of a different technique, another reverse of Bessel’s inequality has been proven, namely:$$\begin{aligned}
0& \leq \left\Vert x\right\Vert ^{2}-\sum_{i\in F}\left\vert \left\langle
x,e_{i}\right\rangle \right\vert ^{2} \label{1.16} \\
& \leq \frac{1}{4}\cdot \sum_{i\in F}\left\vert \phi _{i}-\varphi
_{i}\right\vert ^{2}-\sum_{i\in F}\left\vert \frac{\phi _{i}+\varphi _{i}}{2}%
-\left\langle x,e_{i}\right\rangle \right\vert ^{2} \notag \\
& \leq \frac{1}{4}\cdot \sum_{i\in F}\left\vert \phi _{i}-\varphi
_{i}\right\vert ^{2}, \notag\end{aligned}$$provided that $\left( e_{i}\right) _{i\in I},$ $\left( \phi _{i}\right)
_{i\in I},$ $\left( \varphi _{i}\right) _{i\in I},$ $x$ and $F$ are as above.
Here the constant $\frac{1}{4}$ is sharp in both inequalities.
It has also been shown that the bounds provided by (\[1.15\]) and ([1.16]{}) for the Bessel’s difference $\left\Vert x\right\Vert ^{2}-\sum_{i\in
F}\left\vert \left\langle x,e_{i}\right\rangle \right\vert ^{2}$ cannot be compared in general, meaning that there are examples for which one is smaller than the other [@SSD4].
Finally, we recall another type of reverse for Bessel inequality that has been obtained in [@SSD6]:$$\left\Vert x\right\Vert ^{2}\leq \frac{1}{4}\cdot \frac{\sum_{i\in F}\left(
\left\vert \phi _{i}\right\vert +\left\vert \varphi _{i}\right\vert \right)
^{2}}{\sum_{i\in F}\func{Re}\left( \phi _{i}\overline{\varphi _{i}}\right) }%
\sum_{i\in F}\left\vert \left\langle x,e_{i}\right\rangle \right\vert ^{2};
\label{1.17}$$provided $\left( \phi _{i}\right) _{i\in I},$ $\left( \varphi _{i}\right)
_{i\in I}$ satisfy (\[1.13\]) (or, equivalently (\[1.14\])) and $%
\sum_{i\in F}\func{Re}\left( \phi _{i}\overline{\varphi _{i}}\right) >0.$ Here the constant $\frac{1}{4}$ is also best possible.
An additive version of (\[1.17\]) is $$\begin{aligned}
0& \leq \left\Vert x\right\Vert ^{2}-\sum_{i\in F}\left\vert \left\langle
x,e_{i}\right\rangle \right\vert ^{2} \label{1.18} \\
& \leq \frac{1}{4}\cdot \frac{\sum_{i\in F}\left\{ \left( \left\vert \phi
_{i}\right\vert -\left\vert \varphi _{i}\right\vert \right) ^{2}+4\left[
\left\vert \phi _{i}\varphi _{i}\right\vert -\func{Re}\left( \phi _{i}%
\overline{\varphi _{i}}\right) \right] \right\} }{\sum_{i\in F}\func{Re}%
\left( \phi _{i}\overline{\varphi _{i}}\right) }. \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible.
It is the main aim of the present paper to point out new reverse inequalities to Schwarz’s, triangle and Bessel’s inequalities.
Some results related to Grüss’ inequality in inner product spaces are also pointed out. Natural applications for integrals are also provided.
Some Reverses of Schwarz’s Inequality\[s2\]
===========================================
The following result holds.
\[t2.1\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over the real or complex number field $\mathbb{K}$ $\left( \mathbb{K}=\mathbb{R},\ \mathbb{K}=\mathbb{C}\right) $ and $x,a\in
H, $ $r>0$ are such that$$x\in \overline{B}\left( a,r\right) :=\left\{ z\in H|\left\Vert
z-a\right\Vert \leq r\right\} . \label{2.1}$$
1. If $\left\Vert a\right\Vert >r,$ then we have the inequality$$0\leq \left\Vert x\right\Vert ^{2}\left\Vert a\right\Vert ^{2}-\left\vert
\left\langle x,a\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert
^{2}\left\Vert a\right\Vert ^{2}-\left[ \func{Re}\left\langle
x,a\right\rangle \right] ^{2}\leq r^{2}\left\Vert x\right\Vert ^{2}.
\label{2.2}$$The constant $C=1$ in front of $r^{2}$ is best possible in the sense that it cannot be replaced by a smaller one.
2. If $\left\Vert a\right\Vert =r,$ then$$\left\Vert x\right\Vert ^{2}\leq 2\func{Re}\left\langle x,a\right\rangle
\leq 2\left\vert \left\langle x,a\right\rangle \right\vert . \label{2.3}$$The constant $2$ is best possible in both inequalities.
3. If $\left\Vert a\right\Vert <r,$ then$$\left\Vert x\right\Vert ^{2}\leq r^{2}-\left\Vert a\right\Vert ^{2}+2\func{Re%
}\left\langle x,a\right\rangle \leq r^{2}-\left\Vert a\right\Vert
^{2}+2\left\vert \left\langle x,a\right\rangle \right\vert . \label{2.4}$$Here the constant $2$ is also best possible.
Since $x\in \overline{B}\left( a,r\right) ,$ then obviously $\left\Vert
x-a\right\Vert ^{2}\leq r^{2},$ which is equivalent to $$\left\Vert x\right\Vert ^{2}+\left\Vert a\right\Vert ^{2}-r^{2}\leq 2\func{Re%
}\left\langle x,a\right\rangle . \label{2.5}$$
1. If $\left\Vert a\right\Vert >r,$ then we may divide (\[2.5\]) by $\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}>0$ getting $$\frac{\left\Vert x\right\Vert ^{2}}{\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}%
}+\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}\leq \frac{2\func{Re}\left\langle
x,a\right\rangle }{\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}}. \label{2.6}$$Using the elementary inequality$$\alpha p+\frac{1}{\alpha }q\geq 2\sqrt{pq},\ \ \ \alpha >0,\ \ p,q\geq 0,$$we may state that$$2\left\Vert x\right\Vert \leq \frac{\left\Vert x\right\Vert ^{2}}{\sqrt{%
\left\Vert a\right\Vert ^{2}-r^{2}}}+\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}%
}. \label{2.7}$$Making use of (\[2.6\]) and (\[2.7\]), we deduce$$\left\Vert x\right\Vert \sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}\leq \func{%
Re}\left\langle x,a\right\rangle . \label{2.8}$$Taking the square in (\[2.8\]) and re-arranging the terms, we deduce the third inequality in (\[2.2\]). The others are obvious.
To prove the sharpness of the constant, assume, under the hypothesis of the theorem, that, there exists a constant $c>0$ such that$$\left\Vert x\right\Vert ^{2}\left\Vert a\right\Vert ^{2}-\left[ \func{Re}%
\left\langle x,a\right\rangle \right] ^{2}\leq cr^{2}\left\Vert x\right\Vert
^{2}, \label{2.9}$$provided $x\in \overline{B}\left( a,r\right) $ and $\left\Vert a\right\Vert
>r.$
Let $r=\sqrt{\varepsilon }>0,$ $\varepsilon \in \left( 0,1\right) ,$ $a,e\in
H$ with $\left\Vert a\right\Vert =\left\Vert e\right\Vert =1$ and $a\perp e.$ Put $x=a+\sqrt{\varepsilon }e.$ Then obviously $x\in \overline{B}\left(
a,r\right) ,$ $\left\Vert a\right\Vert >r$ and $\left\Vert x\right\Vert
^{2}=\left\Vert a\right\Vert ^{2}+\varepsilon \left\Vert e\right\Vert
^{2}=1+\varepsilon $, $\func{Re}\left\langle x,a\right\rangle =\left\Vert
a\right\Vert ^{2}=1,$ and thus $\left\Vert x\right\Vert ^{2}\left\Vert
a\right\Vert ^{2}-\left[ \func{Re}\left\langle x,a\right\rangle \right]
^{2}=\varepsilon .$ Using (\[2.9\]), we may write that$$\varepsilon \leq c\varepsilon \left( 1+\varepsilon \right) ,\ \ \varepsilon
>0$$giving $$c+c\varepsilon \geq 1\text{ \ for any }\varepsilon >0 \label{2.10}$$Letting $\varepsilon \rightarrow 0+,$ we get from (\[2.10\]) that $c\geq 1,
$ and the sharpness of the constant is proved.
2. The inequality (\[2.3\]) is obvious by (\[2.5\]) since $%
\left\Vert a\right\Vert =r.$ The best constant follows in a similar way to the above.
3. The inequality (\[2.3\]) is obvious. The best constant may be proved in a similar way to the above. We omit the details.
The following reverse of Schwarz’s inequality holds.
\[t2.2\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over $\mathbb{K}$ and $x,y\in H,$ $\gamma ,\Gamma
\in \mathbb{K}$ such that either$$\func{Re}\left\langle \Gamma y-x,x-\gamma y\right\rangle \geq 0,
\label{2.11}$$or, equivalently,$$\left\Vert x-\frac{\Gamma +\gamma }{2}y\right\Vert \leq \frac{1}{2}%
\left\vert \Gamma -\gamma \right\vert \left\Vert y\right\Vert , \label{2.12}$$holds.
1. If $\func{Re}\left( \Gamma \overline{\gamma }\right) >0,$ then we have the inequalities$$\begin{aligned}
\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}& \leq \frac{1}{4}%
\cdot \frac{\left\{ \func{Re}\left[ \left( \overline{\Gamma }+\overline{%
\gamma }\right) \left\langle x,y\right\rangle \right] \right\} ^{2}}{\func{Re%
}\left( \Gamma \overline{\gamma }\right) } \label{2.13} \\
& \leq \frac{1}{4}\cdot \frac{\left\vert \Gamma +\gamma \right\vert ^{2}}{%
\func{Re}\left( \Gamma \overline{\gamma }\right) }\left\vert \left\langle
x,y\right\rangle \right\vert ^{2}. \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible in both inequalities.
2. If $\func{Re}\left( \Gamma \overline{\gamma }\right) =0,$ then $$\left\Vert x\right\Vert ^{2}\leq \func{Re}\left[ \left( \overline{\Gamma }+%
\overline{\gamma }\right) \left\langle x,y\right\rangle \right] \leq
\left\vert \Gamma +\gamma \right\vert \left\vert \left\langle
x,y\right\rangle \right\vert . \label{2.14}$$
3. If $\func{Re}\left( \Gamma \overline{\gamma }\right) <0,$ then $$\begin{aligned}
\left\Vert x\right\Vert ^{2}& \leq -\func{Re}\left( \Gamma \overline{\gamma }%
\right) \left\Vert y\right\Vert ^{2}+\func{Re}\left[ \left( \overline{\Gamma
}+\overline{\gamma }\right) \left\langle x,y\right\rangle \right]
\label{2.15} \\
& \leq -\func{Re}\left( \Gamma \overline{\gamma }\right) \left\Vert
y\right\Vert ^{2}+\left\vert \Gamma +\gamma \right\vert \left\vert
\left\langle x,y\right\rangle \right\vert . \notag\end{aligned}$$
The proof of the equivalence between the inequalities (\[2.11\]) and ([2.12]{}) follows by the fact that in an inner product space $\func{Re}%
\left\langle Z-x,x-z\right\rangle \geq 0$ for $x,z,Z\in H$ is equivalent with $\left\Vert x-\frac{z+Z}{2}\right\Vert \leq \frac{1}{2}\left\Vert
Z-z\right\Vert $ (see for example [@SSD3]).
Consider, for $y\neq 0,$ $a=\frac{\gamma +\Gamma }{2}y$ and $r=\frac{1}{2}%
\left\vert \Gamma -\gamma \right\vert \left\Vert y\right\Vert ^{2}.$ Then$$\left\Vert a\right\Vert ^{2}-r^{2}=\frac{\left\vert \Gamma +\gamma
\right\vert ^{2}-\left\vert \Gamma -\gamma \right\vert ^{2}}{4}\left\Vert
y\right\Vert ^{2}=\func{Re}\left( \Gamma \overline{\gamma }\right)
\left\Vert y\right\Vert ^{2}.$$
1. If $\func{Re}\left( \Gamma \overline{\gamma }\right) >0,$ then the hypothesis of (i) in Theorem \[t2.1\] is satisfied, and by the second inequality in (\[2.2\]) we have$$\left\Vert x\right\Vert ^{2}\frac{\left\vert \Gamma +\gamma \right\vert ^{2}%
}{4}\left\Vert y\right\Vert ^{2}-\frac{1}{4}\left\{ \func{Re}\left[ \left(
\overline{\Gamma }+\overline{\gamma }\right) \left\langle x,y\right\rangle %
\right] \right\} ^{2}\leq \frac{1}{4}\left\vert \Gamma -\gamma \right\vert
^{2}\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}$$from where we derive$$\frac{\left\vert \Gamma +\gamma \right\vert ^{2}-\left\vert \Gamma -\gamma
\right\vert ^{2}}{4}\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert
^{2}\leq \frac{1}{4}\left\{ \func{Re}\left[ \left( \overline{\Gamma }+%
\overline{\gamma }\right) \left\langle x,y\right\rangle \right] \right\}
^{2},$$giving the first inequality in (\[2.13\]).
The second inequality is obvious.
To prove the sharpness of the constant $\frac{1}{4},$ assume that the first inequality in (\[2.13\]) holds with a constant $c>0,$ i.e., $$\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}\leq c\cdot \frac{%
\left\{ \func{Re}\left[ \left( \overline{\Gamma }+\overline{\gamma }\right)
\left\langle x,y\right\rangle \right] \right\} ^{2}}{\func{Re}\left( \Gamma
\overline{\gamma }\right) }, \label{2.16}$$provided $\func{Re}\left( \Gamma \overline{\gamma }\right) >0$ and either (\[2.11\]) or (\[2.12\]) holds.
Assume that $\Gamma ,\gamma >0,$ and let $x=\gamma y.$ Then (\[2.11\]) holds and by (\[2.16\]) we deduce$$\gamma ^{2}\left\Vert y\right\Vert ^{4}\leq c\cdot \frac{\left( \Gamma
+\gamma \right) ^{2}\gamma ^{2}\left\Vert y\right\Vert ^{4}}{\Gamma \gamma }$$giving$$\Gamma \gamma \leq c\left( \Gamma +\gamma \right) ^{2}\text{ \ for any \ }%
\Gamma ,\gamma >0. \label{2.17}$$Let $\varepsilon \in \left( 0,1\right) $ and choose in (\[2.17\]), $\Gamma
=1+\varepsilon ,$ $\gamma =1-\varepsilon >0$ to get $1-\varepsilon ^{2}\leq
4c$ for any $\varepsilon \in \left( 0,1\right) .$ Letting $\varepsilon
\rightarrow 0+,$ we deduce $c\geq \frac{1}{4},$ and the sharpness of the constant is proved.
\(ii) and (iii) are obvious and we omit the details.
We observe that the second bound in (\[2.13\]) for $\left\Vert
x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}$ is better than the second bound provided by (\[1.5\]).
The following corollary provides a reverse inequality for the additive version of Schwarz’s inequality.
\[c2.3\]With the assumptions of Theorem \[t2.2\] and if $\func{Re}%
\left( \Gamma \overline{\gamma }\right) >0,$ then we have the inequality:$$0\leq \left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2}-\left\vert
\left\langle x,y\right\rangle \right\vert ^{2}\leq \frac{1}{4}\cdot \frac{%
\left\vert \Gamma -\gamma \right\vert ^{2}}{\func{Re}\left( \Gamma \overline{%
\gamma }\right) }\left\vert \left\langle x,y\right\rangle \right\vert ^{2}.
\label{2.18}$$The constant $\frac{1}{4}$ is best possible in (\[2.18\]).
The proof is obvious from (\[2.13\]) on subtracting in both sides the same quantity $\left\vert \left\langle x,y\right\rangle \right\vert ^{2}.$ The sharpness of the constant may be proven in a similar manner to the one incorporated in the proof of (i), Theorem \[t2.2\]. We omit the details.
It is obvious that the inequality (\[2.18\]) is better than (\[1.6\]) obtained in [@SSD3].
For some recent results in connection to Schwarz’s inequality, see [@ADR], [@DM] and [@GH].
Reverses of the Triangle Inequality\[s3\]
=========================================
The following reverse of the triangle inequality holds.
\[p2.4\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over the real or complex number field $\mathbb{K}$ $\left( \mathbb{K}=\mathbb{R},\mathbb{C}\right) $ and $x,a\in H,$ $r>0$ are such that $$\left\Vert x-a\right\Vert \leq r<\left\Vert a\right\Vert . \label{2.19}$$Then we have the inequality$$0\leq \left\Vert x\right\Vert +\left\Vert a\right\Vert -\left\Vert
x+a\right\Vert \leq \sqrt{2}r\cdot \sqrt{\frac{\func{Re}\left\langle
x,a\right\rangle }{\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}\left( \sqrt{%
\left\Vert a\right\Vert ^{2}-r^{2}}+\left\Vert a\right\Vert \right) }}.
\label{2.20}$$
Using the inequality (\[2.8\]), we may write that$$\left\Vert x\right\Vert \left\Vert a\right\Vert \leq \frac{\left\Vert
a\right\Vert \func{Re}\left\langle x,a\right\rangle }{\sqrt{\left\Vert
a\right\Vert ^{2}-r^{2}}},$$giving$$\begin{aligned}
0& \leq \left\Vert x\right\Vert \left\Vert a\right\Vert -\func{Re}%
\left\langle x,a\right\rangle \label{2.21} \\
& \leq \func{Re}\left\langle x,a\right\rangle \frac{\left\Vert a\right\Vert -%
\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}}{\sqrt{\left\Vert a\right\Vert
^{2}-r^{2}}} \notag \\
& =\frac{r^{2}\func{Re}\left\langle x,a\right\rangle }{\sqrt{\left\Vert
a\right\Vert ^{2}-r^{2}}\left( \sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}%
+\left\Vert a\right\Vert \right) }. \notag\end{aligned}$$Since$$\left( \left\Vert x\right\Vert +\left\Vert a\right\Vert \right)
^{2}-\left\Vert x+a\right\Vert ^{2}=2\left( \left\Vert x\right\Vert
\left\Vert a\right\Vert -\func{Re}\left\langle x,a\right\rangle \right) ,$$then by (\[2.21\]), we have$$\begin{aligned}
\left\Vert x\right\Vert +\left\Vert a\right\Vert & \leq \sqrt{\left\Vert
x+a\right\Vert ^{2}+\frac{2r^{2}\func{Re}\left\langle x,a\right\rangle }{%
\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}\left( \sqrt{\left\Vert
a\right\Vert ^{2}-r^{2}}+\left\Vert a\right\Vert \right) }} \\
& \leq \left\Vert x+a\right\Vert +\sqrt{2}r\cdot \sqrt{\frac{\func{Re}%
\left\langle x,a\right\rangle }{\sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}%
\left( \sqrt{\left\Vert a\right\Vert ^{2}-r^{2}}+\left\Vert a\right\Vert
\right) }},\end{aligned}$$giving the desired inequality (\[2.20\]).
The following proposition providing a simpler reverse for the triangle inequality also holds.
\[p2.5\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over $\mathbb{K}$ and $x,y\in H,$ $M>m>0$ such that either$$\func{Re}\left\langle My-x,x-my\right\rangle \geq 0, \label{2.22}$$or, equivalently,$$\left\Vert x-\frac{M+m}{2}\cdot y\right\Vert \leq \frac{1}{2}\left(
M-m\right) \left\Vert y\right\Vert , \label{2.23}$$holds. Then we have the inequality$$0\leq \left\Vert x\right\Vert +\left\Vert y\right\Vert -\left\Vert
x+y\right\Vert \leq \frac{\sqrt{M}-\sqrt{m}}{\sqrt[4]{mM}}\sqrt{\func{Re}%
\left\langle x,y\right\rangle }. \label{2.24}$$
Choosing in (\[2.8\]), $a=\frac{M+m}{2}y,$ $r=\frac{1}{2}\left( M-m\right)
\left\Vert y\right\Vert $ we get$$\left\Vert x\right\Vert \left\Vert y\right\Vert \sqrt{Mm}\leq \frac{M+m}{2}%
\func{Re}\left\langle x,y\right\rangle$$giving $$0\leq \left\Vert x\right\Vert \left\Vert y\right\Vert -\func{Re}\left\langle
x,y\right\rangle \leq \frac{\left( \sqrt{M}-\sqrt{m}\right) ^{2}}{2\sqrt{mM}}%
\func{Re}\left\langle x,y\right\rangle .$$Following the same arguments as in the proof of Proposition \[p2.4\], we deduce the desired inequality (\[2.24\]).
For some results related to triangle inequality in inner product spaces, see [@JBDFTM], [@SMK], [@PMM] and [@DKR].
Some Grüss Type Inequalities\[s4\]
==================================
We may state the following result.
\[t4.1\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over the real or complex number field $\mathbb{K}$ $\left( \mathbb{K}=\mathbb{R},\mathbb{K}=\mathbb{C}\right) $ and $x,y,e\in H$ with $\left\Vert e\right\Vert =1.$ If $r_{1},r_{2}\in \left( 0,1\right) $ and $$\left\Vert x-e\right\Vert \leq r_{1},\ \ \ \ \left\Vert y-e\right\Vert \leq
r_{2}, \label{4.1}$$then we have the inequality$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert \leq r_{1}r_{2}\left\Vert
x\right\Vert \left\Vert y\right\Vert . \label{4.2}$$The inequality (\[4.2\]) is sharp in the sense that the constant $c=1$ in front of $r_{1}r_{2}$ cannot be replaced by a smaller constant.
Apply Schwarz’s inequality in $\left( H;\left\langle \cdot ,\cdot
\right\rangle \right) $ for the vectors $x-\left\langle x,e\right\rangle e,$ $y-\left\langle y,e\right\rangle e,$ to get (see also [@SSD3])$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert ^{2}\leq \left( \left\Vert
x\right\Vert ^{2}-\left\vert \left\langle x,e\right\rangle \right\vert
^{2}\right) \left( \left\Vert y\right\Vert ^{2}-\left\vert \left\langle
y,e\right\rangle \right\vert ^{2}\right) . \label{4.3}$$Using Theorem \[t2.1\] for $a=e,$ we may state that$$\left\Vert x\right\Vert ^{2}-\left\vert \left\langle x,e\right\rangle
\right\vert ^{2}\leq r_{1}^{2}\left\Vert x\right\Vert ^{2},\ \ \ \ \ \
\left\Vert y\right\Vert ^{2}-\left\vert \left\langle y,e\right\rangle
\right\vert ^{2}\leq r_{2}^{2}\left\Vert y\right\Vert ^{2}. \label{4.4}$$Utilizing (\[4.3\]) and (\[4.4\]), we deduce$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert ^{2}\leq
r_{1}^{2}r_{2}^{2}\left\Vert x\right\Vert ^{2}\left\Vert y\right\Vert ^{2},
\label{4.5}$$which is clearly equivalent to the desired inequality (\[4.2\]).
The sharpness of the constant follows by the fact that for $x=y,$ $%
r_{1}=r_{2}=r,$ we get from (\[4.2\])$$\left\Vert x\right\Vert ^{2}-\left\vert \left\langle x,e\right\rangle
\right\vert ^{2}\leq r^{2}\left\Vert x\right\Vert ^{2} \label{4.6}$$provided $\left\Vert e\right\Vert =1$ and $\left\Vert x-e\right\Vert \leq
r<1.$ The inequality (\[4.6\]) is sharp, as shown in Theorem \[t2.1\], and the theorem is thus proved.
Another companion of the Grüss inequality may be stated as well.
\[t4.2\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an inner product space over $\mathbb{K}$ and $x,y,e\in H$ with $%
\left\Vert e\right\Vert =1.$ Suppose also that $a,A,b,B\in \mathbb{K}$ $%
\left( \mathbb{K}=\mathbb{R},\mathbb{C}\right) $ such that $\func{Re}\left( A%
\overline{a}\right) ,$ $\func{Re}\left( B\overline{b}\right) >0.$ If either$$\func{Re}\left\langle Ae-x,x-ae\right\rangle \geq 0,\ \ \func{Re}%
\left\langle Be-y,y-be\right\rangle \geq 0,\ \label{4.7}$$or, equivalently,$$\left\Vert x-\frac{a+A}{2}e\right\Vert \leq \frac{1}{2}\left\vert
A-a\right\vert ,\ \ \left\Vert y-\frac{b+B}{2}e\right\Vert \leq \frac{1}{2}%
\left\vert B-b\right\vert , \label{4.8}$$holds, then we have the inequality$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert \leq \frac{1}{4}\cdot \frac{%
\left\vert A-a\right\vert \left\vert B-b\right\vert }{\sqrt{\func{Re}\left( A%
\overline{a}\right) \func{Re}\left( B\overline{b}\right) }}\left\vert
\left\langle x,e\right\rangle \left\langle e,y\right\rangle \right\vert .
\label{4.9}$$The constant $\frac{1}{4}$ is best possible.
We know, by (\[4.3\]), that$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert ^{2}\leq \left( \left\Vert
x\right\Vert ^{2}-\left\vert \left\langle x,e\right\rangle \right\vert
^{2}\right) \left( \left\Vert y\right\Vert ^{2}-\left\vert \left\langle
y,e\right\rangle \right\vert ^{2}\right) . \label{4.10}$$If we use Corollary \[c2.3\], then we may state that$$\left\Vert x\right\Vert ^{2}-\left\vert \left\langle x,e\right\rangle
\right\vert ^{2}\leq \frac{1}{4}\cdot \frac{\left\vert A-a\right\vert ^{2}}{%
\func{Re}\left( A\overline{a}\right) }\left\vert \left\langle
x,e\right\rangle \right\vert ^{2} \label{4.11}$$and$$\left\Vert y\right\Vert ^{2}-\left\vert \left\langle y,e\right\rangle
\right\vert ^{2}\leq \frac{1}{4}\cdot \frac{\left\vert B-b\right\vert ^{2}}{%
\func{Re}\left( B\overline{b}\right) }\left\vert \left\langle
y,e\right\rangle \right\vert ^{2}. \label{4.12}$$Utilizing (\[4.10\]) – (\[4.12\]), we deduce$$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert ^{2}\leq \frac{1}{16}\cdot \frac{%
\left\vert A-a\right\vert ^{2}\left\vert B-b\right\vert ^{2}}{\func{Re}%
\left( A\overline{a}\right) \func{Re}\left( B\overline{b}\right) }\left\vert
\left\langle x,e\right\rangle \left\langle e,y\right\rangle \right\vert ^{2},$$which is clearly equivalent to the desired inequality (\[4.9\]).
The sharpness of the constant follows from Corollary \[c2.3\], and we omit the details.
With the assumptions of Theorem \[t4.2\] and if $\left\langle
x,e\right\rangle ,\left\langle y,e\right\rangle \neq 0$ (that is actually the interesting case), one has the inequality$$\left\vert \frac{\left\langle x,y\right\rangle }{\left\langle
x,e\right\rangle \left\langle e,y\right\rangle }-1\right\vert \leq \frac{1}{4%
}\cdot \frac{\left\vert A-a\right\vert \left\vert B-b\right\vert }{\sqrt{%
\func{Re}\left( A\overline{a}\right) \func{Re}\left( B\overline{b}\right) }}.
\label{4.13}$$The constant $\frac{1}{4}$ is best possible.
The inequality (\[4.9\]) provides a better bound for the quantity $$\left\vert \left\langle x,y\right\rangle -\left\langle x,e\right\rangle
\left\langle e,y\right\rangle \right\vert$$than (2.3) of [@SSD3].
For some recent results on Grüss type inequalities in inner product spaces, see [@SSD0], [@SSD00] and [@PFR].
Reverses of Bessel’s Inequality\[s5\]
=====================================
Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be a real or complex infinite dimensional Hilbert space and $\left( e_{i}\right) _{i\in
\mathbb{N}}$ an orthornormal family in $H$, i.e., we recall that $%
\left\langle e_{i},e_{j}\right\rangle =0$ if $i,j\in \mathbb{N}$, $i\neq j$ and $\left\Vert e_{i}\right\Vert =1$ for $i\in \mathbb{N}$.
It is well known that, if $x\in H,$ then the sum $\sum_{i=1}^{\infty
}\left\vert \left\langle x,e_{i}\right\rangle \right\vert ^{2}$ is convergent and the following inequality, called *Bessel’s inequality*$$\sum_{i=1}^{\infty }\left\vert \left\langle x,e_{i}\right\rangle \right\vert
^{2}\leq \left\Vert x\right\Vert ^{2}, \label{5.1}$$holds.
If $\ell ^{2}\left( \mathbb{K}\right) :=\left\{ \mathbf{a}=\left(
a_{i}\right) _{i\in \mathbb{N}}\subset \mathbb{K}\left\vert
\sum_{i=1}^{\infty }\left\vert a_{i}\right\vert ^{2}\right. <\infty \right\}
,$ where $\mathbb{K}=\mathbb{C}$ or $\mathbb{K}=\mathbb{R}$, is the Hilbert space of all complex or real sequences that are $2$-summable and $\mathbf{%
\lambda }=\left( \lambda _{i}\right) _{i\in \mathbb{N}}\in \ell ^{2}\left(
\mathbb{K}\right) ,$ then the sum $\sum_{i=1}^{\infty }\lambda _{i}e_{i}$ is convergent in $H$ and if $y:=\sum_{i=1}^{\infty }\lambda _{i}e_{i}\in H,$ then $\left\Vert y\right\Vert =\left( \sum_{i=1}^{\infty }\left\vert \lambda
_{i}\right\vert ^{2}\right) ^{\frac{1}{2}}.$
We may state the following result.
\[t5.1\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an infinite dimensional Hilbert space over the real or complex number field $\mathbb{K}$, $\left( e_{i}\right) _{i\in \mathbb{N}}$ an orthornormal family in $H,$ $\mathbf{\lambda }=\left( \lambda _{i}\right) _{i\in \mathbb{N%
}}\in \ell ^{2}\left( \mathbb{K}\right) $ and $r>0$ with the property that$$\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}>r^{2}.
\label{5.2}$$If $x\in H$ is such that$$\left\Vert x-\sum_{i=1}^{\infty }\lambda _{i}e_{i}\right\Vert \leq r,
\label{5.3}$$then we have the inequality$$\begin{aligned}
\left\Vert x\right\Vert ^{2}& \leq \frac{\left( \sum_{i=1}^{\infty }\func{Re}%
\left[ \overline{\lambda _{i}}\left\langle x,e_{i}\right\rangle \right]
\right) ^{2}}{\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert
^{2}-r^{2}} \label{5.4} \\
& \leq \frac{\left\vert \sum_{i=1}^{\infty }\overline{\lambda _{i}}%
\left\langle x,e_{i}\right\rangle \right\vert ^{2}}{\sum_{i=1}^{\infty
}\left\vert \lambda _{i}\right\vert ^{2}-r^{2}} \notag \\
& \leq \frac{\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}}{%
\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}-r^{2}}%
\sum_{i=1}^{\infty }\left\vert \left\langle x,e_{i}\right\rangle \right\vert
^{2}; \notag\end{aligned}$$and$$\begin{aligned}
0 &\leq &\left\Vert x\right\Vert ^{2}-\sum_{i=1}^{\infty }\left\vert
\left\langle x,e_{i}\right\rangle \right\vert ^{2} \label{5.5} \\
&\leq &\frac{r^{2}}{\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert
^{2}-r^{2}}\sum_{i=1}^{\infty }\left\vert \left\langle x,e_{i}\right\rangle
\right\vert ^{2}.\end{aligned}$$
Applying the third inequality in (\[2.2\]) for $a=\sum_{i=1}^{\infty
}\lambda _{i}e_{i}\in H,$ we have$$\left\Vert x\right\Vert ^{2}\left\Vert \sum_{i=1}^{\infty }\lambda
_{i}e_{i}\right\Vert ^{2}-\left[ \func{Re}\left\langle x,\sum_{i=1}^{\infty
}\lambda _{i}e_{i}\right\rangle \right] ^{2}\leq r^{2}\left\Vert
x\right\Vert ^{2} \label{5.6}$$and since$$\begin{aligned}
\left\Vert \sum_{i=1}^{\infty }\lambda _{i}e_{i}\right\Vert ^{2}&
=\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}, \\
\func{Re}\left\langle x,\sum_{i=1}^{\infty }\lambda _{i}e_{i}\right\rangle &
=\sum_{i=1}^{\infty }\func{Re}\left[ \overline{\lambda _{i}}\left\langle
x,e_{i}\right\rangle \right] ,\end{aligned}$$then by (\[5.6\]) we deduce$$\left\Vert x\right\Vert ^{2}\sum_{i=1}^{\infty }\left\vert \lambda
_{i}\right\vert ^{2}-\left[ \func{Re}\left\langle x,\sum_{i=1}^{\infty
}\lambda _{i}e_{i}\right\rangle \right] ^{2}\leq r^{2}\left\Vert
x\right\Vert ^{2},$$giving the first inequality in (\[5.4\]).
The second inequality is obvious by the modulus property.
The last inequality follows by the Cauchy-Bunyakovsky-Schwarz inequality$$\left\vert \sum_{i=1}^{\infty }\overline{\lambda _{i}}\left\langle
x,e_{i}\right\rangle \right\vert ^{2}\leq \sum_{i=1}^{\infty }\left\vert
\lambda _{i}\right\vert ^{2}\sum_{i=1}^{\infty }\left\vert \left\langle
x,e_{i}\right\rangle \right\vert ^{2}.$$The inequality (\[5.5\]) follows by the last inequality in (\[5.4\]) on subtracting in both sides the quantity $\sum_{i=1}^{\infty }\left\vert
\left\langle x,e_{i}\right\rangle \right\vert ^{2}<\infty .$
The following result provides a generalization for the reverse of Bessel’s inequality obtained in [@SSD6].
\[t5.2\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ and $\left( e_{i}\right) _{i\in \mathbb{N}}$ be as in Theorem \[t5.1\]. Suppose that $\mathbf{\Gamma }=\left( \Gamma _{i}\right) _{i\in \mathbb{N}%
}\in \ell ^{2}\left( \mathbb{K}\right) ,$ $\mathbf{\gamma }=\left( \gamma
_{i}\right) _{i\in \mathbb{N}}\in \ell ^{2}\left( \mathbb{K}\right) $ are sequences of real or complex numbers such that$$\sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}\overline{\gamma _{i}}\right)
>0. \label{5.7}$$If $x\in H$ is such that either$$\left\Vert x-\sum_{i=1}^{\infty }\frac{\Gamma _{i}+\gamma _{i}}{2}%
e_{i}\right\Vert \leq \frac{1}{2}\left( \sum_{i=1}^{\infty }\left\vert
\Gamma _{i}-\gamma _{i}\right\vert ^{2}\right) ^{\frac{1}{2}} \label{5.8}$$or, equivalently,$$\func{Re}\left\langle \sum_{i=1}^{\infty }\Gamma
_{i}e_{i}-x,x-\sum_{i=1}^{\infty }\gamma _{i}e_{i}\right\rangle \geq 0
\label{5.9}$$holds, then we have the inequalities$$\begin{aligned}
\left\Vert x\right\Vert ^{2}& \leq \frac{1}{4}\cdot \frac{\left(
\sum_{i=1}^{\infty }\func{Re}\left[ \left( \overline{\Gamma _{i}}+\overline{%
\gamma _{i}}\right) \left\langle x,e_{i}\right\rangle \right] \right) ^{2}}{%
\sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}\overline{\gamma _{i}}\right)
} \label{5.10} \\
& \leq \frac{1}{4}\cdot \frac{\left\vert \sum_{i=1}^{\infty }\left(
\overline{\Gamma _{i}}+\overline{\gamma _{i}}\right) \left\langle
x,e_{i}\right\rangle \right\vert ^{2}}{\sum_{i=1}^{\infty }\func{Re}\left(
\Gamma _{i}\overline{\gamma _{i}}\right) } \notag \\
& \leq \frac{1}{4}\cdot \frac{\sum_{i=1}^{\infty }\left\vert \Gamma
_{i}+\gamma _{i}\right\vert ^{2}}{\sum_{i=1}^{\infty }\func{Re}\left( \Gamma
_{i}\overline{\gamma _{i}}\right) }\sum_{i=1}^{\infty }\left\vert
\left\langle x,e_{i}\right\rangle \right\vert ^{2}. \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible in all inequalities in ([5.10]{}).
We also have the inequalities:$$0\leq \left\Vert x\right\Vert ^{2}-\sum_{i=1}^{\infty }\left\vert
\left\langle x,e_{i}\right\rangle \right\vert ^{2}\leq \frac{1}{4}\cdot
\frac{\sum_{i=1}^{\infty }\left\vert \Gamma _{i}-\gamma _{i}\right\vert ^{2}%
}{\sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}\overline{\gamma _{i}}%
\right) }\sum_{i=1}^{\infty }\left\vert \left\langle x,e_{i}\right\rangle
\right\vert ^{2}. \label{5.11}$$Here the constant $\frac{1}{4}$ is also best possible.
Since $\mathbf{\Gamma }$, $\mathbf{\gamma }\in \ell ^{2}\left( \mathbb{K}%
\right) ,$ then also $\frac{1}{2}\left( \mathbf{\Gamma }+\mathbf{\gamma }%
\right) \in \ell ^{2}\left( \mathbb{K}\right) ,$ showing that the series$$\sum_{i=1}^{\infty }\left\vert \frac{\Gamma _{i}+\gamma _{i}}{2}\right\vert
^{2},\ \sum_{i=1}^{\infty }\left\vert \frac{\Gamma _{i}-\gamma _{i}}{2}%
\right\vert ^{2}\text{ and}\ \sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}%
\overline{\gamma _{i}}\right)$$are convergent. Also, the series $$\sum_{i=1}^{\infty }\Gamma _{i}e_{i},\text{ }\sum_{i=1}^{\infty }\gamma
_{i}e_{i}\text{ and }\sum_{i=1}^{\infty }\frac{\gamma _{i}+\Gamma _{i}}{2}%
e_{i}$$ are convergent in the Hilbert space $H.$
The equivalence of the conditions (\[5.8\]) and (\[5.9\]) follows by the fact that in an inner product space we have, for $x,z,Z\in H,$ $\func{Re}%
\left\langle Z-x,x-z\right\rangle \geq 0$ is equivalent to $\left\Vert x-%
\frac{z+Z}{2}\right\Vert \leq \frac{1}{2}\left\Vert Z-z\right\Vert ,$ and we omit the details.
Now, we observe that the inequalities (\[5.10\]) and (\[5.11\]) follow from Theorem \[t5.1\] on choosing $\lambda _{i}=\frac{\gamma _{i}+\Gamma
_{i}}{2},$ $i\in \mathbb{N}$ and $r=\frac{1}{2}\left( \sum_{i=1}^{\infty
}\left\vert \Gamma _{i}-\gamma _{i}\right\vert ^{2}\right) ^{\frac{1}{2}}.$
The fact that $\frac{1}{4}$ is the best constant in both (\[5.10\]) and (\[5.11\]) follows from Theorem \[t2.2\] and Corollary \[c2.3\], and we omit the details.
Note that (\[5.10\]) improves (\[1.17\]) and (\[5.11\]) improves ([1.18]{}), that have been obtained in [@SSD6].
For some recent results related to Bessel inequality, see [@SSD01], [SSDJS]{}, [@HXC] and [@GH1].
Some Grüss Type Inequalities for Orthonormal Families\[s6\]
===========================================================
The following result related to Grüss inequality in inner product spaces, holds.
\[t6.1\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an infinite dimensional Hilbert space over the real or complex number field $\mathbb{K}$, and $\left( e_{i}\right) _{i\in \mathbb{N}}$ an orthornormal family in $H.$ Assume that $\mathbf{\lambda }=\left( \lambda
_{i}\right) _{i\in \mathbb{N}},\ \mathbf{\mu }=\left( \mu _{i}\right) _{i\in
\mathbb{N}}\in \ell ^{2}\left( \mathbb{K}\right) $ and $r_{1},r_{2}>0$ with the properties that$$\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}>r_{1}^{2},\ \ \
\sum_{i=1}^{\infty }\left\vert \mu _{i}\right\vert ^{2}>r_{2}^{2}.
\label{6.1}$$If $x,y\in H$ are such that$$\left\Vert x-\sum_{i=1}^{\infty }\lambda _{i}e_{i}\right\Vert \leq r_{1},\ \
\ \ \ \ \left\Vert y-\sum_{i=1}^{\infty }\mu _{i}e_{i}\right\Vert \leq r_{2},
\label{6.2}$$then we have the inequalities$$\begin{aligned}
&&\left\vert \left\langle x,y\right\rangle -\sum_{i=1}^{\infty }\left\langle
x,e_{i}\right\rangle \left\langle e_{i},y\right\rangle \right\vert
\label{6.3} \\
&\leq &\frac{r_{1}r_{2}}{\sqrt{\sum_{i=1}^{\infty }\left\vert \lambda
_{i}\right\vert ^{2}-r_{1}^{2}}\sqrt{\sum_{i=1}^{\infty }\left\vert \mu
_{i}\right\vert ^{2}-r_{2}^{2}}}\cdot \sqrt{\sum_{i=1}^{\infty }\left\vert
\left\langle x,e_{i}\right\rangle \right\vert ^{2}\sum_{i=1}^{\infty
}\left\vert \left\langle y,e_{i}\right\rangle \right\vert ^{2}} \notag \\
&\leq &\frac{r_{1}r_{2}\left\Vert x\right\Vert \left\Vert y\right\Vert }{%
\sqrt{\sum_{i=1}^{\infty }\left\vert \lambda _{i}\right\vert ^{2}-r_{1}^{2}}%
\sqrt{\sum_{i=1}^{\infty }\left\vert \mu _{i}\right\vert ^{2}-r_{2}^{2}}}.
\notag\end{aligned}$$
Applying Schwarz’s inequality for the vectors $x-\sum_{i=1}^{\infty
}\left\langle x,e_{i}\right\rangle e_{i},$ $y-\sum_{i=1}^{\infty
}\left\langle y,e_{i}\right\rangle e_{i},$ we have$$\begin{gathered}
\left\vert \left\langle x-\sum_{i=1}^{\infty }\left\langle
x,e_{i}\right\rangle e_{i},y-\sum_{i=1}^{\infty }\left\langle
y,e_{i}\right\rangle e_{i}\right\rangle \right\vert ^{2} \label{6.4} \\
\leq \left\Vert x-\sum_{i=1}^{\infty }\left\langle x,e_{i}\right\rangle
e_{i}\right\Vert ^{2}\left\Vert y-\sum_{i=1}^{\infty }\left\langle
y,e_{i}\right\rangle e_{i}\right\Vert ^{2}.\end{gathered}$$Since$$\left\langle x-\sum_{i=1}^{\infty }\left\langle x,e_{i}\right\rangle
e_{i},y-\sum_{i=1}^{\infty }\left\langle y,e_{i}\right\rangle
e_{i}\right\rangle =\left\langle x,y\right\rangle -\sum_{i=1}^{\infty
}\left\langle x,e_{i}\right\rangle \left\langle e_{i},y\right\rangle$$and$$\left\Vert x-\sum_{i=1}^{\infty }\left\langle x,e_{i}\right\rangle
e_{i}\right\Vert ^{2}=\left\Vert x\right\Vert ^{2}-\sum_{i=1}^{\infty
}\left\vert \left\langle x,e_{i}\right\rangle \right\vert ^{2},$$then by (\[5.5\]) applied for $x$ and $y,$ and from (\[6.4\]), we deduce the first part of (\[6.3\]).
The second part follows by Bessel’s inequality.
The following Grüss type inequality may be stated as well.
\[t6.2\]Let $\left( H;\left\langle \cdot ,\cdot \right\rangle \right) $ be an infinite dimensional Hilbert space and $\left( e_{i}\right) _{i\in
\mathbb{N}}$ an orthornormal family in $H.$ Suppose that $\left( \Gamma
_{i}\right) _{i\in \mathbb{N}},$ $\left( \gamma _{i}\right) _{i\in \mathbb{N}%
},$ $\left( \phi _{i}\right) _{i\in \mathbb{N}},$ $\left( \Phi _{i}\right)
_{i\in \mathbb{N}}\in \ell ^{2}\left( \mathbb{K}\right) $ are sequences of real and complex numbers such that$$\sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}\overline{\gamma _{i}}\right)
>0,\ \ \ \sum_{i=1}^{\infty }\func{Re}\left( \Phi _{i}\overline{\phi _{i}}%
\right) >0. \label{6.5}$$If $x,y\in H$ are such that either$$\begin{aligned}
\left\Vert x-\sum_{i=1}^{\infty }\frac{\Gamma _{i}+\gamma _{i}}{2}\cdot
e_{i}\right\Vert & \leq \frac{1}{2}\left( \sum_{i=1}^{\infty }\left\vert
\Gamma _{i}-\gamma _{i}\right\vert ^{2}\right) ^{\frac{1}{2}} \label{6.6} \\
\left\Vert y-\sum_{i=1}^{\infty }\frac{\Phi _{i}+\phi _{i}}{2}\cdot
e_{i}\right\Vert & \leq \frac{1}{2}\left( \sum_{i=1}^{\infty }\left\vert
\Phi _{i}-\phi _{i}\right\vert ^{2}\right) ^{\frac{1}{2}} \notag\end{aligned}$$or, equivalently,$$\begin{aligned}
\func{Re}\left\langle \sum_{i=1}^{\infty }\Gamma
_{i}e_{i}-x,x-\sum_{i=1}^{\infty }\gamma _{i}e_{i}\right\rangle & \geq 0,
\label{6.7} \\
\func{Re}\left\langle \sum_{i=1}^{\infty }\Phi
_{i}e_{i}-y,y-\sum_{i=1}^{\infty }\phi _{i}e_{i}\right\rangle & \geq 0,
\notag\end{aligned}$$holds, then we have the inequality$$\begin{aligned}
& \left\vert \left\langle x,y\right\rangle -\sum_{i=1}^{\infty }\left\langle
x,e_{i}\right\rangle \left\langle e_{i},y\right\rangle \right\vert
\label{6.8} \\
& \leq \frac{1}{4}\cdot \frac{\left( \sum_{i=1}^{\infty }\left\vert \Gamma
_{i}-\gamma _{i}\right\vert ^{2}\right) ^{\frac{1}{2}}\left(
\sum_{i=1}^{\infty }\left\vert \Phi _{i}-\phi _{i}\right\vert ^{2}\right) ^{%
\frac{1}{2}}}{\left( \sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}%
\overline{\gamma _{i}}\right) \right) ^{\frac{1}{2}}\left(
\sum_{i=1}^{\infty }\func{Re}\left( \Phi _{i}\overline{\phi _{i}}\right)
\right) ^{\frac{1}{2}}} \notag \\
& \times \left( \sum_{i=1}^{\infty }\left\vert \left\langle
x,e_{i}\right\rangle \right\vert ^{2}\right) ^{\frac{1}{2}}\left(
\sum_{i=1}^{\infty }\left\vert \left\langle y,e_{i}\right\rangle \right\vert
^{2}\right) ^{\frac{1}{2}} \notag \\
& \leq \frac{1}{4}\cdot \frac{\left( \sum_{i=1}^{\infty }\left\vert \Gamma
_{i}-\gamma _{i}\right\vert ^{2}\right) ^{\frac{1}{2}}\left(
\sum_{i=1}^{\infty }\left\vert \Phi _{i}-\phi _{i}\right\vert ^{2}\right) ^{%
\frac{1}{2}}}{\left[ \sum_{i=1}^{\infty }\func{Re}\left( \Gamma _{i}%
\overline{\gamma _{i}}\right) \right] ^{\frac{1}{2}}\left[
\sum_{i=1}^{\infty }\func{Re}\left( \Phi _{i}\overline{\phi _{i}}\right) %
\right] ^{\frac{1}{2}}}\left\Vert x\right\Vert \left\Vert y\right\Vert
\notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible in the first inequality.
Follows by (\[5.11\]) and (\[6.4\]).
The best constant follows from Theorem \[t4.2\], and we omit the details.
We note that the inequality (\[6.8\]) is better than the inequality (3.3) in [@SSD6]. We omit the details.
Integral Inequalities\[s7\]
===========================
Let $\left( \Omega ,\Sigma ,\mu \right) $ be a measurable space consisting of a set $\Omega ,$ a $\sigma -$algebra of parts $\Sigma $ and a countably additive and positive measure $\mu $ on $\Sigma $ with values $\mathbb{R\cup
}\left\{ \infty \right\} .$ Let $\rho \geq 0$ be a $g-$measurable function on $\Omega $ with $\int_{\Omega }\rho \left( s\right) d\mu \left( s\right)
=1.$ Denote by $L_{\rho }^{2}\left( \Omega ,\mathbb{K}\right) $ the Hilbert space of all real or complex valued functions defined on $\Omega $ and $%
2-\rho -$integrable on $\Omega ,$ i.e.,$$\int_{\Omega }\rho \left( s\right) \left\vert f\left( s\right) \right\vert
^{2}d\mu \left( s\right) <\infty . \label{7.1}$$It is obvious that the following inner product$$\left\langle f,g\right\rangle _{\rho }:=\int_{\Omega }\rho \left( s\right)
f\left( s\right) \overline{g\left( s\right) }d\mu \left( s\right) ,
\label{7.2}$$generates the norm $\left\Vert f\right\Vert _{\rho }:=\left( \int_{\Omega
}\rho \left( s\right) \left\vert f\left( s\right) \right\vert ^{2}d\mu
\left( s\right) \right) ^{\frac{1}{2}}$ of $L_{\rho }^{2}\left( \Omega ,%
\mathbb{K}\right) ,$ and all the above results may be stated for integrals.
It is important to observe that, if $$\func{Re}\left[ f\left( s\right) \overline{g\left( s\right) }\right] \geq 0%
\text{ \ for }\mu -\text{a.e. }s\in \Omega , \label{7.3}$$then, obviously,$$\begin{aligned}
\func{Re}\left\langle f,g\right\rangle _{\rho }& =\func{Re}\left[
\int_{\Omega }\rho \left( s\right) f\left( s\right) \overline{g\left(
s\right) }d\mu \left( s\right) \right] \label{7.4} \\
& =\int_{\Omega }\rho \left( s\right) \func{Re}\left[ f\left( s\right)
\overline{g\left( s\right) }\right] d\mu \left( s\right) \geq 0. \notag\end{aligned}$$The reverse is evidently not true in general.
Moreover, if the space is real, i.e., $\mathbb{K=R}$, then a sufficient condition for (\[7.4\]) to hold is:$$f\left( s\right) \geq 0,\ \ g\left( s\right) \geq 0\text{ \ for }\mu -\text{%
a.e. }s\in \Omega . \label{7.5}$$
We provide now, by the use of certain result obtained in Section \[s2\], some integral inequalities that may be used in practical applications.
\[p7.1\]Let $f,g\in L_{\rho }^{2}\left( \Omega ,\mathbb{K}\right) $ and $%
r>0$ with the properties that$$\left\vert f\left( s\right) -g\left( s\right) \right\vert \leq r\leq
\left\vert g\left( s\right) \right\vert \ \text{\ for }\mu -\text{a.e. }s\in
\Omega , \label{7.6}$$and $\int_{\Omega }\rho \left( s\right) \left\vert g\left( s\right)
\right\vert ^{2}d\mu \left( s\right) \neq r.$ Then we have the inequalities$$\begin{aligned}
0& \leq \int_{\Omega }\rho \left( s\right) \left\vert f\left( s\right)
\right\vert ^{2}d\mu \left( s\right) \int_{\Omega }\rho \left( s\right)
\left\vert g\left( s\right) \right\vert ^{2}d\mu \left( s\right) -\left\vert
\int_{\Omega }\rho \left( s\right) f\left( s\right) \overline{g\left(
s\right) }d\mu \left( s\right) \right\vert ^{2} \label{7.7} \\
& \leq \int_{\Omega }\rho \left( s\right) \left\vert f\left( s\right)
\right\vert ^{2}d\mu \left( s\right) \int_{\Omega }\rho \left( s\right)
\left\vert g\left( s\right) \right\vert ^{2}d\mu \left( s\right) \notag \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\left[ \int_{\Omega
}\rho \left( s\right) \func{Re}\left( f\left( s\right) \overline{g\left(
s\right) }\right) d\mu \left( s\right) \right] ^{2} \notag \\
& \leq r^{2}\int_{\Omega }\rho \left( s\right) \left\vert g\left( s\right)
\right\vert ^{2}d\mu \left( s\right) . \notag\end{aligned}$$The constant $c=1$ in front of $r^{2}$ is best possible.
The proof follows by Theorem \[t2.1\] and we omit the details.
\[p7.2\]Let $f,g\in L_{\rho }^{2}\left( \Omega ,\mathbb{K}\right) $ and $%
\gamma ,\Gamma \in \mathbb{K}$ such that $\func{Re}\left( \Gamma \overline{%
\gamma }\right) >0$ and$$\func{Re}\left[ \left( \Gamma g\left( s\right) -f\left( s\right) \right)
\left( \overline{f\left( s\right) }-\overline{\gamma }\overline{g\left(
s\right) }\right) \right] \geq 0\text{ \ for }\mu -\text{a.e. }s\in \Omega .
\label{7.8}$$Then we have the inequalities$$\begin{aligned}
& \int_{\Omega }\rho \left( s\right) \left\vert f\left( s\right) \right\vert
^{2}d\mu \left( s\right) \int_{\Omega }\rho \left( s\right) \left\vert
g\left( s\right) \right\vert ^{2}d\mu \left( s\right) \label{7.9} \\
& \leq \frac{1}{4}\cdot \frac{\left\{ \func{Re}\left[ \left( \overline{%
\Gamma }+\overline{\gamma }\right) \int_{\Omega }\rho \left( s\right)
f\left( s\right) \overline{g\left( s\right) }d\mu \left( s\right) \right]
\right\} ^{2}}{\func{Re}\left( \Gamma \overline{\gamma }\right) } \notag \\
& \leq \frac{1}{4}\cdot \frac{\left\vert \Gamma +\gamma \right\vert ^{2}}{%
\func{Re}\left( \Gamma \overline{\gamma }\right) }\left\vert \int_{\Omega
}\rho \left( s\right) f\left( s\right) \overline{g\left( s\right) }d\mu
\left( s\right) \right\vert ^{2}. \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible in both inequalities.
The proof follows by Theorem \[t2.2\] and we omit the details.
\[c7.3\]With the assumptions of Proposition \[p7.2\], we have the inequality$$\begin{aligned}
0& \leq \int_{\Omega }\rho \left( s\right) \left\vert f\left( s\right)
\right\vert ^{2}d\mu \left( s\right) \int_{\Omega }\rho \left( s\right)
\left\vert g\left( s\right) \right\vert ^{2}d\mu \left( s\right)
\label{7.10} \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\left\vert \int_{\Omega
}\rho \left( s\right) f\left( s\right) \overline{g\left( s\right) }d\mu
\left( s\right) \right\vert ^{2} \notag \\
& \leq \frac{1}{4}\cdot \frac{\left\vert \Gamma -\gamma \right\vert ^{2}}{%
\func{Re}\left( \Gamma \overline{\gamma }\right) }\left\vert \int_{\Omega
}\rho \left( s\right) f\left( s\right) \overline{g\left( s\right) }d\mu
\left( s\right) \right\vert ^{2}. \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible.
If the space is real and we assume, for $M>m>0,$ that$$mg\left( s\right) \leq f\left( s\right) \leq Mg\left( s\right) \text{ \ for }%
\mu -\text{a.e. }s\in \Omega , \label{7.11}$$then, by (\[7.9\]) and (\[7.10\]), we deduce the inequalities$$\begin{gathered}
\int_{\Omega }\rho \left( s\right) \left[ f\left( s\right) \right] ^{2}d\mu
\left( s\right) \int_{\Omega }\rho \left( s\right) \left[ g\left( s\right) %
\right] ^{2}d\mu \left( s\right) \label{7.12} \\
\leq \frac{1}{4}\cdot \frac{\left( M+m\right) ^{2}}{mM}\left[ \int_{\Omega
}\rho \left( s\right) f\left( s\right) g\left( s\right) d\mu \left( s\right) %
\right] ^{2}.\end{gathered}$$and $$\begin{aligned}
0& \leq \int_{\Omega }\rho \left( s\right) \left[ f\left( s\right) \right]
^{2}d\mu \left( s\right) \int_{\Omega }\rho \left( s\right) \left[ g\left(
s\right) \right] ^{2}d\mu \left( s\right) \label{7.13} \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\left[ \int_{\Omega }\rho \left(
s\right) f\left( s\right) g\left( s\right) d\mu \left( s\right) \right] ^{2}
\notag \\
& \leq \frac{1}{4}\cdot \frac{\left( M-m\right) ^{2}}{mM}\left[ \int_{\Omega
}\rho \left( s\right) f\left( s\right) g\left( s\right) d\mu \left( s\right) %
\right] ^{2}. \notag\end{aligned}$$The inequality (\[7.12\]) is known in the literature as Cassel’s inequality.
The following Grüss type integral inequality for real or complex-valued functions also holds.
\[p.7.3\]Let $f,g,h\in L_{\rho }^{2}\left( \Omega ,\mathbb{K}\right) $ with $\int_{\Omega }\rho \left( s\right) \left\vert h\left( s\right)
\right\vert ^{2}d\mu \left( s\right) =1$ and $a,A,b,B\in \mathbb{K}$ such that $\func{Re}\left( A\overline{a}\right) ,\func{Re}\left( B\overline{b}%
\right) >0$ and$$\begin{aligned}
\func{Re}\left[ \left( Ah\left( s\right) -f\left( s\right) \right) \left(
\overline{f\left( s\right) }-\overline{a}\overline{h\left( s\right) }\right) %
\right] &\geq &0, \\
\func{Re}\left[ \left( Ah\left( s\right) -g\left( s\right) \right) \left(
\overline{g\left( s\right) }-\overline{b}\overline{h\left( s\right) }\right) %
\right] &\geq &0\text{,}\end{aligned}$$for $\mu -$a.e. $s\in \Omega .$ Then we have the inequalities$$\begin{aligned}
& \left\vert \int_{\Omega }\rho \left( s\right) f\left( s\right) \overline{%
g\left( s\right) }d\mu \left( s\right) -\int_{\Omega }\rho \left( s\right)
f\left( s\right) \overline{h\left( s\right) }d\mu \left( s\right)
\int_{\Omega }\rho \left( s\right) h\left( s\right) \overline{g\left(
s\right) }d\mu \left( s\right) \right\vert \\
& \leq \frac{1}{4}\cdot \frac{\left\vert A-a\right\vert \left\vert
B-b\right\vert }{\sqrt{\func{Re}\left( A\overline{a}\right) \func{Re}\left( B%
\overline{b}\right) }}\left\vert \int_{\Omega }\rho \left( s\right) f\left(
s\right) \overline{h\left( s\right) }d\mu \left( s\right) \int_{\Omega }\rho
\left( s\right) h\left( s\right) \overline{g\left( s\right) }d\mu \left(
s\right) \right\vert \notag\end{aligned}$$The constant $\frac{1}{4}$ is best possible.
The proof follows by Theorem \[t4.2\].
All the other inequalities in Sections \[s3\] – \[s6\] may be used in a similar way to obtain the corresponding integral inequalities. We omit the details.
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|
---
author:
- 'Shervin Minaee, Yuri Boykov, Fatih Porikli, Antonio Plaza, Nasser Kehtarnavaz, and Demetri Terzopoulos'
title: |
Image Segmentation Using Deep Learning:\
A Survey
---
segmentation is an essential component in many visual understanding systems. It involves partitioning images (or video frames) into multiple segments or objects [@vision_book1]. Segmentation plays a central role in a broad range of applications [@vision_book2], including medical image analysis (e.g., tumor boundary extraction and measurement of tissue volumes), autonomous vehicles (e.g., navigable surface and pedestrian detection), video surveillance, and augmented reality to count a few. Numerous image segmentation algorithms have been developed in the literature, from the earliest methods, such as thresholding [@otsu1979threshold], histogram-based bundling, region-growing [@region_grow], k-means clustering [@seg_clus], watersheds [@najman1994watershed], to more advanced algorithms such as active contours [@Snakes], graph cuts [@graphcut], conditional and Markov random fields [@plath2009multi], and sparsity-based [@seg_sparse]-[@seg_sparse2] methods. Over the past few years, however, deep learning (DL) networks have yielded a new generation of image segmentation models with remarkable performance improvements —often achieving the highest accuracy rates on popular benchmarks— resulting in what many regard as a paradigm shift in the field. For example, Figure \[intro\_sample\_results\] presents sample image segmentation outputs of a prominent deep learning model, DeepLabv3 [@deeplabv3].
![Segmentation results of DeepLabV3 [@deeplabv3] on sample images.[]{data-label="intro_sample_results"}](intro_sample_results.pdf){width="0.98\linewidth"}
Image segmentation can be formulated as a classification problem of pixels with semantic labels (semantic segmentation) or partitioning of individual objects (instance segmentation). Semantic segmentation performs pixel-level labeling with a set of object categories (e.g., human, car, tree, sky) for all image pixels, thus it is generally a harder undertaking than image classification, which predicts a single label for the entire image. Instance segmentation extends semantic segmentation scope further by detecting and delineating each object of interest in the image (e.g., partitioning of individual persons).
Our survey covers the most recent literature in image segmentation and discusses more than a hundred deep learning-based segmentation methods proposed until 2019. We provide a comprehensive review and insights on different aspects of these methods, including the training data, the choice of network architectures, loss functions, training strategies, and their key contributions. We present a comparative summary of the performance of the reviewed methods and discuss several challenges and potential future directions for deep learning-based image segmentation models.
We group deep learning-based works into the following categories based on their main technical contributions:
1. Fully convolutional networks
2. Convolutional models with graphical models
3. Encoder-decoder based models
4. Multi-scale and pyramid network based models
5. R-CNN based models (for instance segmentation)
6. Dilated convolutional models and DeepLab family
7. Recurrent neural network based models
8. Attention-based models
9. Generative models and adversarial training
10. Convolutional models with active contour models
11. Other models
Some the key contributions of this survey paper can be summarized as follows:
- This survey covers the contemporary literature with respect to segmentation problem, and overviews more than 100 segmentation algorithms proposed till 2019, grouped into 10 categories.
- We provide a comprehensive review and an insightful analysis of different aspects of segmentation algorithms using deep learning, including the training data, the choice of network architectures, loss functions, training strategies, and their key contributions.
- We provide an overview of around 20 popular image segmentation datasets, grouped into 2D, 2.5D (RGB-D), and 3D images.
- We provide a comparative summary of the properties and performance of the reviewed methods for segmentation purposes, on popular benchmarks.
- We provide several challenges and potential future directions for deep learning-based image segmentation.
The remainder of this survey is organized as follows: Section \[sec:DNNs\] provides an overview of popular deep neural network architectures that serve as the backbone of many modern segmentation algorithms. Section \[sec:dl-models\] provides a comprehensive overview of the most significant state-of-the-art deep learning based segmentation models, more than 100 till 2019. We also discuss their strengths and contributions over previous works here. Section \[sec:datasets\] reviews some of the most popular image segmentation datasets and their characteristics. Section \[sec:metrics\] reviews popular metrics for evaluating deep-learning-based segmentation models. In Section \[sec:quant\_result\], we report the quantitative results and experimental performance of these models. In Section \[sec:challenges\], we discuss the main challenges and future directions for deep learning-based segmentation methods. Finally, we present our conclusions in Section \[sec:conclusions\].
Overview of Deep Neural Networks {#sec:DNNs}
================================
This section provides an overview of some of the most prominent deep learning architectures used by the computer vision community, including convolutional neural networks (CNNs) [@CNN], recurrent neural networks (RNNs) and long short term memory (LSTM) [@lstm], encoder-decoders [@DL_book], and generative adversarial networks (GANs) [@GAN]. With the popularity of deep learning in recent years, several other deep neural architectures have been proposed, such as transformers, capsule networks, gated recurrent units, spatial transformer networks, etc., which will not be covered here.
Convolutional Neural Networks (CNNs)
------------------------------------
CNNs are among the most successful and widely used architectures in the deep learning community, especially for computer vision tasks. CNNs were initially proposed by Fukushima in his seminal paper on the “Neocognitron” [@neocog], based on the hierarchical receptive field model of the visual cortex proposed by Hubel and Wiesel. Subsequently, Waibel *et al.* [@Waibel] introduced CNNs with weights shared among temporal receptive fields and backpropagation training for phoneme recognition, and LeCun *et al.* [@CNN] developed a CNN architecture for document recognition (Figure \[fig:CNN\_arch\]).
![Architecture of convolutional neural networks. From [@CNN].[]{data-label="fig:CNN_arch"}](Figures.pdf){width="0.98\linewidth"}
CNNs mainly consist of three type of layers: i) convolutional layers, where a kernel (or filter) of weights is convolved in order to extract features; ii) nonlinear layers, which apply an activation function on feature maps (usually element-wise) in order to enable the modeling of non-linear functions by the network; and iii) pooling layers, which replace a small neighborhood of a feature map with some statistical information (mean, max, etc.) about the neighborhood and reduce spatial resolution. The units in layers are locally connected; that is, each unit receives weighted inputs from a small neighborhood, known as the receptive field, of units in the previous layer. By stacking layers to form multi-resolution pyramids, the higher-level layers learn features from increasingly wider receptive fields. The main computational advantage of CNNs is that all the receptive fields in a layer share weights, resulting in a significantly smaller number of parameters than fully-connected neural networks. Some of the most well-known CNN architectures include: AlexNet [@alexnet], VGGNet [@vggnet], ResNet [@resnet], GoogLeNet [@googlenet], MobileNet [@mobilenet], and DenseNet [@densenet].
Recurrent Neural Networks (RNNs) and the LSTM
---------------------------------------------
RNNs [@RNN] are widely used to process sequential data, such as speech, text, videos, and time-series, where data at any given time/position depends on previously encountered data. At each time-stamp the model collects the input from the current time $X_i$ and the hidden state from the previous step $h_{i-1}$, and outputs a target value and a new hidden state (Figure \[fig:RNN\_arch\]).
![Architecture of a simple recurrent neural network.[]{data-label="fig:RNN_arch"}](img/RNN_arch.pdf){width="0.9\linewidth"}
RNNs are typically problematic with long sequences as they cannot capture long-term dependencies in many real-world applications (although they exhibit no theoretical limitations in this regard) and often suffer from gradient vanishing or exploding problems. However, a type of RNNs called Long Short Term Memory (LSTM) [@lstm] is designed to avoid these issues. The LSTM architecture (Figure \[fig:lstm\_model\]) includes three gates (input gate, output gate, forget gate), which regulate the flow of information into and out from a memory cell, which stores values over arbitrary time intervals.
![Architecture of a standard LSTM module. Courtesy of Karpathy.[]{data-label="fig:lstm_model"}](lstm_module.png){width="0.99\linewidth"}
The relationship between input, hidden states, and different gates is given by: $$\begin{aligned}
f_t&= \sigma (\textbf{W}^{(f)} x_t+\textbf{U}^{(f)} h_{t-1}+ b^{(f)}), \hspace{2.73cm} \\
i_t&= \sigma (\textbf{W}^{(i)} x_t+\textbf{U}^{(i)} h_{t-1}+ b^{(i)}), \hspace{2.94cm} \\
o_t&= \sigma (\textbf{W}^{(o)} x_t+\textbf{U}^{(o)} h_{t-1}+ b^{(o)}), \hspace{2.78cm} \\
c_t&= f_t \odot c_{t-1}+ i_t \odot \text{tanh} (\textbf{W}^{(c)} x_t+\textbf{U}^{(c)} h_{t-1}+ b^{(c)} ), \\
h_t&= o_t \odot \text{tanh}(c_t)
\end{aligned}$$ where $x_t \in R^d$ is the input at time-step $t$, and $d$ denotes the feature dimension for each word, $\sigma$ denotes the element-wise sigmoid function (to map the values within $[0,1]$), $\odot$ denotes the element-wise product, and $c_t$ denotes the memory cell designed to lower the risk of vanishing/exploding gradient (and therefore enabling learning of dependencies over larger periods of time, feasible with traditional RNNs). The forget gate, $f_t$, is intended to reset the memory cell. $i_t$ and $o_t$ denote the input and output gates, respectively, and essentially control the input and output of the memory cell.
Encoder-Decoder and Auto-Encoder Models
---------------------------------------
Encoder-Decoder models are a family of models which learn to map data-points from an input domain to an output domain via a two-stage network: The encoder, represented by an encoding function $z=f(x)$, compresses the input into a latent-space representation; the decoder, $y=g(z)$, aims to predict the output from the latent space representation. The latent representation here essentially refers to a feature (vector) representation, which is able to capture the underlying semantic information of the input that is useful for predicting the output. These models are extremely popular in image-to-image translation problems, as well as for sequence models in NLP. Figure \[fig:autoencoder\] illustrates the block-diagram of a simple encoder-decoder model. These models are usually trained by minimizing the reconstruction loss $L(y,\hat{y})$, which measures the differences between the ground-truth output $y$ and the subsequent reconstruction $\hat{y}$. The output here could be an enhanced version of the image (such as in image de-blurring, or super-resolution), or a segmentation map.
![The architecture of a simple encoder-decoder model.[]{data-label="fig:autoencoder"}](encdec.pdf){width="0.99\linewidth"}
Auto-encoders are special case of encoder-decoder models in which the input and output are the same. Several variations of auto-encoders have been proposed. One of the most popular is the stacked denoising auto-encoder (SDAE) [@stac_ae], which stacks several auto-encoders and uses them for image denoising purposes. Another popular variant is the variational auto-encoder (VAE) [@vr_ae], which imposes a prior distribution on the latent representation. VAEs are able to generate realistic samples from a given data distribution. Another variant is adversarial auto-encoders, which introduces an adversarial loss on the latent representation to encourage them to approximate a prior distribution.
Auto-encoders are a family of neural networks for learning efficient data encoding in an unsupervised manner [@DL_book]. They achieve this task by compressing the input data into a latent-space representation, and then reconstructing the output (usually the same as the input) from this representation. Auto-encoders comprise two parts (Figure \[fig:autoencoder\]): The encoder, represented by an encoding function $z=f(x)$, compresses the input into a latent-space representation; the decoder, $y=g(z)$, aims to reconstruct the input from the latent space representation. Auto-encoders are usually trained by minimizing the reconstruction error $L(x,\hat{x})$, which measures the differences between the original input $x$ and the subsequent reconstruction $\hat{x}$. The mean squared error and mean absolute deviation are popular choices for the reconstruction loss.
![architecture of a standard auto-encoder model.[]{data-label="fig:autoencoder"}](AE.pdf){width="0.99\linewidth"}
Several variations of auto-encoders have been proposed. One of the most popular is the stacked denoising auto-encoder (SDAE) [@stac_ae], which stacks several auto-encoders and uses them for image denoising purposes. Another popular variant is the variational auto-encoder (VAE) [@vr_ae], which imposes a prior distribution on the latent representation. VAEs are able to generate realistic samples from a given data distribution. Another variant is adversarial auto-encoders, which introduce an adversarial loss on the latent representation to encourage them to approximate a prior distribution.
Generative Adversarial Networks (GANs)
--------------------------------------
GANs are a newer family of deep learning models [@GAN]. They consist of two networks—a generator and a discriminator (Figure \[fig:gen\_arch\]). The generator network $G= z \rightarrow
y$ in the conventional GAN learns a mapping from noise $z$ (with a prior distribution) to a target distribution $y$, which is similar to the “real” samples. The discriminator network $D$ attempts to distinguish the generated samples (“fakes”) from the “real” ones. The GAN loss function may be written as $\mathcal{L}_\text{GAN}=
\mathbb{E}_{x \sim p_\text{data}(x)}[\log D(x)]+ \mathbb{E}_{z \sim
p_z(z)}[\log(1-D(G(z)))]$. We can regard the GAN as a minimax game between $G$ and $D$, where $D$ is trying to minimize its classification error in distinguishing fake samples from real ones, hence maximizing the loss function, and $G$ is trying to maximize the discriminator network’s error, hence minimizing the loss function. After training the model, the trained generator model would be $G^*=
\text{arg} \ \min_G \max_D \ \mathcal{L}_\text{GAN}$ In practice, this function may not provide enough gradient for effectively training $G$, specially initially (when $D$ can easily discriminate fake samples from real ones). Instead of minimizing $\mathbb{E}_{z \sim
p_z(z)}[\log(1-D(G(z)))]$, a possible solution is to train it to maximize $\mathbb{E}_{z \sim p_z(z)}[\log(D(G(z)))]$.
![Architecture of a generative adversarial network.[]{data-label="fig:gen_arch"}](GANFig.pdf){width="0.7\linewidth"}
Since the invention of GANs, researchers have endeavored to improve/modify GANs several ways. For example, Radford *et al.* [@dc-gan] proposed a convolutional GAN model, which works better than fully-connected networks when used for image generation. Mirza [@con-gan] proposed a conditional GAN model that can generate images conditioned on class labels, which enables one to generate samples with specified labels. Arjovsky *et al.* [@was-gan] proposed a new loss function based on the Wasserstein (a.k.a. earth mover’s distance) to better estimate the distance for cases in which the distribution of real and generated samples are non-overlapping (hence the Kullback–Leiber divergence is not a good measure of the distance). For additional works, we refer the reader to [@GanZoo].
Transfer Learning
-----------------
In some cases the DL-models can be trained from scratch on new applications/datasets (assuming a sufficient quantity of labeled training data), but in many cases there are not enough labeled data available to train a model from scratch and one can use **transfer learning** to tackle this problem. In transfer learning, a model trained on one task is re-purposed on another (related) task, usually by some adaptation process toward the new task. For example, one can imagine adapting an image classification model trained on ImageNet to a different task, such as texture classification, or face recognition. In image segmentation case, many people use a model trained on ImageNet (a larger dataset than most of image segmentation datasets), as the encoder part of the network, and re-train their model from those initial weights. The assumption here is that those pre-trained models should be able to capture the semantic information of the image required for segmentation, and therefore enabling them to train the model with less labeled samples.
DL-Based Image Segmentation Models {#sec:dl-models}
==================================
This section provides a detailed review of more than a hundred deep learning-based segmentation methods proposed until 2019, grouped into 10 categories. It is worth mentioning that there are some pieces that are common among many of these works, such as having encoder and decoder parts, skip-connections, multi-scale analysis, and more recently the use of dilated convolution. Because of this, it is difficult to mention the unique contributions of each work, but easier to group them based on their underlying architectural contribution over previous works.
In recent years many techniques have been developed for image segmentation using deep learning models. Although it is hard to cover all of them in this survey paper, we do our best to provide an overview of more than 100 promising deep learning based works for image segmentation, with a focus on semantic segmentation, and some also a few promising works for instance segmentation, and medical image segmentation. The goal of semantic segmentation is to learn a pixel-wise classification for images. The main advantage of deep learning models is the ability to learn a semantic feature representations which is more discriminative than classical approaches for segmentation, in an end-to-end fashion. This is in contrast with the hand-crafted features developed by computer vision experts, which may work on one dataset, but not necessarily on a different one.
Given the wide range of deep architectures used for image segmentation purposes, we group them into several categories (based on their main contributions) and, for each category, we introduce some of the most promising works developed in recent years (models which do not fall into any of the considered categories are discussed in a separate subsection).
Fully Convolutional Networks
----------------------------
Long *et al.* [@seg_fcn] proposed one of the first deep learning works for semantic image segmentation, using a fully convolutional network (FCN). An FCN (Figure \[fig:FCN\_blk\]) includes only convolutional layers, which enables it to take an image of arbitrary size and produce a segmentation map of the same size. The authors modified existing CNN architectures, such as VGG16 and GoogLeNet, to manage non-fixed sized input and output, by replacing all fully-connected layers with the fully-convolutional layers. As a result, the model outputs a spatial segmentation map instead of classification scores.
![A fully convolutional image segmentation network. The FCN learns to make dense, pixel-wise predictions. From [@seg_fcn].[]{data-label="fig:FCN_blk"}](seg_fcn.pdf){width="0.6\linewidth"}
Through the use of skip connections in which feature maps from the final layers of the model are up-sampled and fused with feature maps of earlier layers (Figure \[fig:FCN\_blk2\]), the model combines semantic information (from deep, coarse layers) and appearance information (from shallow, fine layers) in order to produce accurate and detailed segmentations. The model was tested on PASCAL VOC, NYUDv2, and SIFT Flow, and achieved state-of-the-art segmentation performance.
![Skip connections combine coarse, high-level information and fine, low-level information. From [@seg_fcn].[]{data-label="fig:FCN_blk2"}](seg_fcn2.pdf){width="0.8\linewidth"}
This work is considered a milestone in image segmentation, demonstrating that deep networks can be trained for semantic segmentation in an end-to-end manner on variable-sized images. However, despite its popularity and effectiveness, the conventional FCN model has some limitations—it is not fast enough for real-time inference, it does not take into account the global context information in an efficient way, and it is not easily transferable to 3D images. Several efforts have attempted to overcome some of the limitations of the FCN.
For instance, Liu *et al.* [@seg_parsenet] proposed a model called ParseNet, to address an issue with FCN—ignoring global context information. ParseNet adds global context to FCNs by using the average feature for a layer to augment the features at each location. The feature map for a layer is pooled over the whole image resulting in a context vector. This context vector is normalized and unpooled to produce new feature maps of the same size as the initial ones. These feature maps are then concatenated. In a nutshell, ParseNet is an FCN with the described module replacing the convolutional layers (Figure \[fig:parse\_blk\]).
![ParseNet, showing the use of extra global context to produce smoother segmentation (d) than an FCN (c). From [@seg_parsenet].[]{data-label="fig:parse_blk"}](parsenet.pdf){width="0.9\linewidth"}
FCNs have been applied to a variety of segmentation problems, such as brain tumor segmentation [@fcn_brain], instance-aware semantic segmentation [@fcn_instance], skin lesion segmentation [@fcn_skin], and iris segmentation [@fcn_iris].
Convolutional Models With Graphical Models {#sec:cnn+crf}
------------------------------------------
As discussed, FCN ignores potentially useful scene-level semantic context. To integrate more context, several approaches incorporate probabilistic graphical models, such as Conditional Random Fields (CRFs) and Markov Random Field (MRFs), into DL architectures.
Chen *et al.* [@deeplab_v1] proposed a semantic segmentation algorithm based on the combination of CNNs and fully connected CRFs (Figure \[fig:cnn\_crf\]). They showed that responses from the final layer of deep CNNs are not sufficiently localized for accurate object segmentation (due to the invariance properties that make CNNs good for high level tasks such as classification). To overcome the poor localization property of deep CNNs, they combined the responses at the final CNN layer with a fully-connected CRF. They showed that their model is able to localize segment boundaries at a higher accuracy rate than it was possible with previous methods.
![A CNN+CRF model. The coarse score map of a CNN is up-sampled via interpolated interpolation, and fed to a fully-connected CRF to refine the segmentation result. From [@deeplab_v1].[]{data-label="fig:cnn_crf"}](cnn_crf.pdf){width="0.8\linewidth"}
Schwing and Urtasun [@seg_dsn] proposed a fully-connected deep structured network for image segmentation. They presented a method that jointly trains CNNs and fully-connected CRFs for semantic image segmentation, and achieved encouraging results on the challenging PASCAL VOC 2012 dataset. In [@CRF-RNN], Zheng et al. proposed a similar semantic segmentation approach integrating CRF with CNN.
In another relevant work, Lin *et al.* [@seg_dsn2] proposed an efficient algorithm for semantic segmentation based on contextual deep CRFs. They explored “patch-patch” context (between image regions) and “patch-background” context to improve semantic segmentation through the use of contextual information.
The feature-map network architecture of this work is shown in Figure \[fig:featmap\].
![FeatMap-Net. An input image is first resized into 3 scales. Then, each resized image goes through 6 convolution blocks to output one feature map. The top 5 convolution blocks are shared for all scales. Every scale has a specific convolution block. 2-level sliding pyramid pooling is performed. From [@seg_dsn2].[]{data-label="fig:featmap"}](featmap.pdf){width="0.8\linewidth"}
Liu *et al.* [@seg_dsn3] proposed a semantic segmentation algorithm that incorporates rich information into MRFs, including high-order relations and mixture of label contexts. Unlike previous works that optimized MRFs using iterative algorithms, they proposed a CNN model, namely a Parsing Network, which enables deterministic end-to-end computation in a single forward pass.
Encoder-Decoder Based Models
----------------------------
Another popular family of deep models for image segmentation is based on the convolutional encoder-decoder architecture. Most of the DL-based segmentation works use some kind of encoder-decoder models. We group these works into two categories, encoder-decoder models for general segmentation, and for medical image segmentation (to better distinguish between applications).
Noh *et al.* [@seg_decon1] published an early paper on semantic segmentation based on deconvolution (a.k.a. transposed convolution). Their model (Figure \[fig:seg\_decon\]) consists of two parts, an encoder using convolutional layers adopted from the VGG 16-layer network and a deconvolutional network that takes the feature vector as input and generates a map of pixel-wise class probabilities. The deconvolution network is composed of deconvolution and unpooling layers, which identify pixel-wise class labels and predict segmentation masks. This network achieved promising performance on the PASCAL VOC 2012 dataset, and obtained the best accuracy (72.5%) among the methods trained with no external data at the time.
![Deconvolutional semantic segmentation. Following a convolution network based on the VGG 16-layer net, is a multi-layer deconvolution network to generate the accurate segmentation map. From [@seg_decon1].[]{data-label="fig:seg_decon"}](seg_decon.pdf){width="0.99\linewidth"}
(Figure \[fig:decon\_fcn\]).
![Example semantic segmentations by deconvolution network on PASCAL VOC 2012, and comparison. From [@seg_decon1].[]{data-label="fig:decon_fcn"}](decon_fcn.pdf){width="0.8\linewidth"}
In another promising work known as SegNet, Badrinarayanan *et al.* [@segnet] proposed a convolutional encoder-decoder architecture for image segmentation (Figure \[fig:segnet\_arc\]). Similar to the deconvolution network, the core trainable segmentation engine of SegNet consists of an encoder network, which is topologically identical to the 13 convolutional layers in the VGG16 network, and a corresponding decoder network followed by a pixel-wise classification layer. The main novelty of SegNet is in the way the decoder upsamples its lower resolution input feature map(s); specifically, it uses pooling indices computed in the max-pooling step of the corresponding encoder to perform non-linear up-sampling. This eliminates the need for learning to up-sample. The (sparse) up-sampled maps are then convolved with trainable filters to produce dense feature maps. SegNet is also significantly smaller in the number of trainable parameters than other competing architectures. A Bayesian version of SegNet was also proposed by the same authors to model the uncertainty inherent to the convolutional encoder-decoder network for scene segmentation [@bayes_segnet].
![SegNet has no fully-connected layers; hence, the model is fully convolutional. A decoder up-samples its input using the transferred pool indices from its encoder to produce a sparse feature map(s). From [@segnet].[]{data-label="fig:segnet_arc"}](segnet.pdf){width="0.8\linewidth"}
Several other works adopt transposed convolutions, or encoder-decoders for image segmentation, such as Stacked Deconvolutional Network (SDN) [@SDN], Linknet [@linknet], W-Net [@wnet], and locality-sensitive deconvolution networks for RGB-D segmentation [@cheng2017locality].
There are several models initially developed for medical/biomedical image segmentation, which are inspired by FCNs and encoder-decoder models. U-Net [@unet], and V-Net [@vnet], are two well-known such architectures, which are now also being used outside the medical domain.
Ronneberger *et al.* [@unet] proposed the U-Net for segmenting biological microscopy images. Their network and training strategy relies on the use of data augmentation to learn from the available annotated images more effectively. The U-Net architecture (Figure \[fig:unet\]) comprises two parts, a contracting path to capture context, and a symmetric expanding path that enables precise localization. The down-sampling or contracting part has a FCN-like architecture that extracts features with $3\times3$ convolutions. The up-sampling or expanding part uses up-convolution (or deconvolution), reducing the number of feature maps while increasing their dimensions. Feature maps from the down-sampling part of the network are copied to the up-sampling part to avoid losing pattern information. Finally, a $1\times1$ convolution processes the feature maps to generate a segmentation map that categorizes each pixel of the input image. U-Net was trained on 30 transmitted light microscopy images, and it won the ISBI cell tracking challenge 2015 by a large margin.
![The U-net model. The blue boxes denote feature map blocks with their indicated shapes. From [@unet].[]{data-label="fig:unet"}](unet.pdf){width="0.95\linewidth"}
Various extensions of U-Net have been developed for different kinds of images. For example, Cicek [@unet3d] proposed a U-Net architecture for 3D images. Zhou *et al.* [@unetplus] developed a nested U-Net architecture. U-Net has also been applied to various other problems. For example, Zhang *et al.* [@unet_road] developed a road segmentation/extraction algorithm based on U-Net. V-Net (Figure \[fig:vnet\]) is another well-known, FCN-based model, which was proposed by Milletari *et al.* [@vnet] for 3D medical image segmentation. For model training, they introduced a new objective function based on the Dice coefficient, enabling the model to deal with situations in which there is a strong imbalance between the number of voxels in the foreground and background. The network was trained end-to-end on MRI volumes depicting prostate, and learns to predict segmentation for the whole volume at once.
![The V-net model for 3D image segmentation. From [@vnet].[]{data-label="fig:vnet"}](vnet.pdf){width="0.85\linewidth"}
Some of the other relevant works on medical image segmentation includes Progressive Dense V-net (PDV-Net) *et al.* for fast and automatic segmentation of pulmonary lobes from chest CT images, and the 3D-CNN encoder for lesion segmentation [@brosch2016deep].
Multi-Scale and Pyramid Network Based Models
--------------------------------------------
Multi-scale analysis, a rather old idea in image processing, has been deployed in various neural network architectures. One of the most prominent models of this sort is the Feature Pyramid Network (FPN) proposed by Lin *et al.* [@fpn], which was developed mainly for object detection but was then also applied to segmentation. The inherent multi-scale, pyramidal hierarchy of deep CNNs was used to construct feature pyramids with marginal extra cost. To merge low and high resolution features, the FPN is composed of a bottom-up pathway, a top-down pathway and lateral connections. The concatenated feature maps are then processed by a $3\times3$ convolution to produce the output of each stage. Finally, each stage of the top-down pathway generates a prediction to detect an object. For image segmentation, the authors use two multi-layer perceptrons (MLPs) to generate the masks. Figure \[fig:fpn\] shows how the lateral connections and the top-down pathway are merged via addition.
![A building block illustrating the lateral connection and the top-down pathway, merged by addition. From [@fpn].[]{data-label="fig:fpn"}](fpn.pdf){width="0.6\linewidth"}
Zhao *et al.* [@pspn] developed the Pyramid Scene Parsing Network (PSPN), a multi-scale network to better learn the global context representation of a scene (Figure \[fig:pspn\]). Different patterns are extracted from the input image using a residual network (ResNet) as a feature extractor, with a dilated network. These feature maps are then fed into a pyramid pooling module to distinguish patterns of different scales. They are pooled at four different scales, each one corresponding to a pyramid level and processed by a $1\times1$ convolutional layer to reduce their dimensions. The outputs of the pyramid levels are up-sampled and concatenated with the initial feature maps to capture both local and global context information. Finally, a convolutional layer is used to generate the pixel-wise predictions.
![The PSPN architecture. A CNN produces the feature map and a pyramid pooling module aggregates the different sub-region representations. Up-sampling and concatenation are used to form the final feature representation from which, the final pixel-wise prediction is obtained through convolution. From [@pspn].[]{data-label="fig:pspn"}](pspn.pdf){width="0.9\linewidth"}
Ghiasi and Fowlkes [@laplace] developed a multi-resolution reconstruction architecture based on a Laplacian pyramid that uses skip connections from higher resolution feature maps and multiplicative gating to successively refine segment boundaries reconstructed from lower-resolution maps. They showed that, while the apparent spatial resolution of convolutional feature maps is low, the high-dimensional feature representation contains significant sub-pixel localization information.
![The Laplacian pyramid reconstruction and refinement network. From [@laplace].[]{data-label="fig:laplace"}](laplace.pdf){width="0.8\linewidth"}
Ye *et al.* [@3d_rnn] developed 3D recurrent neural networks with context fusion for point cloud semantic segmentation. First, an efficient point-wise pyramid pooling module is deployed to capture local structures at various densities by taking multi-scale neighborhood into account. Then the two-directional hierarchical RNNs are utilized to explore long-range spatial dependencies. Each recurrent layer takes as input the local features derived from unrolled cells and sweeps the 3D space along two directions successively to incorporate structure knowledge. This model achieved performance improvement over previous state-of-the-art algorithms on indoor and outdoor 3D datasets. The block-diagram of the architecture in [@3d_rnn] is shown in Figure \[fig:3d\_rnn\].
![Point cloud semantic segmentation approach using 3D-RNNs. The architecture takes as input the unstructured point cloud and outputs point-wise semantic labels. Point-features and local cell features are concatenated and passed through the two-direction RNN module along the $x$ and $y$ directions. The outputs of the first RNN are reorganized and fed to the next RNNs (red). From [@3d_rnn].[]{data-label="fig:3d_rnn"}](3d_rnn.pdf){width="0.8\linewidth"}
In another work [@seg_rnn3], Li *et al.* proposed an image segmentation refinement algorithm based on language descriptions, using a recurrent neural network. The block-diagram of this model is given in Figure \[fig:DARNN\].
![The overall architecture of the Recurrent Refinement Network (RRN) designed for referring image segmentation. From [@seg_rnn3].[]{data-label="fig:seg_rnn3"}](seg_rnn3.pdf){width="0.6\linewidth"}
An example aimed at illustrating the amount of improvement that can be obtained by applying refinement is shown in Figure \[fig:seg\_query\].
![Segmentation masks generated by different models for the query “zebra right side”. Notice how the model gradually refines the segmentation mask by combining features at different scales. From [@seg_rnn3].[]{data-label="fig:seg_query"}](seg_query.pdf){width="0.8\linewidth"}
They utilize the feature pyramids inherently present in CNNs to capture the semantics at different scales. To produce a suitable information flow through the feature hierarchy path, they deployed an RNN that takes pyramidal features as input to refine the segmentation mask progressively.
There are other models using multi-scale analysis for segmentation, such as DM-Net (Dynamic Multi-scale Filters Network) [@dmsf], Context contrasted network and gated multi-scale aggregation (CCN) [@CCL], Adaptive Pyramid Context Network (APC-Net) [@apcnet], Multi-scale context intertwining (MSCI) [@MSCI], and salient object segmentation [@salient_seg].
R-CNN Based Models (for Instance Segmentation)
----------------------------------------------
The regional convolutional network (R-CNN) and its extensions (Fast R-CNN, Faster R-CNN, Maksed-RCNN) have proven successful in object detection applications. Some of the extensions of R-CNN have been heavily used to address the instance segmentation problem; i.e., the task of simultaneously performing object detection and semantic segmentation. In particular, the Faster R-CNN [@faster_rcnn] architecture (Figure \[fig:faster\_rcnn\]) developed for object detection uses a region proposal network (RPN) to propose bounding box candidates. The RPN extracts a Region of Interest (RoI), and a RoIPool layer computes features from these proposals in order to infer the bounding box coordinates and the class of the object.
![Faster R-CNN architecture. Each image is processed by convolutional layers and its features are extracted, a sliding window is used in RPN for each location over the feature map, for each location, $k$ ($k=9$) anchor boxes are used (3 scales of 128, 256 and 512, and 3 aspect ratios of 1:1, 1:2, 2:1) to generate a region proposal; A cls layer outputs 2k scores whether there or not there is an object for $k$ boxes; A reg layer outputs $4k$ for the coordinates (box center coordinates, width and height) of $k$ boxes. From [@faster_rcnn].[]{data-label="fig:faster_rcnn"}](faster_rcnn.pdf){width="0.4\linewidth"}
In one extension of this model, He *et al.* [@mask_rcnn] proposed a Mask R-CNN for object instance segmentation, which beat all previous benchmarks on many COCO challenges. This model efficiently detects objects in an image while simultaneously generating a high-quality segmentation mask for each instance. Mask R-CNN is essentially a Faster R-CNN with 3 output branches (Figure \[fig:mask\_rcnn\])—the first computes the bounding box coordinates, the second computes the associated classes, and the third computes the binary mask to segment the object. The Mask R-CNN loss function combines the losses of the bounding box coordinates, the predicted class, and the segmentation mask, and trains all of them jointly. Figure \[fig:mask\_rcnn\_Res\] shows the Mask-RCNN result on some sample images.
![Mask R-CNN architecture for instance segmentation. From [@mask_rcnn].[]{data-label="fig:mask_rcnn"}](mask_rcnn.pdf){width="0.6\linewidth"}
![Mask R-CNN results on sample images from the COCO test set. From [@mask_rcnn].[]{data-label="fig:mask_rcnn_Res"}](mask_rcnn_Res.pdf){width="0.7\linewidth"}
The Path Aggregation Network (PANet) proposed by Liu *et al.* [@seg_pan] is based on the Mask R-CNN and FPN models (Figure \[fig:seg\_pan\]). The feature extractor of the network uses an FPN architecture with a new augmented bottom-up pathway improving the propagation of low-layer features. Each stage of this third pathway takes as input the feature maps of the previous stage and processes them with a $3\times3$ convolutional layer. The output is added to the same stage feature maps of the top-down pathway using a lateral connection and these feature maps feed the next stage. As in the Mask R-CNN, the output of the adaptive feature pooling layer feeds three branches. The first two use a fully connected layer to generate the predictions of the bounding box coordinates and the associated object class. The third processes the RoI with an FCN to predict the object mask.
![The Path Aggregation Network. (a) FPN backbone. (b) Bottom-up path augmentation. (c) Adaptive feature pooling. (d) Box branch. (e) Fully-connected fusion. Courtesy of [@seg_pan].[]{data-label="fig:seg_pan"}](seg_pan.pdf){width="0.7\linewidth"}
Dai *et al.* [@seg_Cascades] developed a multi-task network for instance-aware semantic segmentation, that consists of three networks, respectively differentiating instances, estimating masks, and categorizing objects. These networks form a cascaded structure, and are designed to share their convolutional features. Hu *et al.* [@seg_learn] proposed a new partially-supervised training paradigm, together with a novel weight transfer function, that enables training instance segmentation models on a large set of categories, all of which have box annotations, but only a small fraction of which have mask annotations.
Chen *et al.* [@masklab] developed an instance segmentation model, MaskLab (Figure \[fig:masklab\]), by refining object detection with semantic and direction features based on Faster R-CNN. This model produces three outputs, box detection, semantic segmentation, and direction prediction. Building on the Faster-RCNN object detector, the predicted boxes provide accurate localization of object instances. Within each region of interest, MaskLab performs foreground/background segmentation by combining semantic and direction prediction.
![The MaskLab model. MaskLab generates three outputs—refined box predictions (from Faster R-CNN), semantic segmentation logits for pixel-wise classification, and direction prediction logits for predicting each pixel’s direction toward its instance center. From [@masklab].[]{data-label="fig:masklab"}](masklab.pdf){width="0.9\linewidth"}
Another interesting model is Tensormask, proposed by Chen *et al.* [@TensorMask], which is based on dense sliding window instance segmentation. They treat dense instance segmentation as a prediction task over 4D tensors and present a general framework that enables novel operators on 4D tensors. They demonstrate that the tensor view leads to large gains over baselines and yields results comparable to Mask R-CNN. TensorMask achieves promising results on dense object segmentation (Figure \[fig:TensorMask\]).
![The predicted segmentation map of a sample image by TensorMask. From [@TensorMask].[]{data-label="fig:TensorMask"}](TensorMask.pdf){width="0.55\linewidth"}
Many other instance segmentation models have been developed based on R-CNN, such as those developed for mask proposals, including R-FCN [@RFCN], DeepMask [@DeepMask], SharpMask [@SharpMask], PolarMask [@polarmask], and boundary-aware instance segmentation [@Hayer]. It is worth noting that there is another promising research direction that attempts to solve the instance segmentation problem by learning grouping cues for bottom-up segmentation, such as Deep Watershed Transform [@bai2017deep], and Semantic Instance Segmentation via Deep Metric Learning [@fathi2017semantic].
Dilated Convolutional Models and DeepLab Family
-----------------------------------------------
Dilated convolution (a.k.a. “atrous” convolution) introduces another parameter to convolutional layers, the dilation rate. The dilated convolution (Figure \[fig\_dilation\]) of a signal $x(i)$ is defined as $y_i= \sum_{k=1}^K x[i+rk] w[k]$, where $r$ is the dilation rate that defines a spacing between the weights of the kernel $w$. For example, a $3\times3$ kernel with a dilation rate of 2 will have the same size receptive field as a $5\times5$ kernel while using only 9 parameters, thus enlarging the receptive field with no increase in computational cost. Dilated convolutions have been popular in the field of real-time segmentation, and many recent publications report the use of this technique. Some of most important include the DeepLab family [@deeplab], multi-scale context aggregation [@multi_cont_agg], dense upsampling convolution and hybrid dilatedconvolution (DUC-HDC) [@UCS], densely connected Atrous Spatial Pyramid Pooling (DenseASPP) [@Denseaspp], and the efficient neural network (ENet) [@Enet].
![Dilated convolution. A $3\times3$ kernel at different dilation rates.[]{data-label="fig_dilation"}](fig_dilation.pdf){width="0.7\linewidth"}
DeepLabv1 [@deeplab_v1] and DeepLabv2 [@deeplab] are among some of the most popular image segmentation approaches, developed by Chen *et al.*. The latter has three key features. First is the use of dilated convolution to address the decreasing resolution in the network (caused by max-pooling and striding). Second is Atrous Spatial Pyramid Pooling (ASPP), which probes an incoming convolutional feature layer with filters at multiple sampling rates, thus capturing objects as well as image context at multiple scales to robustly segment objects at multiple scales. Third is improved localization of object boundaries by combining methods from deep CNNs and probabilistic graphical models. The best DeepLab (using a ResNet-101 as backbone) has reached a 79.7% mIoU score on the 2012 PASCAL VOC challenge, a 45.7% mIoU score on the PASCAL-Context challenge and a 70.4% mIoU score on the Cityscapes challenge. Figure \[deeplab\] illustrates the Deeplab model, which is similar to [@deeplab_v1], the main difference being the use of dilated convolution and ASPP.
![The DeepLab model. A CNN model such as VGG-16 or ResNet-101 is employed in fully convolutional fashion, using dilated convolution. A bilinear interpolation stage enlarges the feature maps to the original image resolution. Finally, a fully connected CRF refines the segmentation result to better capture the object boundaries. From [@deeplab][]{data-label="deeplab"}](deeplab.pdf){width="0.7\linewidth"}
Subsequently, Chen *et al.* [@deeplabv3] proposed DeepLabv3, which combines cascaded and parallel modules of dilated convolutions. The parallel convolution modules are grouped in the ASPP. A $1\times1$ convolution and batch normalisation are added in the ASPP. All the outputs are concatenated and processed by another $1\times1$ convolution to create the final output with logits for each pixel. In 2018, Chen *et al.* [@deeplabv3plus] released Deeplabv3+, which uses an encoder-decoder architecture (Figure \[deeplabplus\]), including atrous separable convolution, composed of a depthwise convolution (spatial convolution for each channel of the input) and pointwise convolution ($1\times1$ convolution with the depthwise convolution as input). They used the DeepLabv3 framework as encoder. The most relevant model has a modified Xception backbone with more layers, dilated depthwise separable convolutions instead of max pooling and batch normalization. The best DeepLabv3+ pretrained on the COCO and the JFT datasets has obtained a 89.0% mIoU score on the 2012 PASCAL VOC challenge.
![The DeepLabv3+ model. From [@deeplabv3plus].[]{data-label="deeplabplus"}](deeplabplus.pdf){width="0.9\linewidth"}
Recurrent Neural Network Based Models {#sec:RNN}
-------------------------------------
While CNNs are a natural fit for computer vision problems, they are not the only possibility. RNNs are useful in modeling the short/long term dependencies among pixels to (potentially) improve the estimation of the segmentation map. Using RNNs, pixels may be linked together and processed sequentially to model global contexts and improve semantic segmentation. One challenge, though, is the natural 2D structure of images.
Visin *et al.* [@Reseg] proposed an RNN-based model for semantic segmentation called ReSeg. This model is mainly based on another work, ReNet [@renet], which was developed for image classification. Each ReNet layer (Figure \[fig:renet\]) is composed of four RNNs that sweep the image horizontally and vertically in both directions, encoding patches/activations, and providing relevant global information. To perform image segmentation with the ReSeg model (Figure \[fig:reseg\]), ReNet layers are stacked on top of pre-trained VGG-16 convolutional layers that extract generic local features. ReNet layers are then followed by up-sampling layers to recover the original image resolution in the final predictions. Gated Recurrent Units (GRUs) are used because they provide a good balance between memory usage and computational power.
![A single-layer ReNet. From [@renet].[]{data-label="fig:renet"}](renet.pdf){width="0.24\linewidth"}
![The ReSeg model. The pre-trained VGG-16 feature extractor network is not shown. From [@Reseg].[]{data-label="fig:reseg"}](reseg.pdf){width="0.9\linewidth"}
In another work, Byeon *et al.* [@seg_lstm] developed a pixel-level segmentation and classification of scene images using long-short-term-memory (LSTM) network. They investigated two-dimensional (2D) LSTM networks for images of natural scenes, taking into account the complex spatial dependencies of labels. In this work, classification, segmentation, and context integration are all carried out by 2D LSTM networks, allowing texture and spatial model parameters to be learned within a single model. The block-diagram of the proposed 2D LSTM network for image segmentation in [@seg_lstm] is shown in Figure \[fig:seg\_lstm\].
![The 2D-LSTM model for semantic segmentation. The input image is divided into non-overlapping windows. Each window with RGB channels ($3\times N\times N$) is fed into four separate LSTM memory blocks. The current window of LSTM block is connected to its surrounding directions $x$ and $y$; i.e., left-top, left-bottom, right-top, and right-bottom; it propagates surrounding contexts. The output of each LSTM block is then passed to the feedforward layer, that sums all directions and applies hyperbolic tangent. In the final layer, the outputs of the final LSTM blocks are summed up and sent to the softmax layer. From [@seg_lstm].[]{data-label="fig:seg_lstm"}](seg_lstm.pdf){width="0.9\linewidth"}
Liang *et al.* [@graph_lstm] proposed a semantic segmentation model based on the Graph Long Short-Term Memory (Graph LSTM) network, a generalization of LSTM from sequential data or multidimensional data to general graph-structured data. Instead of evenly dividing an image to pixels or patches in existing multi-dimensional LSTM structures (e.g., row, grid and diagonal LSTMs), they take each arbitrary-shaped superpixel as a semantically consistent node, and adaptively construct an undirected graph for the image, where the spatial relations of the superpixels are naturally used as edges. Figure \[fig:graph\_lstm\] presents a visual comparison of the traditional pixel-wise RNN model and graph-LSTM model. To adapt the Graph LSTM model to semantic segmentation (Figure \[fig:seg\_graph\_lstm\]), LSTM layers built on a super-pixel map are appended on the convolutional layers to enhance visual features with global structure context. The convolutional features pass through $1\times1$ convolutional filters to generate the initial confidence maps for all labels. The node updating sequence for the subsequent Graph LSTM layers is determined by the confidence-drive scheme based on the initial confidence maps, and then the Graph LSTM layers can sequentially update the hidden states of all superpixel nodes.
![Comparison between the graph-LSTM model and traditional pixel-wise RNN models. From [@graph_lstm].[]{data-label="fig:graph_lstm"}](graph_lstm.pdf){width="0.8\linewidth"}
![The graph-LSTM model for semantic segmentation. From [@graph_lstm].[]{data-label="fig:seg_graph_lstm"}](seg_graph_lstm.pdf){width="0.9\linewidth"}
Xiang and Fox [@DARNN] proposed Data Associated Recurrent Neural Networks (DA-RNNs), for joint 3D scene mapping and semantic labeling. DA-RNNs use a new recurrent neural network architecture (Figure \[fig:DARNN\]) for semantic labeling on RGB-D videos. The output of the network is integrated with mapping techniques such as Kinect-Fusion in order to inject semantic information into the reconstructed 3D scene.
![The DA-RNN architecture. From [@DARNN].[]{data-label="fig:DARNN"}](DARNN.pdf){width="0.7\linewidth"}
Hu *et al.* [@seg_lstm_cnn] developed a semantic segmentation algorithm based on natural language expression, using a combination of CNN to encode the image and LSTM to encode its natural language description. This is different from traditional semantic segmentation over a predefined set of semantic classes, as, e.g., the phrase “two men sitting on the right bench” requires segmenting only the two people on the right bench and no one standing or sitting on another bench. To produce pixel-wise segmentation for language expression, they propose an end-to-end trainable recurrent and convolutional model that jointly learns to process visual and linguistic information (Figure \[fig:lstm\_cnn\_blk\]). In the considered model, a recurrent LSTM network is used to encode the referential expression into a vector representation, and an FCN is used to extract a spatial feature map from the image and output a spatial response map for the target object. An example segmentation result of this model (for the query “people in blue coat”) is shown in Figure \[fig:seg\_query1\].
![The CNN+LSTM architecture for segmentation from natural language expressions. From [@seg_lstm_cnn].[]{data-label="fig:lstm_cnn_blk"}](lstm_cnn_blk.pdf){width="0.95\linewidth"}
![Segmentation masks generated for the query “people in blue coat”. From [@seg_lstm_cnn].[]{data-label="fig:seg_query1"}](seg_lstm_cnn.pdf){width="0.9\linewidth"}
Attention-Based Models
----------------------
Attention mechanisms have been persistently explored in computer vision over the years, and it is therefore not surprising to find publications that apply such mechanisms to semantic segmentation.
Chen *et al.* [@seg_att1] proposed an attention mechanism that learns to softly weight multi-scale features at each pixel location. They adapt a powerful semantic segmentation model and jointly train it with multi-scale images and the attention model (Figure \[fig:seg\_att1\]). The attention mechanism outperforms average and max pooling, and it enables the model to assess the importance of features at different positions and scales.
![Attention-based semantic segmentation model. The attention model learns to assign different weights to objects of different scales; e.g., the model assigns large weights on the small person (green dashed circle) for features from scale 1.0, and large weights on the large child (magenta dashed circle) for features from scale 0.5. From [@seg_att1].[]{data-label="fig:seg_att1"}](seg_att1.pdf){width="0.8\linewidth"}
In contrast to other works in which convolutional classifiers are trained to learn the representative semantic features of labeled objects, Huang *et al.* [@seg_att2] proposed a semantic segmentation approach using reverse attention mechanisms. Their Reverse Attention Network (RAN) architecture (Figure \[fig:seg\_att2\]) trains the model to capture the opposite concept (i.e., features that are not associated with a target class) as well. The RAN is a three-branch network that performs the direct, and reverse-attention learning processes simultaneously.
![The reverse attention network for segmentation. From [@seg_att2].[]{data-label="fig:seg_att2"}](seg_att2.pdf){width="0.7\linewidth"}
Li *et al.* [@seg_att3] developed a Pyramid Attention Network for semantic segmentation. This model exploits the impact of global contextual information in semantic segmentation. They combined attention mechanisms and spatial pyramids to extract precise dense features for pixel labeling, instead of complicated dilated convolutions and artificially designed decoder networks. More recently, Fu *et al.* [@seg_DAN] proposed a dual attention network for scene segmentation, which can capture rich contextual dependencies based on the self-attention mechanism. Specifically, they append two types of attention modules on top of a dilated FCN which models the semantic inter-dependencies in spatial and channel dimensions, respectively. The position attention module selectively aggregates the feature at each position by a weighted sum of the features at all positions. The architecture of the dual attention network is shown in Figure \[fig:seg\_DAN\].
![The dual attention network for semantic segmentation. Courtesy of [@seg_DAN].[]{data-label="fig:seg_DAN"}](seg_DAN.pdf){width="0.95\linewidth"}
Various other works explore attention mechanisms for semantic segmentation, such as OCNet [@OCNet] which proposed an object context pooling inspired by self-attention mechanism, Expectation-Maximization Attention (EMANet) [@EMAnet], Criss-Cross Attention Network (CCNet) [@CcNet], end-to-end instance segmentation with recurrent attention [@seg_rec_att], a point-wise spatial attention network for scene parsing [@seg_att5], and a discriminative feature network (DFN) [@seg_att6], which comprises two sub-networks: a Smooth Network (that contains a Channel Attention Block and global average pooling to select the more discriminative features) and a Border Network (to make the bilateral features of the boundary distinguishable).
Generative Models and Adversarial Training
------------------------------------------
Since their introduction, GANs have been applied to a wide range tasks in computer vision, and have been adopted for image segmentation too.
Luc *et al.* [@seg_gan1] proposed an adversarial training approach for semantic segmentation. They trained a convolutional semantic segmentation network (Figure \[fig:seg\_gan1\]), along with an adversarial network that discriminates ground-truth segmentation maps from those generated by the segmentation network. They showed that the adversarial training approach leads to improved accuracy on the Stanford Background and PASCAL VOC 2012 datasets.
![The proposed adversarial model for semantic segmentation. The segmentation network (left) inputs an RGB image and produces per-pixel class predictions. The adversarial network (right) inputs the label map and produces class labels (1=ground truth or 0=synthetic). From [@seg_gan1].[]{data-label="fig:seg_gan1"}](seg_gan1.pdf){width="0.9\linewidth"}
Figure \[fig:seg\_gan1\_res\] shows the improvement brought up by adversarial training on one example image from Stanford Background dataset.
![Segmentation result on a sample image from Stanford Background with and without adversarial training. From [@seg_gan1].[]{data-label="fig:seg_gan1_res"}](seg_gan1_res.pdf){width="0.8\linewidth"}
Souly *et al.* [@seg_gan2] proposed semi-weakly supervised semantic segmentation using GANs. It consists of a generator network providing extra training examples to a multi-class classifier, acting as discriminator in the GAN framework, that assigns sample a label y from the $K$ possible classes or marks it as a fake sample (extra class).
The main idea is that adding large fake visual data forces real samples to be close in feature space, enabling a bottom-up clustering process which, in turn, improves multi-class pixel classification. The block diagram of the semi-supervised convolutional GAN architecture by is shown in Figure \[fig:seg\_gan2\_res\].
![The semi-supervised convolutional GAN. Noise is used by the Generator to generate an image. The Discriminator uses generated data, unlabeled data and labeled data to learn class confidences and produces confidence maps for each class as well as a label for a fake data sample. From [@seg_gan2].[]{data-label="fig:seg_gan2_res"}](seg_gan2_blk.pdf){width="0.8\linewidth"}
In another work, Hung *et al.* [@seg_gan3] developed a framework for semi-supervised semantic segmentation using an adversarial network. They designed an FCN discriminator to differentiate the predicted probability maps from the ground truth segmentation distribution, considering the spatial resolution. The considered loss function of this model contains three terms: cross-entropy loss on the segmentation ground truth, adversarial loss of the discriminator network, and semi-supervised loss based on the confidence map; i.e., the output of the discriminator. The architecture of the model by Hung and colleagues is shown in Figure \[fig:seg\_gan3\].
![A semi-supervised segmentation framework. From [@seg_gan3].[]{data-label="fig:seg_gan3"}](seg_gan3.pdf){width="0.9\linewidth"}
Xue *et al.* [@seg_gan4] proposed an adversarial network with multi-scale L1 Loss for medical image segmentation. They used an FCN as the segmentor to generate segmentation label maps, and proposed a novel adversarial critic network with a multi-scale L1 loss function to force the critic and segmentor to learn both global and local features that capture long and short range spatial relationships between pixels. The block-diagram of the segmentor and critic networks are shown in Figure \[fig:seg\_gan4\].
![The proposed adversarial network with multi-scale L1 Loss for semantic segmentation. From[@seg_gan4].[]{data-label="fig:seg_gan4"}](seg_gan4.pdf){width="0.8\linewidth"}
The segmentor and critic networks are trained in an alternating fashion in a min-max game: the critic takes as input a pair of images, (original image - predicted label map, original image - ground truth label map), and is trained by maximizing a multi-scale loss function; The segmentor is trained only with gradients passed along by the critic, with the aim of minimizing the multi-scale loss function. This model was tested on datasets from the MICCAI BRATS brain tumor segmentation challenge. The authors demonstrate the effectiveness of SegAN with multi-scale loss: on BRATS 2013 SegAN gives performance comparable to the state-of-the-art for whole tumor and tumor core segmentation while achieving better precision and sensitivity for Gd-enhance tumor core segmentation;
Various other publications report on segmentation models based on adversarial training, such as Cell Image Segmentation Using GANs [@seg_gan5], and segmentation and generation of the invisible parts of objects [@seg_gan8].
CNN Models With Active Contour Models
-------------------------------------
The exploration of synergies between FCNs and Active Contour Models (ACMs) [@Snakes] has recently attracted research interest. One approach is to formulate new loss functions that are inspired by ACM principles. For example, inspired by the global energy formulation of [@chan2001active], Chen *et al.* [@chen2019learning] proposed a supervised loss layer that incorporated area and size information of the predicted masks during training of an FCN and tackled the problem of ventricle segmentation in cardiac MRI. Similarly, Gur *et al.* [@gur2019unsupervised] presented an unsupervised loss function based on morphological active contours without edges [@marquez2014morphological] for microvascular image segmentation.
A different approach initially sought to utilize the ACM merely as a post-processor of the output of an FCN and several efforts attempted modest co-learning by pre-training the FCN. One example of an ACM post-processor for the task of semantic segmentation of natural images is the work by Le *et al.* [@le2018reformulating] in which level-set ACMs are implemented as RNNs. Deep Active Contours by Rupprecht *et al.* [@deepacm], is another example. For medical image segmentation, Hatamizadeh *et al.* [@hatamizadeh2019deep] proposed an integrated Deep Active Lesion Segmentation (DALS) model that trains the FCN backbone to predict the parameter functions of a novel, locally-parameterized level-set energy functional. In another relevant effort, Marcos *et al.* [@marcos2018learning] proposed Deep Structured Active Contours (DSAC), which combines ACMs and pre-trained FCNs in a structured prediction framework for building instance segmentation (albeit with manual initialization) in aerial images. For the same application, Cheng *et al.* [@cheng2019darnet] proposed the Deep Active Ray Network (DarNet), which is similar to DSAC, but with a different explicit ACM formulation based on polar coordinates to prevent contour self-intersection. A truly end-to-end backpropagation trainable, fully-integrated FCN-ACM combination was recently introduced by Hatamizadeh *et al.* [@hatamizadeh2019end], dubbed Deep Convolutional Active Contours (DCAC).
Other Models
------------
In addition to the above models, there are several other popular DL architectures for segmentation, such as the following: Context Encoding Network (EncNet) that uses a basic feature extractor and feeds the feature maps into a Context Encoding Module [@EncNet]. RefineNet [@Refinenet], which is a multi-path refinement network that explicitly exploits all the information available along the down-sampling process to enable high-resolution prediction using long-range residual connections. “Object-Contextual Representations” (OCR) [@hrrocr], which learns object regions under the supervision of the ground-truth, and computes the object region representation, and the relation between each pixel and each object region, and augment the representation pixels with the object-contextual representation. Seednet [@Seednet], which introduced an automatic seed generation technique with deep reinforcement learning that learns to solve the interactive segmentation problem, Feedforward-Net [@seg_feedforward] which maps image super-pixels to rich feature representations extracted from a sequence of nested regions of increasing extent and exploits statistical structures in the image and in the label space without setting up explicit structured prediction mechanisms.
{width="0.95\linewidth"}
Yet additional models include BoxSup [@Boxsup], Graph convolutional networks [@GCN], Wide ResNet [@wideresnet], Exfuse (enhancing low-level and high-level features fusion) [@Exfuse], dual image segmentation (DIS) [@DIS], FoveaNet (Perspective-aware scene parsing) [@Foveanet], Ladder DenseNet [@ladder], Bilateral segmentation network (BiSeNet) [@BiSeNet], Semantic Prediction Guidance for Scene Parsing (SPGNet) [@SPGNet], Gated shape CNNs [@GSCNN], Adaptive context network (AC-Net) [@ACnet], Dynamic-structured semantic propagation network (DSSPN) [@DSSPN], symbolic graph reasoning (SGR) [@SGR], CascadeNet [@ADE20k], Scale-adaptive convolutions (SAC) [@SAC], Unified perceptual parsing (UperNet) [@uper].
Panoptic segmentation [@kirillov2019panoptic] is also another interesting (and newer) segmentation problem with rising popularity, and there are already several interesting works on this direction, including Panoptic Feature Pyramid Network [@pfpn], attention-guided network for Panoptic segmentation [@seg_att4], and Seamless Scene Segmentation [@porzi2019seamless].
Figure \[seg\_timeline\] illustrates the timeline of popular DL-based works for semantic segmentation, as well as instance segmentation since 2014. Given the large number of works developed in the last few years, we only show some of the most representative ones.
A main contribution of this work is the “zoom-out feature fusion”, which casts category-level segmentation of an image as classifying a set of superpixels. The main idea of the zoom-out architecture (Figure \[fig:zoomout\]) is to allow features extracted from different levels of spatial context around the superpixel to contribute to labeling decision at that superpixel. The proposed model achieved 69.6% average accuracy on the PASCAL VOC 2012 test set.
![Zoom-out feature extraction procedure for a simple network with 3 convolutional and 2 pooling layers. From [@seg_feedforward].[]{data-label="fig:zoomout"}](feedforward.pdf){width="0.7\linewidth"}
Image Segmentation Datasets {#sec:datasets}
===========================
In this section we provide a summary of some of the most widely used image segmentation datasets. We group these datasets into 3 categories—2D images, 2.5D RGB-D (color+depth) images, and 3D images—and provide details about the characteristics of each dataset. The listed datasets have pixel-wise labels, which can be used for evaluating model performance.
It is worth mentioning that some of these works, use **data augmentation** to increase the number of labeled samples, specially the ones which deal with small datasets (such as in medical domain). Data augmentation serves to increase the number of training samples by applying a set of transformation (either in the data space, or feature space, or sometimes both) to the images (i.e., both the input image and the segmentation map). Some typical transformations include translation, reflection, rotation, warping, scaling, color space shifting, cropping, and projections onto principal components. Data augmentation has proven to improve the performance of the models, especially when learning from limited datasets, such as those in medical image analysis. It can also be beneficial in yielding faster convergence, decreasing the chance of over-fitting, and enhancing generalization. For some small datasets, data augmentation has been shown to boost model performance more than 20%.
2D Datasets
-----------
The majority of image segmentation research has focused on 2D images; therefore, many 2D image segmentation datasets are available. The following are some of the most popular:
**PASCAL Visual Object Classes (VOC)** [@pascal_voc2010] is one of most popular datasets in computer vision, with annotated images available for 5 tasks—classification, segmentation, detection, action recognition, and person layout. Nearly all popular segmentation algorithms reported in the literature have been evaluated on this dataset. For the segmentation task, there are 21 classes of object labels—vehicles, household, animals, aeroplane, bicycle, boat, bus, car, motorbike, train, bottle, chair, dining table, potted plant, sofa, TV/monitor, bird, cat, cow, dog, horse, sheep, and person (pixel are labeled as background if they do not belong to any of these classes). This dataset is divided into two sets, training and validation, with 1,464 and 1,449 images, respectively. There is a private test set for the actual challenge. Figure \[fig:pascal\_voc2010\] shows an example image and its pixel-wise label.
![An example image from the PASCAL VOC dataset. From [@pascal_voc2010].[]{data-label="fig:pascal_voc2010"}](pascal_voc2010.pdf){width="0.8\linewidth"}
**PASCAL Context** [@pascal_context] is an extension of the PASCAL VOC 2010 detection challenge, and it contains pixel-wise labels for all training images. It contains more than 400 classes (including the original 20 classes plus backgrounds from PASCAL VOC segmentation), divided into three categories (objects, stuff, and hybrids). Many of the object categories of this dataset are too sparse and; therefore, a subset of 59 frequent classes are usually selected for use. Figure \[fig:pascal\_context\] shows the segmentation map of three sample images of this dataset.
![Three sample images and segmentation maps from the PASCAL context dataset. From [@pascal_context].[]{data-label="fig:pascal_context"}](pascal_context.pdf){width="0.8\linewidth"}
**Microsoft Common Objects in Context (MS COCO)** [@mscoco] is another large-scale object detection, segmentation, and captioning dataset. COCO includes images of complex everyday scenes, containing common objects in their natural contexts. This dataset contains photos of 91 objects types, with a total of 2.5 million labeled instances in 328k images. It has been used mainly for segmenting individual object instances. Figure \[fig:mscoco\] shows the difference between MS COCO labels and the previous datasets for a given sample image. The detection challenge includes more than 80 classes, providing more than 82k images for training, 40.5k images for validation, and more than 80k images for its test set.
![A sample image and its segmentation map in COCO, and its comparison with previous datasets. From [@mscoco].[]{data-label="fig:mscoco"}](mscoco.pdf){width="0.8\linewidth"}
**Cityscapes** [@Cityscapes] is a large-scale database with a focus on semantic understanding of urban street scenes. It contains a diverse set of stereo video sequences recorded in street scenes from 50 cities, with high quality pixel-level annotation of 5k frames, in addition to a set of 20k weakly annotated frames. It includes semantic and dense pixel annotations of 30 classes, grouped into 8 categories—flat surfaces, humans, vehicles, constructions, objects, nature, sky, and void. Figure \[fig:Cityscapes\] shows four sample segmentation maps from this dataset.
![Three sample images with their corresponding segmentation maps from the Cityscapes dataset. From [@Cityscapes].[]{data-label="fig:Cityscapes"}](Cityscapes.pdf){width="0.99\linewidth"}
**ADE20K/MIT Scene Parsing (SceneParse150)** offers a standard training and evaluation platform for scene parsing algorithms. The data for this benchmark comes from the ADE20K dataset [@ADE20k], which contains more than 20K scene-centric images exhaustively annotated with objects and object parts. The benchmark is divided into 20K images for training, 2K images for validation, and another batch of images for testing. There are 150 semantic categories in this dataset. Figure \[fig:ADE20k\] shows three sample images and their segmentation maps.
![Three sample images (with their segmentation maps) from MIT Schene Parsing dataset. From [@ADE20k].[]{data-label="fig:ADE20k"}](mit_scene.pdf){width="0.8\linewidth"}
**SiftFlow** [@SiftFlow] includes 2,688 annotated images from a subset of the LabelMe database. The $256\times256$ pixel images are based on 8 different outdoor scenes, among them streets, mountains, fields, beaches, and buildings. All images belong to one of 33 semantic classes.
**Stanford background** [@stan_back] contains outdoor images of scenes from existing datasets, such as LabelMe, MSRC, and PASCAL VOC. It contains 715 images with at least one foreground object. The dataset is pixel-wise annotated, and can be used for semantic scene understanding. Semantic and geometric labels for this dataset were obtained using Amazon’s Mechanical Turk (AMT).
**Berkeley Segmentation Dataset (BSD)** [@BSD] contains 12,000 hand-labeled segmentations of 1,000 Corel dataset images from 30 human subjects. It aims to provide an empirical basis for research on image segmentation and boundary detection. Half of the segmentations were obtained from presenting the subject a color image and the other half from presenting a grayscale image. The public benchmark based on this data consists of all of the grayscale and color segmentations for 300 images. The images are divided into a training set of 200 images and a test set of 100 images.
**Youtube-Objects** [@youtube_video] contains videos collected from YouTube, which include objects from ten PASCAL VOC classes (aeroplane, bird, boat, car, cat, cow, dog, horse, motorbike, and train). The original dataset did not contain pixel-wise annotations (as it was originally developed for object detection, with weak annotations). However, Jain *et al.* [@jain_youtube] manually annotated a subset of 126 sequences, and then extracted a subset of frames to further generate semantic labels. In total, there are about 10,167 annotated 480x360 pixel frames available in this dataset.
**KITTI** [@kitti] is one of the most popular datasets for mobile robotics and autonomous driving. It contains hours of videos of traffic scenarios, recorded with a variety of sensor modalities (including high-resolution RGB, grayscale stereo cameras, and a 3D laser scanners). The original dataset does not contain ground truth for semantic segmentation, but researchers have manually annotated parts of the dataset for research purposes. For example, Alvarez *et al.* [@Alvarez] generated ground truth for 323 images from the road detection challenge with 3 classes, road, vertical, and sky. **Other Datasets** are available for image segmentation purposes too, such as **Semantic Boundaries Dataset (SBD)** [@sbd], **PASCAL Part** [@pascal_part], **SYNTHIA** [@synthia], and **Adobe’s Portrait Segmentation** [@adobe].
2.5D Datasets
-------------
With the availability of affordable range scanners, RGB-D images have became popular in both research and industrial applications. The following RGB-D datasets are some of the most popular:
**NYU-D V2** [@nyuv2] consists of video sequences from a variety of indoor scenes, recorded by the RGB and depth cameras of the Microsoft Kinect. It includes 1,449 densely labeled pairs of aligned RGB and depth images from more than 450 scenes taken from 3 cities. Each object is labeled with a class and an instance number (e.g., cup1, cup2, cup3, etc.). It also contains 407,024 unlabeled frames. This dataset is relatively small compared to other existing datasets. Figure \[fig:nyuv2\] shows a sample image and its segmentation map.
![A sample image from the NYU V2 dataset. From left: the RGB image, pre-processed depth, and set of labels. From [@nyuv2].[]{data-label="fig:nyuv2"}](nyuv2.pdf){width="0.95\linewidth"}
**SUN-3D** [@sun3d] is a large-scale RGB-D video dataset that contains 415 sequences captured for 254 different spaces in 41 different buildings; 8 sequences are annotated and more will be annotated in the future. Each annotated frame comes with the semantic segmentation of the objects in the scene, as well as information about the camera pose.
**SUN RGB-D** [@sunrgbd] provides an RGB-D benchmark for the goal of advancing the state-of-the-art in all major scene understanding tasks. It is captured by four different sensors and contains 10,000 RGB-D images at a scale similar to PASCAL VOC. The whole dataset is densely annotated and includes 146,617 2D polygons and 58,657 3D bounding boxes with accurate object orientations, as well as the 3D room category and layout for scenes. Figure \[fig:sunrgbd\] shows two example images (with annotations).
![Two example images (with annotations) from SUN RGB-D dataset. From [@sunrgbd].[]{data-label="fig:sunrgbd"}](sunrgbd2.pdf){width="0.9\linewidth"}
**UW RGB-D Object Dataset** [@uw_rgbd] contains 300 common household objects recorded using a Kinect style 3D camera. The objects are organized into 51 categories, arranged using WordNet hypernym-hyponym relationships (similar to ImageNet). This dataset was recorded using a Kinect style 3D camera that records synchronized and aligned $640\times480$ pixel RGB and depth images at 30Hz. This dataset also includes 8 annotated video sequences of natural scenes, containing objects from the dataset (the UW RGB-D Scenes Dataset).
**ScanNet** [@scannet] is an RGB-D video dataset containing 2.5 million views in more than 1,500 scans, annotated with 3D camera poses, surface reconstructions, and instance-level semantic segmentations. To collect these data, an easy-to-use and scalable RGB-D capture system was designed that includes automated surface reconstruction, and the semantic annotation was crowd-sourced. Using this data helped achieve state-of-the-art performance on several 3D scene understanding tasks, including 3D object classification, semantic voxel labeling, and CAD model retrieval.
3D Datasets
-----------
3D image datasets are popular in robotic, medical image analysis, 3D scene analysis, and construction applications. Three dimensional images are usually provided via meshes or other volumetric representations, such as point clouds. Here, we mention some of the popular 3D datasets.
**Stanford 2D-3D:** This dataset provides a variety of mutually registered modalities from 2D, 2.5D and 3D domains, with instance-level semantic and geometric annotations [@stan3d], and is collected in 6 indoor areas. It contains over 70,000 RGB images, along with the corresponding depths, surface normals, semantic annotations, global XYZ images as well as camera information. **ShapeNet Core:** ShapeNetCore is a subset of the full ShapeNet dataset [@shapenet] with single clean 3D models and manually verified category and alignment annotations [@shapenetcore]. It covers 55 common object categories with about 51,300 unique 3D models. **Sydney Urban Objects Dataset:** This dataset contains a variety of common urban road objects, collected in the central business district of Sydney, Australia. There are 631 individual scans of objects across classes of vehicles, pedestrians, signs and trees [@sydney3d].
3D Datasets
-----------
Three dimensional images are usually provided via meshes or other volumetric representations, such as point clouds. While point clouds can be directly rendered and inspected, they are often converted to polygon mesh or triangle mesh models, NURBS surface models, or CAD models (through a process commonly referred to as surface reconstruction). Generating large-scale 3D datasets for segmentation is more costly/difficult than 2D images, and also not many deep learning methods are able to process very high-dimensional data. For these reasons, 3D datasets are not as popular as 2D and RGB-D datasets in this context as of yet. Here, we mention three popular datasets.
**Stanford 2D-3D:** This dataset provides a variety of mutually registered modalities from 2D, 2.5D and 3D domains, with instance-level semantic and geometric annotations [@stan3d]. This dataset is collected in 6 large-scale indoor areas that originate from 3 different buildings of mainly educational and office use. It contains over 70,000 RGB images, along with the corresponding depths, surface normals, semantic annotations, global XYZ images (all in forms of both regular and 30$^\circ$ equi-rectangular images) as well as camera information. It also includes registered raw and semantically annotated 3D meshes and point clouds. The dataset contains colored point clouds and textured meshes for each scanned area. The annotations were initially performed on the point cloud, and then projected onto the closest surface on the 3D mesh model.
**ShapeNet Core:** ShapeNetCore is a subset of the full ShapeNet dataset [@shapenet] with single clean 3D models and manually verified category and alignment annotations [@shapenetcore]. It covers 55 common object categories with about 51,300 unique 3D models. The 12 object categories of PASCAL 3D+, a popular computer vision 3D benchmark dataset, are all covered by ShapeNetCore.
**Sydney Urban Objects Dataset:** This dataset contains a variety of common urban road objects scanned with a Velodyne HDL-64E LIDAR, collected in the central business district of Sydney, Australia. There are 631 individual scans of objects across classes of vehicles, pedestrians, signs and trees [@sydney3d]. This dataset aims to provide non-ideal sensing conditions that are representative of practical urban sensing systems, with a large variability in viewpoint and occlusion.
Performance Review {#sec:performance}
==================
In this section, we first provide a summary of some of the popular metrics used in evaluating the performance of segmentation models, and then we provide the quantitative performance of the promising DL-based segmentation models on popular datasets.
Metrics For Segmentation Models {#sec:metrics}
-------------------------------
Ideally, a model should be evaluated in multiple respects, such as quantitative accuracy, speed (inference time), and storage requirements (memory footprint). Measuring speed can be tricky, as it depends on the hardware and experimental conditions, but it is an important factor in real-time applications, as is the memory footprint if a model is intended for small devices with limited memory capacity. However, most of the research works so far, focus on the metrics for evaluating the model accuracy. Below we summarize the most popular metrics for assessing the accuracy of segmentation algorithms. Although quantitative metrics are used to compare different models on benchmarks, the visual quality of model outputs is also important in deciding which model is best (as human is the final consumer of many of the models developed for computer vision applications).
**Pixel accuracy** simply finds the ratio of pixels properly classified, divided by the total number of pixels. For $K+1$ classes ($K$ foreground classes and the background) pixel accuracy is defined as Eq \[eq\_PA\]: $$\text{PA}= \frac{\sum_{i=0}^K p_{ii}} {\sum_{i=0}^K \sum_{j=0}^K p_{ij}},
\label{eq_PA}$$ where $p_{ij}$ is the number of pixels of class $i$ predicted as belonging to class $j$.
**Mean Pixel Accuracy (MPA)** is the extended version of PA, in which the ratio of correct pixels is computed in a per-class manner and then averaged over the total number of classes, as in Eq \[eq\_MPA\]: $$\text{MPA}= \frac{1}{K+1} \sum_{i=0}^K \frac{p_{ii}}{\sum_{j=0}^K p_{ij}}.
\label{eq_MPA}$$
**Intersection over Union (IoU)** or the **Jaccard Index** is one of the most commonly used metrics in semantic segmentation. It is defined as the area of intersection between the predicted segmentation map and the ground truth, divided by the area of union between the predicted segmentation map and the ground truth: $$\text{IoU}= J(A,B) = \frac{|A \cap B|}{|A \cup B|},$$ where $A$ and $B$ denote the ground truth and the predicted segmentation maps, respectively. It ranges between 0 and 1.
**Mean-IoU** is another popular metric, which is defined as the average IoU over all classes. It is widely used in reporting the performance of modern segmentation algorithms.
IoU can also be defined based on the $p_{ij}$ (defined previously) as $$\text{IoU}= J(A,B)= \frac{|A \cap B|}{|A \cup B|}
\label{eq_iou2}$$ That IoU is computed on a per-class basis and then averaged.
**Precision/Recall/F1 score** are popular metrics for reporting the accuracy of many of the classical image segmentation models. Precision and recall can be defined for each class, as well as at the aggregate level, as follows: $$\begin{aligned}
\text{Precision}&= \frac{\text{TP}}{\text{TP}+\text{FP}},
\ \ \text{Recall}&= \frac{\text{TP}}{\text{TP}+\text{FN}},
\end{aligned}
\label{prec_rec}$$ where TP refers to the true positive fraction, FP refers to the false positive fraction, and FN refers to the false negative fraction. Usually we are interested into a combined version of precision and recall rates. A popular such a metric is called the F1 score, which is defined as the harmonic mean of precision and recall: $$\text{F1-score}= \frac{2 \ \text{Prec} \ \text{Rec}}{ \text{Prec}+\text{Rec} }.
\label{f1_score}$$
**Dice coefficient** is another popular metric for image segmentation, which can be defined as twice the overlap area of predicted and ground-truth maps, divided by the total number of pixels in both images. The Dice coefficient is very similar to the IoU: $$\text{Dice}= \frac{ 2| A \cap B |}{|A|+ |B|}.
\label{eq_dice}$$ When applied to boolean data (e.g., binary segmentation maps), and referring to the foreground as a positive class, the Dice coefficient is essentially identical to the F1 score, defined as Eq \[eq\_dice2\]: $$\text{Dice}= \frac{ 2 \text{TP}}{ 2\text{TP}+ \text{FP}+\text{FN} }=
\text{F1}.
\label{eq_dice2}$$ The Dice coefficient and IoU are positively correlated.
Quantitative Performance of DL-Based Models {#sec:quant_result}
-------------------------------------------
In this section we tabulate the performance of several of the previously discussed algorithms on popular segmentation benchmarks. It is worth mentioning that although most models report their performance on standard datasets and use standard metrics, some of them fail to do so, making across-the-board comparisons difficult. Furthermore, only a small percentage of publications provide additional information, such as execution time and memory footprint, in a reproducible way, which is important to industrial applications of segmentation models (such as drones, self-driving cars, robotics, etc.) that may run on embedded consumer devices with limited computational power and storage, making fast, light-weight models crucial.
Method Backbone mIoU
----------------------------------- --------------- -------
FCN [@seg_fcn] VGG-16 62.2
CRF-RNN [@CRF-RNN] - 72.0
CRF-RNN$^{*}$ [@CRF-RNN] - 74.7
BoxSup\* [@Boxsup] - 75.1
Piecewise [@seg_dsn2] - 75.3
Piecewise$^{*}$ [@seg_dsn2] - 78.0
DPN [@seg_dsn3] - 74.1
DPN$^{*}$ [@seg_dsn3] - 77.5
DeepLab-CRF [@deeplab] ResNet-101 79.7
GCN$^{*}$ [@GCN] ResNet-152 82.2
RefineNet [@Refinenet] ResNet-152 84.2
Wide ResNet [@wideresnet] WideResNet-38 84.9
OCR [@hrrocr] ResNet-101 84.3
OCR [@pspn] HRNetV2-W48 84.5
PSPNet [@pspn] ResNet-101 85.4
DeeplabV3 [@deeplabv3] ResNet-101 85.7
PSANet [@seg_att5] ResNet-101 85.7
EncNet [@EncNet] ResNet-101 85.9
DFN [@seg_att6] ResNet-101 82.7
DFN$^{*}$ [@seg_att6] ResNet-101 86.2
Exfuse [@Exfuse] ResNet-101 86.2
SDN [@SDN] DenseNet-161 83.5
SDN\* [@SDN] DenseNet-161 86.6
DIS [@DIS] ResNet-101 86.8
DM-Net [@dmsf] ResNet-101 84.4
DM-Net$^{*}$ [@dmsf] ResNet-101 87.06
APC-Net [@apcnet] ResNet-101 84.2
APC-Net$^{*}$ [@apcnet] ResNet-101 87.1
EMANet [@EMAnet] ResNet-101 87.7
DeeplabV3+ [@deeplabv3plus] Xception-71 87.8
Exfuse [@Exfuse] ResNeXt-131 87.9
MSCI [@MSCI] ResNet-152 88.0
EMANet [@EMAnet] ResNet-152 88.2
DeeplabV3+$^{*}$ [@deeplabv3plus] Xception-71 89.0
: Accuracies of segmentation models on the PASCAL VOC test set. (\* Refers to the model pre-trained on another dataset, such as MS-COCO, ImageNet, or JFT-300M.)[]{data-label="table_voc"}
Method Backbone mIoU
-------------------------------- --------------------- ------
SegNet basic [@segnet] - 57.0
FCN-8s [@seg_fcn] - 65.3
DPN [@seg_dsn3] - 66.8
Dilation10 [@multi_cont_agg] - 67.1
DeeplabV2 [@deeplab] ResNet-101 70.4
RefineNet [@Refinenet] ResNet-101 73.6
FoveaNet [@Foveanet] ResNet-101 74.1
Ladder DenseNet [@ladder] Ladder DenseNet-169 73.7
GCN [@GCN] ResNet-101 76.9
DUC-HDC [@UCS] ResNet-101 77.6
Wide ResNet [@wideresnet] WideResNet-38 78.4
PSPNet [@pspn] ResNet-101 85.4
BiSeNet [@BiSeNet] ResNet-101 78.9
DFN [@seg_att6] ResNet-101 79.3
PSANet [@seg_att5] ResNet-101 80.1
DenseASPP [@Denseaspp] DenseNet-161 80.6
SPGNet [@SPGNet] 2xResNet-50 81.1
DANet [@seg_DAN] ResNet-101 81.5
CCNet [@CcNet] ResNet-101 81.4
DeeplabV3 [@deeplabv3] ResNet-101 81.3
DeeplabV3 [@deeplabv3plus] Xception-71 82.1
AC-Net [@ACnet] ResNet-101 82.3
OCR [@hrrocr] ResNet-101 82.4
GS-CNN [@GSCNN] WideResNet 82.8
HRNetV2+OCR (w/ASPP) [@hrrocr] HRNetV2-W48 83.7
: Accuracies of segmentation models on the Cityescapes dataset.[]{data-label="table_cityscape"}
The following tables summarize the performances of several of the prominent DL-based segmentation models on different datasets. Table \[table\_voc\] focuses on the PASCAL VOC test set. Clearly, there has been much improvement in the accuracy of the models since the introduction of the FCN, the first DL-based image segmentation model.[^1] Table \[table\_cityscape\] focuses on the Cityscape test dataset. The latest models feature about 23% relative gain over the initial FCN model on this dataset. Table \[table\_mscoco\] focuses on the MS COCO stuff test set. This dataset is more challenging than PASCAL VOC, and Cityescapes, as the highest mIoU is approximately 40%. Table \[table\_ADE20k\] focuses on the ADE20k validation set. This dataset is also more challenging than the PASCAL VOC and Cityescapes datasets. Finally, Table \[table\_NYUv2\] summarizes the performance of several prominent models for RGB-D segmentation on the NYUD-v2 and SUN-RGBD datasets.
Method Backbone mIoU
------------------------ --------------------- ------
RefineNet [@Refinenet] ResNet-101 33.6
CCN [@CCL] Ladder DenseNet-101 35.7
DANet [@seg_DAN] ResNet-50 37.9
DSSPN [@DSSPN] ResNet-101 37.3
EMA-Net [@EMAnet] ResNet-50 37.5
SGR [@SGR] ResNet-101 39.1
OCR [@hrrocr] ResNet-101 39.5
DANet [@seg_DAN] ResNet-101 39.7
EMA-Net [@EMAnet] ResNet-50 39.9
AC-Net [@ACnet] ResNet-101 40.1
OCR [@hrrocr] HRNetV2-W48 40.5
: Accuracies of segmentation models on the MS COCO stuff dataset.[]{data-label="table_mscoco"}
Method Backbone mIoU
------------------------------ ------------- -------
FCN [@seg_fcn] - 29.39
DilatedNet [@multi_cont_agg] - 32.31
CascadeNet [@ADE20k] - 34.9
RefineNet [@Refinenet] ResNet-152 40.7
PSPNet [@pspn] ResNet-101 43.29
PSPNet [@pspn] ResNet-269 44.94
EncNet [@EncNet] ResNet-101 44.64
SAC [@SAC] ResNet-101 44.3
PSANet [@seg_att5] ResNet-101 43.7
UperNet [@uper] ResNet-101 42.66
DSSPN [@DSSPN] ResNet-101 43.68
DM-Net [@dmsf] ResNet-101 45.5
OCR [@hrrocr] HRNetV2-W48 45.6
AC-Net [@ACnet] ResNet-101 45.9
: Accuracies of segmentation models on the ADE20k validation dataset.[]{data-label="table_ADE20k"}
--------------------------------- ------- ------- ------- -------
Method m-Acc m-IoU m-Acc m-IoU
Mutex [@deng2015semantic] - 31.5 - -
MS-CNN [@rgbd_multiscale] 45.1 34.1 - -
FCN [@seg_fcn] 46.1 34.0 - -
Joint-Seg [@mousavian2016joint] 52.3 39.2 - -
SegNet [@segnet] - - 44.76 31.84
Structured Net [@seg_dsn2] 53.6 40.6 53.4 42.3
B-SegNet [@bayes_segnet] - - 45.9 30.7
3D-GNN [@3dgnn] 55.7 43.1 57.0 45.9
LSD-Net [@cheng2017locality] 60.7 45.9 58.0 -
RefineNet [@Refinenet] 58.9 46.5 58.5 45.9
D-aware CNN [@wang2018depth] 61.1 48.4 53.5 42.0
RDFNet [@RDFNet] 62.8 50.1 60.1 47.7
G-Aware Net [@Geometry-Aware] 68.7 59.6 74.9 54.5
--------------------------------- ------- ------- ------- -------
: Performance of segmentation models on the NYUD-v2, and SUN-RGBD datasets, in terms of mIoU, and mean Accuracy (mAcc).[]{data-label="table_NYUv2"}
To summarize the tabulated data, there has been significant progress in the performance of deep segmentation models over the past 5–6 years, with a relative improvement of 25%-42% in mIoU on different datasets. However, some publications suffer from lack of reproducibility for multiple reasons—they report performance on non-standard benchmarks/databases, or they report performance only on arbitrary subsets of the test set from a popular benchmark, or they do not adequately describe the experimental setup and sometimes evaluate the model performance only on a subset of object classes. Most importantly, many publications do not provide the source-code for their model implementations. However, with the increasing popularity of deep learning models, the trend has been positive and many research groups are moving toward reproducible frameworks and open-sourcing their implementations.
Challenges and Opportunities {#sec:challenges}
============================
There is not doubt that image segmentation has benefited greatly from deep learning, but several challenges lie ahead. We will next introduce some of the promising research directions that we believe will help in further advancing image segmentation algorithms.
More Challenging Datasets
-------------------------
Several large-scale image datasets have been created for semantic segmentation and instance segmentation. However, there remains a need for more challenging datasets, as well as datasets for different kinds of images. For still images, datasets with a large number of objects and overlapping objects would be very valuable. This can enable training models that are better at handling dense object scenarios, as well as large overlaps among objects as is common in real-world scenarios.
With the rising popularity of 3D image segmentation, especially in medical image analysis, there is also a strong need for large-scale 3D images datasets. These datasets are more difficult to create than their lower dimensional counterparts. Existing datasets for 3D image segmentation available are typically not large enough, and some are synthetic, and therefore larger and more challenging 3D image datasets can be very valuable.
Interpretable Deep Models
-------------------------
While DL-based models have achieved promising performance on challenging benchmarks, there remain open questions about these models. For example, what exactly are deep models learning? How should we interpret the features learned by these models? What is a minimal neural architecture that can achieve a certain segmentation accuracy on a given dataset? Although some techniques are available to visualize the learned convolutional kernels of these models, a concrete study of the underlying behavior/dynamics of these models is lacking. A better understanding of the theoretical aspects of these models can enable the development of better models curated toward various segmentation scenarios.
Weakly-Supervised and Unsupervised Learning
-------------------------------------------
Weakly-supervised (a.k.a. few shot learning) and unsupervised learning are becoming very active research areas. These techniques promise to be specially valuable for image segmentation, as collecting labeled samples for segmentation problem is problematic in many application domains, particularly so in medical image analysis. The transfer learning approach is to train a generic image segmentation model on a large set of labeled samples (perhaps from a public benchmark), and then fine-tune that model on a few samples from some specific target application. Self-supervised learning is another promising direction that is attracting much attraction in various fields. There are many details in images that that can be captured to train a segmentation models with far fewer training samples, with the help of self-supervised learning. Models based on reinforcement learning could also be another potential future direction, as they have scarcely received attention for image segmentation.
Real-time Models for Various Applications
-----------------------------------------
In many applications, accuracy is the most important factor; however, there are applications in which it is also critical to have segmentation models that can run in near real-time, or at least near common camera frame rates (at least 25 frames per second). This is useful for computer vision systems that are, for example, deployed in autonomous vehicles. Most of the current models are far from this frame-rate; e.g., FCN-8 takes roughly 100ms to process a low-resolution image. Models based on dilated convolution help to increase the speed of segmentation models to some extent, but there is still plenty of room for improvement.
Memory Efficient Models
-----------------------
Many modern segmentation models require a significant amount of memory even during the inference stage. So far, much effort has been directed towards improving the accuracy of such models, but in order to fit them into specific devices, such as mobile phones, the networks must be simplified. This can be done either by using simpler models, or by using model compression techniques, or even training a complex model and then using knowledge distillation techniques to compress it into a smaller, memory efficient network that mimics the complex model.
3D Point-Cloud Segmentation
---------------------------
Numerous works have focused on 2D image segmentation, but much fewer have addressed 3D point-cloud segmentation. However, there has been an increasing interest in point-cloud segmentation, which has a wide range of applications, in 3D modeling, self-driving cars, robotics, building modeling, etc. Dealing with 3D unordered and unstructured data such as point clouds poses several challenges. For example, the best way to apply CNNs and other classical deep learning architectures on point clouds is unclear. Graph-based deep models can be a potential area of exploration for point-cloud segmentation, enabling additional industrial applications of these data.
Conclusions {#sec:conclusions}
===========
We have surveyed more than 100 recent image segmentation algorithms based on deep learning models, which have achieved impressive performance in various image segmentation tasks and benchmarks, grouped into ten categories such as: CNN and FCN, RNN, R-CNN, dilated CNN, attention-based models, generative and adversarial models, among others. We summarized quantitative performance analyses of these models on some popular benchmarks, such as the PASCAL VOC, MS COCO, Cityscapes, and ADE20k datasets. Finally, we discussed some of the open challenges and potential research directions for image segmentation that could be pursued in the coming years.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Tsung-Yi Lin from Google Brain, and Jingdong Wang and Yuhui Yuan from Microsoft Research Asia, for reviewing this work, and providing very helpful comments and suggestions.
[Shervin Minaee]{} is a machine learning scientist at Expedia Group. He received his PhD in Electrical Engineering and Computer Science from New York University, in 2018. His research interest includes computer vision, image segmentation, biometrics recognition, and unsupervised learning. He has published more than 40 papers and patents during his PhD. He has also previously worked as a research scientist at Samsung Research, AT&T Labs, and Huawei. He is a reviewer for more than 20 computer vision related journals from IEEE, ACM, and Elsevier, including IEEE Transactions on Image Processing, and International Journal of Computer Vision.
-5pt plus -1fil
[Yuri Boykov]{} is a Professor at Cheriton School of Computer Science at the University of Waterloo. He is also an adjunct Professor of Computer Science at Western University. His research is concentrated in the area of computer vision and biomedical image analysis with focus on modeling and optimization for structured segmentation, restoration, registration, stereo, motion, model fitting, recognition, photo-video editing and other data analysis problems. He is an editor for the International Journal of Computer Vision (IJCV). His work was listed among 10 most influential papers in IEEE Transactions of Pattern Analysis and Machine Intelligence (TPAMI Top Picks for 30 years). In 2017 Google Scholar listed his work on segmentation as a “classic paper in computer vision and pattern recognition” (from 2006). In 2011 he received Helmholtz Prize from IEEE and Test of Time Award by the International Conference on Computer Vision. The Faculty of Science at the University of Western Ontario recognized his work by awarding Distinguished Research Professorship in 2014 and Florence Bucke Prize in 2008.
-1pt plus -1fil
[Fatih Porikli]{} is an IEEE Fellow and a Professor in the Research School of Engineering, Australian National University. He is acting as the Chief Scientist at Huawei, Santa Clara. He received his Ph.D. from New York University in 2002. His research interests include computer vision, pattern recognition, manifold learning, image enhancement, robust and sparse optimization and online learning with commercial applications in video surveillance, car navigation, intelligent transportation, satellite, and medical systems.
-1pt plus -1fil
[Antonio Plaza]{} is a professor at the Department of Technology of Computers and Communications, University of Extremadura, where he received the M.Sc. degree in 1999 and the PhD degree in 2002, both in Computer Engineering. He has authored more than 600 publications, including 263 JCR journal papers (more than 170 in IEEE journals), 24 book chapters, and over 300 peer-reviewed conference proceeding papers. Prof. Plaza is a Fellow of IEEE “for contributions to hyperspectral data processing and parallel computing of Earth observation data.” He is a recipient of the recognition of Best Reviewers of the IEEE Geoscience and Remote Sensing Letters (in 2009) and a recipient of the recognition of Best Reviewers of the IEEE Transactions on Geoscience and Remote Sensing (in 2010), for which he served as Associate Editor in 2007-2012. He is a recipient of the Best Column Award of the IEEE Signal Processing Magazine in 2015, the 2013 Best Paper Award of the JSTARS journal, and the most highly cited paper (2005-2010) in the Journal of Parallel and Distributed Computing. He is included in the 2018 and 2019 Highly Cited Researchers List.
-5pt plus -1fil
[Nasser Kehtarnavaz]{} is a Distinguished Professor at the Department of Electrical and Computer Engineering at the University of Texas at Dallas, Richardson, TX. His research interests include signal and image processing, machine learning, and real-time implementation on embedded processors. He has authored or co-authored ten books and more than 390 journal papers, conference papers, patents, manuals, and editorials in these areas. He is a Fellow of SPIE, a licensed Professional Engineer, and Editor-in-Chief of Journal of Real-Time Image Processing.
-5pt plus -1fil
[Demetri Terzopoulos]{} is a Distinguished Professor of Computer Science at the University of California, Los Angeles, where he directs the UCLA Computer Graphics & Vision Laboratory. He is also Co-Founder and Chief Scientist of , Inc. He graduated from McGill University in Honours Electrical Engineering and received the PhD degree in Artificial Intelligence from the Massachusetts Institute of Technology (MIT) in 1984. He is or was a Guggenheim Fellow, a Fellow of the ACM, IEEE, Royal Society of Canada, and Royal Society of London, and a Member of the European Academy of Sciences, the New York Academy of Sciences, and Sigma Xi. Among his many awards are an Academy Award from the Academy of Motion Picture Arts and Sciences for his pioneering work on physics-based computer animation, and the inaugural Computer Vision Distinguished Researcher Award from the IEEE for his pioneering and sustained research on deformable models and their applications. ISI and other indexes have listed him among the most highly-cited authors in engineering and computer science, with more than 400 published research papers and several volumes.He has given over 500 invited talks around the world about his research, including more than 100 distinguished lectures and keynote/plenary addresses. He joined UCLA in 2005 from New York University, where he held the Lucy and Henry Moses Endowed Professorship in Science and was Professor of Computer Science and Mathematics at NYU’s Courant Institute of Mathematical Sciences. Previously, he was Professor of Computer Science and Professor of Electrical and Computer Engineering at the University of Toronto. Before becoming an academic in 1989, he was a Program Leader at Schlumberger corporate research centers in California and Texas.
[^1]: Note that some works report two versions of their models: one which is only trained on PASCAL VOC and another that is pre-trained on a different dataset (such as MS-COCO, ImageNet, or JFT-300M) and then fine-tuned on VOC.
|
Introduction
============
Assume that we are given a sequence of real-valued supermartingale differences $(\xi _i,\mathcal{F}_i)_{i=0,...,n}$ defined on some probability space $(\Omega ,\mathcal{F},\mathbf{P})$, where $\xi _0=0 $ and $\{\emptyset, \Omega\}=\mathcal{F}_0\subseteq ...\subseteq \mathcal{F}_n\subseteq \mathcal{F}$ are increasing $\sigma$-fields. So we have $\mathbf{E}(\xi_{i}|\mathcal{F}_{i-1})\leq 0, \ i=1,...,n, $ by definition. Set $$\label{matingal}
S_k=\sum_{i=1}^k\xi _i,\quad k=1,...,n.$$ Then $S=(S_k ,\mathcal{F}_k)_{k=1,...,n}$ is a supermartingale. Let $\left\langle S\right\rangle $ and $[S]$ be respectively the quadratic characteristic and the squared variation of the supermartingale $S:$ $$\left\langle S\right\rangle
_k=\sum_{i=1}^k\mathbf{E}(\xi _i^2|\mathcal{F} _{i-1})\ \ \ \ \ \textrm{and} \ \ \ \ \ \ \ [S]_k=\sum_{i=1}^{k}\xi_i^2.$$ The following exponential inequality for supermartingales can be found in Freedman [@FR75].\
**Theorem A. ** Suppose $\xi_{i} \leq \epsilon$ for a positive constant $\epsilon$. Then, for all $x, v > 0,$ $$\begin{aligned}
\mathbf{P}\Big(S_k \geq x\ \mbox{and}\ \left\langle S\right\rangle_k\leq v^2\ \mbox{for some}\ k \Big)
&\leq& B_{2}(x, \epsilon, v) \label{freedmanxf2} \\
&:=& \exp \left\{-\frac{x^2}{ 2(v^2+ x \epsilon )} \right\}. \nonumber\end{aligned}$$ After Freedman’s seminal work, many interesting exponential inequalities for martingales have been established. For continuous-time martingales with bounded jumps, inequality (\[freedmanxf2\]) has been established by Shorack and Wellner [@SW86]. By imposing certain moment conditions, [@V95] relaxed the condition of Shorack and Wellner and generalized inequality (\[freedmanxf2\]) for martingales with non-bounded jumps. Under the following conditional Bernstein condition: for a positive constant $\epsilon$, $$\begin{aligned}
\mathbf{E} (|\xi_{i}|^l|\mathcal{F} _{i-1}) \leq \frac{1}{2}\, l !\, \epsilon^{l-2}\mathbf{E} (\xi _i^2|\mathcal{F} _{i-1}),\ \ \ \mbox{for all} \ \ l\geq 2, \label{Bernstein}\end{aligned}$$ de la Peña [@D99] have obtained the following Bernstein type inequality for martingales, for all $x, v > 0,$ $$\begin{aligned}
\mathbf{P}(S_k \geq x\ \mbox{and}\ \langle S\rangle_{k}\leq v^2\ \mbox{for some}\ k ) &\leq& B_{1}(x, \epsilon, v) \label{djknss}\\
&:=& \exp \bigg\{-\frac{x^2}{v^2(1+\sqrt{1+2 x \epsilon/v^2})+x \epsilon } \bigg\}\nonumber \\
&\leq& B_{2}(x, \epsilon, v). \label{djkss}\end{aligned}$$ Inequality (\[djkss\]) has also been obtained by [@V95]. In particular, when $(\xi_i)_{i=1,...,n}$ are independent, the inequalities (\[djknss\]) and (\[djkss\]) reduce, respectively, to the inequalities of Bennett [@B62] and Bernstein [@B27]. Many other generalizations of Freedman’s inequality can be found in Haeusler [@H84], Pinelis [@P94a], Dzhaparidze and van Zanten [@Dz01], Delyon [@De09] and Khan [@K09].
Following the work of Freedman [@FR75], Shorack and Wellner [@SW86], van de Geer [@V95] and de la Peña [@D99], we develop some new methods, based on changes of probability measure, for establishing some general exponential inequalities for supermartingales. The methods are user-friendly and efficient.
In Theorem \[th5\], we obtain two exponential inequalities for supermartingales under a very general condition. Assume that $$\mathbf{E}(\exp\left\{\lambda \xi_i - g(\lambda)\xi_i^2 \right\}|\mathcal{F}_{i-1}) \leq 1+ f(\lambda)V_{i-1}$$ for some $\lambda \in (0, \infty),$ for two non-negative functions $ f(\lambda)$ and $g(\lambda)$, and for some non-negative and $\mathcal{F}_{i-1}$-measureable random variables $V_{i-1}$. Then, for all $ x, v, \omega > 0$, $$\begin{aligned}
&& \mathbf{P}\Big( S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w \ \mbox{for some}\ k \in [1, n] \Big) \nonumber\\
&\leq& \exp \left\{ -\lambda x + g(\lambda)v^2+n\log\left( 1+ \frac{f(\lambda)}{n} w \right)\right\} \label{gnhs1}\\
&\leq& \exp \Big\{ -\lambda x + g(\lambda)v^2 +f(\lambda) w \Big\} . \label{gnhs2}\end{aligned}$$
If $\xi_i \geq -\epsilon$ for a positive constant $\epsilon$, then our result (\[gnhs2\]) implies that, for all $x, v >0$, $$\begin{aligned}
\mathbf{P}\left( S_k \geq x\ \mbox{and}\ [S]_k\leq v^2 \ \mbox{for some}\ k \right)
\leq B_2\left(x, \epsilon, v\right).\label{ghssfyl2}\end{aligned}$$ This inequality is similar to the one of Freedman (\[freedmanxf2\]). To highlight the differences between (\[freedmanxf2\]) and (\[ghssfyl2\]), notice that the conditions $\xi_i \leq \epsilon$ and conditional variance $\langle S\rangle_k$ in Freedman’s inequality (\[freedmanxf2\]) are respectively replaced by the condition $\xi_i\geq -\epsilon$ and squared variation $[S]_k$ in our inequality (\[ghssfyl2\]). Moreover, inequality (\[ghssfyl2\]) completes Freedman’s inequality (\[freedmanxf2\]) by giving an estimation of deviation probabilities on the left side: if the martingale differences $ (\xi _i,\mathcal{F}_i)_{i=1,..., n }$ satisfy $\xi_i \leq \epsilon$ for all $i$, then, for all $x, v >0$, $$\begin{aligned}
\mathbf{P}\left( S_k \leq -x\ \mbox{and}\ [S]_k\leq v^2 \ \mbox{for some}\ k \right) \leq B_2\left(x, \epsilon, v\right).\end{aligned}$$
If the martingale differences verifies canonical assumption (which means $g(\lambda)=\lambda^2/2$ and $f(\lambda)=0$), then (\[gnhs2\]) implies the following de la Peña inequality [@D99], for all $x, v>0$, $$\begin{aligned}
\label{dlpieq}
\mathbf{P}\Big( S_{k} \geq x\ \mbox{and}\ [S]_k \leq v^2\ \mbox{for some}\ k \Big) \leq \exp \left\{ - \frac{x^2}{2\,v^2} \right\}.\end{aligned}$$ Moreover, we find that (\[dlpieq\]) implies the following self-normalized deviation result associated with independent and symmetric random variables, for all $x>0$, $$\begin{aligned}
\mathbf{P}\bigg( \max_{1\leq k \leq n} \frac{ S_{k}}{\sqrt{[S]_n} } \geq x \bigg) \leq \exp \left\{ - \frac{x^2}{2} \right\}.\end{aligned}$$
If $\mathbf{E} |\xi _i|^3 < \infty$, then (\[gnhs2\]) implies the following Bernstein type inequality, for all $x, v, w>0$, $$\begin{aligned}
\mathbf{P}\Big( S_k \geq x,\ [S]_k \leq v^2\ \mbox{and}\ \Upsilon(S_k)\leq w \ \mbox{for some}\ k \Big) &\leq& B_1\left(x, \frac{w }{3v^2}, v\right) \label{fineq2}\\
&\leq& B_2\left(x, \frac{w }{3v^2}, v\right), \label{fineq3}\end{aligned}$$ where $$\Upsilon(S_k) =\sum_{i=1}^k \mathbf{E}(|\xi_i|^3 |\mathcal{F}_{i-1});$$ see Corollary \[co9\]. Compared to the inequalities (\[djknss\]) and (\[djkss\]), the advantage of the last two inequalities (\[fineq2\]) and (\[fineq3\]) is that we do not assume the existence of moments of all orders.
Assume that $\mathbf{E}(e^{\lambda \xi_{i}}|\mathcal{F}_{i-1}) \leq 1+ f(\lambda)\mathbf{E}(\xi_i^2 |\mathcal{F}_{i-1})$ for some $\lambda \in (0, \infty)$ and a positive function $f(\lambda).$ Then Theorem \[th5\] implies that, for all $ x, v > 0$, $$\begin{aligned}
&& \mathbf{P}\left( S_k \geq x\ \mbox{and}\ \langle S\rangle_{k}\leq v^2\ \mbox{for some}\ k \in [1, n] \right) \nonumber\\
&\leq& \exp \left\{ -\lambda x +n\log\left( 1+ \frac{f(\lambda)}{n} v^2\right)\right\} \ \ \ \label{ff07} \\
&\leq& \exp \Big\{ -\lambda x +f(\lambda) v^2 \Big\} \label{ff08}.\end{aligned}$$ In particular, if $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ satisfies condition (\[Bernstein\]), then it holds $$\mathbf{E}(e^{\overline{\lambda} \xi_{i}}|\mathcal{F}_{i-1}) \leq 1+ f(\overline{\lambda})\mathbf{E}(\xi_i^2 |\mathcal{F}_{i-1}),$$ where $$\overline{\lambda} = \frac{2x/v^2}{2x\epsilon/v^2+1+\sqrt{1+2x\epsilon/v^2}} \ \ \ \textrm{and} \ \ \ f(\lambda)=\frac{\lambda^2 }{2 (1-\lambda\epsilon)}.$$ Inequality (\[ff08\]) reduces to inequality (\[djknss\]) with $\lambda = \overline{\lambda}$. Hence, our bound (\[ff07\]) with $\lambda = \overline{\lambda}$ improves inequality (\[djknss\]). In the i.i.d. case, bound (\[ff07\]) significantly improves the large deviation bound (\[djknss\]) on large deviation tail probabilities $\mathbf{P}(S_n \geq nx)$ by adding a factor with exponentially decay rate $\exp\{ -n c_{x}\},$ where $c_{x}>0$ does not depend on $n$. In the applications for linear regression models, we find that such type refinements are useful; see Theorem \[thlin\].
In Theorem \[th3\], we consider the case that supermartingale has sub-Gaussian differences. Assume that $ \mathbf{E}(e^{\lambda \xi_{i}}|\mathcal{F}_{i-1}) \leq \exp \{ f(\lambda) V_{i-1} \}$ for some $\lambda \in (0, \infty),$ for a positive function $ f(\lambda)$ and for some $\mathcal{F}_{i-1}$-measurable random variables $V_{i-1}$. Then, for all $ x, v > 0$, $$\begin{aligned}
\mathbf{P}\Big( S_{k} \geq x\ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq v^2\ \mbox{for some}\ k \Big)
\leq \exp \Big\{ -\lambda x +f(\lambda) v^2 \Big\}. \label{fsat9}\end{aligned}$$ In particular, when the function $f(\lambda)=\lambda^2/2$ for all $\lambda>0$ and $(V_{i})_{i=1,..,n}$ are constants, inequality (\[fsat9\]) reduces to Fuk’s inequality with $\lambda= \lambda(x):=x/\sum_{i=1}^n V_i^2$ (cf. Theorem 4 of [@F73]). Thus (\[fsat9\]) is a generalization of Fuk’s inequality [@F73] for supermartingales. If $V_{i-1}=\mathbf{E}(\xi_i^2|\mathcal{F}_{i-1})$ is the conditional variance, inequality (\[fsat9\]) reduces to Theorem 4.2 of Khan [@K09]. Inequality (\[fsat9\]) implies the following result, where $V_{i-1}$ is not the conditional variance. If $\xi_{i} \leq U_{i-1}$ for some $\mathcal{F}_{i-1}$-measurable random variables $U_{i-1}$, then, for all $x, v >0$, $$\begin{aligned}
\label{fint3}
\mathbf{P}\bigg( S_k \geq x\ \mbox{and}\ \sum_{i=1}^k C_{i-1}^2 \leq v^2\ \mbox{for some}\ k \bigg) \leq \exp\left\{- \frac{ x^2}{ 2\, v^2 } \right\},\end{aligned}$$ where $$\begin{aligned}
\label{f1}
C_{i-1}^2 = \left\{ \begin{array}{ll}
\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) , & \textrm{\ \ \ \ \ if $\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) \geq U^2_{i-1}$ }, \\
\displaystyle\frac{1}{4}\left( U_{i-1} + \frac{\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) }{ U_{i-1} }\right)^2, & \textrm{\ \ \ \ \ otherwise}.
\end{array} \right.\end{aligned}$$ Then we show that (\[fint3\]) implies a generalization of Azuma-Hoeffding’s inequality for martingales due to van de Geer [@V02]. Moreover, we also show that (\[fint3\]) significantly improves some recent inequalities of [@Be03] and Pinelis [@P06; @P06b] by adding an exponential decay factor in the case of $||\sum_{i=1}^n C_{i-1}^2||_{\infty}$ $ <
\sum_{i=1}^n ||C_{i-1}^2||_{\infty}$; see (\[mspi\]) and *Example* 1 for details. We find that such improvements are important in the applications for linear regression models and autoregressive processes; see Remarks \[fsfdf\] and \[fsfdf1\].
The paper is organized as follows. We present our theoretical results in Section \[sec2\], give the applications of our results in Section \[sec2.5\] and devote to the proofs of our results in Sections \[sec6\] - \[endsec\]. The proofs of the theorems and their corollaries are in the same sections.
Main results {#sec2}
============
Our first result is given under a very general condition.
\[th5\] Assume that $V_{i-1}, i \in [1, n],$ are non-negative and $\mathcal{F}_{i-1}$-measureable random variables. Suppose that $$\label{condimenme1}
\mathbf{E}(\exp\left\{\lambda \xi_i - g(\lambda)\xi_i^2 \right\}|\mathcal{F}_{i-1}) \leq 1+ f(\lambda)V_{i-1}$$ for some $\lambda\in (0, \infty)$, for two non-negative functions $f(\lambda)$ and $g(\lambda)$, and for all $i \in [1, n]$. Then, for all $ x, v, \omega > 0$, $$\begin{aligned}
&& \mathbf{P}\bigg( S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w \ \mbox{for some}\ k \in [1, n] \bigg) \nonumber\\
&\leq& \exp \left\{ -\lambda x + g(\lambda)v^2+n\log\left( 1+ \frac{f(\lambda)}{n} w \right)\right\} \label{fhgsa1} \\
&\leq& \exp \Big\{ -\lambda x + g(\lambda)v^2 +f(\lambda) w \Big\} \label{fhgsa2}.\end{aligned}$$
Notice that when $g(\lambda)=\lambda^2/2$ and $f(\lambda)\equiv 0$, condition (\[condimenme1\]) is called canonical assumption considered by de la Peña et al. [@D04; @D07]. In particular, when $V_{i-1}$ is a constant and $g(\lambda)\equiv 0$, condition (\[condimenme1\]) reduces to the condition considered by Rio [@R13b].
Next we show that Theorem \[th5\] is very useful for obtaining the concentration inequalities for supermartingales. Introducing the third moments of the supermartingale differences, we have the following Bernstein type inequalities.
\[co9\]Assume $\mathbf{E} (\xi _i^-)^3 < \infty$ for all $i \in [1, n]$. Denote by $
\langle\langle S \rangle\rangle_k=\sum_{i=1}^k \mathbf{E}((\xi_i^-)^3 |\mathcal{F}_{i-1})
$ for all $k \in [1, n].$ Then, for all $x, v, w>0$, $$\begin{aligned}
&&\mathbf{P}\Big( S_k \geq x,\ [S]_k \leq v^2\ \mbox{and}\ \langle\langle S \rangle\rangle_k\leq w \ \mbox{for some}\ k\in[1,n]\Big) \nonumber \\
&\leq& \exp\left\{-\overline{\lambda} x + \frac12 \overline{\lambda}^2v^2 + \frac{1}3\overline{\lambda}^3w \right\} \label{fhgst2} \\
&\leq& B_1\left(x, \frac{w }{3v^2}, v\right)\label{dktgsdt1}\\
& \leq& B_2\left(x, \frac{w }{3v^2}, v\right),\label{dktgsdt2}\end{aligned}$$ where $\overline{\lambda}=2x/(v^2+\sqrt{v^4+4w x})$.
Since $\langle\langle S \rangle\rangle_k \leq \Upsilon(S_k)$, the inequalities (\[fhgst2\]), (\[dktgsdt1\]) and (\[dktgsdt2\]) hold true when $\langle\langle S \rangle\rangle_k$ is replaced by $\Upsilon(S_k)$. To the best of our knowledge, such inequalities have not been established for the sums of independent random variables.
Notice that (\[dktgsdt1\]) and (\[dktgsdt2\]) are respectively the bounds of Bennett and Bernstein. Compared to the conditional Bernstein condition (\[Bernstein\]), the condition of Corollary \[co9\] does not assume the existence of the moments of all orders.
For supermartingales with differences bounded from below, we still have the following Bernstein type inequality.
\[th7\] Assume $\xi_i \geq -1$ for all $i \in [1, n]$. Then, for all $x, v>0$, $$\begin{aligned}
\mathbf{P}\Big( S_k \geq x\ \mbox{and}\ [S]_k\leq v^2 \ \mbox{for some}\ k\in[1,n]\Big)
&\leq & \left(1+ \frac{x}{v^2}\right)^{v^2}e^{-x} \nonumber\\
&\leq& B_1\left(x, 1, v\right) \nonumber \\
&\leq& B_2\left(x, 1, v\right).\label{frie2}\end{aligned}$$
Inequality (\[frie2\]) is similar to Freedman’s inequality (\[freedmanxf2\]). However, there are two differences between (\[frie2\]) and (\[freedmanxf2\]). First, we assume $\xi_i$ bounded from below instead of $\xi_i$ bounded from above. Second, the quadratic characteristic $\langle S\rangle_k$ in Freedman’s inequality is replaced by the squared variation $[S]_k$ in our inequality (\[frie2\]). Such inequality could be useful for estimating the tail probabilities when the variances of $(\xi_i)$ do not exist.
Under the conditional Bernstein condition, we have
\[co2\] Assume, for a constant $\epsilon \in (0, \infty)$, $$\begin{aligned}
\label{br2}
\mathbf{E}(\xi_{i}^{l} | \mathcal{F}_{i-1}) \leq \frac12\, l!\, \epsilon^{l-2} \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}) \ \ a.s.\ \ \mbox{for all}\ \ l\geq 2\ \mbox{and all}\ i \in [1,n].\end{aligned}$$ Then, for all $ x, v> 0,$ $$\begin{aligned}
&& \mathbf{P}\Big( S_k \geq x\ \mbox{and}\ \langle S\rangle_{k}\leq v^2\ \mbox{for some}\ k \in [1,n] \Big) \nonumber\\
&\leq& B_{1,n}(x, \epsilon, v):=\exp \left\{- \overline{\lambda} x + n\log \left(1+ \frac{\overline{\lambda}^2 v^2 }{2n(1-\overline{\lambda}\epsilon)}\right)\right\} \label{ie1g5}\\
&\leq& B_{1}(x, \epsilon, v) , \label{ie16}\end{aligned}$$ where $$\overline{\lambda}= \frac{2x/v^2}{2x\epsilon/v^2+1+\sqrt{1+2x\epsilon/v^2}} \, \in (0, \epsilon^{-1}).$$
Notice that $B_{1}(x, \epsilon, v) =\exp \left\{- \overline{\lambda} x + \frac{\overline{\lambda}^2 v^2 }{2 (1-\overline{\lambda}\epsilon)} \right\}.$ In the independent case, inequality (\[ie16\]) is known as Bennett’s inequality [@B62]. To highlight how the bound $B_{1,n}(x, \epsilon, v)$ improves Bennett’s bound $B_{1}(x, \epsilon, v)$, we rewrite $$\begin{aligned}
B_{1,n}\left(x, \epsilon, v \right) = B_{1}(x, \epsilon, v) \exp\left\{-n \psi\left(\frac{\overline{\lambda}^2 v^2 }{2n(1-\overline{\lambda}\epsilon)}\right)\right\},\nonumber\end{aligned}$$ where $\psi(t)=t-\log(1+t)$ is a nonnegative convex function in $t\geq 0$. It is easy to see that, in the i.i.d. case with $v^2=n\sigma_1^2$ (or more generally when $\frac{\epsilon}{v}=\frac{\sigma_1}{\sqrt{n}}$ for a constant $\sigma_1>0$), we have $$\begin{aligned}
\label{fbnb}
\ \ \ \ \ \ \ \ \ B_{1,n}\left(nx, \epsilon, \sqrt{n} \sigma_1 \right) = B_{1}\left(nx, \epsilon, \sqrt{n} \sigma_1 \right) \exp \left\{ - n \, c_{x,\sigma_1,\epsilon} \right\},\end{aligned}$$ where $c_{x,\sigma_1,\epsilon}=\psi\left(\frac{\overline{\lambda}^2 v^2 }{2 (1-\overline{\lambda}\epsilon)}\right)>0$ does not depend on $n$. Thus Bennett’s bound $B_{1}\left(nx, \epsilon, \sqrt{n} \sigma_1 \right)$ on tail probabilities $\mathbf{P}\left( S_n \geq nx \right)$ is strengthened by adding a factor with exponential decay rate $\exp \left\{ -n \, c_{x,\sigma_1,\epsilon} \right\}$ as $n\rightarrow \infty$. Since the conditional Bernstein condition (\[Bernstein\]) implies condition (\[br2\]), inequality (\[ie1g5\]) strengthen de la Peña’s inequality (\[djknss\]).
One calls $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ *conditionally symmetric*, if $\mathbf{E}(\xi_i >y |\mathcal{F}_{i-1})= \mathbf{E}(\xi_i < -y |\mathcal{F}_{i-1})$ for all $i \in [1,n]$ and for any $y \geq0$; see Hitczenko [@H90], de la Peña [@D99] and Bercu and Touati [@BT08]. It is obvious that if $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ are conditionally symmetric, then, for any $y> 0$, $(\xi _i\mathbf1_{\{ |\xi_i|> y \}},\mathcal{F}_i)_{i=1,...,n}$ are also conditionally symmetric. In particular, the conditionally symmetric martingale differences satisfy the canonical assumption $ \mathbf{E}(\exp\left\{\lambda \xi_i - \lambda^2\xi_i^2/2 \right\}|\mathcal{F}_{i-1})\leq 1$ for all $\lambda\geq 0$; see [@D99; @D04; @D07]. Thus, by Theorem \[th5\] and optimizing on $\lambda$, inequality (\[fhgsa2\]) implies de la Peña’s inequality (\[dlpieq\]).
The following result is a Fuk-Nagaev type inequality [@F73; @N79] for martingales with conditionally symmetric differences. Its proof is based on a truncation argument on martingale differences.
\[co1\] Assume that $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ are conditionally symmetric. Let $$V_k^2(y)=\sum_{i=1}^{k}\mathbf{E}(\xi_i^2\textbf{\emph{1}}_{\{|\xi_i|\leq y\}} |\mathcal{F}_{i-1}), \ \ \ k\in [1,n].$$ Then, for all $x, y, v> 0$ and $v^2\leq ny^2$, $$\begin{aligned}
&& \mathbf{P}\Big( S_k \geq x\ \mbox{and}\ V_k^2(y)\leq v^2\ \mbox{for some}\ k \in [1,n] \Big) \nonumber\\
&\leq& \exp \left\{- \underline{\lambda} x + n\log \left(1+ \frac{ v^2 }{ n\,y^2}\left( \cosh(\underline{\lambda}\, y) -1\right) \right)\right\} +\, \mathbf{P}\left( \max_{1\leq i \leq n} \xi_i > y \right) \label{hdgh1} \\
&\leq& \exp \left\{- \overline{\lambda} x + \frac{ v^2 }{ y^2}\left( \cosh(\overline{\lambda}\, y)-1\right) \right\} +\, \mathbf{P}\left( \max_{1\leq i \leq n} \xi_i > y \right), \label{hdgh2}\end{aligned}$$ where $$\underline{\lambda}=\frac{1}{y} \log\left( \frac{\frac{xy}{v^2}-\frac{x}{ny}+\sqrt{1+\frac{(xy)^2}{v^4} -2\frac{x^2}{nv^2} } }{1-\frac{x}{ny}} \,\right) \ \ \ \mbox{and}\ \ \ \ \overline{\lambda}=\frac{1}{y} \log \left( \frac{xy}{v^2} +\sqrt{1+\frac{(xy)^2}{v^4} } \,\right).$$
Inequality (\[hdgh1\]) is the best possible that can be obtained from the exponential Markov inequality $\mathbf{P}\left( S_n \geq x \right) \leq \inf_{\lambda\geq 0} \mathbf{E}e^{\lambda(S_n- x)}$ under the present assumption. Indeed, if $(\xi_i)_{i=1,...,n}$ are i.i.d. and satisfy the following distribution $$\begin{aligned}
\mathbf{P}(\xi_i=y )= \mathbf{P}(\xi_i= -y )= \frac{v^2}{2ny^2} \ \ \ \ \mbox{and} \ \ \ \mathbf{P}(\xi_i= 0 )= 1- \frac{v^2}{ ny^2},\end{aligned}$$ then the bound (\[hdgh1\]) equals to $\inf_{\lambda\geq 0} \mathbf{E}e^{\lambda(S_n- x)}$. In this sense, inequality (\[hdgh1\]) is a version of Hoeffding’s inequality (cf. (2.8) of [@Ho63]) for martingales with conditionally symmetric differences.
For martingales with bounded conditionally symmetric differences, Sason [@S12] has obtained (\[hdgh1\]) under the conditions $|\xi_i|\leq y$ and $\mathbf{E}(\xi_i^2 |\mathcal{F}_{i-1})\leq v^2/n$. He has also obtained (\[hdgh2\]) under the assumption $|\xi_i|\leq y$. Thus (\[hdgh1\]) improves and generalizes the Sason’s inequalities under a more general condition.
For the martingales with square integrable differences, several Nagaev type inequalities based on the truncation arguments on martingale differences can be found in Haeusler [@H84] and Courbot [@Co99]. For optimal exponential convergence speed of such type bounds, we refer to Lesigne and Volný [@LV01] and Fan et al. [@F12; @Fx1].
Consider the case that the differences $ (\xi _i,\mathcal{F}_i)_{i=1,...,n}$ are sub-Gaussian. We have the following very general result.
\[th3\] Assume that $V_{i-1}, i \in [1,n], $ are positive and $\mathcal{F}_{i-1}$-measureable random variables. Suppose $ \mathbf{E}(e^{\lambda \xi_{i}}|\mathcal{F}_{i-1}) \leq \exp \{ f(\lambda) V_{i-1} \}$ for all $i \in [1,n]$ and for a positive function $ f(\lambda)$ for some $\lambda \in (0, \infty).$ Then, for all $ x, v > 0$, $$\begin{aligned}
\mathbf{P}\Big( S_{k} \geq x\ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq v^2 \ \mbox{for some}\ k \in [1,n] \Big)
\leq \exp \Big\{ -\lambda x +f(\lambda) v^2 \Big\} .\label{f1sd}\end{aligned}$$
In the particular case where $v^2=\sum_{i=1}^{n} ||V_{i-1}||_{\infty}$ and $f(\lambda)=\lambda^2/2$, Theorem \[th3\] reduces to Theorem 4 of Fuk [@F73] after optimizing on $\lambda$. If $V_{i-1}=\mathbf{E}(\xi_i^2|\mathcal{F}_{i-1})$, Theorem \[th3\] reduces to Theorem 4.2 of Khan [@K09]. Thus (\[f1sd\]) can be regarded as a generalization of the inequalities of Fuk [@F73] and Khan [@K09].
Using Theorem \[th3\], we extend Azuma-Hoeffding’s inequality (cf. [@A67; @Ho63]) to the case that the differences are only bounded from above.
\[co4\] Assume that $U_{i-1}, i \in [1,n],$ are nonnegative and $\mathcal{F}_{i-1}$-measureable random variables. Denote by $$\begin{aligned}
\label{f1}
C_{i-1}^2 = \left\{ \begin{array}{ll}
\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) , & \textrm{\ \ \ \ if \ $ \mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) \geq U^2_{i-1}$ }, \\
\displaystyle\frac{1}{4}\left( U_{i-1} + \frac{\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) }{ U_{i-1} }\right)^2, & \textrm{\ \ \ \ otherwise}.
\end{array} \right.\end{aligned}$$ If $\xi_{i} \leq U_{i-1}$ for all $i \in [1,n]$, then, for all $\lambda >0$, $$\begin{aligned}
\mathbf{E}(e^{ \lambda \xi_i}|\mathcal{F}_{i-1}) \leq \exp\left\{ \frac{\lambda^2}{2} C_{i-1}^2 \right\} ,\label{gnlm}\end{aligned}$$ and, for all $ x, v >0$, $$\begin{aligned}
\label{f3}
\mathbf{P}\bigg( S_k \geq x\ \mbox{and}\ \sum_{i=1}^k C_{i-1}^2 \leq v^2\ \mbox{for some}\ k \in [1, n] \bigg) \leq \exp\left\{- \frac{ x^2}{ 2\, v^2 } \right\}.\end{aligned}$$ In particular, if $\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) \geq U^2_{i-1}$ for all $i\in [1,n]$, then, for all $x,v \geq0$, $$\begin{aligned}
\label{f3saf}
\mathbf{P}\left( \max_{1\leq k \leq n} S_k \geq x\ \mbox{and}\ \langle S\rangle_n \leq v^2\right) \leq \exp\left\{- \frac{ x^2}{ 2\, v^2 } \right\}.\end{aligned}$$
Notice that if $(\xi_i)_{i=1,...,n}$ are independent and satisfy the conditions $\xi_i \leq c_i$ and $\mathbf{E}\xi_i^2\geq c_i^2$ for some constants $(c_i)_{i=1,...,n}$, then (\[f3saf\]) is a gaussian bound with $v^2= \sum_{i=1}^n\mathbf{E}\xi_i^2$. It is obvious that the Rademacher random variables satisfy this assumption.
For martingale differences $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$, inequality (\[f3\]) generalizes the following inequality due to van de Geer (cf.Theorem 2.5 of [@V02]): if $L_{i-1} \leq \xi_{i} \leq U_{i-1}$ for some $\mathcal{F}_{i-1}$-measureable random variables $L_{i-1}$ and $U_{i-1}$, then, for all $x,v > 0$, $$\begin{aligned}
\label{fgr}
\mathbf{P}\bigg( S_k \geq x\ \mbox{and}\ \frac14 \sum_{i=1}^k \left( U_{i-1} -L_{i-1}\right)^2 \leq v^2\ \mbox{for some}\ k \in [1, n] \bigg) \leq \exp\left\{- \frac{ x^2}{ 2\, v^2 } \right\}.\end{aligned}$$ Indeed, since $$\begin{aligned}
\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1})= \mathbf{E}((\xi_{i}-L_{i-1}) \xi_{i}|\mathcal{F}_{i-1}) \leq \mathbf{E}((\xi_{i}-L_{i-1}) U_{i-1}|\mathcal{F}_{i-1}) \leq -L_{i-1}U_{i-1}\ ,\end{aligned}$$ we have $$\begin{aligned}
\sum_{i=1}^k C_{i-1}^2 \leq \sum_{i=1}^k \frac{1}{4}\left( U_{i-1} + \frac{\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) }{ U_{i-1} }\right)^2 \leq \frac{1}{4} \sum_{i=1}^k \left( U_{i-1} -L_{i-1}\right)^2 \label{fsav}\end{aligned}$$ and $$\left\{ \frac{1}{4} \sum_{i=1}^k \left( U_{i-1} -L_{i-1}\right)^2 \leq v^2 \right\} \subseteq \left\{ \sum_{i=1}^k C_{i-1}^2 \leq v^2 \right\},$$ which together with (\[f3\]) implies (\[fgr\]).
Under the assumption of Corollary \[co4\], Pinelis [@P06; @P06b] (see also Bentkus [@Be03]) proved the following inequality, for all $x>0$, $$\begin{aligned}
\mathbf{P}\bigg( \max_{1\leq k \leq n} S_k \geq x
\bigg) &\leq& c\left(1-\Phi(\frac{x}{\hat{v}})\right) \label{pio}\\
&=& O\left( \frac{1}{1+ x/ \hat{v} } \exp\left\{- \frac{ x^2}{ 2\, \hat{v}^2 } \right\} \right),\ \ \ \ x\rightarrow \infty, \nonumber\end{aligned}$$ where $c$ is an absolute constant and $$\hat{v}^2= \sum_{i=1}^n \left|\left|\frac{1}{4}\left( U_{i-1} + \frac{\mathbf{E}(\xi_{i}^2|\mathcal{F}_{i-1}) }{ U_{i-1} }\right)^2\right|\right|_{\infty}.$$ Notice that $\hat{v}^2 \geq \left|\left|\sum_{i=1}^n C_{i-1}^2\right|\right|_{\infty}.$ If $\hat{v}^2 = \left|\left|\sum_{i=1}^n C_{i-1}^2\right|\right|_{\infty}$, Pinelis’ inequality (\[pio\]) is better than ours (\[f3\]) by adding a factor $\frac{O(1)}{1+ x/ \hat{v} }$. Otherwise $\hat{v}^2 > \left|\left|\sum_{i=1}^n C_{i-1}^2\right|\right|_{\infty}$, our inequality (\[f3\]) improves Pinelis’ inequality (\[pio\]) by adding an exponential decay factor of order $$\begin{aligned}
\label{mspi}
\left(1+ \frac{x}{\hat{v}} \right) \exp\left\{ -\frac{x^2}{2}\delta \right\}, \ \ \ \ \ x\rightarrow \infty,\end{aligned}$$ where $$\delta=\frac{ \hat{v}^2-\left|\left|\sum_{i=1}^n C_{i-1}^2\right|\right|_{\infty}}{ \hat{v}^2 \left|\left|\sum_{i=1}^n C_{i-1}^2\right|\right|_{\infty} } > 0.$$ To illustrate this factor, consider the following example. For a much more significant improvement, we refer to Remark \[fsfdf\].
*Example 1*: Assume that $(\varepsilon_{i})_{i=1,...,n}$ is a sequence of Rademacher random variables, and that $\mathcal{N}$ is a random variable independent of $(\varepsilon_{i})_{i=1,...,n}$. Set $$\begin{aligned}
\xi_i = \left(\frac{ \varepsilon_{i}}{\sqrt{n}} \sin \mathcal{N}\,\right) \mathbf{1}_{\{i\, \textrm{ is odd}\}} + \left( \frac{ \varepsilon_{i}}{\sqrt{n}} \cos \mathcal{N}\, \right) \mathbf{1}_{\{i\, \textrm{ is even}\}},\end{aligned}$$ $\mathcal{F}_0=\sigma\{\mathcal{N} \}$ and $\mathcal{F}_i=\sigma\{\mathcal{N} ,$ $ \varepsilon_{j}, 1\leq j \leq i \}$. So we have $$\begin{aligned}
\xi_i \leq U_{i} := \left(\frac{ 1}{\sqrt{n}} | \sin \mathcal{N} |\,\right) \mathbf{1}_{\{i\, \textrm{ is odd}\}} + \left(\frac{ 1}{\sqrt{n}} | \cos \mathcal{N} |\,\right)\, \mathbf{1}_{\{i\, \textrm{ is even}\}}\end{aligned}$$ and $$\begin{aligned}
\sum_{i=1}^n C_{i-1}^2 = \langle S\rangle_n=\sum_{i=1}^n \left( \frac{ \sin^2 \mathcal{N}}{ n } \, \mathbf{1}_{\{i\, \textrm{ is odd}\}} + \frac{ \cos^2 \mathcal{N}}{ n } \, \mathbf{1}_{\{i\, \textrm{ is even}\}} \right).\end{aligned}$$ Hence, for any even number $n$, it is easy to see that $\hat{v}^2=1 > \frac{1}{2}= \sum_{i=1}^n C_{i-1}^2=\langle S\rangle_n $. Then Pinelis’ inequality (\[pio\]) shows that: $$\mathbf{P}\left( \max_{1\leq k \leq n} S_k \geq x
\right) = O\bigg( \frac{1}{1+ x } \exp\left\{- \frac{ \, x^2}{ 2 } \right\} \bigg),\ \ \ \ x \rightarrow \infty,$$ while our inequality (\[f3\]) implies that: $$\mathbf{P}\Big( \max_{1\leq k \leq n} S_k \geq x
\Big) \leq \exp\Big\{- x^2 \Big\} .$$ Thus our inequality (\[f3\]) improves Pinelis’ inequality (\[pio\]) by adding a factor with the exponential decay rate $(1+ x)\exp\left\{- \frac{x^2}{ 2} \right\} .$
\[eton\] Corollary \[co4\] implies a simple proof of the following self-normalized deviation inequality. Assume that $(\xi _i)_{i=1,...,n}$ are independent and symmetric. Then, for all $x > 0$, $$\begin{aligned}
\mathbf{P}\left( \max_{1\leq k \leq n} \frac{S_k}{\sqrt{[S]_n}} \geq x \right)
\leq \exp\left\{- \frac{x^2}{ 2 } \right\}, \label{fdgfgfdgh}\end{aligned}$$ where by convention $\frac00=0.$ A similar result can be found in Hitczenko [@H90]. Hitczenko has obtained the same upper bound on tail probabilities $\mathbf{P}\left( S_n \geq x ||\sqrt{[S]_n} ||_{\infty}\right).$ For more precise results, we refer to Wang and Jing [@WJ99]. In particular, the Cramér type large deviations have been established by Jing, Shao and Wang [@JSW03] without assuming that $(\xi _i)_{i=1,...,n}$ are symmetric (or $(\xi _i)_{i=1,...,n}$ have exponential moments).
Applications to statical estimation {#sec2.5}
===================================
The exponential concentration inequalities for martingales certainly have many applications. [@M] and Rio [@R13a] applied such type inequalities to estimate the concentration of separately Lipschhitz functions. Van de Geer [@V95] found that such inequalities can be used for maximum likelihood estimation for counting processes. Liu and Watbled [@Liu09a] considered the free energy of directed polymers in a random environment via martingale inequalities. Dedecker and Fan [@DF12] gave an application of these inequalities to the Wasserstein distance between the empirical measure and the invariant distribution. We refer to Bercu [@B08] for more interesting applications of the concentration inequalities for martingales.
In the sequel, we discuss how to apply our results to linear regression models, autoregressive processes and branching processes. We find these models in Liptser and Spokoiny [@Ls01] and Bercu and Touati [@BT08].
**1. Linear regression models.** Consider the stochastic linear regression models given, for all $k \in [1, n],$ by $$\label{ine29}
X_{k}=\theta \phi_k + \varepsilon_{k}$$ where $X_k, \phi_k$ and $\varepsilon_{k}$ are the observations, the regression variables and the driven noises, respectively. We assume that $(\phi_k)$ is a sequence of independent random variables. We also assume that $(\varepsilon_k)$ is a sequence of independent and identically distributed (i.i.d.) random variables, with mean zero and variation $\sigma^2>0.$ Moreover, we suppose that $(\phi_k)$ and $(\varepsilon_k)$ are independent. Our interest is to estimate the unknown parameter $\theta.$ The well-known least-squares estimator $\theta_n$ is given below $$\label{ine30}
\theta_n = \frac{\sum_{k=1}^n \phi_{k} X_k}{\sum_{k=1}^n \phi_{k}^2}.$$ When $(\phi_k)$ and $(\varepsilon_k)$ are sub-Gaussian, exponential inequalities on the convergence of $\theta_n -\theta$ have been established by Bercu and Touati [@BT08]. When $(\varepsilon_k)$ are the normal random variables, Liptser and Spokoiny [@Ls01] have established the following estimation: for all $x\geq 1,$ $$\begin{aligned}
\label{ls21}
\mathbf{P}\left( \pm \, (\theta_n -\theta)\sqrt{ \Sigma _{k=1}^n \phi_{k}^2} \geq x \sigma \right) \ \leq\ \sqrt{\frac2 \pi} \, \frac{1}{x} \, \exp\bigg\{ - \frac{x^2}{2 } \bigg\}.\end{aligned}$$ Here, we would like to give a generalization of this inequality. Consider the case that the random variables $(\varepsilon_k)$ satisfy the Bernstein condition.
\[thlin\] Assume $ |\phi_{k}|/\sqrt{\sum_{k=1}^n \phi_{k}^2} \leq \epsilon_1$ and $$|\mathbf{E} \varepsilon_{i}^{k} | \leq \frac12 k!\epsilon_2^{k-2} \mathbf{E} \varepsilon_{i}^2 ,\ \ \ \ \ \textrm{for all}\ k\geq 2\ \textrm{and all}\ i \in [1, n],$$ for two positive numbers $\epsilon_1$ and $\epsilon_2$. Let $\epsilon = \epsilon_1 \epsilon_2/\sigma.$ Then, for all $x \geq 0$, $$\begin{aligned}
\label{th5ineq}
\mathbf{P}\left( \pm \, (\theta_n -\theta)\sqrt{ \Sigma _{k=1}^n \phi_{k}^2} \geq x \sigma \right) \ \leq\ B_{1,n}(x, \epsilon, 1) \ \leq \ \exp\bigg\{ - \frac{x^2}{2(1+ x\epsilon) } \bigg\}.\end{aligned}$$
Since $ |\phi_{k}|/\sqrt{\sum_{k=1}^n \phi_{k}^2} \leq 1,$ the condition imposed on $(\phi_{k})$ of Theorem \[thlin\] can be dropped by taking $\epsilon_1 =1$. It is interesting to see that by taking $\epsilon_1 =1,$ bound (\[th5ineq\]) does not depend on the distribution of the regression variables $(\phi_{k})$. This is a big advantage in practice.
If $a\leq |\phi_{k}| \leq b $ for two positive constants $a$ and $b$, then the condition of Theorem \[th5\] is satisfied with $\epsilon_1 = \frac{b\, }{a \sqrt{n } }.$ Indeed, it is easy to see that $$\frac{|\phi_{k}|}{\sqrt{\sum_{i=1}^n \phi_{i}^2} } \leq \frac{b}{\sqrt{n a^2} } = \epsilon_1.$$ In this case, bound (\[th5ineq\]) behaviors like $\exp\{-x^2/2\}$ when $x=o( \sqrt{n})$ as $n\rightarrow \infty$. When $x$ is large, bound (\[th5ineq\]) behaviors like $\exp\{-x \}$.
If $(\varepsilon_k)$ are bounded from above, we have the following sub-Gaussian tail bound from Corollary \[co4\].
\[dssaf\] If $ \varepsilon_k \leq \epsilon$ for all $k \in [1, n],$ then, for all $x \geq 0$, $$\begin{aligned}
\label{tsfsdfd}
\mathbf{P}\left( (\theta_n -\theta)\sqrt{ \Sigma _{k=1}^n \phi_{k}^2} \geq x \sigma \right) \ \leq \ \exp \bigg\{ -\frac{x^2 }{2 C_n } \bigg\},\end{aligned}$$ where $$C_n = \frac14 \Big( \frac{ \epsilon}{ \sigma} + \frac{\sigma }{\epsilon} \Big)^2 .$$ In particular, if $ |\varepsilon_k| \leq \epsilon,$ bound (\[tsfsdfd\]) holds true on the tail probabilities $$\mathbf{P}\left( \pm \, (\theta_n -\theta)\sqrt{ \Sigma _{k=1}^n \phi_{k}^2} \geq x \sigma \right).$$
\[fsfdf\] If $ |\varepsilon_k| \leq \epsilon,$ we can obtain some similar bounds by using van de Geer’s inequality (\[fgr\]) or Pinelis’ inequality (\[pio\]). However, those bounds are less tight than (\[tsfsdfd\]). Indeed, by van de Geer’s inequality, we can obtain the bound (\[tsfsdfd\]) with a larger $C_n=(\epsilon/\sigma)^2.$ If we make use of Pinelis’ inequality (or Bentkus’ inequality [@Be03]), the bound will be as large as $$\frac{O(1)}{x} \exp \bigg\{ -\frac{x^2 }{2 n C_n } \bigg\}.$$
Next, consider the tail probabilities of $(\theta_n -\theta) \sum_{k=1}^n \phi_{k}^2.$ It seems that our inequalities fit well to such type estimations.
\[thds\] Assume that there exist $ \alpha \in (1, 2]$ and $c> 0$ such that $$\mathbf{E}e^{ \lambda \varepsilon_{i} } \leq e^{c |\lambda|^\alpha} \ \ \ \ \ \textrm{for all } i\in [1, n] \textrm{ and all } \lambda \in \mathbf{R} .$$ Then, for all $x, v \geq 0$, $$\begin{aligned}
\label{sfcscs}
\mathbf{P}\Big( \pm (\theta_n -\theta) \sum_{k=1}^n \phi_{k}^2 \geq x \ \textrm{and} \ \sum_{k=1}^n |\phi_{k}|^\alpha \leq v^\alpha \Big) &\leq& \exp\left\{-C(\alpha) \left(\frac{x}{v} \right)^\frac{\alpha}{\alpha-1} \right\},\end{aligned}$$ where $$C(\alpha)=(c\, \alpha )^{\frac1{1-\alpha}} \left(1- \alpha^{-1} \right).$$
When the condition of Theorem \[thds\] is verified with $\alpha=2,$ then $(\varepsilon_i)$ are known as sub-Gaussian random variables. It is known that the bounded random variables and the normal random variables are all sub-Gaussian random variables. In particular, if $(\varepsilon_i)$ are the standard normal random variables, then bound (\[sfcscs\]) is valid with $\alpha=2$ and $c=C(2)=1/2$.
**2. Autoregressive processes.** The model of autoregressive can be stated as follows: for all $k \in [1, n],$ $$\label{scfso1}
X_{k} = \theta X_{k-1} + \varepsilon_k \, ,$$ where $(X_k)$ and $(\varepsilon_k)$ are the observations and driven noises, respectively. We assume that $(\varepsilon_k)$ is a sequence of i.i.d. centered random variables with variation $\sigma^2>0.$ The process is said to be stable if $|\theta|\leq 1,$ unstable if $|\theta|=1$ and explosive if $|\theta|>1.$ We can estimate the unknown parameter $\theta$ by the least-squares estimator given by, for all $n\geq 1$, $$\label{scfso2}
\theta'_n = \frac{\sum_{k=1}^n X_{k}X_{k-1}}{\sum_{k=1}^n X_{k-1}^2} .$$ When $X_0$ and $(\varepsilon_k)$ are the normal random variables, the convergence rate of $\theta_n '-\theta$ has been established by Bercu and Touati [@BT08]. Here, we would like to give an almost sure convergence rate of $(\theta_n' -\theta) \sum_{k=1}^n X_{k-1}^2 .$
By an argument similar to that of Theorem \[thds\], we have the following result.
Assume the condition of Theorem \[thds\]. Then bound (\[sfcscs\]) holds true on the tail probabilities $$\begin{aligned}
\label{sfcsc2s}
\mathbf{P}\Big( \pm (\theta_n' -\theta) \sum_{k=1}^n X_{k-1}^2 \geq x \ \textrm{and} \ \sum_{k=1}^n | X_{k-1}|^\alpha \leq v^\alpha \Big).\end{aligned}$$
If $(\varepsilon_{i})$ are bounded, then we have
\[th6\] Assume $|\varepsilon_{i}| \leq \epsilon$ for all $i \in [1, n].$ Then, for all $x, v > 0$, $$\begin{aligned}
\label{tsfsdfdog}
\mathbf{P}\left(\pm \, (\theta_n '-\theta) \sum _{k=1}^n X_{k-1}^2 \geq x \ \textrm{and} \ L_n \leq v^2 \right) \ \leq\ \exp \bigg\{ -\frac{x^2 }{2\, v^2} \bigg\},\end{aligned}$$ where $$L_n = \frac14 \Big( \epsilon + \frac{\sigma^2}{\epsilon} \Big)^2 \sum _{k=1}^n X_{k-1}^2.$$
\[fsfdf1\] We can obtain some similar bounds by using Corollary 2.3 or van de Geer’s inequality. However, those bounds are less tight than (\[tsfsdfdog\]). For instance, by van de Geer’s inequality, we can obtain the bound (\[tsfsdfdog\]) with a larger $L_n= \epsilon^2
\sum _{k=1}^n X_{k-1}^2.$
**3. Branching processes.** Consider the Galton-Watson process stating from $X_0=1$ and given, for all $n\geq 1$, by $$X_n = \sum_{k=1}^{X_{n-1}} Y_{n, k} \, ,$$ where $(Y_{n, k})$ is a sequence i.i.d. and nonnegative integer-valued random variables. The distribution of $(Y_{n, k}),$ with finite mean $m$ and variance $\sigma^2,$ is commonly called the offspring or reproduction distribution. We are interested in the estimation of the offspring mean $m.$ The Lotka-Nagaev estimator is given by $$m_n= \frac{X_{n}}{X_{n-1}} \,.$$ Assume $X_{n}> 0$ a.s. such that the Lotka-Nagaev estimator $m_n$ is always well defined. Our goal is to establish exponential inequalities for $m_n.$ Denote by $$\xi_{n, k} = Y_{n, k} -m .$$ Then $$(m_n -m ) X_{n-1} = X_n -mX_{n-1} = \sum_{k=1}^{X_{n-1}}\xi_{n, k} \, .$$ Thus $(m_n -m ) X_{n-1}$ is a sum of independent random variables by given $X_{n-1}$. By Corollary \[co2\], we easily obtain the following exponential inequalities.
Assume, for a constant $\epsilon \in (0, \infty)$, $$\begin{aligned}
|\mathbf{E} \xi_{n, k}^{l} | \leq \frac12\, l!\, \epsilon^{l-2} \mathbf{E} \xi_{n, k}^2 \ \ \ \ \mbox{for all}\ \ l\geq 2\ \mbox{and all}\ k \in [1,X_{n-1}].\end{aligned}$$ Then, for all $x, v > 0$, it holds $$\begin{aligned}
\mathbf{P}\Big( |m_n -m | X_{n-1} \geq x \ \textrm{and} \ X_{n-1}\sigma^2 \leq v^2 \, \Big| \, X_{n-1} \Big) & \leq& 2 \, B_{1,n}(x, \epsilon, v) \\
&\leq& 2 \exp\bigg\{ - \frac{x^2}{2( v^2+ x\epsilon) } \bigg\}.\end{aligned}$$ In particular, it implies that, for all $x > 0,$ $$\begin{aligned}
\mathbf{P}\Big( |m_n -m | \geq x \Big) \ \leq \ 2\, \mathbf{E}\bigg( \exp \bigg\{ - X_{n-1} \frac{x^2 }{2\, (\sigma^2+ x\epsilon) } \bigg\} \, \bigg).\end{aligned}$$
Since $\xi_{n, k} \geq -m$, we have the following one side sub-Gaussian bound by Corollary \[co4\]. This bound cannot be obtained from Azuma-Hoefding’s inequality.
For all $x, v > 0$, it holds $$\begin{aligned}
\mathbf{P}\Big( (m_n -m ) X_{n-1} \leq - x, M X_{n-1} \leq v^2 \, \Big| \, X_{n-1} \Big) \ \leq \ \exp \bigg\{ -\frac{x^2 }{2\, v^2 } \bigg\} \, ,\end{aligned}$$ where $$\begin{aligned}
\label{f1}
M = \left\{ \begin{array}{ll}
\sigma^2 , & \textrm{\ \ \ \ \ if $\sigma \geq m$, } \\
\displaystyle\frac{1}{4}\left( m + \frac{\sigma^2}{ m }\right)^2, & \textrm{\ \ \ \ \ if $\sigma < m$.}
\end{array} \right.\end{aligned}$$ In particular, it implies that, for all $x > 0,$ $$\begin{aligned}
\mathbf{P}\Big( (m_n -m ) \leq - x \Big) \ \leq \ \mathbf{E}\bigg( \exp \bigg\{ - X_{n-1} \frac{x^2 }{2\, M } \bigg\} \, \bigg).\end{aligned}$$
More generale estimations on the tail probabilities $\mathbf{P}\left( |m_n -m | \geq x \right),$ we refer to Bercu and Touati [@BT08]. In particular, Bercu and Touati have established the Bernstein bounds associated with the cumulant generating function of $\xi_{n, k}.$
Proof of Theorem \[th5\] {#sec6}
==========================
Suppose $ \mathbf{E}(\exp\left\{\lambda \xi_i - g(\lambda)\xi_i^2 \right\}|\mathcal{F}_{i-1}) \leq 1+ f(\lambda)V_{i-1}$ for a constant $ \lambda \in (0, \infty)$ and all $i \in [1, n].$ Define the *exponential multiplicative martingale* $Z(\lambda )=(Z_k(\lambda ),\mathcal{F}_k)_{k=0,...,n},$ where $$Z_k(\lambda )=\prod_{i=1}^{ k}\frac{\exp\left\{\lambda \xi_i -g (\lambda)\xi_i^2 \right\}}{\mathbf{E}\left(\exp\left\{\lambda \xi_i -g(\lambda)\xi_i^2 \right\} |
\mathcal{F}_{i-1} \right)}, \quad \quad \quad Z_0(\lambda )=1. \label{C-1}$$ If $T$ is a stopping time, then $Z_{T\wedge k}(\lambda )$ is also a martingale, where $$Z_{T\wedge k}(\lambda )=\prod_{i=1}^{T\wedge k}\frac{\exp\left\{\lambda \xi_i -g (\lambda)\xi_i^2 \right\}}{\mathbf{E}\left(\exp\left\{\lambda \xi_i -g(\lambda)\xi_i^2 \right\} |
\mathcal{F}_{i-1} \right)}, \quad \quad Z_0(\lambda )=1.$$ Thus, the random variable $Z_{T\wedge k}(\lambda ) $ is a probability density on $(\Omega ,\mathcal{F},\mathbf{P})$, i.e. $$\int Z_{T\wedge k}(\lambda) d \mathbf{P} = \mathbf{E}(Z_{T\wedge k}(\lambda))=1.$$ Define the *conjugate probability measure* $$d\mathbf{P}_\lambda =Z_{T\wedge n}(\lambda )d\mathbf{P}. \label{chmeasure3}$$ Denote $\mathbf{E}_{\lambda}$ the expectation with respect to $\mathbf{P}_{\lambda}.$
*Proof of Theorem \[th5\].* For any $x, v, w>0$, define the stopping time $$T(x,v,w)=\min\left\{k\in [1, n]: S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w \right\},$$ with the convention that $\min{\emptyset}=0$. Then $$\textbf{1}_{\{ S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w\ \mbox{for some}\ k\in[1,n] \}} = \sum_{k=1}^{n} \textbf{1}_{\{ T(x,v,w)=k\}}.$$ By the change of measure (\[chmeasure3\]), we deduce that, for all $x,\lambda, v,w>0$, $$\begin{aligned}
&& \mathbf{P}\left( S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w\ \mbox{for some}\ k\in[1,n] \right) \nonumber\\
&=& \mathbf{E}_{\lambda}\Big( Z_{T\wedge n}(\lambda)^{-1}\textbf{1}_{\{S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w\ \mbox{for some}\ k\in[1,n]\}}\Big) \nonumber \\
&=& \sum_{k=1}^{n}\mathbf{E}_{\lambda}\Big( \exp\{-\lambda S_{k}+g(\lambda) [S]_{k} + \Xi_{k}(\lambda) \} \textbf{1}_{\{T(x,v,w)=k\}}\Big) , \label{ghnda}\end{aligned}$$ where $$\Xi_{k}(\lambda)= \sum_{i=1}^k \log \mathbf{E} \left(\exp\left\{\lambda \xi_i -g(\lambda)\xi_i^2 \right\} |\mathcal{F}_{i-1}\right).$$ Using Jensen’s inequality and the condition $ \mathbf{E}(\exp\left\{\lambda \xi_i - g(\lambda)\xi_i^2 \right\}|\mathcal{F}_{i-1}) \leq 1+ f(\lambda)V_{i-1}$, we have $$\begin{aligned}
\Xi_{k}(\lambda) &\leq& k \log\left( \frac{1}{k} \sum_{i=1}^k \mathbf{E} \left(\exp\left\{\lambda \xi_i -g(\lambda)\xi_i^2 \right\} |\mathcal{F}_{i-1}\right) \right) \nonumber \\
&\leq& k \log\left( 1+ \frac{1}{k} f(\lambda)\sum_{i=1}^k V_{i-1} \right).\end{aligned}$$ Thus (\[ghnda\]) implies that, for all $x,\lambda, v, w>0$, $$\begin{aligned}
&& \mathbf{P}\left( S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w\ \mbox{for some}\ k\in[1,n] \right) \nonumber\\
&\leq& \sum_{k=1}^{n}\mathbf{E}_{\lambda} \Bigg( \exp \left\{-\lambda S_k +g(\lambda)[S]_{k} + k \log\left( 1+ \frac{1}{k} f(\lambda)\sum_{i=1}^k V_{i-1} \right) \right\} \textbf{1}_{\{T(x,v,w)=k\}}\Bigg) .\ \ \ \\end{aligned}$$ By the fact $S_{k}\geq x$, $[S]_{k}\leq v^2$ and $\sum_{i=1}^k V_{i-1}\leq w$ on the set $\{T(x,v,w)=k\}$, we find that, for all $x,\lambda, v, w>0$, $$\begin{aligned}
&& \mathbf{P}\left( S_{k} \geq x,\ [S]_{k} \leq v^2 \ \mbox{and}\ \sum_{i=1}^{k} V_{i-1} \leq w\ \mbox{for some}\ k\in[1,n] \right) \nonumber\\
&\leq& \exp \left\{-\lambda x +g(\lambda)v^2 + k \log\left( 1+ \frac{1}{k} f(\lambda)w \right) \right\} \mathbf{E}_{\lambda} \Big(\sum_{k=1}^{n}\textbf{1}_{\{T(x,v,w)=k\}}\Big)\nonumber\\
&\leq& \exp \left\{-\lambda x +g(\lambda)v^2 + n \log\left( 1+ \frac{1}{n} f(\lambda)w \right) \right\} \label{sthv1}\\
&\leq& \exp \left\{-\lambda x +g(\lambda)v^2 + f(\lambda)w \right\} \label{sthv2}.\end{aligned}$$ This gives the desired inequalities (\[fhgsa1\]) and (\[fhgsa2\]), and completes the proof of Theorem \[th5\].
*Proof of Corollary \[co9\].* To prove Corollary \[co9\], we should use the following basic inequality: $$\exp\left\{x-\frac{1}{2}x^2\right\} \leq 1 + x +\frac{1}{3}(x^-)^3 ,\ \ \ \ x \in \mathbf{R}.$$ By the last inequality, it follows that, for all $\lambda>0$, $$\begin{aligned}
\mathbf{E}\left(\exp\left\{\lambda \xi_i -\frac{1}{2} (\lambda\xi_i)^2 \right\} \bigg|\mathcal{F}_{i-1} \right) \leq 1+ \frac{1}{3}\lambda^3 \mathbf{E}\left( (\xi_i^- )^3 |\mathcal{F}_{i-1} \right).\end{aligned}$$ Applying the inequalities (\[fhgsa1\]) and (\[fhgsa2\]) with $g(\lambda)=\frac{\lambda^2}{2}$, $f(\lambda)=\frac{\lambda^3}{3}$ and $V_{i-1}=\mathbf{E}\left( (\xi_i^- )^3 |\mathcal{F}_{i-1} \right),$ we get (\[fhgst2\]) by noting the fact that $$\begin{aligned}
\label{ineq385}
\inf_{\lambda>0}\exp\left\{-\lambda x + \frac12 \lambda^2v^2 + \frac{1}3\lambda^3w \right\} = \exp\left\{-\overline{\lambda} x + \frac12 \overline{\lambda}^2v^2 + \frac{1}3\overline{\lambda}^3w \right\},\end{aligned}$$ where $\overline{\lambda}=2x/(v^2+\sqrt{v^4+4wx}).$ By a simple calculation, we find that, for all $v, w >0$ and all $0<\lambda < \frac{3v^2}{w}$, $$\begin{aligned}
\frac12 \lambda^2v^2 + \frac{1}3\lambda^3w \leq \frac{\lambda^2v^2}{2( 1-\frac{\lambda w}{3 v^2})}.\end{aligned}$$ Thus, for all $x, v, w >0$, $$\begin{aligned}
\exp\left\{-\overline{\lambda} x + \frac12 \overline{\lambda}^2v^2 + \frac{1}3\overline{\lambda}^3w \right\} &\leq& \inf_{0< \lambda < \frac{3v^2}{w}}\exp\left\{-\lambda x + \frac{\lambda^2v^2}{2( 1-\frac{\lambda w}{3 v^2})} \right\} \nonumber\\
&=& B_1\left(x, \frac{w}{3v^2}, v\right)\nonumber\\
&\leq& B_2\left(x, \frac{w}{3v^2}, v\right).\nonumber\end{aligned}$$ Combining this inequality with (\[fhgst2\]), we obtain the desired inequalities (\[dktgsdt1\]) and (\[dktgsdt2\]) of the corollary.
To prove Corollary \[th7\], we need the following lemma.
\[lemma17\] If $\xi$ is a random variable such that $\xi \geq -1$ and $\mathbf{E}\xi
\leq 0$, then, for all $\lambda \in [0, 1),$ $$\begin{aligned}
\mathbf{E}\Big( \exp\left\{\lambda \xi +(\lambda+\log(1-\lambda))\xi^2\right\}\Big) &\leq& 1.\end{aligned}$$
*Proof.* Assume $\xi \geq -1$ and $\lambda \in [0, 1)$. Then $\lambda \xi \geq -\lambda> -1$. Since the function $$f(x)=\frac{\log(1+x)-x}{ x^2/2 }, \ \ \ \ \ \ \ x > -1,$$ is increasing in $x$, we have $$\begin{aligned}
\log(1+\lambda \xi )&\geq & \lambda \xi +\frac{1}{2}(\lambda \xi )^2 f(-\lambda) \nonumber\\
&=& \lambda \xi +\xi^2(\lambda+\log(1-\lambda)).\label{sksn}\end{aligned}$$ Thus $$\begin{aligned}
\exp\left\{\lambda \xi +\xi^2(\lambda+\log(1-\lambda))\right\}
\leq 1+ \lambda \xi .\end{aligned}$$ Since $\mathbf{E} \xi \leq 0$, it follows that $$\begin{aligned}
\mathbf{E}\Big( \exp\left\{\lambda \xi +\xi^2(\lambda+\log(1-\lambda))\right\} \Big) \leq 1,\end{aligned}$$ which gives the desired inequality.
*Proof of Corollary \[th7\].* Let $T=\min\{ k \in [1, n]: S_k\geq x \ \mbox{and} \ [S]_k\leq v^2 \}.$ Applying inequality (\[fhgsa2\]) with $g(\lambda)=-(\lambda+\log(1-\lambda))$ and $f(\lambda)=0,$ from Lemma \[lemma17\], we obtain, for all $x, v >0$ and all $\lambda \in [0, 1)$, $$\begin{aligned}
&& \mathbf{P}( S_k\geq x \ \mbox{and} \ [S]_k\leq v^2\ \mbox{for some}\ k\in[1,n]) \nonumber\\
&\leq& \exp\{-\lambda x- (\lambda+\log(1-\lambda))v^2 \} . \label{fmula}\end{aligned}$$ It is easy to see that bound (\[fmula\]) attains its minimum at $$\begin{aligned}
\lambda = \lambda(x)= \frac{x}{v^2+ x}. \label{lanbda1}\end{aligned}$$ Substituting $\lambda=\lambda(x)$ in (\[fmula\]), we get, for all $x, v >0$, $$\begin{aligned}
&&\mathbf{P}\left( S_k \geq x\ \mbox{and}\ [S]_k\leq v^2 \ \ \mbox{for some}\ \ k\in[1,n]\right) \nonumber \\
&\leq& \inf_{ \lambda \in [0, 1)} \exp\{-\lambda x- (\lambda+\log(1-\lambda))v^2 \} \label{fnmsx}\\
&= & \left(1+ \frac{x}{v^2}\right)^{v^2}e^{-x}. \label{fnmsv}\end{aligned}$$ Using Taylor’s expansion, we deduce that, for all $\lambda \in [0, 1)$, $$\begin{aligned}
\lambda+\log(1-\lambda) &=&-\frac{ \lambda^2}{2} \left(1 + \frac23 \lambda + \frac24 \lambda^2+...\right) \nonumber \\
&\geq& -\frac{ \lambda^2}{2}\left(1+\lambda+\lambda^2+...\right) \nonumber \\
&=& -\frac{ \lambda^2 }{2(1-\lambda)}.\end{aligned}$$ Thus we have, for all $x, v >0$, $$\begin{aligned}
\inf_{ \lambda \in [0, 1)} \exp\{-\lambda x- (\lambda+\log(1-\lambda))v^2 \} &\leq & \inf_{ \lambda \in [0, 1)} \exp\left\{-\lambda x+ \frac{ \lambda^2v^2 }{2(1-\lambda)}\right\} \nonumber \\
&=& B_{1}(x, 1, v) \label{copie1}\\
&\leq& B_{2}(x, 1, v). \label{copie2}\end{aligned}$$ Combining (\[fnmsv\]), (\[copie1\]) and (\[copie2\]) together, we obtain the desired inequalities of Corollary \[th7\].
*Proof of Corollary \[co2\].* Assume $ \mathbf{E}(\xi_{i}^{l} | \mathcal{F}_{i-1}) \leq \frac12 l!\epsilon^{l-2} \mathbf{E}(\xi_i^2 | \mathcal{F}_{i-1})$ for all $ l\geq 2$ and a constant $\epsilon \in (0, \infty)$. Then, for all $0\leq \lambda < \epsilon^{-1}$, $$\begin{aligned}
\mathbf{E}(e^{\lambda \xi _i}|\mathcal{F}_{i-1})-1 &=& \sum_{k=2}^{+\infty}\frac{\lambda^{k}}{k !} \mathbf{E}(\xi_{i}^{k} |\mathcal{F}_{i-1}) \nonumber\\
&\leq & \frac{\lambda^2}{2}\mathbf{E}(\xi_{i}^{2} |\mathcal{F}_{i-1})\sum_{k=2}^{\infty}(\lambda\epsilon)^{k-2} \nonumber\\
&= &\frac{\lambda^2}{2(1-\lambda\epsilon)}\mathbf{E}(\xi_{i}^{2} |\mathcal{F}_{i-1}).\nonumber\end{aligned}$$ Using Theorem \[th5\], we obtain the desired inequality (\[ie1g5\]) with $\lambda= \overline{\lambda}$. Since $n\log(1+\frac{t}{n}) \leq t$ for all $t\geq0,$ it follows that, for all $x,v>0$, $$\begin{aligned}
B_{1,n}(x, \epsilon, v) \leq \exp \left\{- \overline{\lambda} x + \frac{\overline{\lambda}^2 v^2 }{2 (1-\overline{\lambda}\epsilon)} \right\}
= B_1(x, \epsilon, v).\end{aligned}$$ This completes the proof of Corollary \[co2\].
*Proof of Corollary \[co1\].* Assume that $(\xi_{i}, \mathcal{F}_{i})_{i=1,...,n}$ are conditionally symmetric. For any $y>0$, let $\eta_i=\xi_{i}\textbf{1}_{\{|\xi_{i}|\leq y \}}$. Then $(\eta_i, \mathcal{F}_{i})_{i=1,...,n}$ is a sequence of bounded and conditionally symmetric martingale differences. Using Taylor’s expansion, we obtain the following estimation of the moment generating function of $\eta_i$, $$\begin{aligned}
\mathbf{E}(e^{\lambda \eta_i}|\mathcal{F}_{i-1}) &=& \mathbf{E}\left(\frac{e^{\lambda \eta_i}+e^{-\lambda \eta_i}}{2}\ \bigg|\ \mathcal{F}_{i-1} \right) \nonumber\\
&=& 1+ \sum_{k=1}^{\infty}\frac{ \lambda^{2k}}{(2k)!} \mathbf{E}\left( \eta_i^{2k} \ |\ \mathcal{F}_{i-1} \right).\end{aligned}$$ Since $|\eta_i|\leq y$, it follows that $\mathbf{E}\left( \eta_i^{2k} \ |\ \mathcal{F}_{i-1} \right) \leq y^{2k-2}\mathbf{E}\left( \eta_i^{2} \ |\ \mathcal{F}_{i-1} \right)$ and that $$\begin{aligned}
\mathbf{E}(e^{\lambda \eta_i}|\mathcal{F}_{i-1}) &\leq& 1+ \frac{\mathbf{E}\left( \eta_i^{2} \ |\ \mathcal{F}_{i-1} \right)}{y^2} \sum_{k=1}^{\infty}\frac{ (\lambda y)^{2k}}{(2k)!} \nonumber\\
&\leq& 1+ \frac{\mathbf{E}\left( \eta_i^{2} \ |\ \mathcal{F}_{i-1} \right)}{y^2} \left( \cosh(\lambda y)-1\right).\label{kdagfd}\end{aligned}$$ Set $V_k^2(y)=\sum_{i=1}^{k}\mathbf{E}(\eta_i^2 |\mathcal{F}_{i-1})$ for all $k \in [1,n]$. Using Theorem \[th5\], we obtain, for all $ x, v> 0,$ $$\begin{aligned}
P_1&:=& \mathbf{P}\left( \sum_{i=1}^k\eta_i \geq x\ \mbox{and}\ V_k^2(y) \leq v^2\ \mbox{for some}\ k \in [1,n] \right) \nonumber\\
&\leq& \inf_{ \lambda \geq 0} \exp \left\{- \lambda x + n\log \left(1+ \frac{ v^2 }{ n\,y^2}\left( \cosh(\lambda y)-1\right) \right)\right\} \label{fg25}\\
&\leq& \inf_{ \lambda \geq 0} \exp \left\{- \lambda x + \frac{ v^2 }{ y^2}\left( \cosh(\lambda y)-1\right) \right\}.\label{fg26}\end{aligned}$$ By some simple calculations, we find that (\[fg25\]) and (\[fg26\]) attain their minimums at $\underline{\lambda}$ and $\overline{\lambda}$ of Corollary \[co1\], respectively. It is easy to see that $$\begin{aligned}
&& \mathbf{P}\left( S_k \geq x\ \mbox{and}\ V_k^2(y)\leq v^2\ \mbox{for some}\ k \in [1,n] \right) \nonumber\\
&\leq& \mathbf{P}\left( \sum_{i=1}^k\left(\eta_i + \xi_{i}\textbf{1}_{\{ \xi_{i} < - y \}}\right)\geq x\ \mbox{and}\ V_k^2(y) \leq v^2\ \mbox{for some}\ k \in [1,n] \right) \nonumber\\
&&+ \,\mathbf{P}\left( \sum_{i=1}^k \xi_{i}\textbf{1}_{\{ \xi_{i} > y \}}> 0\ \mbox{and}\ V_k^2(y) \leq v^2\ \mbox{for some}\ k \in [1,n] \right) \nonumber\\
&\leq& P_1\ + \, \mathbf{P}\left( \max_{1\leq i \leq n} \xi_i > y \right) . \label{fghgh}\end{aligned}$$ Implementing (\[fg25\]) and (\[fg26\]) into (\[fghgh\]), we get the desired inequalities (\[hdgh1\]) and (\[hdgh2\]).
Proof of Theorem \[th3\] and its corollaries {#sec5}
============================================
The proof of Theorem \[th3\] is similar to the argument of Theorem \[th5\].
*Proof of Theorem \[th3\].* Let $T=\min\{ k \in [1, n]: S_k\geq x \ \mbox{and} \ \sum_{i=1}^{k}V_{i-1}\leq v^2 \}.$ According to (\[ghnda\]) with $g(\lambda)\equiv 0$, we have the following estimation, for all $ x, v>0$, $$\begin{aligned}
\mathbf{P}\bigg( S_{k} \geq x\ \mbox{and}\ \sum_{i=1}^{k}V_{i-1} \leq v^2\ \textrm{for some}\ k \in [1,n] \bigg)\ \leq \ \sum_{k=1}^n \mathbf{E}_\lambda \Big(\exp \left\{ -\lambda x +\Psi _k(\lambda )\right\} \mathbf{1}_{\left\{ T = k \right\}}\Big),\end{aligned}$$ where $$\begin{aligned}
\Psi _k(\lambda )= \sum_{i=1}^k \log \mathbf{E}(e^{\lambda \xi_{i}}|\mathcal{F}_{i-1}) .\end{aligned}$$ Using the condition $ \mathbf{E}(e^{\lambda \xi_{i}}|\mathcal{F}_{i-1})$ $\leq $$\exp \{ f(\lambda) V_{i-1} \}$ and the fact $\sum_{i=1}^{k}V_{i-1}\leq v^2$ on the set $\left\{ T = k \right\}$, we obtain $$\begin{aligned}
&& \mathbf{P}\bigg( S_{k} \geq x\ \mbox{and}\ \sum_{i=1}^{k}V_{i-1} \leq v^2\ \textrm{for some}\ k \in [1,n] \bigg) \nonumber \\
&\leq & \sum_{k=1}^n \mathbf{E}_\lambda \Bigg(\exp \left\{ -\lambda x +f(\lambda) \sum_{i=1}^{k}V_{i-1} \right\} \mathbf{1}_{\left\{ T = k \right\}} \Bigg) \\
&\leq& \exp \left\{ -\lambda x +f(\lambda) v^2 \right\} ,\end{aligned}$$ which gives (\[f1sd\]) of Theorem \[th3\].
In the proof of Corollary \[co4\], we shall need the following two lemmas.
\[lemma5\] If $\xi$ is a random variable satisfying $\xi \leq 1$, $\mathbf{E}\xi
\leq 0$ and $\mathbf{E}\xi^2=\sigma^2$, then, for all $\lambda > 0,$ $$\mathbf{E} e^{\lambda \xi} \leq \frac{1}{1+\sigma^2} \exp\left\{-\lambda \sigma^2
\right\} + \frac{\sigma^2}{1+\sigma^2 }\exp\{\lambda\} .$$
A proof can be found in Fan, Grama and Liu [@F12].
\[lemma6\] Assume that $\xi$ is a random variable satisfying $\mathbf{E}\xi
\leq 0$, $\xi \leq b$ for a constant $b> 0$ and $\mathbf{E}\xi^2=\sigma^2$. Set $$\begin{aligned}
s^2 = \left\{ \begin{array}{ll}
\sigma^2 , & \textrm{\ \ \ \ \ if $\sigma \geq b $}, \\
\frac{1}{4}\left(b + \frac{\sigma^2}{b }\right)^2, & \textrm{\ \ \ \ \ if $\sigma < b $}.
\end{array} \right.\end{aligned}$$ Then, for all $\lambda > 0,$ $$\label{ghsdfd}
\mathbf{E} e^{\lambda \xi} \leq \exp\left\{ \frac{\lambda^2s^2}{2} \right\}.$$
*Proof.* If $\sigma \geq b$, by Lemma \[lemma5\], then, for all $t\geq0$, $$\mathbf{E}e^{t \xi/\sigma} \leq \frac{1}{2}\Big(e^{-t}+ e^{ t}\Big)\leq \exp\left\{ \frac{t^2}{2} \right\}.$$ Taking $t =\lambda \sigma \geq 0$, we have $$\begin{aligned}
\label{fie1}
\mathbf{E}e^{\lambda \xi } \leq \exp\left\{ \frac{ \lambda^2 \sigma^2}{2} \right\}=\exp\left\{\frac{\lambda^2s^2}{2} \right\}.\end{aligned}$$ If $\sigma < b $, by Lemma \[lemma5\], we get, for all $t\geq0$, $$\begin{aligned}
\mathbf{E}e^{t \xi/b} &\leq& \frac{1}{1+\sigma^2/b^2} \exp\left\{-t \sigma^2/b^2
\right\} + \frac{\sigma^2/b^2}{1+\sigma^2/b^2 }\exp\{t\} \\
&=& \exp\left\{ f(z)\right\},\end{aligned}$$ where $z=t(1+\sigma^2/b^2)$ and $f(z)=- z p + \log(1-p + p e^z)$ with $p=\frac{\sigma^2/b^2}{1+\sigma^2/b^2}$. Since $f(0)=f'(0)=0$, $$f'(z)=-p+\frac{p}{p+(1-p)e^{-z}}$$ and $$f''(z)=\frac{p(1-p)e^{-z}}{(p+(1-p)e^{-z})^2} \leq \frac{1}{4},$$ we have $$f(z)\leq \frac{1}{8}z^2= \frac{ t^2}{8}\left(1+\frac{\sigma^2 }{ b^2} \right)^2 \ \ \ \ \mbox{and}\ \ \ \ \ \mathbf{E}e^{t \xi/b} \leq \exp\left\{ \frac{ t^2}{8}\left(1+\frac{\sigma^2 }{ b^2} \right)^2 \right\}.$$ Taking $t =\lambda b\geq 0$, we obtain $$\begin{aligned}
\mathbf{E}e^{\lambda \xi } \leq \exp\left\{ \frac{\lambda^2}{8} \left( b +\frac{\sigma^2 }{ b } \right)^2 \right\}=\exp\left\{\frac{\lambda^2s^2}{2} \right\}. \label{fie2}\end{aligned}$$ Combining (\[fie1\]) and (\[fie2\]) together, we obtain (\[ghsdfd\]).
*Proof of Corollary \[co4\].* Inequality (\[gnlm\]) follows immediately from Lemma \[lemma6\]. Using Theorem \[th3\], we obtain, for all $ x, \lambda, v>0$, $$\begin{aligned}
\mathbf{P}\left( S_k \geq x\ \mbox{and}\ \sum_{i=1}^k C_{i-1}^2 \leq v^2\ \mbox{for some}\ k \in [1, n] \right)
\leq \exp\left\{- \lambda x + \frac{ \lambda^2v^2}{2} \right\}.\end{aligned}$$ Minimizing the right hand side of the last inequality with respect to $\lambda\geq 0$, we easily obtain (\[f3\]).
*Proof of Remark \[eton\].* Assume that $(\xi _i)_{i=1,...,n}$ are independent and symmetric. Set $$\mathcal{F}_{i}=\sigma \left\{\xi_{k}, k\leq i, \xi_j^2, 1\leq j \leq n \right\}.$$ Since $\xi _i$ is symmetric, we deduce that $$\mathbf{E}(\xi_i> y|\, \mathcal{F}_{i-1} ) = \mathbf{E}(\xi_i> y|\, \xi_i^2 ) =\mathbf{E}( -\xi_i > y|\, (-\xi_i)^2 )=\mathbf{E}( -\xi_i > y|\, \mathcal{F}_{i-1} ).$$ Thus $\Big(\frac{\xi_i}{\sqrt{[S]_n}},\mathcal{F}_i\Big)_{i=1,...,n}$ are conditionally symmetric martingale differences. For all $1\leq i\leq n$, we have $$\begin{aligned}
\mathbf{E}\left(\exp\left\{ \lambda \frac{\xi_i}{\sqrt{[S]_n}} \right\} \Bigg| \mathcal{F}_{i-1} \right) = \frac12 \ \mathbf{E}\left(\exp\left\{ \lambda \frac{\xi_i}{\sqrt{[S]_n}} \right\} + \exp\left\{ -\lambda \frac{\xi_i}{\sqrt{[S]_n}} \right\} \Bigg| \mathcal{F}_{i-1} \right).\end{aligned}$$ Using the inequality $\frac12(e^t+e^{-t}) \leq e^{t^2/2}$, we obtain, for all $\lambda\geq0$, $$\begin{aligned}
\mathbf{E}\left(\exp\left\{ \lambda \frac{\xi_i}{\sqrt{[S]_n}} \right\} \Bigg| \mathcal{F}_{i-1} \right)
\leq \exp\left\{\frac{\lambda^2\xi_i^2}{2\, [S]_n } \right\}.\end{aligned}$$ Since $\xi_i^2$ is measurable with respect to $\mathcal{F}_{i-1},$ it follows that $$\sum_{i=1}^{k}\mathbf{E}\left(\frac{\xi_i^2}{\,[S]_n} \Big| \mathcal{F}_{i-1} \right)= \sum_{i=1}^{k} \frac{\xi_i^2}{\,[S]_n} \leq 1$$ for all $k \in [1, n].$ By Theorem \[th3\] with $V_{i-1}=\frac{\xi_i^2}{\,[S]_n},$ it follows that, for all $x, \lambda\geq0$, $$\begin{aligned}
\mathbf{P}\left( \max_{ 1\leq k \leq n } \frac{S_k}{\sqrt{[S]_n} } \geq x \right)
\leq \exp \left\{ -\lambda x +\frac{\lambda^2}{2} \right\}. \label{fjna}\end{aligned}$$ The right hand side of the last inequality attends its minimum at $\lambda=x$. Substituting $\lambda=x$ into (\[fjna\]), we easily get (\[fdgfgfdgh\]) of Remark \[eton\].
Proof of Theorems \[thlin\] - \[th6\] {#endsec}
=====================================
We make use of Corollary \[co2\] to prove Theorem \[thlin\].\
*Proof of Theorem \[thlin\].* From (\[ine29\]) and (\[ine30\]), it is easy to see that $$\label{sdgvf1}
\theta_n -\theta = \sum_{k=1}^n\frac{ \phi_{k} \varepsilon_k}{\sum_{k=1}^n \phi_{k}^2}. \nonumber$$ For any $i=1,...,n$, set $$\begin{aligned}
\label{sdgvf2}
\xi_i= \frac{ \phi_{i} \varepsilon_i}{ \sigma \sqrt{\sum_{k=1}^n \phi_{k}^2}}\ \ \ \ \textrm{and} \ \ \ \ \mathcal{F}_{i} = \sigma \Big( \phi_{k}, \varepsilon_k, 1\leq k\leq i,\ \phi_{k}^2, 1\leq k\leq n \Big).\end{aligned}$$ Then $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ is a sequence of martingale differences and satisfies $$\frac{(\theta_n -\theta)\sqrt{\sum_{k=1}^n \phi_{k}^2} } { \sigma} =\sum_{i=1}^n\xi_i.$$ Notice that $$\langle S\rangle_n = \sum_{i=1}^{n} \frac{ \phi_{i}^2 }{ \sigma^2 (\sum_{k=1}^n \phi_{k}^2)} \mathbf{E}(\varepsilon_i^2 | \mathcal{F}_{i-1} ) =\sum_{i=1}^{n} \frac{ \phi_{i}^2 }{ \sum_{k=1}^n \phi_{k}^2 } = 1,$$ and that $$\begin{aligned}
\label{fgdsasas}
|\mathbf{E}(\xi_i^k | \mathcal{F}_{i-1} )| &=& \frac{ \phi_{i}^2 }{ \sigma^k (\sum_{k=1}^n \phi_{k}^2)} \mathbf{E}\bigg( \Big(\frac{ \phi_{i} }{ \sqrt{\sum_{k=1}^n \phi_{k}^2}}\Big)^{k-2} \varepsilon_i^k \bigg| \mathcal{F}_{i-1} \bigg) \nonumber\\
&\leq& \frac{ \phi_{i}^2 \epsilon_1^{k-2} }{ \sigma^k (\sum_{k=1}^n \phi_{k}^2)} \mathbf{E} \Big( \varepsilon_i^k \Big| \mathcal{F}_{i-1} \Big) \nonumber\\
&\leq& \frac12 k! \, \epsilon_2^{k-2} \frac{ \phi_{i}^2 \epsilon_1^{k-2} }{ \sigma^{k-2}(\sum_{k=1}^n \phi_{k}^2)} \nonumber\\
&=& \frac12 k! \, \epsilon ^{k-2} \mathbf{E}(\xi_i^2 | \mathcal{F}_{i-1} ). \nonumber
\end{aligned}$$ Applying Corollary \[co2\] to $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$, we obtain the claim of Theorem \[thlin\].
*Proof of Theorem \[dssaf\].* It is easy to see that the martingale differences $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$, defined by (\[sdgvf2\]), satisfy $$\begin{aligned}
\xi_i \leq U_{i-1}:= \frac{ |\phi_{i}| \epsilon }{ \sigma \sqrt{\sum_{k=1}^n \phi_{k}^2}}\ \ \ \ \ \textrm{and} \ \ \ \ \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1})=\frac{ \phi_{i}^2 }{ \sum_{k=1}^n \phi_{k}^2 }.\end{aligned}$$ Applying Corollary \[co4\] to $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$, we obtain the desired inequality.
*Proof of Theorem \[thds\].* From (\[ine29\]) and (\[ine30\]), it is easy to see that $$\label{sdgvf1}
(\theta_n -\theta )\sum_{k=1}^n \phi_{k}^2 = \sum_{k=1}^n\phi_{k} \varepsilon_k . \nonumber$$ For any $i=1,...,n$, set $$\begin{aligned}
\xi_i=\phi_{i} \varepsilon_i \ \ \ \textrm{and} \ \ \ \ \ \mathcal{F}_{i} = \sigma \Big( \phi_{k}, \varepsilon_k, 1\leq k\leq i, \phi_{i+1} \Big).\end{aligned}$$ Then $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ is a sequence of martingale differences and satisfies $$\mathbf{E}(e^{ \lambda \xi_{i} }|\mathcal{F}_{i-1}) \leq e^{c |\lambda \phi_{i}|^\alpha }\ \textrm{ for all }i \in [1, n] .$$ Applying Theorem \[th3\] to $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$, we obtain, for all $x, \lambda, v \geq 0$, $$\begin{aligned}
\label{scsasd}
\mathbf{P}\Big( \pm (\theta_n -\theta) \sum_{k=1}^n \phi_{k}^2 \geq x \ \textrm{and} \ \sum_{k=1}^n |\phi_{k}|^\alpha \leq v^\alpha \Big) &\leq& \exp\Big\{ - \lambda x + c \lambda^\alpha v^\alpha \Big\}.\end{aligned}$$ The right hand side of the last inequality takes its minimum at $$\lambda= \lambda(x)=\Big(\frac{x}{c \, \alpha \, v^\alpha } \Big)^{\frac{ 1}{\alpha -1}} .$$ Substituting $\lambda= \lambda(x)$ into (\[scsasd\]), we obtain the desired inequality.
*Proof of Theorem \[th6\].* From (\[scfso1\]) and (\[scfso2\]), it is easy to see that $$\label{sdgvf1}
(\theta_n' -\theta) \sum_{k=1}^n X_{k-1}^2= \sum_{k=1}^n X_{k-1} \varepsilon_k . \nonumber$$ For any $i=1,...,n$, set $$\begin{aligned}
\xi_i=X_{i-1} \varepsilon_i \ \ \ \textrm{and} \ \ \ \ \ \mathcal{F}_{i} = \sigma \Big( X_{0}, \varepsilon_k, 1\leq k\leq i \Big).\end{aligned}$$ Then $(\xi _i,\mathcal{F}_i)_{i=1,...,n}$ is a sequence of martingale differences and satisfies $$|\xi_i| \leq U_{i-1}:= X_{i-1}\epsilon \ \ \ \ \ \textrm{and} \ \ \ \ \ \mathbf{E} (( X_{k-1} \varepsilon_{i})^2|\mathcal{F}_{i-1} ) = X_{k-1}^2 \mathbf{E} \varepsilon_{i}^2 \leq X_{k-1}^2 \sigma^2 .$$ Applying Corollary \[co4\] to $(\xi_i, \mathcal{F}_{i})$, we obtain the desired inequality.
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Wang, Q. Y. and Jing, B. Y., 1999. An exponential nonuniform Berry-Esseen bound for self-normalized sums, *Ann. Probab.*, **27**, No. 4, 2068-2088.
|
---
author:
- '[^1]'
title: Cosmogenic photons strongly constrain UHECR source models
---
Introduction {#intro}
============
Recently the Fermi-LAT collaboration updated their measurements on the isotropic diffuse gamma-ray background (IGRB) and extended it up to 820 GeV [@Ackermann:2014usa]. A possible source for part of the IGRB is secondary electromagnetic cascades initiated by interactions of ultra-high-energy cosmic rays (UHECRs) with the cosmic microwave background (CMB) or the extragalactic background light (EBL). In these same interactions secondary neutrinos can be produced. These neutrinos could possibly contribute to the astrophysical neutrino flux as measured by IceCube [@Aartsen:2015zva].
The UHECR energy spectrum has been measured with unprecedented statistics by the Pierre Auger [@Aab:2015bza; @ThePierreAuger:2015rha] (Auger) and Telescope Array [@Jui:2015tac] (TA) collaborations. The UHECR mass measured by these two collaborations can, however, still be interpreted in different ways. While the measurements of Auger show a depth of the shower maximum, $X_{\mathrm{max}}$, indicating an increasingly heavier mass composition for $E \gtrsim10^{18.3}$ eV [@Porcelli:2015pac], TA results in the same energy range are consistent with a pure proton composition [@Fujii:2015tac]. Despite these differences the $X_{\mathrm{max}}$ measurements are in good agreement with each other [@Unger:2015ptc]. The predictions of different air shower simulation models, however, leave room for varying interpretations of the data. Therefore many UHECR composition models are still viable.
However, as shown e.g. in Refs. [@Gavish:2016tfl; @Berezinsky:2016jys; @Supanitsky:2016gke], the parameter range of possible pure proton models can be constrained when taking into account the secondary gamma-ray and neutrino production during the propagation of UHECRs from their sources to Earth. Ref. [@Heinze:2015hhp] even claims that the proton dip model is challenged at more than $95\%$ C.L. by the cosmogenic neutrino flux alone.
To obtain the predicted cosmogenic gamma-ray and neutrino flux for a certain UHECR model the propagation of UHECRs through the universe, including all relevant interactions with the CMB and EBL, has to be simulated. Here the newest version of our UHECR propagation code, CRPropa version 3 [@Batista:2016yrx], is used to simulate the cosmic ray, electromagnetic cascade and neutrino propagation and obtain predictions for the cosmogenic gamma-ray and neutrino fluxes. CRPropa is a full simulation framework for Monte Carlo UHECR propagation including all relevant interactions for protons as well as for heavier nuclei (photo-meson production, pair production, photodisintegration, nuclear decay and energy reduction due to the adiabatic expansion of the universe). For the electromagnetic cascade propagation the specialized code DINT [@Lee:1996fp], interfaced and shipped with CRPropa 3, is used. DINT solves the one-dimensional transport equations for electromagnetic cascades initiated by electrons, positrons or photons and includes single, double and triplet pair production, inverse-Compton scattering and synchrotron radiation.
Simulation setup
================
[\[sec:SimSetup\]]{}
All the simulations done here assume a homogeneous distribution of identical sources. The cosmic rays are injected at the sources following a spectrum of $$\label{eq:PowerLawInjection}
\frac{\text{d}N}{\text{d}E} \propto (E/E_0)^{-\alpha}\exp(-E/E_{\text{cut}})~,$$ with $E$ the energy of the particles, $E_0$ an arbitrary normalization energy, $\alpha$ the spectral index at injection and $E_{\text{cut}}$ the cutoff energy. A minimum energy of cosmic rays at the sources of $E_{\text{min}} = 0.1$ EeV is assumed. The EBL used for the cosmic ray simulations is the Gilmore 2012 model [@Gilmore:2011ks]. In Fig. \[fig:Ev\] (solid lines) a reference scenario is given with pure proton injection at the sources for which $\alpha = 2.5$ and $E_{\text{cut}} = 200$ EeV. In this case a co-moving source evolution up to a maximum redshift of $z_{\text{max}}=6$ is implemented. Unless stated otherwise these simulation parameters have been used for the other scenarios as well.
These parameters are not optimized to fit the cosmic ray spectrum perfectly as the systematic uncertainty of the energy spectrum measurements is much larger than the statistical uncertainty for both Auger and TA. Furthermore the assumption made here, a homogeneous distribution of identical pure proton sources with a power law injection spectrum with exponential cutoff, may not correctly depict the real situation. Additionally it is convenient to change only one parameter at a time in order to see the effect of only that parameter on the predicted spectra. Fitting a pure proton model to the TA spectrum would lead to a strong source evolution and a relatively hard injection spectrum and would be strongly constrained by both the Fermi-LAT IGRB data and the IceCube data [@Heinze:2015hhp; @Supanitsky:2016gke].
Source evolution dependence
===========================
[\[sec:EvDep\]]{}
The evolution in time (redshift) of the sources of UHECRs influences the expected cosmogenic gamma-ray and neutrino flux significantly. As the sources of UHECRs are unknown, a wide range of possible source evolutions could be applicable. Here the effect of the source evolution on the UHECR spectrum and the expected cosmogenic neutrino and gamma-ray fluxes is investigated by taking the reference scenario and adjusting the source evolution by a factor of $(1+z)^m$ with $z$ the redshift of the source and $-6 \leq m \leq 6$. A value of $m = -6$ (negative source evolution) would, for instance, correspond roughly to the source evolution of High Synchrotron Peaked (HSP) BL Lacs [@Gavish:2016tfl]. All the other parameters of the simulation are kept the same in order to clearly see the effect of the source evolution alone. In Fig. \[fig:Ev\] the results are given for $m = 0$, $m = -6$ and $m = 6$. The simulated cosmic ray spectrum is normalized to the flux measured by Auger [@Aab:2015bza] at $E = 10^{18.85}$ eV and the simulated cosmogenic neutrino and gamma-ray spectra are normalized accordingly. For comparison the spectrum measured by TA [@Jui:2015tac] is given as well. The simulated neutrino flux is compared with IceCube data [@Aartsen:2015zva] while the simulated gamma-ray flux is compared with Fermi-LAT IGRB data [@Ackermann:2014usa], using galactic foreground model A with foreground model uncertainties and IGRB intensity uncertainties added in quadrature.
From Fig. \[fig:EvCRs\] can be seen that the cosmic ray spectrum is affected by the source evolution most strongly at the lower energy range ($E \lesssim 5\times10^{18}$ eV). This is expected as in this energy range the particles can reach us from very large distances while at higher energy, due to the different energy-loss processes, the cosmic rays can only come from relatively nearby sources. Note here that this lower energy range might also be the regime where a Galactic contribution to the cosmic ray spectrum starts playing a role, which could possibly compensate for an underprediction of the cosmic ray spectrum. Fig. \[fig:EvNeutrinos\] shows that a strong source evolution is clearly constrained by the astrophysical neutrino flux as measured by IceCube, but a co-moving or negative source evolution are still allowed when only the neutrino flux is considered. However, Fig. \[fig:EvPhotons\] indicates that especially the non-detection at the highest energy bin (580-820 GeV) of the IGRB measured by Fermi LAT is very constraining. Only sources with a number density strongly peaked at recent times are still allowed by this limit. Note furthermore that there are even many other sources expected to contribute to the IGRB (see e.g. Ref. [@TheFermi-LAT:2015ykq]), which are not taken into account here.
Instead of a co-moving source evolution multiplied by $(1+z)^m$, specific functions for certain UHECR source candidates can be tested in the same way. For this procedure the source evolution parametrizations listed in Ref. [@Gavish:2016tfl] for gamma-ray bursts (GRBs), high luminosity active galactic nuclei (HLAGNs), medium high luminosity AGNS (MHLAGNs), medium low luminosity AGNs (MLLAGNs) and sources following the star formation rate (SFR) are implemented. The results (with all other parameters of the simulations kept the same) are shown in Fig. \[fig:EvSpec\]. From Fig. \[fig:EvSpecPhotons\] can be seen that all these source classes exceed the IGRB data measured by Fermi LAT.
Conclusions
===========
[\[sec:Conclusions\]]{}
Figs. \[fig:Ev\] and \[fig:EvSpec\] show that the IGRB measured by Fermi LAT is more constraining for UHECR models than the IceCube neutrino measurements. The bin with the highest energy (580-820 GeV) of the IGRB especially provides a strong constraint for pure proton UHECR models. For the scenarios investigated here, only in the case of source densities that are strongly decreasing with redshift (for instance in the case of HSP BL Lacs) is it possible to get a gamma-ray spectrum in agreement with that highest energy bin. With a few more years of data and, perhaps, an extension to even higher energies, Fermi LAT might be able to rule out all realistic UHECR pure proton models.
I want to thank Rafael Alves Batista and Jörg Hörandel for helpful discussions and the Auger PC for their suggested corrections. I acknowledge financial support from the NWO Astroparticle Physics grant WARP.
M. Ackermann et al. (Fermi-LAT Collaboration), Astrophys. J. **799**, 86 (2015)
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C. Jui for the Telescope Array Collaboration, PoS (ICRC2015) 035
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T. Fujii for the Telescope Array Collaboration, PoS (ICRC2015) 320
M. Unger for the Pierre Auger and the Telescope Array Collaboration, PoS (ICRC2015) 307
E. Gavish and D. Eichler, Astrophys. J. **822**, 56 (2016)
V. Berezinsky, A. Gazizov and O. Kalashev, arXiv:1606.09293 (2016)
A. D. Supanitsky, arXiv:1607.00290 (2016)
J. Heinze et al., Astrophys. J. **825**, 122 (2016)
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[^1]:
|
---
abstract: |
[This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem: $$\left \{ \begin{array}{l}
-M\left(\displaystyle\int_\Omega \Phi(\mid\nabla u\mid)dx\right)div\big(\phi(\mid\nabla u\mid)\nabla u\big) -\phi(|u|)u\in \partial F(u) \ \mbox{in}\ \Omega,\\
u\in W_0^1L_\Phi(\Omega), \end{array}
\right.$$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain, $N\geq 2$, $M$ is continuous function, $\Phi$ is an N-function with $\Phi(t)=\displaystyle\int^{|t|}_{0}\phi(s)s \ ds$ and $\partial
F(t)$ is a generalized gradient of $F(t)$. We use genus theory to obtain the main result.]{}
author:
- |
Giovany M. Figueiredo[^1]\
Universidade Federal do Pará, Faculdade de Matemática,\
CEP: 66075-110, Belém - Pa, Brazil\
e-mail: giovany@ufpa.br\
Jefferson A. Santos [^2]\
Universidade Federal de Campina Grande,\
Unidade Acadêmica de Matemática e Estatística,\
CEP:58109-970, Campina Grande - PB, Brazil\
e-mail: jefferson@dme.ufcg.edu.br\
title: ' On a nonlocal multivalued problem in an Orlicz-Sobolev space via Krasnoselskii’s genus '
---
10000
Introduction
============
The purpose of this article is investigate the multiplicity of nontrivial solutions to the multivalued elliptic problem $$\left \{ \begin{array}{l}
-M\left(\displaystyle\int_\Omega \Phi(\mid\nabla u\mid)dx\right)div\big(\phi(\mid\nabla u\mid)\nabla u\big)-\phi(|u|)u\in \partial F(u) \ \mbox{in}\ \Omega,\\
u\in W_0^1L_\Phi(\Omega), \end{array}
\right.\leqno{(P)}$$ where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain with $N\geq 2$, $F(t)= \displaystyle\int_0^t f(s)ds$ and $$\partial F(t)=\left\{s\in\mathbb{R}; F^0(t;r)\geq sr, \ r\in \mathbb{R}\right\}.$$ Here $F^0(t;r)$ denotes the generalized directional derivative of $t\mapsto F(t)$ in direction of $r$, that is, $$F^0(t;r)=\displaystyle \limsup_{h\rightarrow
t,s\downarrow0}\frac{F(h+sr)-F(h)}{s}.$$ We shall assume in this work that $f(t)$ is locally bounded in $\mathbb{R}$ and $$\underline{f}(t)=\displaystyle \lim_{\epsilon\downarrow 0}\mbox{ess
inf}\left\{ f(s);|s-t|<\epsilon\right\} \mbox{ and }
\overline{f}(t)=\lim_{\epsilon\downarrow 0}\mbox{ess sup}\left\{
f(s);|s-t|<\epsilon\right\} .$$ It is well known that $$\partial F(t)=[\underline{f}(t),\overline{f}(t)], \mbox{ (see
\cite{chang})},$$ and that, if $f(t)$ is continuous then $\partial F(t)=\{f(t)\}$.
Problem $(P)$ with $\phi(t)=2$, that is, $$\left \{ \begin{array}{l}
-M\left(\displaystyle\int_\Omega \mid\nabla u\mid^{2} dx\right)\Delta u - u \in \partial F(u) \ \mbox{in}\ \Omega,\\
u\in H^{1}_{0}(\Omega) \end{array}
\right.\leqno{(*)}$$ is called nonlocal because of the presence of the term $M\left(\dis\int_{\Omega}|\nabla u|^{2} dx \right)$ which implies that the equation $(*)$ is no longer a pointwise identity.
The reader may consult $\cite{alvescorrea}$, $\cite{alvescorreama}$, $\cite{GJ}$ and the references therein, for more information on nonlocal problems.
On the other hand, in this study, the nonlinearity $f$ can be discontinuous. There is by now an extensive literature on multivalued equations and we refer the reader to [@Goncalves], [@Nascimento], [@AlvesNascimento], [@Santos], [@Carvalho], and references therein. The interest in the study of nonlinear partial differential equations with discontinuous nonlinearities has increased because many free boundary problems arising in mathematical physics may be stated in this form.
Among these problems, we have the obstacle problem, the seepage surface problem, and the Elenbaas equation, see for example [@chang], [@chang1] and [@chang2].
For enunciate the main result, we need to give some hypotheses on the functions $M, \phi$ and $f$.
The hypotheses on the function $\phi:\mathbb{R}^{+}\rightarrow
\mathbb{R}^{+}$ of $C^{1}$ class are the following:
($\phi_1$)
: For all $t>0$, $$\phi(t)>0 \ \ \mbox{and} \ \ (\phi(t)t)'>0.$$
($\phi_2$)
: There exist $l, m \in (1,N)$, $l\leq m < l^{*}=
\displaystyle\frac{lN}{N-l}$ such that $$l\leq \frac{\phi(t)t^{2}}{\Phi(t)}\leq m,$$ for $t> 0$, where $\Phi(t)=\displaystyle\int^{|t|}_{0}\phi(s)s ds$.
The hypothesis on the continuous function $M:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is the following:
($M_1$)
: There exist $k_{0},k_{1},\alpha, q_{0},q_{1}>0$ and $b:\mathbb{R}\rightarrow \mathbb{R}$ of $C^{1}$ class such that $$k_0t^{\alpha}\leq M(t)\leq k_1t^{\alpha},$$ $\alpha >\frac{q_1}{l}$, where $$m< q_0\leq \frac{b(t)t^2}{B(t)}\leq q_1 <l^*,$$ for all $t>0$ with $$(b(t)t)'>0, \ t>0$$ and $$B(t)=\int_0^tb(s)sds.$$
The hypotheses on the function $f:\mathbb{R}\rightarrow \mathbb{R}$ are the following:
($f_1$)
: For all $t \in \mathbb{R}$,
$$f(t)=-f(-t).$$
($f_2$)
: There exist $b_{0},b_{1}>0$ and $a_0\geq 0$ such that $$b_0b(t)t\leq f(t)\leq b_1b(t)t,\ |t|\geq a_0.$$
($f_3$)
: There exists $a_0\geq 0$ such that $$f(t)=0,\ |t|\leq a_0.$$
The main result of this paper is:
\[teorema1\] Assume that conditions $(\phi_{1})$, $(\phi_{2})$, $(M_{1})$, $(f_{1})-(f_{3})$ hold. Then for $a_0>0$ sufficiently small (or $a_0=0$), the problem $(P)$ has infinitely many solutions.
Below we show two graphs of functions that satisfy the hypotheses $(f_{1})-(f_{3})$. Note that the second graph corresponds to a function that has an enumerable number of points of discontinuity.
{width="40.00000%"} {width="40.00000%"}
In the last twenty years the study on nonlocal problems of the type $$\left \{ \begin{array}{l}
-M\left(\displaystyle\int_\Omega \mid\nabla u\mid^{2} dx\right)\Delta u= f(x,u) \ \mbox{in}\ \Omega,\\
u\in H^{1}_{0}(\Omega) \end{array}
\right.\leqno{(K)}$$ grew exponentially. That was, probably, by the difficulties existing in this class of problems and that do not appear in the study of local problems, as well as due to their significance in applications. Without hope of being thorough, we mention some articles with multiplicity results and that are related with our main result. We will restrict our comments to the works that have emerged in the last four years
The problem $(K)$ was studied in [@GJ]. The version with p-Laplacian operator was studied in [@correa]. In both cases, the authors showed a multiplicity result using genus theory. In [@Xiao] the authors showed a multiplicity result for the problem $(K)$ using the Fountain theorem and the Symmetric Mountain Pass theorem. In all these articles the nonlinearity is continuous. The case discontinuous was studied in [@Nascimento]. With a nonlinearity of the Heaviside type the authors showed a existence of two solutions via Mountain Pass Theorem and Ekeland’s Variational Principle.
In this work we extend the studies found in the papers above in the following sense:
a\) We cannot use the classical Clark’s Theorem for $C^1$ functional (see [@Davi Theorem 3.6]), because in our case, the energy functional is only locally Lipschitz continuous. Thus, in all section \[finalissimo\] we adapt for nondifferentiable functionals an argument found in [@GP].\
b) Unlike [@Nascimento], we show a result of multiplicity using genus theory considering a nonlinearity that can have a number enumerable of discontinuities.\
c) Problem $(P)$ possesses more complicated nonlinearities, for example:
\(i) $\Phi(t)=t^{p_{0}}+ t^{p_{1}}$, $1< p_{0} < p_{1}< N$ and $ p_1\in (p_0,p^{*}_{0})$.
\(ii) $\Phi(t)=(1+t^{2})^{\gamma}-1$, $\gamma\in
(1,\frac{N}{N-2})$.
\(iii) $\Phi(t)=t^{p}\log(1+t)$ with $1<p_0<p<N-1$, where $p_0=\frac{-1+\sqrt{1+4N}}{2}$.
\(iv) $\Phi(t)=\int_0^ts^{1-\alpha}\left(\sinh^{-1}s\right)^\beta ds, \ 0\leq\alpha\leq 1, \ \beta>0.$\
d) We work with Orlicz-Sobolev spaces and some different estimates from those found in the papers above are necessary. For example, the Lemma \[LipLoc1\] is a version for Orlicz-Sobolev spaces of a well-known result of Chang (see [@chang], [@clarke] and [@Nascimento Lemma 3.3]). In the Lemma \[PS1\] one different estimate was necessary because of the presence of the nonlocal term.
The paper is organized as follows. In the next section we present a brief review on Orlicz-Sobolev spaces. In section \[Section results for the discontinuous\] we recall some definitions and basic results on the critical point theory of locally Lipschitz continuous functionals. We also present variational tools which we will prove the main result of this paper. Furthermore, in this chapter, we prove the Lemma \[LipLoc1\], which is a version for Orlicz-Sobolev spaces of a well-known result of Chang (see [@chang], [@clarke] and [@Nascimento Lemma 3.3]). In Section \[Section Genus\] we present just some preliminary results involving genus theory that will be used in this work. In the Section \[finalissimo\] we prove Theorem \[teorema1\].
A brief review on Orlicz-Sobolev spaces {#Section
Orlicz}
=======================================
Let $\phi$ be a real-valued function defined $[0,\infty)$ and having the following properties:
$a)$ $\phi(0)=0$, $\phi(t)>0$ if $t>0$ and $\displaystyle\lim_{t\rightarrow \infty}\phi(t)=\infty$.
$b)$ $\phi$ is nondecreasing, that is, $s>t$ implies $\phi(s) \geq \phi(t)$.
$c)$ $\phi$ is right continuous, that is, $\displaystyle\lim_{s\rightarrow t^{+}}\phi(s)=\phi(t)$.
Then, the real-valued function $\Phi$ defined on $\mathbb{R}$ by $$\Phi(t)= \displaystyle\int^{|t|}_{0}\phi(s) \ ds$$ is called an N-function. For an N-function $\Phi$ and an open set $\Omega \subseteq \mathbb{R}^{N}$, the Orlicz space $L_{\Phi}(\Omega)$ is defined (see [@adams]). When $\Phi$ satisfies $\Delta_{2}$-condition, that is, when there are $t_{0}\geq
0$ and $K>0$ such that $\Phi(2t)\leq K\Phi(t)$, for all $t\geq
t_{0}$, the space $L_{\Phi}(\Omega)$ is the vectorial space of the measurable functions $u: \Omega \to \mathbb{R}$ such that $$\displaystyle\int_{\Omega}\Phi(|u|) \ dx < \infty.$$ The space $L_{\Phi}(\Omega)$ endowed with Luxemburg norm, that is, the norm given by $$|u|_{\Phi}= \inf \biggl\{\lambda >0:
\int_{\Omega}\Phi\Big(\frac{|u|}{\lambda}\Big)\ dx\leq 1\biggl\},$$ is a Banach space. The complement function of $\Phi$, denoted by $\widetilde{\Phi}$, is given by the Legendre transformation, that is $$\widetilde{\Phi}(s)=\displaystyle\max_{t \geq 0}\{st -\Phi(t)\} \ \
\mbox{for} \ \ s \geq 0.$$ These $\Phi$ and $\widetilde{\Phi}$ are complementary each other. Involving the functions $\Phi$ and $\widetilde{\Phi}$, we have the Young’s inequality given by $$st \leq \Phi(t) + \widetilde{\Phi}(s).$$ Using the above inequality, it is possible to prove the following Hölder type inequality $$\biggl|\displaystyle\int_{\Omega}u v \ dx \biggl|\leq
2|u|_{\Phi}|v|_{\widetilde{\Phi}}\,\,\, \forall \,\, u \in
L_{\Phi}(\Omega) \,\,\, \mbox{and} \,\,\, v \in
L_{\widetilde{\Phi}}(\Omega).$$
Hereafter, we denote by $W^{1}_{0}L_{\Phi}(\Omega)$ the Orlicz-Sobolev space obtained by the completion of $C^{\infty}_{0}(\Omega)$ with norm $$\|u\|_\Phi=|u|_\Phi+|\nabla u|_\Phi.$$
When $\Omega$ is bounded, there is $c>0$ such that $$|u|_\Phi \leq c |\nabla u|_\Phi.$$
In this case, we can consider
$$\|u\|_{\Phi}=|\nabla u|_{\Phi}.$$
Another important function related to function $\Phi$, is the Sobolev conjugate function $\Phi_{*}$ of $\Phi$ defined by $$\Phi^{-1}_{*}(t)=\displaystyle\int^{t}_{0}\displaystyle\frac{\Phi^{-1}(s)}{s^{(N+1)/N}}ds, \ t>0.$$ The function $\Phi_{*}$ is very important because it is related to some embedding involving $W^{1}_{0}L_{\Phi}(\Omega)$.
We say that $\Psi$ increases essentially more slowly than $\Phi_{*}$ near infinity when $$\displaystyle \lim_{t \rightarrow
\infty}\frac{\Psi(kt)}{\Phi_*(t)}=0, \ \text{ for all } k>0.$$
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{N}$. If $\Psi$ is any N-function increasing essentially more slowly than $\Phi_{*}$ near infinity, then the imbedding $W_0^{1}L_{\Phi}(\Omega)\hookrightarrow L_\Psi(\Omega)$ exists and is compact (see [@adams]).
The hypotheses $(\phi_{1})-(\phi_{2})$ implies that $\Phi$, $\widetilde{\Phi}$, $\Phi_{*}$ and $\widetilde{\Phi}_{*}$ satisfy $\Delta_{2}$-condition. This condition allows us conclude that:
1\) $u_{n}\rightarrow 0$ in $L_{\Phi}(\Omega)$ if, and only if, $\displaystyle\int_{\Omega}\Phi(u_{n})\ dx\rightarrow 0$.
2\) $L_{\Phi}(\Omega)$ is separable and $\overline{C^{\infty}_{0}(\Omega)}^{|.|_{\Phi}}=L_{\Phi}(\Omega)$.
3\) $L_{\Phi}(\Omega)$ is reflexive and its dual is $L_{\widetilde{\Phi}}(\Omega)$(see [@adams]).
Under assumptions $(\phi_{1})-(\phi_{2})$, some elementary inequalities listed in the following lemmas are valid. For the proofs, see [@fukagai].
\[desigualdadeimportantes\] Let $\xi_{0}(t)=\min\{t^{l},t^{m}\}$, $\xi_{1}(t)=\max\{t^{l},t^{m}\}$, $\xi_{2}(t)=\min\{t^{l^{*}},t^{m^{*}}\}$, $\xi_{3}(t)=\max\{t^{l^{*}},t^{m^{*}}\}$, $t\geq 0$. Then $$\xi_{0}(\|u\|_{\Phi})\leq \displaystyle\int_{\Omega}\Phi(|\nabla u|)
\ dx \ \leq\xi_{1}(\|u\|_{\Phi}),$$ $$\xi_{2}(|u|_{\Phi_{*}})\leq \displaystyle\int_{\Omega}\Phi_{*}(|u|)
\ dx \ \leq\xi_{3}(|u|_{\Phi_{*}})$$ and $$\Phi_{*}(t)\geq \Phi_{*}(1)\xi_{2}(t).$$
\[desigualdadeimportantes1\] Let $\eta_{0}(t)=\min\{t^{q_0},t^{q_1}\},
\eta_{1}(t)=\max\{t^{q_0},t^{q_1}\}, t\geq 0$. Then $$\eta_{0}(|u|_{B}) \leq \displaystyle\int_{\Omega}B(|u|)dx \leq
\eta_{1}(|u|_{B})$$ and $$B(1)\eta_{0}(t) \leq B(t) \leq B(1)\eta_{1}(t), \ t \in \mathbb{R}.$$
\[DESIGUALD\] $\widetilde{\Phi}(\frac{\Phi(s)}{s})\leq \Phi(s),\ s>0.$
The next result is a version of Brezis-Lieb’s Lemma [@Brezis_Lieb] for Orlicz-Sobolev spaces and the proof can be found in [@Gossez].
\[brezislieb\] Let $\Omega\subset \mathbb{R}^N$ open set and $\Phi:\mathbb{R}\rightarrow [0,\infty)$ an N-function satisfies $\Delta_2-$condition. If the complementary function $\widetilde{\Phi}$ satisfies $\Delta_2-$condition, $(f_n)$ is bounded in $L_\Phi(\Omega)$, such that $$f_n(x)\rightarrow f(x)\ \text{a.s }x\in\Omega,$$ then $$f_n\rightharpoonup f \ \text{in }L_\Phi(\Omega).$$
\[imersao\] The imbedding $W_0^{1}L_{\Phi}(\Omega)\hookrightarrow L_{B}(\Omega)$ exists and is compact.
**Proof:** It is sufficiently to show that $B$ increasing essentially more slowly than $\Phi_{*}$ near infinity. Indeed, $$\frac{B(kt)}{\Phi_*(t)}\leq
\frac{B(1)B(kt)}{\xi_2(t)}=B(1)k^{q_1}t^{q_1-l^*}, \ k>0.$$ Since $q_1<l^*$, we get $$\displaystyle\lim_{t\rightarrow +\infty}\frac{B(kt)}{\Phi_*(t)}= 0.$$
------------------------------------------------------------------------
Technical results on locally Lipschitz functional and variational framework {#Section results for the
discontinuous}
===========================================================================
In this section, for the reader’s convenience, we recall some definitions and basic results on the critical point theory of locally Lipschitz continuous functionals as developed by Chang [@chang], Clarke [@clarke; @Clarke] and Grossinho & Tersian [@grossinho].
Let $X$ be a real Banach space. A functional $J:X \rightarrow {\mathbb{R}}$ is locally Lipschitz continuous, $J \in
Lip_{loc}(X, {\mathbb{R}})$ for short, if given $u \in X$ there is an open neighborhood $V := V_u \subset X$ and some constant $K =
K_V > 0$ such that $$\mid J(v_2) - J(v_1) \mid \leq K \parallel v_2-v_1 \parallel,~ v_i
\in V,~ i = 1,2.$$
The directional derivative of $J$ at $u$ in the direction of $v
\in X$ is defined by
$$J^0(u;v)=\displaystyle \displaystyle \limsup_{h \to 0,~\sigma
\downarrow 0} \frac{J(u+h+ \sigma v)-I(u+h)}{\sigma}.$$ The generalized gradient of $J$ at $u$ is the set $$\partial J(u)=\big\{\mu\in X^*; \langle \mu,v\rangle\leq J^0(u;v), \ v\in
X\big \}.$$ Since $J^0(u;0) = 0$, $\partial J(u)$ is the subdifferential of $J^0(u;0)$. Moreover, $J^0(u;v)$ is the support function of $\partial J(u)$ because $$J^0(u;v)=\max\{\langle
\xi,v\rangle; \xi\in \partial J(u)\}.$$
The generalized gradient $\partial J(u)\subset X^*$ is convex, non-empty and weak\*-compact, and $$m^{J}(u) = \min \big\{\parallel\mu\parallel_{X^*};\mu \in
\partial J(u) \big \}.$$ Moreover, $$\partial J(u) = \big \{J'(u) \big \}, \mbox{if}\ J \in C^1(X,{\mathbb{R}}).$$ A critical point of $J$ is an element $u_0 \in X$ such that $0\in
\partial J(u_0)$ and a critical value of $J$ is a real number $c$ such that $J(u_0)=c$ for some critical point $u_0 \in X$.
About variational framework, we say that $u \in
W^{1}_{0}L_{\Phi}(\Omega)$ is a weak solution of the problem $(P)$ if it verifies $$M\left(\dis\int_\Omega\Phi(\mid\nabla u\mid )\ dx
\right)\dis\int_\Omega\phi(\mid\nabla u\mid)\nabla u\nabla v \
dx-\int_\Omega\phi(u)uvdx-\int_\Omega \rho v \ dx=0,$$ for all $v \in W^{1}_{0}L_{\Phi}(\Omega)$ and for some $\rho \in
L_{\widetilde{B}}(\Omega)$ with $$\underline{f}(u(x))\leq \rho(x) \leq \overline{f}(u(x)) \ \
\mbox{a.e
in} \ \ \Omega,$$ and moreover the set $\{x\in \Omega; \mid u\mid\geq a_0\}$ has positive measure. Thus, weak solutions of $(P)$ are critical points of the functional
$$J(u) =\widehat{M}\left({\displaystyle}\int_{\Omega}\Phi(|\nabla u|) dx\right)
-\int_\Omega \Phi(u)dx-
{\displaystyle}\int_{\Omega}F(u)\ dx,$$ where $\widehat{M}(t)=\displaystyle\int^{t}_{0}M(s) ds$. In order to use variational methods, we first derive some results related to the Palais-Smale compactness condition for the problem $(P)$.
We say that a sequence $(u_{n})\subset W^{1}_{0}L_{\Phi}(\Omega)$ is a Palais-Smale sequence for the locally lipschitz functional $J$ associated of problem $(P)$ if $$\begin{aligned}
\label{****}
J(u_{n})\rightarrow c \ \mbox{and} \ m^{J}(u_{n})\rightarrow 0 \
\mbox{in} \ (W^{1}_{0}L_{\Phi}(\Omega))^*,\end{aligned}$$ where $$c = \displaystyle\inf_{\eta \in \Gamma}
\displaystyle\max_{t \in [0,1]} J(\eta(t))>0$$ and $$\Gamma := \{ \eta \in C([0,1],X) : \eta(0)=0, ~I(\eta(1)) < 0\}.$$
If (\[\*\*\*\*\]) implies the existence of a subsequence $(u_{n_{j}})
\subset (u_{n})$ which converges in $W^{1}_{0}L_{\Phi}(\Omega)$, we say that these one functionals satisfies the nonsmooth $(PS)_{c}$ condition.
Note that $J \in Lip_{loc}(W^{1}_{0}L_{\Phi}(\Omega), {\mathbb{R}})$ and from convex analysis theory, for all $ w \in \partial J(u)$, $$\langle w,v\rangle=M\left({\displaystyle}\int_{\Omega}\Phi(|\nabla u|) \
dx\right) {\displaystyle}\int_{\Omega}\phi(|\nabla u|)\nabla u \nabla v \ dx -
\int_\Omega \phi(u)uvdx-
\langle\rho,v\rangle,$$ for some $\rho \in \partial \Psi(u)$, where $\Psi(u)=\displaystyle\int_{\Omega}F(u) dx$. We have $\Psi \in
Lip_{loc}(L_{B}(\Omega), {\mathbb{R}})$, $\partial \Psi(u) \in
L_{\widetilde{B}}(\Omega)$.
The next result is a version for Orlicz-Sobolev spaces of a well-known result of Chang (see [@chang], [@clarke] and [@Nascimento Lemma 3.3]).
\[LipLoc1\] Suppose that $M_{1}$, $(f_{2})$ and $(f_{3})$ hold. For each $u \in
L_{B}(\Omega)$, if $\rho \in
\partial \Psi(u)$, then $$\underline{f}(u(x))\leq \rho(x) \leq \overline{f}(u(x)) \ a.e \ x
\in \Omega,$$ and if $a_0>0$ $$\rho(x)=0 \ a.e \ x\in \{x\in \Omega; \mid u(x)\mid < a_0\}.$$
**Proof:** Considering $u,v \in L_{B}(\Omega)$, from definition $$\begin{aligned}
\Psi^0(u;v)&=&\displaystyle\limsup_{h\rightarrow0,t\rightarrow0^+}\frac{\Psi(u+h+tv)-\Psi(u+h)}{t}\\
&=&\displaystyle\limsup_{h\rightarrow0,t\rightarrow0^+}\frac{1}{t}\int_\Omega
\left(F(u+h+tv)-F(u+h)\right)dx.\end{aligned}$$
We set $(h_{n}) \subset L_{B}(\Omega)$ and $(t_{n}) \subset
\mathbb{R}_{+}$ such that $h_{n}\rightarrow 0$ in $L_{B}(\Omega)$ and $t_{n}\rightarrow 0^{+}$. Thus, $$\label{eq 3.1}
\Psi^0(u;v)=\displaystyle\limsup_{n\rightarrow+\infty}\int_\Omega\frac{F(u+h_n+t_nv)-F(u+h_n)}{t_n}dx.$$
Note that from the Mean Value Theorem, $(M_{1})$, $(f_2)$ and $(f_{3})$ that, $$F_{n}(u,v):=\frac{F(u+h_n+t_nv)-F(u+h_n)}{t_n}\leq c
b(|\theta_{n}(x)|)|\theta_{n}(x)||v|,$$ where $$\theta_{n}(x) \in \biggl[\min\bigl\{u+h_{n}+t_{n}v,
u+h_{n}\bigl\},\max\bigl\{u+h_{n}+t_{n}v, u+h_{n}\bigl\} \biggl], \
\ x \in \Omega.$$
Using monotonicity of $b(t)t$ we get $$|F_{n}(u,v)|\leq c b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v||v| + c
b(|u+h_{n}|)|u+h_{n}||v|.$$
On the other hand, by lemma \[DESIGUALD\] we have $$\widetilde{B}(b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v|) \leq C
B(|u+h_{n}+t_{n}v|) \leq
C\bigl(B(u)+B(h_{n})+\eta_{1}(t_{n})B(v)\bigl)$$ and $$\widetilde{B}(b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v|)\rightarrow
\widetilde{B}(b(|u|)|u|) \ \ a.e \ \ \mbox{in} \ \ \Omega,$$ where $\widetilde{B}(b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v| \leq c B(u)
\in L^{1}(\Omega)$.
By Lebesgue’s Theorem we obtain
$$\displaystyle\int_{\Omega}\widetilde{B}(b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v|)
dx \rightarrow \displaystyle\int_{\Omega}\widetilde{B}(b(|u|)|u|)
dx.$$
From \[brezislieb\] we conclude that
$$\displaystyle\int_{\Omega}\widetilde{B}\left(b(|u+h_{n}+t_{n}v|)|u+h_{n}+t_{n}v|
- b(|u|)|u| \right) dx \rightarrow 0.$$
Moreover, with obvious changes, we can prove that
$$\displaystyle\int_{\Omega}\widetilde{B}\left(b(|u+h_{n}|)|u+h_{n}| -
b(|u|)|u| \right) dx \rightarrow 0.$$
Thus, by Fatou’s lemma that $$\begin{aligned}
\label{G}
\displaystyle\limsup
\int_{\Omega} F_{n}(u,v) \ dx \leq \displaystyle\int_{\Omega}
\limsup F_{n}(u,v) \ dx.\end{aligned}$$
From (\[eq 3.1\]) and (\[G\]) we get $$\Psi^{0}(u,v) \leq \displaystyle\int_{\Omega} F^{0}(u,v) \ dx =
\displaystyle\int_{\Omega}\max \{\langle\xi,v\rangle; \xi \in
\partial F(u)\}dx.$$
Consider $\widehat{\rho} \in \partial \Psi(u) \subset L_{B}(\Omega)^* \equiv
L_{\widetilde{B}}(\Omega)$ with $u \in L_{B}(\Omega)$. Then, there is $\rho \in L_{\widetilde{B}}(\Omega)$ such that $$\langle\widehat{\rho}, v\rangle= \displaystyle\int_{\Omega} \rho v \
dx, \ \ v
\in L_{B}(\Omega).$$
We claim that $$\rho(x) \geq \underline{f}(u(x)) \ \ a.e \ \ \mbox{in} \ \ \Omega.$$
Arguing, by contradiction, we suppose that there is $A \subset
\Omega$ with $|A|>0$ such that $\rho(x) < \underline{f}(u(x))$. Hence, $$\begin{aligned}
\label{G1}
\displaystyle\int_{A}\rho(x) \ dx<
\displaystyle\int_{A}\underline{f}(u(x)) \ dx.\end{aligned}$$
Let $v=-\chi_{A}$ be a function in $ L_{B}(\Omega)$, where $\chi_{A}$ is characteristic function of set $A$. Thus, $$-\displaystyle\int_{A}\rho \ dx =\displaystyle\int_{\Omega}\rho v \
dx \leq \Psi^{0}(u,v) \leq
\displaystyle\int_{\Omega}\underline{f}(u(x)) v dx = -
\displaystyle\int_{A}\underline{f}(u(x)) \ dx,$$ with is a contradiction with $(\ref{G1})$. Thus $$\rho(x) \geq \underline{f}(u(x)) \ \ a.e \ \ \mbox{in} \ \ \Omega.$$
The inequality $$\rho(x) \leq \overline{f}(u(x)) \ \ a.e \ \ \mbox{in} \ \ \Omega$$ follows the same argument.
------------------------------------------------------------------------
Results involving genus {#Section Genus}
=======================
We will start by considering some basic notions on the Krasnoselskii genus that we will use in the proof of our main results.
Let $E$ be a real Banach space. Let us denote by $\mathfrak{A}$ the class of all closed subsets $A\subset E\setminus \{0\}$ that are symmetric with respect to the origin, that is, $u\in A$ implies $-u\in A$.
Let $A\in \mathfrak{A}$. The Krasnoselskii genus $\gamma(A)$ of $A$ is defined as being the least positive integer $k$ such that there is an odd mapping $\phi \in C(A,\mathbb{R}^{k})$ such that $\phi(x)\neq 0$ for all $x\in A$. If $k$ does not exist we set $\gamma(A)=\infty$. Furthermore, by definition, $\gamma(\emptyset)=0$.
In the sequel we will establish only the properties of the genus that will be used through this work. More information on this subject may be found in the references by [@Ambrosetti], [@Castro], [@Davi] and [@Kranolseskii].
Let $E={\mathbb{R}}^{N}$ and $\partial\Omega$ be the boundary of an open, symmetric and bounded subset $\Omega \subset {\mathbb{R}}^{N}$ with $0 \in \Omega$. Then $\gamma(\partial\Omega)=N$.
\[esfera\] $\gamma(\mathcal{S}^{N-1})=N$ where $\mathcal{S}^{N-1}$ is a unit sphere of ${\mathbb{R}}^{N}$.
\[paracompletar\] If $K \in \mathfrak{A}$, $0 \notin K$ and $\gamma(K) \geq 2$, then $K$ has infinitely many points.
Proof of Theorem \[teorema1\] {#finalissimo}
=============================
The plan of the proof is to show that the set of critical points of the functional $J$ is compact, symmetric, does not contain the zero and has genus more than $2$. Thus, our main result is a consequence of Proposition \[paracompletar\].
In the proof of the Theorem \[teorema1\] we shall need the followings technical results:
\[Gl\] The functional $J$ is coercive.
**Proof:** Using $(M_{1})$ and $(f_{2})$ we get $$\begin{aligned}
J(u)&\geq& k_0\int_0^{\displaystyle\int_\Omega\Phi(\mid\nabla
u\mid) \ dx}s^\alpha ds-\int_\Omega\Phi(u)dx- b_1\displaystyle\int_\Omega B(u)\ dx\\
&\geq&\frac{k_0}{\alpha+1}\left(\int_\Omega \Phi(\mid\nabla
u\mid)dx\right)^{\alpha +1}-\int_\Omega\Phi(u)dx-b_1\int_\Omega B(u)
\ dx .
\end{aligned}$$
From Lemmas \[desigualdadeimportantes\] and \[desigualdadeimportantes1\] we obtain $$J(u)\geq \frac{k_0}{\alpha
+1}\xi_0(\|u\|_\Phi)^{\alpha+1}-\xi_1(\mid u\mid_\Phi)-b_1\eta_1(|
u|_B).$$
Using now Corollary \[imersao\] we get the continuous imbedding $W_0^{1}L_{\Phi}(\Omega)\hookrightarrow
L_{B}(\Omega),L_\Phi(\Omega)$ hold. Hence, there are positive constants $C_{1},C_{2}$ and $C_{3}$ such that, for $|\nabla
u|_{\Phi}\geq 1$, we have $$\begin{aligned}
J(u)&\geq& C_{1}\| u\|_{\Phi}^{l(\alpha+1)}-C_{2}\| u\|_{\Phi}^m-
C_3\| u\|_{\Phi}^{q_1}.\end{aligned}$$ Since $l(\alpha+1)>q_1>m$, we conclude that $J$ is coercive.
------------------------------------------------------------------------
Now we prove that $J$ satisfies the nonsmooth $(PS)_{c}$ condition.
\[PS1\] The functional $J$ satisfies the nonsmooth $(PS)_{c}$ condition, for all $c \in \mathbb{R}$.
**Proof:** Let $(u_{n})$ be a sequence in $W^{1}_{0}L_{\Phi}(\Omega)$ such that $$J(u_{n})\rightarrow c \ \ \mbox{and} \ \ m^{J}(u_{n})\rightarrow 0.$$
From now we consider $(w_{n})\subset\partial J(u_n)\subset
(W^{1}_{0}L_{\Phi}(\Omega))^{*}$ such that $$m^{J}(u_{n})=\|w_{n}\|_{*}=o_{n}(1)$$ and $$\langle w_{n},v\rangle=
M\left(\displaystyle\int_{\Omega}\Phi(|\nabla u_{n}|) \
dx\right)\displaystyle\int_{\Omega}\phi(|\nabla u_{n}|)\nabla
u_{n}\nabla v \ dx-\int_\Omega \phi(u)uvdx
-\langle\rho_{n},v\rangle,$$ with $\rho_{n} \in \partial\Psi(u_{n})$.
Note that from Lemma \[LipLoc1\] we have $$\underline{f}(u_{n}) \leq \rho_{n} \leq \overline{f}(u_{n}) \ \
\mbox{a.e in} \ \ \Omega.$$
On the other hand, since $J$ is coercive, we derive that $(u_{n})$ is bounded in $W^{1}_{0}L_{\Phi}(\Omega)$. Thus, passing to a subsequence, if necessary, we have $$u_{n}\rightharpoonup u \ \ \mbox{in} \ \ W^{1}_{0}L_{\Phi}(\Omega),$$ $$\frac{\partial u_{n}}{\partial x_{i}}\rightharpoonup \frac{\partial
u}{\partial x_{i}} \ \ \mbox{in} \ \ L_{\Phi}(\Omega),$$ $$u_{n}\rightarrow u \ \ \mbox{in} \ \ L_{B}(\Omega) \mbox{ and } L_\Phi(\Omega),$$ and $$\displaystyle\int_{\Omega}\Phi (|\nabla u_{n}|) \ dx \rightarrow
t_{0} \geq 0.$$
If $t_{0}=0$, then from Lemma \[desigualdadeimportantes\] we obtain $$| \nabla u_n|_\Phi\leq \xi_0^{-1}\left(\displaystyle\int_\Omega
\Phi(\mid\nabla u_n\mid) \ dx \right) \rightarrow 0$$ and the proof is finished.
If $t_{0}>0$, since $M$ is a continuous function, we get $$M\left(\displaystyle\int_\Omega\Phi(\mid\nabla u_n\mid) \ dx
\right)\rightarrow M(t_0).$$
Thus, from $(M_{1})$ and for $n$ sufficiently large, $$\begin{aligned}
\label{limitacaoporbaixo1}
M\left(\int_\Omega\Phi (\mid\nabla u_n\mid) \ dx \right)\geq
k_0t_{0}^{\alpha}>0.\end{aligned}$$
Now we proof that $(\rho_{n})$ is bounded in $L_{\widetilde{B}}(\Omega)$. Note that from $(f_{2})$, $(f_{3})$ and $(M_{1})$ that $$\overline{f}(t) \leq c b(t)t.$$
Since $$\overline{f}(t)=-\underline{f}(-t)$$ we get from Lemmas \[desigualdadeimportantes1\] and \[DESIGUALD\] that $$\begin{aligned}
\displaystyle\int_{\Omega}\widetilde{B}(u_{n}) \ dx &\leq & C
\displaystyle\int_{[u_{n}\geq 0]}B(u_{n}) \ dx+
\displaystyle\int_{[u_{n}< 0]}\widetilde{B}(\underline{f}(- u_{n}))
\ dx \\
&\leq & \displaystyle\int_{\Omega}\widetilde{B}(u_{n}) \ dx \leq
C\eta (|u_{n}|_{B}) \leq \overline{C}(\|u_{n}\|_{\Phi}),\end{aligned}$$ which implies that $(\rho_{n})$ is bounded in $L_{\widetilde{B}}(\Omega)$.
Then $$\begin{aligned}
\label{convergrho}
\displaystyle\int_{\Omega}\rho_{n}(u_{n}-u) \ dx \rightarrow 0.\end{aligned}$$
From definition of $(u_{n})$ we have $$\begin{aligned}
o_{n}(1)= \langle w_{n}, u_{n}-u\rangle&= &
M\left(\displaystyle\int_{\Omega}\Phi(|\nabla u_{n}|) \
dx\right)\displaystyle\int_{\Omega}\phi(|\nabla u_{n}|)\nabla
u_{n}\nabla (u_{n}- u) \ dx\nonumber\\
&&-\int_\Omega\phi(u_n)u_n(u_n-u)dx- \displaystyle\int_{\Omega}\rho_{n} (u_{n}- u) \ dx.\end{aligned}$$ Since $|u_n-u|_\Phi$ goes to $0$ and $(\phi(u_n)u_n)$ is bounded in $L_{\widetilde{\Phi}}(\Omega)$, have that $$\begin{aligned}
\label{der01}
\int_\Omega\phi(u_n)u_n(u_n-u)dx\rightarrow 0.\end{aligned}$$ We get from (\[convergrho\]) and (\[der01\]) that $$M\left(\displaystyle\int_{\Omega}\Phi(|\nabla u_{n}|) \
dx\right)\displaystyle\int_{\Omega}\phi(|\nabla u_{n}|)\nabla
u_{n}\nabla (u_{n}- u) \ dx\rightarrow 0.$$
From (\[limitacaoporbaixo1\]) and the last convergence implies that $$\displaystyle\int_{\Omega}\phi(|\nabla u_{n}|)\nabla u_{n}\nabla
(u_{n}- u) \ dx\rightarrow 0.$$ Setting $\beta:{\mathbb{R}}^N\rightarrow{\mathbb{R}} ^N$ by $$\beta(x)=\phi(\mid \nabla x\mid)\nabla x, \ x\in{\mathbb{R}}^N,$$ the last limit imply that for some subsequence, still denoted by itself, $$\left(\beta(\nabla u_n(x))-\beta(\nabla u(x))\right)(\nabla
u_n(x)-\nabla u(x))\to 0 \ \mbox{a.e in } \,\,\, \Omega.$$ Applying a result found in Dal Maso and Murat [@Maso], it follows that $$\nabla u_n(x)\to \nabla u(x) \ \mbox{a.e in } \,\,\,\Omega.$$ Then $$u_{n}\rightarrow u \ \ \mbox{in} \ \ W^{1}_{0}L_{\Phi}(\Omega).$$
------------------------------------------------------------------------
Let $K_{c}$ be the set of critical points of $J$. More precisely $$K_{c}=\{u \in W^{1}_{0}L_{\Phi}(\Omega): 0 \in \partial J(u) \ \
\mbox{and} \ \ J(u)=c\}.$$
Since $J$ is even, we have that $K_{c}$ is symmetric. The next result is important in our arguments and allows we conclude that $K_{c}$ is compact. The proof can be found in [@chang].
\[KcCompacto\] If $J$ satisfies the nonsmooth $(PS)_{c}$ condition, then $K_{c}$ is compact.
To prove that $K_{c}$ does not contain zero, we construct a special class of the levels $c$.
For each $k \in \mathbb{N}$, we define the set $$\Gamma_{k}=\{C \subset W^{1}_{0}L_{\Phi}(\Omega): C \ \ \mbox{is
closed}, C=-C \ \ \mbox{and} \ \ \gamma(C) \geq k\},$$ and the values $$c_{k}=\displaystyle\inf_{C\in \Gamma_{k}}\displaystyle\sup_{u \in
C}J(u).$$
Note that $$-\infty\leq c_{1}\leq c_{2}\leq c_{3}\leq ...\leq
c_{k}\leq ...$$ and, once that $J$ is coercive and continuous, $J$ is bounded below and, hence, $c_{1} > -\infty$. In this case, arguing as in [@BWW Proposition 3.1], we can prove that each $c_{k}$ is a critical value for the functional $J$.
\[minimax\] Given $k \in \mathbb{N}$, there exists $\epsilon = \epsilon(k)>0$ such that $$\gamma(J^{-\epsilon}) \geq k,$$ where $J^{-\epsilon}=\{u \in W^{1}_{0}L_{\Phi}(\Omega): J(u) \leq
-\epsilon\}$.
**Proof:** Fix $k \in \mathbb{N}$, let $X_{k}$ be a k-dimensional subspace of $W^{1}_{0}L_{\Phi}(\Omega)$. Thus, there exists $C_k>0$ such that $$-C_k|\nabla u|_{\Phi}\geq - \displaystyle|u|_{\Phi},$$ for all $u \in X_{k}$.
We now use the inequality above, $(M_{1})$, $(f_{2})$, $(f_{3})$, Lemmas \[desigualdadeimportantes\] and \[desigualdadeimportantes1\] to conclude that $$\begin{aligned}
J(u)\leq \frac{k_1}{\alpha+1}\xi_1(\|
u\|_\Phi)^{\alpha+1}-\xi_0(C_k\|u\|_\Phi).\end{aligned}$$
For $\|u\|_{\Phi} \leq 1$ we get $$J(u)\leq \| u \|_\Phi^{m}\left(\frac{k_1}{\alpha+1}\|
u\|_\Phi^{(\alpha+1)l-m}-C_k^{m}\right).$$ Considering $R>0$ such that $$R<min\left\{1,
\left(\frac{\alpha+1}{k_1}C_k^{m}\right)^{\frac{1}{(\alpha+1)l-m}}\right\},$$ there exists $\epsilon=\epsilon(R)>0$ such that $$J(u)<-\epsilon < 0,$$ for all $u\in {\mathcal{S}_R}=\{u\in X_k; |\nabla u |_\Phi=R \}$. Since $X_k$ and $\mathbb{R}^k$ are isomorphic and $\mathcal{S}_R$ and $S^{k-1}$ are homeomorphic, we conclude from Corollary \[esfera\] that $\gamma(\mathcal{S}_R)=\gamma(S^{k-1})=k$. Moreover, once that ${\mathcal{S}_R} \subset J^{-\epsilon}$ and $J^{-\epsilon}$ is symmetric and closed, we have $$k= \gamma ({\mathcal{S}_R})\leq \gamma( J^{-\epsilon}).$$
------------------------------------------------------------------------
\[minimax1\] Given $k \in \mathbb{N}$, the number $c_{k}$ is negative.
**Proof:** From Lemma \[minimax\], for each $k\in
\mathbb{N}$ there exists $\epsilon >0$ such that $\gamma(J^{-\epsilon}) \geq k$. Moreover, $ 0 \notin J^{-\epsilon}$ and $J^{-\epsilon}\in \Gamma_{k}$. On the other hand $$\displaystyle\sup_{u\in J^{-\epsilon}}J(u)\leq -\epsilon.$$
Hence, $$-\infty < c_{k}=\displaystyle\inf_{C\in
\Gamma_{k}}\displaystyle\sup_{u \in C}J(u) \leq
\displaystyle\sup_{u\in J^{-\epsilon}}J(u) \leq -\epsilon <0.$$
------------------------------------------------------------------------
A direct consequence of the last Lemma is that $0 \notin K_{c_{k}}$. The next result is also important in our arguments and the proof can be found in [@chang].
\[Deformacao\] Suppose that $X$ is a reflexive Banach space and $J$ is even and a locally Lipschitz function, satisfying the $(PS)_{c}$ condition. If $U$ is any neighborhood of $K_{c}$, then for any $\epsilon_0>0$ there exist $\epsilon \in (0,\epsilon_{0})$ and a odd homeomorphism $\eta :
X\rightarrow X$ such that:\
$a)$ $\eta (x)=x$ for $x\notin
J^{c+\epsilon}\backslash J^{c-\epsilon}$\
$b)$ $\eta(J^{c+\epsilon} \backslash U)\subset J^{c-\epsilon}$\
$c)$ If $K_{c}=\emptyset$, then $\eta(J^{c+\epsilon})\subset J^{c-\epsilon}$.
\[minimax2\] If $c_{k}=c_{k+1}=...=c_{k+r}$ for some $r \in \mathbb{N}$, then $$\gamma(K_{c_{k}})\geq r+1.$$
**Proof:** Suppose, by contradiction, that $\gamma(K_{c_k})\leq r$. Since $K_{c_{k}}$ is compact and symmetric, there exists a closed and symmetric set $U$ with $ K_{c_{k}}\subset
U$ such that $\gamma(U)= \gamma(K_{c_{k}}) \leq r$. Note that we can choose $U\subset J^{0}$ because $c_{k}<0$. By the deformation lemma \[Deformacao\] we have an odd homeomorphism $ \eta:
W^{1}_{0}L_{\Phi}(\Omega)\rightarrow W^{1}_{0}L_{\Phi}(\Omega)$ such that $\eta(J^{c_{k}+\delta}-U)\subset J^{c_{k}-\delta}$ for some $\delta
> 0$ with $0<\delta < -c_{k}$. Thus, $J^{c_{k}+\delta}\subset J^{0}$ and by definition of $c_{k}=c_{k+r}$, there exists $A \in \Gamma_{k+r}$ such that $\displaystyle\sup_{u \in A}J(u) < c_{k}+\delta$, that is, $A \subset J^{c_{k}+\delta}$ and $$\begin{aligned}
\label{estrela1}
\eta(A-U) \subset \eta ( J^{c_{k}+\delta}-U)\subset
J^{c_{k}-\delta}.\end{aligned}$$ But $\gamma(\overline{A-U})\geq \gamma(A)-\gamma(U) \geq k$ and $\gamma(\eta(\overline{A-U}))\geq \gamma(\overline{A-U})\geq k$. Then $\eta(\overline{A-U}) \in \Gamma_{k}$ and this contradicts (\[estrela1\]). Hence, this lemma is proved.
------------------------------------------------------------------------
Proof of Theorem \[teorema1\] {#proof-of-theorem-teorema1}
-----------------------------
If $-\infty< c_{1} < c_{2} < ...< c_{k}< ...<0$ and since each $c_{k}$ critical value of $J$, then we obtain infinitely many critical points of $J$ and hence, the problem $(P)$ has infinitely many solutions.
On the other hand, if there are two constants $c_{k}=c_{k+r}$, then $c_{k}=c_{k+1}=...=c_{k+r}$ and from Lemma \[minimax2\], we have $$\gamma(K_{c_{k}})\geq r+1 \geq 2.$$ From Proposition \[paracompletar\], $K_{c_{k}}$ has infinitely many points.
Let $(u_{k})$ critical points of $J$. Now we show that, for $$\begin{aligned}
\label{setmeasure}
a_0 < \xi_1^{-1}\left(\frac{k_0 l}{\Phi(1)\mid \Omega\mid
m^2}\xi_0(C)^{\alpha+1}\right),\end{aligned}$$ we have that $$\bigl\{x \in \Omega: \mid u_{k}(x)\mid \geq a_0\bigl\}$$ has positive measure. Thus every critical points of $J$, are solutions of $(P)$. Suppose, by contradiction, that this set has null measure. Thus $$\begin{aligned}
0&=&M\left(\int_\Omega\Phi(\mid \nabla u_k\mid )dx\right)\int_\Omega
\phi(\mid\nabla u_k\mid)\mid\nabla u_k\mid^2dx-\int_\Omega \phi(\mid
u_k\mid)\mid
u_k\mid^2 dx \\
&\geq & k_0 l\left(\int_\Omega\Phi(\mid\nabla
u_k\mid)dx\right)^{\alpha+1}-m\int_\Omega \Phi(u_k)dx\\
&\geq & k_0 l\xi_0(\parallel
u_k\parallel_\Phi)^{(\alpha+1)}-m\Phi(a_0)\mid \Omega\mid,\
\end{aligned}$$ where we conclude $$\begin{aligned}
\label{setmeasure1}
k_0 l\xi_0(\parallel u_k\parallel_\Phi)^{(\alpha+1)}\leq
m\xi_1(a_0)\mid \Omega\mid\Phi(1).\end{aligned}$$
Since $c_{k}\leq -\epsilon<0$, there exists $C>0$ such that $\|u_{k}\|\geq C>0$. Hence $$\begin{aligned}
a_0 \geq \xi_1^{-1}\left(\frac{k_0 l}{\Phi(1)\mid \Omega\mid
m^2}\xi_0(C)^{\alpha+1}\right),\end{aligned}$$ which contradicts (\[setmeasure\]). Then, $$\bigl\{x \in \Omega:
\mid u_{k}(x)\mid \geq a_0\bigl\}$$ has positive measure.
------------------------------------------------------------------------
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[^1]: Partially supported by CNPq/PQ 301242/2011-9 and 200237/2012-8
[^2]: Partially supported by CNPq-Brazil grant Casadinho/Procad 552.464/2011-2
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abstract: |
We present the results of extensive multi-waveband monitoring of the blazar 3C 279 between 1996 and 2007 at X-ray energies (2-10 keV), optical R band, and 14.5 GHz, as well as imaging with the Very Long Baseline Array (VLBA) at 43 GHz. In all bands the power spectral density corresponds to “red noise” that can be fit by a single power law over the sampled time scales. Variations in flux at all three wavebands are significantly correlated. The time delay between high and low frequency bands changes substantially on time scales of years. A major multi-frequency flare in 2001 coincided with a swing of the jet toward a more southerly direction, and in general the X-ray flux is modulated by changes in the position angle of the jet near the core. The flux density in the core at 43 GHz—increases in which indicate the appearance of new superluminal knots—is significantly correlated with the X-ray flux.
We decompose the X-ray and optical light curves into individual flares, finding that X-ray leads optical variations (XO) in 6 flares, the reverse occurs in 3 flares (OX), and there is essentially zero lag in 4 flares. Upon comparing theoretical expectations with the data, we conclude that (1) XO flares can be explained by gradual acceleration of radiating electrons to the highest energies; (2) OX flares can result from either light-travel delays of the seed photons (synchrotron self-Compton scattering) or gradients in maximum electron energy behind shock fronts; and (3) events with similar X-ray and optical radiative energy output originate well upstream of the 43 GHz core, while those in which the optical radiative output dominates occur at or downstream of the core.
author:
- 'Ritaban Chatterjee, Svetlana G. Jorstad, Alan P. Marscher, Haruki Oh, Ian M. McHardy, Margo F. Aller, Hugh D. Aller, Thomas J. Balonek, H. Richard Miller, Wesley T. Ryle, Gino Tosti, Omar Kurtanidze, Maria Nikolashvili, Valeri M. Larionov, and Vladimir A. Hagen-Thorn'
title: 'Correlated Multi-Waveband Variability in the Blazar 3C 279 from 1996 to 2007'
---
Introduction
============
Blazars form a subclass of active galactic nuclei (AGN) characterized by violent variability on time scales from hours to years across the electromagnetic spectrum. It is commonly thought that at radio, infrared, and optical frequencies the variable emission of blazars originates in relativistic jets and is synchrotron in nature [@imp88; @mar98]. The X-ray emission is consistent with inverse Compton (IC) scattering of these synchrotron photons, although seed photons from outside the jet cannot be excluded [@mau96; @rom97; @cop99; @bla00; @sik01; @chi02; @arb05]. The models may be distinguished by measuring time lags between the seed-photon and Compton-scattered flux variations. Comparison of the amplitudes and times of peak flux of flares at different wavebands is another important diagnostic. For this reason, long-term multi-frequency monitoring programs are crucially important for establishing a detailed model of blazar activity and for constraining the physics of relativistic jets. Here we report on such a program that has followed the variations in emission of the blazar 3C 279 with closely-spaced observations over a time span of $\sim$10 years.
The quasar 3C 279 [redshift=0.538; @bur65] is one of the most prominent blazars owing to its high optical polarization and variability of flux across the electromagnetic spectrum. Very long baseline interferometry (VLBI) reveals a one-sided radio jet featuring bright knots (components) that are “ejected” from a bright, presumably stationary “core” [@jor05]. The measured apparent speeds of the knots observed in the past range from $4c$ to $16c$ [@jor04], superluminal motion that results from relativistic bulk velocities and a small angle between the jet axis and line of sight. Relativistic Doppler boosting of the radiation increases the apparent luminosity to $\sim 10^4$ times the value in the rest frame of the emitting plasma.
Characterization of the light curves of 3C 279 includes the power spectral density (PSD) of the variability at all different wavebands.The PSD corresponds to the power in the variability of emission as a function of time scale. @law87 and @mch87 have found that the PSDs of many Seyfert galaxies, in which the continuum emission comes mainly from the central engine, are simple “red noise” power laws, with slopes between $-1$ and $-2$. More recent studies indicate that some Seyfert galaxies have X-ray PSDs that are fit better by a broken power law, with a steeper slope above the break frequency [@utt02; @mch04; @mar03; @ede99; @pou01]. This property of Seyferts is similar to that of Galactic black hole X-ray binaries (BHXRB), whose X-ray PSDs contain one or more breaks [@bel90; @now99]. However, in blazars most of the X-rays are likely produced in the jets rather than near the central engine as in BHXRBs and Seyferts. It is unclear [*a priori*]{} what the shape of the PSD of nonthermal emission from the jet should be, a question that we answer with the dataset presented here.
The raw PSD calculated from a light curve combines two aspects of the dataset: (1) the intrinsic variation of the object and (2) the effects of the temporal sampling pattern of the observations. In order to remove the latter, we apply a Monte-Carlo type algorithm based on the “Power Spectrum Response Method” (PSRESP) of @utt02 to determine the intrinsic PSD (and its associated uncertainties) of the light curve of 3C 279 at each of three wavebands. Similar complications affect the determination of correlations and time lags of variable emission at different wavebands. Uneven sampling, as invariably occurs, can cause the correlation coefficients to be artificially low. In addition, the time lags can vary across the years owing to physical changes in the source. In light of these issues, we use simulated light curves, based on the underlying PSD, to estimate the significance of the derived correlation coefficients.
In [§]{}2 we present the observations and data reduction procedures, while in [§]{}3 we describe the power spectral analysis and its results. In [§]{}4 we cross-correlate the light curves to determine the relationship of the emission at different wavebands. Finally, in [§]{}5 we discuss and interpret the results, focusing on the implications regarding the location of the nonthermal radiation at different frequencies, as well as the physical processes in the jet that govern the development of flux outbursts in blazars.
Observations and Data Analysis
==============================
Monitoring of the X-ray, Optical, and Radio Flux Density
--------------------------------------------------------
Table \[data\] summarizes the intervals of monitoring at different frequencies for each of the three wavebands in our program. We term the entire light curve “monitor data”; shorter segments of more intense monitoring are described below.
The X-ray light curves are based on observations of 3C 279 with the Rossi X-ray Timing Explorer (RXTE) from 1996 to 2007. We observed 3C 279 in 1222 separate pointings with the RXTE, with a typical spacing of 2-3 days. The exposure time varied, with longer on-source times—typically 1-2 ks—after 1999 as the number of fully functional detectors decreased, and shorter times at earlier epochs. For each exposure, we used routines from the X-ray data analysis software FTOOLS and XSPEC to calculate and subtract an X-ray background model from the data and to model the source spectrum from 2.4 to 10 keV as a power law with low-energy photoelectric absorption by intervening gas in our Galaxy. For the latter, we used a hydrogen column density of $8\times 10^{20}$ atoms cm$^{-2}$. There is a $\sim$ 1-year gap in 2000 and annual 8-week intervals when the quasar is too close to the Sun’s celestial position to observe. A small part of the X-ray data is given in Table \[xdatatable\]. The whole dataset will be available in the electronic version of ApJ.
In 1996 December and 1997 January we obtained, on average, two measurements per day for almost two months. We refer to these observations as the “longlook" data. Between 2003 November and 2004 September, we obtained 127 measurements over 300 days (the “medium" data). Figure \[xdata\] presents these three datasets. The X-ray spectral index $\alpha_x$, defined by $f_{\rm x} \propto \nu^{\alpha_{\rm x}}$, where $f_{\rm x}$ is the X-ray flux density and $\nu$ is the frequency, has an average value of $-0.8$ with a standard deviation of 0.2 over the $\sim 10$ years of observation, and remained negative throughout.
We monitored 3C 279 in the optical $R$ band over the same time span as the X-ray observations. The majority of the measurements between 1996 and 2002 are from the 0.3 m telescope of the Foggy Bottom Observatory, Colgate University, in Hamilton, New York. Between 2004 and 2007, the data are from the 2 m Liverpool Telescope (LT) at La Palma, Canary Islands, Spain, supplemented by observations at the 1.8 m Perkins Telescope of Lowell Observatory, Flagstaff, Arizona, 0.4 m telescope of the University of Perugia Observatory, Italy, 0.7 m telescope at the Crimean Astrophysical Observatory, Ukraine, the 0.6 m SMARTS consortium telescope at the Cerro Tololo Inter-American Observatory, Chile, and the 0.7 m Meniscus Telescope of Abastumani Astrophysical Observatory in Abastumani, Republic of Georgia. We checked the data for consistency using overlapping measurements from different telescopes, and applied corrections, if necessary, to adjust to the LT system. We processed the data from the LT, Perkins Telescope, Crimean Astrophysical Observatory, and Abastumani Astrophysical Observatory in the same manner, using comparison stars 2, 7, and 9 from @gon01 to determine the magnitudes in R band. The frequency of optical measurements over the $\sim 10$-year span presented here is, on average, 2-3 observations per week. Over a three-month period between 2005 March and June, we obtained about 100 data points, i.e., almost one per day (“longlook" data). Another subset (“medium”) contains $\sim 100$ points over 200 days between 2004 January and July. Figure \[opdata\] displays these segments along with the entire 10-year light curve. A small part of the optical data is given in Table \[opdatatable\]. The whole dataset will be available in the electronic version of ApJ.
We have compiled a 14.5 GHz light curve (Fig. \[raddata\]) with data from the 26 m antenna of the University of Michigan Radio Astronomy Observatory. Details of the calibration and analysis techniques are described in @all85. The flux scale is set by observations of Cassiopeia A [@baa77]. The sampling frequency was usually of order once per week. An exception is a span of about 190 days between 2005 March and September when we obtained 60 measurements, averaging one observation every $\sim 3$ days (“medium" data). A small part of the radio data is given in Table \[raddatatable\]. The whole dataset will be available in the electronic version of ApJ.
Ultra-high Resolution Images with the Very Long Baseline Array
--------------------------------------------------------------
Starting in 2001 May, we observed 3C 279 with the Very Long Baseline Array (VLBA) at roughly monthly intervals, with some gaps of 2-4 months. The sequence of images from these data (Fig. \[vlbaimage1\] to Fig. \[vlbaimage6\]) provides a dynamic view of the jet at an angular resolution $\sim 0.1$ milliarcseconds (mas). We processed the data in the same manner as described in @jor05. For epochs from 1995 to 2001, we use the images and results of @lis98, @weh01, and @jor01 [@jor05]. We model the brightness distribution at each epoch with multiple circular Gaussian components using the task MODELFIT of the software package DIFMAP [@she97]. At each of the 80 epochs of VLBA observation since 1996, this represents the jet emission downstream of the core by a sequence of knots (also referred to as “components”), each characterized by its flux density, FWHM diameter, and position relative to the core. Figure \[distepoch\] plots the distance vs. epoch for all components brighter than 100 mJy within 2.0 mas of the core. We use the position vs. time data to determine the projected direction on the sky of the inner jet, as well as the apparent speeds and birth dates of new superluminal knots.
We define the inner-jet position angle (PA) $\theta_{\rm jet}$ with respect to the core as that of the brightest component within 0.1-0.3 mas of the core. As seen in Figure \[xoppa\], $\theta_{\rm jet}$ changes significantly ($\sim 80\degr$) over the 11 years of VLBA monitoring. Figure \[vlba12\] displays a sampling of the VLBA images at epochs corresponding to the circled points in the lower panel of Figure \[xoppa\].
We determine the apparent speed $\beta_{app}$ of the moving components using the same procedure as defined in @jor05. The ejection time $T_0$ is the extrapolated epoch of coincidence of a moving knot with the position of the (presumed stationary) core in the VLBA images. In order to obtain the most accurate values of $T_0$, given that non-ballistic motions may occur [@jor04; @jor05], we use only those epochs when a component is within 1 mas of the core, inside of which we assume its motion to be ballistic. $\theta_{\rm jet}$, $T_0$, and $\beta_{app}$ between 1996 and 2007 are shown in Table \[ejecflare\]. As part of the modeling of the images, we have measured the flux density of the unresolved core in all the images, and display the resulting light curve in Figure \[corelc\].
Power Spectral Analysis and Results
===================================
For all three wavebands, we used the monitor, medium, and longlook data to calculate the PSD at the low, intermediate, and high frequencies of variation, respectively. Since we do not have any longlook data at 14.5 GHz, we determine the radio PSD up to the highest variational frequency that can be achieved with the medium data.
Calculation of the PSD of the Observed Light Curve
--------------------------------------------------
We follow @utt02 to calculate the PSD of a discretely sampled light curve $f(t_i)$ of length $N$ points using the formula\
$$|F_{N}(\nu)|^{2} = \left[\sum_{i=1}^{N}f(t_{i})\cos(2\pi\nu t_{i})\right]^{2} + \left[\sum_{i=1}^{N}f(t_{i})\sin(2\pi\nu t_{i})\right]^{2}.\\$$ This is the square of the modulus of the discrete Fourier transform of the (mean subtracted) light curve, calculated for evenly spaced frequencies between $\nu_{\rm min}$ and $\nu_{\rm max}$, i.e., $\nu_{\rm min}$, 2$\nu_{\rm min}$, . . ., $\nu_{\rm max}$. Here, $\nu_{\rm min}$=1/$T$ ($T$ is the total duration of the light curve, $t_{N}-t_{1}$) and $\nu_{\rm max}$=$N$/2$T$ equals the Nyquist frequency $\nu_{\rm Nyq}$. We use the following normalization to calculate the final PSD:\
$$P(\nu) = \frac{2T}{\mu^{2}N^{2}}|F_{N}(\nu)|^{2},$$ where $\mu$ is the average flux density over the light curve.
We bin the data in time intervals $\Delta T$ ranging from 0.5 to 25 days, as listed in Table \[lcprop\], averaging all data points within each bin to calculate the flux. For short gaps in the time coverage, we fill empty bins through linear interpolation of the adjacent bins in order to avoid gaps that would distort the PSD. We account for the effects of longer gaps, such as sun-avoidance intervals and the absence of X-ray data in 2000, by inserting in each of the simulated light curves the same long gaps as occur in the actual data.
PSD Results: Presence/absence of a Break
----------------------------------------
We use a variant of PSRESP [@utt02] to determine the intrinsic PSD of each light curve. The method is described in the Appendix. PSRESP gives both the best-fit PSD model and a “success fraction” that indicates the goodness of fit of the model.
The PSDs of the blazar 3C 279 at X-ray, optical, and radio frequencies show red noise behavior, i.e., there is higher amplitude variability on longer than on shorter timescales. The X-ray PSD is best fit with a simple power law of slope $-2.3\pm0.3$, for which the success fraction is 45%. The slope of the optical PSD is $-1.7\pm0.3$ with success fraction 62%, and for the radio PSD it is $-2.3\pm0.5$ with success fraction 96%. The observed PSDs and their best-fit models are shown in Figure \[psd\]. The error bars on the slope represent the FWHM of the success fraction vs. slope curve (Figure \[psdprob\]). The rejection confidence, equal to one minus the success fraction, is much less than 90% in all three cases (55%, 38%, and 4% in the X-ray, optical, and radio wavebands, respectively). This implies that a simple power-law model provides an acceptable fit to the PSD at all three wavebands.
We also fit a broken power-law model to the X-ray PSD, setting the low-frequency slope at $-1.0$ and allowing the break frequency and the slope above the break over a wide range of parameters ($10^{-9}$ to $10^{-6}$ Hz and $-1.0$ to $-2.5$, respectively) while calculating the success fractions (@mch06). Although this gives lower success fractions than the simple power-law model for the whole paramater space, a break at a frequency $\lesssim 10^{-8}$ Hz with a high frequency slope as steep as $-2.4$ cannot be rejected at the 95% confidence level.
Cross-correlation Analysis and Results
======================================
We employ the discrete cross-correlation function [DCCF; @ede88] method to find the correlation between variations at pairs of wavebands. We bin the light curves of all three wavebands in 1-day intervals before performing the cross-correlation. In order to determine the significance of the correlation, we perform the following steps :\
1. Simulation of $M$ (we use $M$=100) artificial light curves generated with a Monte-Carlo algorithm based on the shape and slope of the PSD as determined using PSRESP for both wavebands (total of 2$M$ light curves).\
2. Resampling of the light curves with the observed sampling function.\
3. Correlation of random pairs of simulated light curves (one at each waveband).\
4. Identification of the peak in each of the $M$ random correlations.\
5. Comparison of the peak values from step 4 with the peak value of the real correlation between the observed light curves. For example, if 10 out of 100 random peak values are greater than the maximum of the real correlation, we conclude that there is a 10% chance of finding the observed correlation by chance. Therefore, if this percentage is low, then the observed correlation is significant even if the correlation coefficient is substantially lower than unity.
As determined by the DCCF (Figure \[xopradcor\]), the X-ray variations are correlated with those at both optical and radio wavelengths in 3C 279. The peak X-ray vs. optical DCCF is 0.66, which corresponds to a 98% significance level. The peak X-ray vs. radio DCCF is relatively modest (0.42), with a significance level of 79%. The radio-optical DCCF has a similar peak value (0.45) at a 62% significance level. The cross-correlation also indicates that the optical variations lead the X-ray by $20\pm15$ and the radio by $260^{+60}_{-110}$ days, while X-rays lead the radio by $240^{+50}_{-100}$ days. The uncertainties in the time delays are the FWHM of the peaks in the correlation function.
Variation of X-ray/optical Correlation with Time
------------------------------------------------
The X-ray and the optical light curves are correlated at a very high significance level. However, the uncertainty in the X-ray-optical time delay is comparable to the delay itself. To characterize the variation of the X-ray/optical time lag over the years, we divide both light curves into overlapping two-year intervals, and repeat the DCCF analysis on each segment. The result indicates that the correlation function varies significantly with time (Fig. \[tw1\]) over the 11 years of observation. Of special note are the following trends:\
1. During the first four years of our program (96-97, 97-98, 98-99) the X-ray variations lead the optical (negative time lag).\
2. There is a short interval of weak correlation in 1999-2000.\
3. In 2000-01, the time delay shifts such that the optical leads the X-ray variations (positive time lag). This continues into the next interval (2001-02).\
4. In 2002-03, there is another short interval of weak correlation (not shown in the figure).\
5. In the next interval (2003-04), the delay shifts again to almost zero.\
6. Over the next 3 years the correlation is relatively weak and the peak is very broad, centered at a slightly negative value.\
This change of time lag over the years is the main reason why the peak value of the overall DCCF is significantly lower than unity. We discuss the physical cause of the shifts in cross-frequency time delay in $\S$5.2.
Correlation of X-ray Flux and Position Angle of the Inner Jet
-------------------------------------------------------------
We find a significant correlation (maximum DCCF=0.6) between the PA of the jet and the X-ray flux (see Figure \[xposangle\]). The changes in the position angle lead those in the X-ray flux by $80\pm150$ days. The large uncertainty in the time delay results from the broad, nearly flat peak in the DCCF. This implies that the jet direction modulates rather gradual changes in the X-ray flux instead of causing specific flares. This is as expected if the main consequence of a swing in jet direction is an increase or decrease in the Doppler beaming factor on a time scale of one or more years.
Comparison of X-ray and Optical Flares
--------------------------------------
We follow @val99 by decomposing the X-ray and optical light curves into individual flares, each with exponential rise and decay. Our goal is to compare the properties of the major long-term flares present in the X-ray and optical lightcurves. Before the decomposition, we smooth the light curve using a Gaussian function with a 10-day FWHM smoothing time. We proceed by first fitting the highest peak in the smoothed light curve to an exponential rise and fall, and then subtracting the flare thus fit from the light curve. We do the same to the “reduced" light curve, i.e., we fit the next highest peak. This reduces confusion created by a flare already rising before the decay of the previous flare is complete. We fit the entire light curve in this manner with a number of individual (sometimes overlapping) flares, leaving a residual flux much lower than the original flux at all epochs. We have determined the minimum number of flares required to adequately model the lightcurve to be 19 (X-ray) and 20 (optical), i.e., using more than 19 flares to model the X-ray light curve does not change the residual flux significantly.
Figure \[modelfit\] compares the smoothed light curves with the summed flux (sum of contributions from all the model flares at all epochs). We identify 13 X-ray/optical flare pairs in which the flux at both wavebands peaks at the same time within $\pm 50$ days. Since both light curves are longer than 4200 days and there are only about 20 significant flares during this time, it is highly probable that each of these X-ray/optical flare pairs corresponds to the same physical event. There are some X-ray and optical flares with no significant counterpart at the other waveband. We note that this does not imply complete absence of flaring activity at the other wavelength, rather that the corresponding increase of flux was not large enough to be detected in our decomposition of the smoothed light curve.
We calculate the area under the curve for each flare to represent the total energy output of the outburst. In doing so, we multiply the R-band flux density by the central frequency ($4.7\times10^{14}$ Hz) to estimate the integrated optical flux. For each of the flares, we determine the time of the peak, width (defined as the mean of the rise and decay times), and area under the curve from the best fit model. Table \[xopcomp\] lists the parameters of each flare pair, along with the ratio $\zeta_{\rm XO}$ of X-ray to optical energy output. The time delays of the flare pairs can be divided into three different classes: X-ray significantly leading the optical peak (XO, 6 out of 13), optical leading the X-ray (OX, 3 out of 13), and nearly coincident (by $< 10$ days, the smoothing length) X-ray and optical maxima (C, 4 out of 13). The number of events of each delay classification is consistent with the correlation analysis (Figure \[tw1\]). XO flares dominate during the first and last segments of our program, but OX flares occur in the middle. In both the DCCF and flare analysis, there are some cases just before and after the transition in 2001 when variations in the two wavebands are almost coincident (C flares).
The value of $\zeta_{\rm XO}$ $\approx 1$ in 5 out of 13 cases; in one flare pair $\zeta_{\rm XO}=1.4$. In all the other cases it is less than unity by a factor of a few. In all the C flares $\zeta_{\rm XO}$ $\approx 1$, while in the 3 OX cases the ratio $\ll 1$. In the C pairs, the width of the X-ray flare profile $\sim 2$ times that of the optical, but in the other events the X-ray and optical widths are comparable.
Flare-Ejection Correlation
--------------------------
The core region on VLBI images becomes brighter as a new superluminal knot passes through it [@sav02]; hence, maxima in the 43 GHz light curve of the core indicate the times of ejection of knots. We find that the core (Figure \[corelc\]) and X-ray light curves are well correlated (correlation coefficient of 0.6), with changes in the X-ray flux leading those in the radio core by $130^{+70}_{-45}$ days (see Figure \[xcore\]). The broad peak in the cross-correlation function suggests that the flare-ejection time delay varies over a rather broad range. This result is consistent with the finding of @lin06 that high-energy flares generally occur during the rising portion of the 37 GHz light curve of 3C 279.
Since some flares can be missed owing to overlapping declines and rises of successive events, we can determine the times of superluminal ejections more robustly from the VLBA data. Table \[ejecflare\] lists the ejection times and apparent speeds of the knots identified by our procedure. We cannot, however, associate an X-ray flare with a particular superluminal ejection without further information, since flux peaks and ejection times are disparate quantities. We pursue this in a separate paper that uses light curves at five frequencies between 14.5 and 350 GHz to analyze the relationship between superluminal knots and flares in 3C 279.
Discussion
==========
Red Noise Behavior and Absence of a Break in the PSD
----------------------------------------------------
The PSDs at all three wavebands are best fit with a simple power law which corresponds to red noise. The red noise nature—greater amplitudes on shorter time scales—of the flux variations at all three wavebands revealed by the PSD analysis is also evident from visual inspection of the light curves of 3C 279 (Figures \[xdata\], \[opdata\], and \[raddata\]). The PSD break frequency in BHXRBs and Seyferts scales with the mass of the black hole [@utt02; @mch04; @mch06; @mar03; @ede99]. Using the best-fit values and uncertainties in the relation between break timescale, black-hole mass, and accretion rate obtained by @mch06, we estimate the expected value of the break frequency in the X-ray PSD of 3C 279 to be $10^{-7.6\pm0.7}$ Hz, which is just within our derived lower limit of $10^{-8.5}$ Hz. Here we use a black-hole mass of $10^9$ M$_\sun$ [@woo02; @liu06] for 3C 279 and a bolometric luminosity of the big blue bump of $4\times 10^{45}$ ergs/s [@har96]. If we follow @mch06 and set the low-frequency slope of the X-ray PSD at $-1.0$ and allow the high-frequency slope to be as steep as $-2.4$, a break at a frequency $\lesssim 10^{-8}$ Hz cannot be rejected at the 95% confidence level. An even longer light curve is needed to place more stringent limits on the the presence of a break at the expected frequency.
Correlation between Light Curves at Different Wavebands
-------------------------------------------------------
The cross-frequency time delays uncovered by our DCCF analysis relate to the relative locations of the emission regions at the different wavebands, which in turn depend on the physics of the jet and the high-energy radiation mechanism(s). If the X-rays are synchrotron self-Compton (SSC) in nature, their variations may lag the optical flux changes owing to the travel time of the seed photons before they are up-scattered. As discussed in @sok04, this is an important effect provided that the angle ($\theta_{\rm obs}$) between the jet axis and the line of sight in the observer’s frame is sufficiently small, $\lesssim 1.2\degr\pm0.2\degr$, in the case of 3C 279, where we have adopted the bulk Lorentz factor ($\Gamma = 15.5\pm2.5)$ obtained by @jor05. According to @sok04, if the emission region is thicker (thickness $\sim$ radius), the allowed angle increases to $2\degr\pm0.4\degr$. X-rays produced by inverse Compton scattering of seed photons from outside the jet (external Compton, or EC, process) may lag the low frequency emission for any value of $\theta_{\rm obs}$ between $0\degr$ and $90\degr$. However, in this case we expect to see a positive X-ray spectral index over a significant portion of a flare if $\theta_{\rm obs}$ is small, and the flares should be asymmetric, with much slower decay than rise [@sok05]. This is because the electrons that up-scatter external photons (radiation from a dusty torus, broad emission-line clouds, or accretion disk) to X-rays have relatively low energies, and therefore have long radiative cooling times. Expansion cooling quenches such flares quite slowly, since the EC flux depends on the total number of radiating electrons (rather than on the number density), which is relatively weakly dependent on the size of the emitting region.
Time delays may also be produced by frequency stratification in the jet. This occurs when the electrons are energized along a surface (e.g., a shock front) and then move away at a speed close to $c$ as they lose energy via synchrotron and IC processes [@mar85]. This causes the optical emission to be radiated from the region immediately behind the surface, with the IR emission arising from a somewhat thicker region and the radio from an even more extended volume. An optical to radio synchrotron flare then begins simultaneously (if opacity effects are negligible), but the higher-frequency peaks occur earlier. On the other hand, SSC (and EC) X-rays are produced by electrons having a range of energies that are mostly lower than those required to produce optical synchrotron emission [@mch99]. Hence, X-rays are produced in a larger region than is the case for the optical emission, so that optical flares are quenched faster and peak earlier. Flatter PSD in the optical waveband is consistent with this picture.
In each of the above cases, the optical variations lead those at X-ray energies. But in majority of the observed flares, the reverse is true. This may be explained by another scenario, mentioned by @bot07, in which the acceleration time scale of the highest-energy electrons is significantly longer than that of the lower-energy electrons, and also longer than the travel time of the seed photons and/or time lags due to frequency stratification. In this case, X-ray flares can start earlier than the corresponding optical events.
We thus have a working hypothesis that XO (X-ray leading) flares are governed by gradual particle acceleration. OX events can result from either (1) light-travel delays, since the value of $\theta_{\rm obs}$ determined by @jor05 ($2.1\degr\pm 1.1\degr$) is close to the required range and could have been smaller in 2001-03 when OX flares were prevalent, or (2) frequency stratification. One way to test this further would be to add light curves at $\gamma$-ray energies, as will be possible with the upcoming [*Gamma Ray Large Area Space Telescope*]{} (GLAST). If the X and $\gamma$ rays are produced by the same mechanism and the X-ray/optical time lag is due to light travel time, we expect the $\gamma$-ray/optical time lag to be similar. If, on the other hand, the latter delay is caused by frequency stratification, then it will be shorter, since IC $\gamma$ rays and optical synchrotron radiation are produced by electrons of similarly high energies.
If the synchrotron flare and the resultant SSC flare are produced by a temporary increase in the Doppler factor of the jet (due to a change in the direction, Lorentz factor, or both), then the variations in flux should be simultaneous at all optically thin wavebands. It is possible for the C flare pairs to be produced in this way. Alternatively, the C events could occur in locations where the size or geometry are such that the time delays from light travel and frequency stratification are very short compared to the durations of the flares. As is discussed in [§]{}4.2, the X-ray/optical time delay varies significantly over the observed period (see Table \[xopcomp\] and Fig. \[tw1\]). As we discuss above, XO flares can be explained by gradual acceleration of electrons. The switch in the time delay from XO to OX at the onset of the highest amplitude multi-waveband outburst, which occurred in 2001 (MJD 2000 to 2200), might then have resulted from a change in the jet that shortened the acceleration time significantly. This could have caused the time delay from light travel time of the SSC seed photons and/or the effects of gradients in maximum electron energy to become a more significant factor than the acceleration time in limiting the speed at which flares developed during the outburst period. However, the flux from the 43 GHz core also reached its maximum value in early 2002, and the apparent speed of the jet decreased from $\sim 17c$ in 2000 to $\sim 4-7c$ in 2001-2003. This coincided with the onset of a swing toward a more southerly direction of the trajectories of new superluminal radio knots. We hypothesize that the change in direction also reduced the angle between the jet and the line of sight, so that the Doppler factor $\delta$ of the jet increased significantly, causing the elevated flux levels and setting up the conditions for major flares to occur at all wavebands. The pronounced variations in flux during the 2001-2002 outburst cannot be explained solely by fluctuations in $\delta$, however, since the time delay switched to OX rather than C. Instead, a longer-term switch to a smaller viewing angle would have allowed the SSC light-travel delay to become important, causing the switch from XO to OX flares.
Quantitative Comparison of X-ray and Optical Flares
---------------------------------------------------
The relative amplitude of synchrotron and IC flares depends on which physical parameters of the jet control the flares. The synchrotron flux is determined by the magnetic field $B$, the total number of emitting electrons $N_{\rm e}$, and the Doppler factor of the flow $\delta$. The IC emission depends on the density of seed photons, number of electrons available for scattering $N_{\rm e}$, and $\delta$. An increase solely in $N_{\rm e}$ would enhance the synchrotron and EC flux by the same factor, while the corresponding SSC flare would have a higher relative amplitude owing to the increase in both the density of seed photons and number of scattering electrons. If the synchrotron flare were due solely to an increase in $B$, the SSC flare would have a relative amplitude similar to that of the synchrotron flare, since $B$ affects the SSC output only by increasing the density of seed photons. In this case, there would be no EC flare at all. Finally, if the synchrotron flare were caused solely by an increase in $\delta$, the synchrotron and SSC flux would rise by a similar factor, while the EC flare would be more pronounced, since the density and energy of the incoming photons in the plasma frame would both increase by a factor $\delta.$ Table \[theory\] summarizes these considerations.
The location of the emission region should also have an effect on the multi-waveband nature of the flares. The magnetic field and electron energy density parameter $N_0$ both decrease with distance $r$ from the base of the jet: $B\sim r^{-b}$ and $N_0\sim r^{-n}$; we adopt $n=2$ and $b=1$ and assume a conical geometry. The cross-sectional radius $R$ of the jet expands with $r$, $R \propto r$. We have performed a theoretical calculation of the energy output of flares that includes the dependence on the location of the emission region. We use a computer code that calculates the synchrotron and SSC radiation from a source with a power-law energy distribution of electrons $N(\gamma)=N_0\gamma^{-s}$ within a range $\gamma_{\rm min}$ to $\gamma_{\rm max}$, where $\gamma$ is the energy in units of the electron rest mass. We introduce time variability of the radiation with an exponential rise and decay in $B$ and/or $N_0$ with time. In addition, we increase $\gamma_{max}$ with time in some computations in order to simulate the gradual acceleration of the electrons. The synchrotron emission coefficient is given by\
$$j_\nu(\nu) = \frac{\nu N_0}{k^\prime}\int_{\gamma_{min}}^{\gamma_{max}} \gamma^{-s}(1-\gamma kt)^{s-2}\, d\gamma\ \int_{\frac{\nu}{k^\prime\gamma^2}}^{+\infty} K_{\frac{5}{3}}(\xi)\, d\xi,$$ where $K_{\frac{5}{3}}$ is the modified Bessel function of the second kind of order $\frac{5}{3}$. We adopt $s=2.5$, consistent with the mean X-ray spectral index. The critical frequency, near which most of the synchrotron luminosity occurs, is given by $\nu_c= k^\prime \gamma^2$, while the synchrotron energy loss rate is given by $dE/dt=-k\gamma^2$. Both $k$ and $k^\prime$ are functions of $B$ and are given by $k=1.3\times10^{-9}B^2$ and $k^\prime=4.2\times10^{6}B$. The inverse-Compton (SSC in this case) emission coefficient is given by\
$$j_\nu^C = \int_{\nu}\int_{\gamma}\frac{\nu_f}{\nu_i}j_\nu(\nu_i)R\sigma(\epsilon_i,\epsilon_f,\gamma) N(\gamma)\,d\gamma d\nu_i,$$ where we approximate that the emission/scattering region is uniform (i.e., we ignore frequency stratification) and spherical with radius $R$. The Compton cross-section $\sigma$ is a function of $\gamma$ as well as the incident ($\nu_i$) and scattered $(\nu_f)$ frequencies of the photons: $$\sigma(\nu_i,\nu_f,\gamma)=\frac{3}{32}\sigma_T\frac{1}{\nu_i\gamma^2}[8+2x-x^2+4x \ln (\frac{x}{4})],$$ where $x\equiv\nu_f/(\nu_i \gamma^2)$ and $\sigma_T$ is the Thompson cross-section. In Figure \[flcomp1\], the top left panel shows the synchrotron and SSC flares at a given distance $r$ from the base of the jet, where $r$ is a constant times a factor $a_{\rm fac}$ ($a_{\rm fac}$=1). The right panel shows the same at a farther distance ($a_{\rm fac}=5$). The flares are created by an exponential rise and decay in the $B$ field, while the value of $\gamma_{\rm max}$ is held constant. The total time of the flares is fixed at $10^{7}$ seconds (120 days). We can see from Figure \[flcomp1\] that at larger distances (right panel) the SSC flare has a lower amplitude than the synchrotron flare even though at smaller values of $r$ (left panel) they are comparable. The bottom panels of Figure \[flcomp1\] are the same as above except that the value of $\gamma_{max}$ increases with time, as inferred from the forward time delay that occurs in some events. We see the same effect as in the top panels. In Figure \[flcomp2\], the top and bottom panels show similar results, but in these cases the flares are created by an exponential rise and fall in the value of $N_0.$ Again, the results are similar. Figure \[flcompreal\] shows segments of the actual X-ray and optical light curves, which are qualitatively similar to the simulated ones. The energy output of both synchrotron and SSC flares decreases with increasing $r$, but more rapidly in the latter. As a result, the ratio of SSC to synchrotron energy output $\zeta_{XO}$ decreases with $r$. The case $\zeta_{XO} \ll 1$ is therefore a natural consequence of gradients in $B$ and $N_0$.
From Table \[xopcomp\], we can see that in 7 out of 13 of the flare pairs $\zeta_{XO} \ll 1$. In one pair, $\zeta_{XO} >\;1$, and in all other pairs, $\zeta_{XO} \approx 1$. Our theoretical calculation suggests that the pairs where the ratio is less than 1 are produced at a larger distance from the base of the jet than those where the ratio $\gtrsim$ 1. The size of the emission region should be related to the cross-frequency time delay, since for all three explanations of the lag a larger physical size of the emission region should lead to a longer delay. We can then predict that the X-ray/optical time delay of the latter flares should be smaller than for the pairs with ratio $<\;1$. Indeed, inspection of Table \[xopcomp\] shows that, for most of the pairs, shorter time delays correspond to larger $\zeta_{XO}$, as expected. The smaller relative width of the optical C flares supports the conclusion that these occur closer to the base of the jet than the other flare pairs.
Conclusions
===========
This paper presents well-sampled, decade-long light curves of 3C 279 between 1996 and 2007 at X-ray, optical, and radio wavebands, as well as monthly images obtained with the VLBA at 43 GHz. We have applied an algorithm based on a method by @utt02 to obtain the broadband PSD of nonthermal radiation from the jet of 3C 279. Cross-correlation of the light curves allows us to infer the relationship of the emission across different wavebands, and we have determined the significance of the correlations with simulated light curves based on the PSDs. Analysis of the VLBA data yields the times of superluminal ejections and reveals time variations in the position angle of the jet near the core. We have identified 13 associated pairs of X-ray and optical flares by decomposing the light curves into individual flares. Comparison of the observed radiative energy output of contemporaneous X-ray and optical flares with theoretical expectations has provided a quantitative evaluation of synchrotron and SSC models. We have discussed the results by focusing on the implications regarding the location of the nonthermal radiation at different frequencies, physical processes in the jet, and the development of disturbances that cause outbursts of flux density in blazars. Our main conclusions are as follows:\
(1) The X-ray, optical, and radio PSDs of 3C 279 are of red noise nature, i.e., there is higher amplitude variability at longer time scales than at shorter time scales. The PSDs can be described as power laws with no significant break, although a break in the X-ray PSD at a variational frequency $\lesssim 10^{-8}$ Hz cannot be excluded at a 95% confidence level.\
(2) X-ray variations correlate with those at optical and radio wavebands, as expected if nearly all of the X-rays are produced in the jet. The X-ray flux correlates with the projected jet direction, as expected if Doppler beaming modulates the mean X-ray flux level.\
(3) X-ray flares are associated with superluminal knots, with the times of the latter indicated by increases in the flux of the core region in the 43 GHz VLBA images. The correlation has a broad peak at a time lag of $130^{+70}_{-45}$ days, with X-ray variations leading.\
(4) Analysis of the X-ray and optical light curves and their interconnection indicates that the X-ray flares are produced by SSC scattering and the optical flares by the synchrotron process. Cases of X-ray leading the optical peaks can be explained by an increase in the time required to accelerate electrons to the high energies needed for optical synchrotron emission. Time lags in the opposite sense can result from either light-travel delays of the SSC seed photons or gradients in maximum electron energy behind the shock fronts.\
(5) The switch to optical-leading flares during the major multi-frequency outburst of 2001 coincided with a decrease in the apparent speeds of knots from 16-17$c$ to 4-7$c$ and a swing toward the south of the projected direction of the jet near the core. This behavior, as well as the high amplitude of the outburst, can be explained if the redirection of the jet (only a 1$\degr$-2$\degr$ change is needed) caused it to point closer to the line of sight than was the case before and after the 2001-02 outburst.\
(6) Contemporaneous X-ray and optical flares with similar radiative energy output originate closer to the base of the jet, where the cross-section of the jet is smaller, than do flares in which the optical energy output dominates. This is supported by the longer time delay in the latter case. This effect is caused by the lower electron density and magnetic field and larger cross-section of the jet as the distance from the base increases.
Further progress in our understanding of the physical structures and processes in compact relativistic jets can be made by increasing the number of wavebands subject to intense monitoring. Expansion of such monitoring to a wide range of $\gamma$-ray energies will soon be possible when GLAST scans the entire sky several times each day. When combined with similar data at lower frequencies as well as VLBI imaging, more stringent tests on models for the nonthermal emission from blazars will be possible.
We thank P. Uttley for many useful discussions. The research at Boston University was funded in part by the National Science Foundation (NSF) through grant AST-0406865 and by NASA through several RXTE Guest Investigator Program grants, most recently NNX06AG86G, and Astrophysical Data Analysis Program grant NNX08AJ64G. The University of Michigan Radio Astronomy Observatory was supported by funds from the NSF, NASA, and the University of Michigan. The VLBA is an instrument of the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias, with financial support from the UK Science and Technology Facilities Council.
[*Facilities:*]{} VLBA, RXTE, Liverpool:2m, Perkins
Power Spectrum Response Method (PSRESP)
=======================================
In this study, we determine the shape and slope of the PSD of the light curves of 3C 279 using the PSRESP method [@utt02]. This involves the following steps:\
1. Calculation of the PSD of the observed light curve (PSD$_{\rm obs})$ with formulas (1) and (2).\
2. Simulation of $M$ artificial light curves of red noise nature with a trial shape (simple power law, broken power law, bending power law, etc.) and slope. We use $M$ = 100.\
3. Resampling of the simulated light curves with the observed sampling function.\
4. Calculation of the PSD of each of the resampled simulated light curves (PSD$_{\rm sim,i}$, i=1, $M$). The resampling with the observed sampling function (which is irregular) adds the same distortions to the simulated PSDs that are present in the real PSD $(\rm PSD_{\rm obs})$.\
5. Calculation of two functions similar to $\chi^2$:\
$$\chi^2_{\rm obs}=\sum_{\nu=\nu_{\rm min}}^{\nu_{\rm max}}{\frac{(PSD_{\rm obs}-\overline{\rm PSD}_{\rm sim})^2}{(\Delta PSD_{\rm sim})^2}}$$ and $$\chi^2_{\rm dist,i}=\sum_{\nu=\nu_{\rm min}}^{\nu_{(\rm max}}{\frac{(PSD_{\rm sim,i}-\overline{\rm PSD}_{\rm sim})^2}{(\Delta PSD_{\rm sim})^2}},$$ where $\overline{\rm PSD}_{\rm sim}$ is the average of $(\rm PSD_{\rm sim,i})$ and $\Delta \rm PSD_{\rm sim}$ is the standard deviation of $(\rm PSD_{\rm sim,i})$, with i=1, $M$.\
6. Comparison of $\chi^2_{\rm obs}$ with the $\chi^2_{\rm dist}$ distribution. Let $m$ be the number of $\chi^2_{\rm dist,i}$ for which $\chi^2_{\rm obs}$ is smaller than $\chi^2_{\rm dist,i}$ . Then $m$/$M$ is the success fraction of that trial shape and slope, a measure of its success at representing the shape and slope of the intrinsic PSD.\
7. Repetition of the entire procedure (steps 2 to 6) for a set of trial shapes and slopes of the initial simulated PSD to determine the shape and slope that gives the highest success fraction. We scan a range of trial slopes from $-1.0$ to $-2.5$ in steps of $0.1$ for the simple power-law fit.
We perform a few additional steps to overcome the distorting effects of finite length and discontinuous sampling of the light curves. These steps are implicitly included in the light curve simulation (step 2). The light curve of an astronomical source is essentially infinitely long, but we have sampled a 10-year long interval of it and are calculating the PSD based on that interval. As a result, power from longer (than observed) time scales “leaks" into the shorter time scales and hence distorts the observed PSD. This effect, called “red noise leak" (RNL), can be accounted for in PSRESP. We overcome this by simulating light curves that are more than 100 times longer than the observed light curve. As a consequence, the resampled simulated light curves are a small subset of the originally simulated ones, and similar RNL distortions are included in $\rm PSD_{\rm sim,i}$ that are present in $\rm PSD_{\rm obs}$.
On the other hand, if a light curve is not continuously sampled, power from frequencies higher than the Nyquist frequency ($\nu_{\rm Nyq}$) is shifted or “aliased" to frequencies below $\nu_{\rm Nyq}$. The observed PSD in that case will be distorted by the aliased power, which is added to the observed light curve from timescales as small as the exposure time ($T_{exp}$) of the observation (about 1000 seconds for the X-ray light curve). Ideally, we should account for this by simulating light curves with a time-resolution as small as 1000 seconds so that the same amount of aliasing occurs in the simulated data. This involves excessive computing time for decade-long light curves. To avoid this, we follow @utt02 by simulating light curves with a resolution 10$T_{exp}$ . To calculate the aliasing power from timescales from $T_{exp}$ to 10$T_{exp}$, we use an analytic approximation of the level of power added to all frequencies by the aliasing, given by\
$$P_{C} = \frac{1}{\nu_{\rm Nyq} - \nu_{\rm min}}\int_{\nu_{\rm Nyq}}^{(2T_{\rm exp})^{-1}} P(\nu)d\nu.$$ We use PSRESP to account for aliasing at frequencies lower than $(10T_{exp})^{-1}$.
We also add Poisson noise to the simulated light curves$\colon$\
$$P_{\rm noise}=\frac{\sum_{i=1}^{N}(\sigma(i))^2}{N(\nu_{\rm Nyq} - \nu_{\rm min})},$$ where $\sigma(i)$ are observational uncertainties. For details of the noise processes, readers should refer to @utt02
The goal of adding the noise and resampling with the observed sampling function is to simulate a dataset that has the same properties, including the imperfections, as the observed one. This provides a physically meaningful comparison of the observed PSD with the distribution of the simulated PSDs.
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[ccccccc]{}\
& &&\
Dataset & Start & End & Start & End & Start & End\
Longlook &Dec 96 &Jan 97 & Mar 05 & Jun 05\
Medium &Nov 03 &Sep 04 & Jan 04 & Jul 04& Mar 05 & Sep 05\
Monitor &Jan 96 &Jun 07 & Jan 96 & Jun 07& Jan 96 & Sep 07\
[cccccccc]{} 105.1275 & 688 & 1.314e-11 & 1.818e-12 & 0.466 & 0.226 & 2.766e+00 & 1.722e-01\
106.2820 & 720 & 1.343e-11 & 1.648e-12 & 0.489 & 0.206 & 2.810e+00 & 1.602e-01\
107.2621 & 672 & 1.150e-11 & 1.829e-12 & 0.425 & 0.246 & 2.393e+00 & 1.625e-01\
108.2144 & 672 & 1.299e-11 & 1.568e-12 & 0.544 & 0.221 & 2.747e+00 & 1.670e-01\
109.2619 & 624 & 1.173e-11 & 1.440e-12 & 0.597 & 0.245 & 2.474e+00 & 1.688e-01\
... & ... & ... & ... & ... & ... & ... & ...\
[ccccc]{} 46.9514 & 14.383 & 0.007 & 5.837 & 0.038\
71.7330 & 15.170 & 0.032 & 2.827 & 0.082\
86.6356 & 15.386 & 0.032 & 2.317 & 0.067\
91.6813 & 14.740 & 0.032 & 4.201 & 0.122\
100.6226 & 14.878 & 0.032 & 3.700 & 0.107\
... & ... & ... & ... & ...\
[ccc]{} 97.00 & 17.51 & 0.20\
98.00 & 17.74 & 0.19\
101.00 & 18.62 & 0.15\
103.00 & 17.54 & 0.44\
104.00 & 18.28 & 0.22\
... & ... & ...\
[cccccccc]{}\
Knot & $T_0$& $T_0$ (MJD) & $\beta_{\rm app}$ & $\theta$ (deg)\
C8 & 1996.09$\pm$ 0.10 & 115$\pm$ 36 & 5.4$\pm$ 0.7 & $-130\pm$ 3\
C9 & 1996.89$\pm$ 0.12 & 407$\pm$ 44 & 12.9$\pm$ 0.3 & $-131\pm$ 5\
C10 & 1997.24$\pm$ 0.16 & 536$\pm$ 58 & 9.9$\pm$ 0.5 & $-132\pm$ 6\
C11 & 1997.59$\pm$ 0.11 & 662$\pm$ 40 & 10.1$\pm$ 1.2 & $-135\pm$ 4\
C12 & 1998.56$\pm$ 0.09 & 1016$\pm$ 33 & 16.9$\pm$ 0.4 & $-129\pm$ 3\
C13 & 1998.98$\pm$ 0.07 & 1174$\pm$ 26 & 16.4$\pm$ 0.5 & $-130\pm$ 4\
C14 & 1999.50$\pm$ 0.09 & 1360$\pm$ 33 & 18.2$\pm$ 0.7 & $-135\pm$ 6\
C15 & 1999.85$\pm$ 0.05 & 1487$\pm$ 18 & 17.2$\pm$ 2.3 & $-131\pm$ 7\
C16 & 2000.27$\pm$ 0.05 & 1642$\pm$ 18 & 16.9$\pm$ 3.5 & $-140\pm$ 8\
C17 & 2000.96$\pm$ 0.12 & 1895$\pm$ 44 & 6.2$\pm$ 0.5 & $-133\pm$ 12\
C18 & 2001.40$\pm$ 0.16 & 2054$\pm$ 58 & 4.4$\pm$ 0.7 & $-150\pm$ 8\
C19 & 2002.97$\pm$ 0.12 & 2648$\pm$ 44 & 6.6$\pm$ 0.6 & $-133\pm$ 7\
C20 & 2003.39$\pm$ 0.10 & 2781$\pm$ 44 & 6.0$\pm$ 0.5 & $-155\pm$ 10\
C21 & 2004.75$\pm$ 0.05 & 3280$\pm$ 18 & 16.7$\pm$ 0.3 & $-147\pm$ 7\
C22 & 2005.18$\pm$ 0.06 & 3434$\pm$ 22 & 12.4$\pm$ 1.2 & $-102\pm$17\
C23 & 2006.41$\pm$ 0.15 & 3888$\pm$ 55 & 16.5$\pm$ 2.3 & $-114\pm$ 5\
[ccccccc]{}\
&Dataset & T (days) & $\Delta$T (days) & log($f_{\rm min}$) & log($f_{\rm max}$) & N$_{\rm points}$\
& Longlook &55.0 &0.5 &-6.67 &-4.93& 111\
X-ray & Medium &301.0 &5.0 &-7.40 &-5.93& 127\
& Monitor &4150.0 &25.0 &-8.55 &-6.63& 1213\
& Longlook &86.0 &1.0 &-6.86 &-5.23& 94\
Optical & Medium &185.0 &5.0 &-7.17 &-5.92& 77\
& Monitor &4225.0 &25.0 &-8.55 &-6.63& 995\
Radio & Medium &189.0 &4.0 &-7.18 &-5.83& 59\
& Monitor &3984.0 &25.0 &-8.53 &-6.63& 609\
[cccccccccc]{}\
ID& &&$\Delta T$ & TDC &$\zeta_{XO}$\
&Time & Area & Width& Time & Area& Width& (days)\
1& 119 (1996.10)& 33.2& 6.0& 134 (1996.14) & 76.3& 12.0& -15 & XO & 0.44\
2& 717 (1997.74)& 371.8& 90.0& 744 (1997.81) & 929.5& 97.5& -27 & XO & 0.40\
3& 920 (1998.30)& 172.8& 50.0& 944 (1998.36) & 278.2& 35.0& -24 & XO & 0.62\
4& 1050 (1998.65)& 338.7& 70.0& 1099 (1998.79) & 502.8& 57.5& -49 & XO & 0.67\
5& 1263 (1999.24)& 160.7& 77.5& 1266 (1999.24) & 168.9& 42.5& -3 & C & 0.95\
6& 1509 (1999.91)& 699.8& 202.5& 1517 (1999.93) & 715.4& 90.0& -8 & C & 0.98\
7& 2045 (2001.38)& 103.7& 60.0& 2029 (2001.33) & 568.3& 65.0& 16 & OX & 0.18\
8& 2151 (2001.67)& 453.6& 62.5& 2126 (2001.60) & 1188.3& 57.5& 25 & OX & 0.38\
9& 2419 (2002.40)& 193.5& 80.0& 2422 (2002.41) & 222.6& 40.0& -3 & C & 0.87\
10& 3185 (2004.50)& 191.8& 92.5& 3191 (2004.52) & 218.6& 55.0& -6 & C & 0.88\
11& 3416 (2005.13)& 217.7& 70.0& 3444 (2005.21)& 198.7& 50.0& -28 & XO & 1.09\
12& 3792 (2006.16)& 525.3& 120.0& 3814 (2006.22)& 375.6& 67.5& -22 & XO & 1.40\
13& 4035 (2006.83)& 324.2& 105.0& 4008 (2006.76)& 1068.4& 75.0& 27 & OX & 0.30\
[ccc]{}\
Parameter varied& SSC/Synch & EC/Synch\
$\delta$ &$\approx 1$ & $>1$\
$N_{\rm e}$ &$>1$ &$\approx 1$\
B &$\approx 1$ &$<1$\
|
---
author:
- Longjie ZHANG
date:
-
-
- |
December, 2015\
Corresponding author University:Graduate School of Mathematical Sciences, The University of Tokyo. Address:3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan. Email:zhanglj@ms.u-tokyo.ac.jp, zhanglj919@gmail.com
title: On curvature flow with driving force under Neumann boundary conditon in the plane
---
[**Abstract:**]{} We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the half space. We impose a boundary condition that the curves are perpendicular to the boundary for $t>0$, however, the initial curve intersects the boundary tangentially. In other words, the initial curve is oriented singularly. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
[**Keywords and phrases:**]{} mean curvature flow, driving force, Neumann boundary, level set method, singularity, fattening.
[**2010MSC:**]{} 35A01, 35A02, 35K55, 53C44.
$$$$
Introduction
============
This paper studies the planar curvature flow with driving force and Neumann boundary condition of the form $$\label{eq:cur}
V=-\kappa+A\, \ \textrm{on}\ \Gamma(t)\subset \Omega ,$$ $$\label{eq:Neum1}
\Gamma(t)\perp\partial \Omega,$$ $$\label{eq:initial1}
\Gamma(0)=\Lambda_0,$$ where $\Omega=\{(x,y)\in \mathbb{R}^2\mid x\geq 0\}$, $V$ is the outer normal velocity of $\Gamma(t)$, $\kappa$ is the curvature of $\Gamma(t)$ and the sign is chosen such that the problem is parabolic. $A$ called driving force is a positive constant.
In this paper, we consider the initial curve $\Lambda_0$ is closed, smooth and given by $$\Lambda_0=\{(x,y)\in \mathbb{R}^2\mid |y|=u_0(x), 0\leq x\leq b_0\},$$ for $u_0(x)\in C[0,b_0]\cap C^{\infty}(0,b_0)$. By the assumption of $\Lambda_0$, there hold $$u_0(x)>0,\ 0<x<b_0$$ and $$u_0(0)=u_0(b_0)=0,\ u_{0}^{\prime}(0)=-u_{0}^{\prime}(b_0)=\infty.$$
![Initial curve[]{data-label="fig:u0"}](chuzhi.pdf){height="6.0cm"}
Before giving our main results, we first consider another problem. $$V=-\kappa+A\, \ \textrm{on}\ \Lambda(t)\subset \mathbb{R}^2,\tag{\ref{eq:cur}*}$$ $$\Lambda(0)=\Lambda_0,\tag{\ref{eq:initial1}*}$$ We consider this problem by level set method. Seeing the theory in [@G], there exists unique viscosity solution $\phi$ of the following level set equation $$\left\{
\begin{array}{lcl}
\dis{\phi_t=|\nabla \phi|\textmd{div}(\frac{\nabla \phi}{|\nabla \phi|})+A|\nabla \phi|}\ \textrm{in}\ \mathbb{R}^2\times(0,T),\\
\phi(x,y,0)=a_1(x,y),
\end{array}
\right.$$ where $a_1(x,y)$ satisfies $\Lambda_0=\{(x,y)\mid a_1(x,y)=0\}$. The results in appendix show that the zero set of $\phi$ is not fattening. Indeed, thanks to Theorem \[thm:gu\], the zero set of $\phi$ can be written into $$\Lambda(t)=\{(x,y)\mid \phi(x,y,t)=0\}=\{(x,y)\in\mathbb{R}^2\mid |y|=v(x,t), a_*(t)\leq x\leq b_*(t)\},\ 0<t<T.$$ Moreover, $(v,a_*,b_*)$ is the solution of the following free boundary problem $$\left\{
\begin{array}{lcl}
\dis{u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2}},\ x\in(a_*(t),b_*(t)),\ 0<t< \delta,\\
u(a_*(t),t)=0,\ u(b_*(t),t)=0,\ 0\leq t< \delta,\\
u_x(a_*(t),t)=\infty,\ u_x(b_*(t),t)=-\infty,\ 0\leq t<\delta,\\
u(x,0)=u_0(x),\ 0\leq x\leq b_0.
\end{array}
\right.\tag{*}$$ And $a_*$ and $b_*$ are called the end points of $\Lambda(t)$.
Here we give our main results.
\[thm:exist\] If there exists $\delta$ such that $a_*(t)<0$, for $0<t<\delta$, then there exist $T_1>0$ and a unique smooth family of smooth curves $\Gamma(t)$ satisfying (\[eq:cur\]), (\[eq:Neum1\]), $0\leq t<T_1$, and satisfying (\[eq:initial1\]) in the sense that $\lim\limits_{t\rightarrow0^+}d_H(\Gamma(t),\Lambda_0)=0$. Moreover, $\Gamma(t)$ can be written into $\Gamma(t)=\{(x,y)\in\Omega\mid |y|=u(x,t),\ 0\leq x\leq b(t)\}$. And $(u,b)$ is the unique solution of the following free boundary problem $$\label{eq:1graph}
u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2},,\ 0<t<T_1,\ 0<x<b(t),$$ $$\label{eq:1bounday}
u(b(t),t)=0,\ u_x(b(t),t)=-\infty,\ u_x(0,t)=0,\ 0<t<T_1,$$ $$\label{eq:1initial}
u(x,0)=u_0(x),\ 0\leq x\leq b_0.$$
Here $d_H(A,B)$ denotes the Hausdorff distance defined as $$d_H(A,B)=\max\{\sup\limits_{x\in A}\inf\limits_{y\in B}d(x,y),\sup\limits_{y\in B}\inf\limits_{x\in A}d(x,y)\}.$$
Let $T$ be the maximal smooth time given by $$T=\sup\{t\mid \Gamma(s)\ \text{is}\ \text{smooth}, 0<s<t\}.$$
\[thm:threecondition\] (Classification) Under the same assumptions of Theorem \[thm:exist\], denote
$h(t)=\max\limits_{0\leq x\leq b(t)}u(x,t)$.Then $\Gamma(t)$ must fulfill one of the following situations.
(1). (Expanding) The existence time $T=\infty$ and both $h(t)$ and $b(t)$ tend to $\infty$, as $t\rightarrow\infty$.
(2). (Bounded) The existence time $T=\infty$ and both $h(t)$ and $b(t)$ are bounded from above and below by two positive constants, as $t\rightarrow\infty$.
(3). (Shrinking) The existence time $T<\infty$ and both $h(t)$ and $b(t)$ tend to 0, as $t\rightarrow T$.
\[thm:asym\] (Asymptotic behavior) Under the same assumptions of Theorem \[thm:exist\]. Then $\Gamma(t)$ must fulfill one of following three conditions.
(1). (Expanding) Assume that $T=\infty$ and that both $h(t)$ and $b(t)$ tend to $\infty$ as $t\rightarrow\infty$, there exist $t_0>0$, $R_1(t)$, $R_2(t)$ such that $$B_{R_1(t)}((0,0))\cap\{x>0\}\subset U(t) \subset B_{R_2(t)}((0,0))\cap\{x>0\},\ t>t_0$$ where $U(t)=\{(x,y)\in\mathbb{R}^2\mid |y|<u(x,t),\ x>0\}$. Moreover $\lim\limits_{t\rightarrow\infty}R_1(t)/t=\lim\limits_{t\rightarrow\infty}R_2(t)/t=A$.
(2). (Bounded) Assume that $T=\infty$ and that both $h(t)$ and $b(t)$ are bounded from above and below by two positive constants for $t>0$. Then $\lim\limits_{t\rightarrow\infty}d_H(\Gamma(t),\partial B_{1/A}((0,0))\cap\{x\geq0\})=0$.
(3). (Shrinking) Assume that $T<\infty$ and that both $h(t)$ and $b(t)$ tend to 0 as $t\rightarrow T$. Then the flow $\Gamma(t)$ shrinks to a point at $t=T$.
We extend $\Gamma(t)$ by even and still denote the extended curve by $\Gamma(t)$. Then problem (\[eq:cur\]), (\[eq:Neum1\]), (\[eq:initial1\]) is equivalent to the following problem in whole space $$\label{eq:cureven}
V=-\kappa+A,\ \text{on}\ \Gamma(t)\subset \mathbb{R}^2,$$ $$\label{eq:initialeven}
\Gamma(0)=\Gamma_0=\Lambda_0\cup\{(-x,y)\in\mathbb{R}^2\mid (x,y)\in\Lambda_0\}.$$ If we extends $u_0$ by even(still denoted by $u_0$), then obviously $$\label{eq:ineven}
\Gamma_0=\{(x,y)\in\mathbb{R}^2\mid |y|=u_0(x)\}.$$ In this paper we consider the problem (\[eq:cureven\]), (\[eq:initialeven\]) instead of the problem (\[eq:cur\]), (\[eq:Neum1\]), (\[eq:initial1\]).
We next give a sufficient result to have a fattening phenomenon. The definition of interface evolution and fattening are given in section 2.
\[thm:fattening1\] (Fattening)
If there exists $\delta$ such that $a_*(t)\geq0$, for $0<t<\delta$, the interface evolution $\Gamma(t)$ for (\[eq:cureven\]) with initial data $\Gamma_0$ is fattening.
Theorem \[thm:exist\] and Theorem \[thm:fattening1\] can be explained by Figure \[fig:exist\] and \[fig:fattening1\]. $\varphi$ in Figure \[fig:exist\] and \[fig:fattening1\] is given by the unique viscosity solution of $$\left\{
\begin{array}{lcl}
\dis{\varphi_t=|\nabla \varphi|\textmd{div}(\frac{\nabla \varphi}{|\nabla \varphi|})+A|\nabla \varphi|}\ \textrm{in}\ \mathbb{R}^2\times(0,T),\\
\varphi(x,y,0)=a_2(x,y),
\end{array}
\right.$$ where $a_2(x,y)$ satisfies $\Gamma_0=\{(x,y)\mid a_2(x,y)=0\}$. Let $\Gamma(t)=\{(x,y)\mid \varphi(x,y,t)=0\}$.
![$a_*(t)<0$ in Theorem \[thm:exist\][]{data-label="fig:exist"}](maintheorem.pdf){height="7.0cm"}
![$a_*(t)\geq0$ in Theorem \[thm:fattening1\][]{data-label="fig:fattening1"}](fatteningth.pdf){height="7.0cm"}
The assumptions for $a_*(t)$ in Theorem \[thm:exist\] and \[thm:fattening1\] seem not to be understood easily. Here we explain the assumptions by giving some sufficient conditions.
Denoting $\kappa(O)$ as the curvature of $\Lambda_0$ at origin, it is easy to see that $$\kappa(O)=-\lim\limits_{x\rightarrow0^{+}}u_0^{\prime\prime}/(1+(u_0^{\prime})^2)^{3/2}.$$ Then we can prove that if
(a). $\kappa(O)<A$, there holds $a_*(t)<0$, for $t$ small;
(b). $\kappa(O)>A$, there holds $a_*(t)>0$, for $t$ small.\
Since $$a_*^{\prime}(0)=\kappa(O)-A.$$
[**The role of $a_*(t)$ in main theorem.**]{} Let $(u,b)$ is the unique solution of the free boundary problem (\[eq:1graph\]), (\[eq:1bounday\]), (\[eq:1initial\]). Obviously, the flow $\Gamma^*(t)=\{(x,y)\mid |y|=u(x,t),\ 0\leq x\leq b(t)\}$ satisfies (\[eq:cur\]), (\[eq:Neum1\]), (\[eq:initial1\]) naturally.
Let $(v,a_*,b_*)$ be the solution of the problem (\*). If $a_*(t)\geq0$, $0<t<\delta$, the curves $$\Lambda(t)=\{(x,y)\mid|y|=v(x,t),\ a_*(t)\leq x\leq b_*(t)\}$$ are located in $\{x\geq0\}$.
Note that $\Gamma^*(0)=\Lambda(0)=\Lambda_0$. However, $\Gamma^*(t)$ are perpendicular to $y$-axis and the family $\{\Lambda(t)\}$ evolves freely. This means that there exist two types of flows $\Gamma^*(t)$ and $\Lambda(t)$ evolving by $V=-\kappa+A$ with the same initial curve $\Lambda_0$. This can be considered as non-uniqueness. Indeed, seeing the proof of Theorem \[thm:fattening1\], the flow given by extending $\Gamma^*(t)$ evenly is the boundary of closed evolution and the flow given by extending $\Lambda(t)$ evenly is the boundary of open evolution.
If $a_*(t)<0$, $0<t<\delta$, the problem (\*) will not make sense in the half space $\{x\ge0\}$. But the solution given by (\*) plays the role of a sub-solution (in the proof of Lemma \[lem:closebou\]). Using this sub-solution, the boundaries of the open evolution and closed evolution are away from the $x$-axis. By the uniqueness result(Proposition 5.4), we can prove they are the same.
In the curve shortening flow——$A=0$, since $a_*(t)\geq0$ always holds, the interface evolution is fattening.
[**Motivation.**]{} This research is motivated by [@MNL], the mean curvature flow with driving force under the Neumann boundary condition in a two-dimensional cylinder with periodically undulating boundary.
![After touching[]{data-label="fig:intro2"}](sectionintrod1.pdf){width="10cm"}
![After touching[]{data-label="fig:intro2"}](sectionintrod2.pdf){width="10cm"}
In [@MNL], they only consider the condition that for initial curve $\{(x,y)\in\mathbb{R}^2\mid y=u_0(x)\}$ with $|u_0^{\prime}(x)|<M$ for some $M$. They show that the interior point of $\Gamma(t)=\{(x,y)\in\mathbb{R}^2\mid y=u(x,t)\}$ never touches the boundary and $\Gamma(t)$ remains graph. Therefore, the problem can be studied by the classical parabolic theory. If removing the assumption $|u_0^{\prime}(x)|<M$, when $u(x,t)$ touches the boundary, the singularity will develop(Figure \[fig:intro1\]). Noting Figure \[fig:intro2\], after touching, $\Gamma(t)$ will possibly separate into two parts and become non-graph($\Gamma(t)$ can’t be represented by $y=u(x,t)$). This makes us analyze what will happen after touching boundary. Since $\Gamma(t)$ may become non-graph, we want to use the level set method established by [@CGG]; see also Evans and Spruck [@ES2] for the mean curvature flow, where fattening phenomenon is first observed. Therefore the main task in this paper is to study whether the interface evolution is fattening or not. The notion of the level set solution is introduced in Section 2.
[**A short review for mean curvature flow.**]{} For the classical mean curvature flow: $A=0$ in (\[eq:cur\]), there are many results. Concerning this problem, Huisken [@H] shows that any solution that starts out as a convex, smooth, compact surface remains so until it shrinks to a “round point” and its asymptotic shape is a sphere just before it disappears. He proves this result for hypersurfaces of $\mathbb{R}^{n+1}$ with $n\geq2$, but Gage and Hamilton [@GH] show that it still holds when $n=1$, the curves in the plane. Gage and Hamilton also show that embedded curve remains embedded, i.e. the curve will not intersect itself. Grayson [@Gr] proves the remarkable fact that such family must become convex eventually. Thus, any embedded curve in the plane will shrink to “round point” under curve shortening flow. But in higher dimensions it is not true. Grayson [@Gr2] also shows that there exists a smooth flow that becomes singular before shrinking to a point. His example consisted of a barbell: two spherical surfaces connected by a sufficiently thin “neck”. In this example, the inward curvature of the neck is so large that it will force the neck to pinch before shrinking. In [@AAG], A. Altschuler, S. B. Angenent and Y. Giga study the flow whose initial hypersurface is a compact, rotationally symmetric hypersurface but pinching on $x$-axis by level set method. They proved the hypersurface will separate into two smooth hypersurfaces after pinching.
[**Main method.**]{} In this paper, one of the most important tools is the intersection number principle. It was also used in [@AAG] that the intersection number between two families evolving by mean curvature flow is non-increasing. But for the problem with driving force, their intersection number may increase. In [@GMSW], they give the extended intersection number principle for the following free boundary problem called (Q) $$\left\{
\begin{array}{lcl}
\dis{u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2}},\ x\in(a(t),b(t)),\ 0<t< T,\\
u(a(t),t)=0,\ u(b(t),t)=0,\ 0\leq t< T,\\
u_x(a(t),t)=\tan\theta_-(t),\ u_x(b(t),t)=-\tan\theta_+(t),\ 0\leq t< T,\\
u(x,0)=u_0(x),\ a(0)\leq x\leq b(0),
\end{array}
\right.\tag{Q}$$ where $0<\theta_{\pm}<\pi/2$.
If $(u_1,a_1,b_1)$, $(u_2,a_2,b_2)$ are the solutions of (Q) with $\theta_{\pm}^1$, $\theta_{\pm}^2$, $u_0^1$ and $u_0^2$, the intersection number between $$\{(x,y)\mid y=u_1(x,t),a_1(t)\leq x\leq b_1(t)\}$$ and $$\{(x,y)\mid y=u_2(x,t),a_2(t)\leq x\leq b_2(t)\}$$ will increase possibly, by some simple examples(here we omit it). In [@GMSW], they find a non-increasing quantity. If $u_1$ and $u_2$ are extended by straight line, such that the extended functions $u_1^{*}$, $u_2^{*}$ are in $C^1(\mathbb{R})$, then the intersection number between $u_1^{*}$ and $u_2^{*}$ is non-increasing provided that $\theta_{\pm}^1 \neq \theta_{\pm}^2$. If $\theta_{+}^1= \theta_{+}^2$, the intersection number will not increase provided that $b_1(t)\neq b_2(t)$ and decrease at $t_0$ satisfying $b_1(t_0)=b_2(t_0)$. Similarly for $a(t)$. These results are called “extended intersection number principle”.
As we observing before, in this paper, the curve symmetric around $x$-axis is considered. Under this condition, $\theta_{\pm}$ in problem (Q) satisfy $\theta_{\pm}=\pi/2$. If we extend $u_1$ and $u_2$ by vertical straight line, it will be seen that the extended $C^1$ curve $\gamma_1(t)$ and $\gamma_2(t)$(Since $\theta_{\pm}=\pi/2$, the extended curve will not be graph) will intersect with each other even if the intersection number between $\gamma_1(0)$ and $\gamma_2(0)$ is zero. We will investigate the intersection number in Section 4.
The rest of this paper is organized as follows. In Section 2, we introduce the level set method established by [@CGG]. The definition of open evolution, closed evolution, fattening and the basic knowledge including comparison principle, monotone convergence theorem and so on are given in this section. In Section 3, we prove the Evans-Spruck estimate(also called gradient interior estimate). In Section 4, the results of intersection number are investigated and their application are given. In Section 5, we give the proof of Theorem \[thm:exist\] and Theorem \[thm:fattening1\]. In Section 6, we study the possible formations of singularity. In Section 7, we classify the solution given by Theorem \[thm:exist\] and prove the asymptotic behavior in each category(Theorem \[thm:threecondition\] and \[thm:asym\]). In Section 8, we give another non-fattening result in $(n+1)$-dimension with a type of initial hypersurfaces.
Level set method
================
Since the initial curve $\Gamma_0$ given in (\[eq:initialeven\]) has singularity at $(0,0)$, the equation $V=-\kappa+A$ does not make sense at $t=0$. Therefore, we want to apply the level set method to our problem. In this section, we introduce the level set method in $\mathbb{R}^N$. For $\Gamma(t)$ being a smooth family of smooth, closed, compact hypersurfaces in $\mathbb{R}^{N}$. Assume there exists $\psi(x,t)$ such that $\Gamma(t)=\{x|\psi(x,t)=0,x\in\mathbb{R}^N\}$. If $\Gamma(t)$ evolves by (\[eq:cureven\]), we can derive $\psi(x,t)$ satisfying $$\label{eq:level}
\dis{\psi_t=|\nabla \psi |\textmd{div}(\frac{\nabla \psi}{|\nabla \psi|})+A|\nabla \psi|\ \textrm{in}\ \mathbb{R}^N\times(0,T)}.$$ Equation (\[eq:level\]) is called the level set equation of (\[eq:cureven\]). Theorem 4.3.1 in [@G] gives the existence and uniqueness of the viscosity solution for (\[eq:level\]) with $\psi(x,0)=\psi_0(x)$. Where $\psi_0(x)$ is a bounded and uniform continuous function.
![Level set method in $\mathbb{R}^2$[]{data-label="fig:Levelsetmethod"}](levelsetmethod.pdf){height="5cm"}
[**Level set method**]{} Using the solution of level set equation, we introduce the level set method.
\[def:evo\] (1) Let $D_0$ be a bounded open set in $\mathbb{R}^N$. A family of open sets $\{D(t)\mid D(t)\subset \mathbb{R}^N\}_{0<t<T}$ is called an $(generlized)$ $open$ $evolution$ of (\[eq:cureven\]) with initial data $D_0$ if there exists a viscosity solution $\psi$ of (\[eq:level\]) that satisfies $$D(t)=\{x\in\mathbb{R}^N \mid \psi(x,t)>0\},\ D_0=\{x\in\mathbb{R}^N\mid \psi(x,0)>0\}.$$
\(2) Let $E_0$ be a bounded closed set in $\mathbb{R}^N$. A family of closed sets $\{E(t)\mid E(t)\subset \mathbb{R}^N\}_{0<t<T}$ is called a $(generlized)$ $closed$ $evolution$ of (\[eq:cureven\]) with initial data $E_0$ if there exists a viscosity solution $\psi$ of (\[eq:level\]) that satisfies $$E(t)=\{x\in\mathbb{R}^N \mid \psi(x,t)\geq0\},\ E_0=\{x\in\mathbb{R}^N\mid \psi(x,0)\geq0\}.$$
The set $\Gamma(t)=E(t)\setminus D(t)$ is called an *(generalized) interface evolution* of (\[eq:cureven\]) with initial data $\Gamma_0=E_0\setminus D_0$.
\[rem:lev\] (1) For open set $D_0$, we often choose $$\psi(x,0)=\max\{\textrm{sd}(x,\partial D_0),-1\}$$ where $$\textrm{sd}(x,\partial D_0)=\left\{
\begin{array}{lcl}
\textrm{dist}(x,\partial D_0), \ x\in D_0,\\
-\textrm{dist}(x,\partial D_0), \ x\notin D_0.
\end{array}\right.$$
\(2) Seeing that the choice of $\psi(x,0)$ isn’t unique, but by the Theorem 4.2.8 in [@G], the open evolution $D(t)$ and closed evolution $E(t)$ are both independent on the choice of $\psi(x,0)$.
\(3) In generally, even if $E_0=\overline{D_0}$, we can not guarantee $E(t)=\overline{D(t)}$. If $E(t)\setminus D(t)$ has interior points for some $t$, we call the interface evolution is fattening. Respectively, if $E(t)=\overline{D(t)}$, for all $0<t<T$, we say the interface evolution is regular. Therefore, in the proof of Theorem \[thm:exist\], it is sufficient to prove $\partial D(t)=\partial E(t)$. In the proof of Theorem \[thm:fattening1\], it is sufficient to prove that there exists a ball $B$ such that $B\subset E(t)\setminus D(t)$.
We now list some fundamental properties of open evolution and closed evolution of (\[eq:cureven\])(Chapter 4 in [@G]).
\[thm:semi\] (Semigroups)[@G]. Denote $U(t)$ and $M(t)$ being the operators such that $U(t)D_0=D(t)$ and $M(t)E_0=E(t)$, for $t>0$. Then we have $U(t)D(s)=D(t+s)$ and $M(t)E(s)=E(t+s)$, for any $t>0$, $s>0$.
\[thm:order\](Order preserving property)[@G]. Let $D_0$, $D_0^{\prime}$ be two open sets in $\mathbb{R}^N$ and let $E_0$, $E_0^{\prime}$ be two closed sets in $\mathbb{R}^N$. Then
\(1) $U(t)D_0\subset U(t)D_0^{\prime}$, if $D_0\subset D_0^{\prime}$;
\(2) $M(t)E_0\subset M(t)E_0^{\prime}$, if $E_0\subset E_0^{\prime}$;
\(3) $U(t)D_0\subset M(t)E_0^{\prime}$, if $D_0\subset E_0^{\prime}$;
\(4) $E_0\subset D_0$ and $\textrm{dist}(E_0,\partial D_0)>0$, then $M(t)E_0\subset U(t)D_0$.
\[thm:mon\](Monotone convergence)[@G].
\(1) Let $D(t)$ and $\{D_j(t)\}$ be open evolutions with initial data $D_0$ and $D_{j0}$ respectively. If $D_{j0}\uparrow D_0$, then $D_j(t)\uparrow D(t)$, $t>0$, i.e., $\bigcup\limits_{j\geq1}D_j(t)=D(t)$;
\(2) Let $E(t)$ and $\{E_j(t)\}$ be closed evolutions with initial data $E_0$ and $E_{j0}$ respectively. If $E_{j0}\downarrow E_0$, then $E_j(t)\downarrow E(t)$, $t>0$, i.e., $\bigcap\limits_{j\geq1}E_j(t)=E(t)$.
\[thm:conti\](Continuity in time)[@G]. Let $D(t)$ and $E(t)$ be open and closed evolutions, respectively.
(1a) $D(t)$ is a lower semicontinuous function of $t\in[0,T)$, in the sense that for any $t_0\geq 0$, and sequence $x_n\in (D(t_n))^c$ with $x_n\rightarrow x_0$, $t_n\rightarrow t_0$, the limit $x_0\in (D(t_0))^c$. If $D(0)$ is bounded so that $\mathcal{C}_{\epsilon}(D(t_0))$ is compact, this implies that for any $t_0\geq0$, $\epsilon>0$ there is a $\delta>0$ such that $|t-t_0|<\delta$ implies $D(t)\supset\mathcal{C}_{\epsilon}(D(t_0))$.
(1b) $E(t)$ is an upper semicontinuous function of $t\in[0,T)$, in the sense that for any $t_0\geq 0$, and sequence $x_n\in E(t_n)$ with $x_n\rightarrow x_0$, $t_n\rightarrow t_0$, the limit $x_0\in E(t_0)$. If $E(0)$ is bounded so that $\mathcal{N}_{\epsilon}(E(t_0))$ is compact, this implies that for any $t_0\geq0$, $\epsilon>0$ there is a $\delta>0$ such that $|t-t_0|<\delta$ implies $E(t)\subset\mathcal{N}_{\epsilon}(E(t_0))$.
(2a) $D(t)$ is a left upper semicontinuous in $t$ in the sense that for any $t_0\in(0,T)$, $x_0\in (D(t_0))^c$ there is a sequence $x_n\rightarrow x_0$ and $t_n\uparrow t_0$ with $x_n\in (D(t_n))^c$. Moreover, for any $t_0\in(0,T)$, $\epsilon>0$ there exists a $\delta>0$ such that $t_0-\delta<t<t_0$ implies $\mathcal{C}_{\epsilon}(D(t))\subset D(t_0)$.
(2b) $E(t)$ is a left lower semicontinuous in $t$ in the sense that for any $t_0\in(0,T)$, $x_0\in E(t_0)$ there is a sequence $x_n\rightarrow x_0$ and $t_n\uparrow t_0$ with $x_n\in E(t_0)$. Moreover, for any $t_0\in(0,T)$, $\epsilon>0$ there exists a $\delta>0$ such that $t_0-\delta<t<t_0$ implies $\mathcal{N}_{\epsilon}(E(t))\supset E(t_0)$.
Where $N_{\epsilon}(A)=\{x\in \mathbb{R}^N\mid d(x,A)<\epsilon\}$, for $A$ is a closed subset in $\mathbb{R}^N$ and $C_{\epsilon}(A)=N_{\epsilon}(A^c)^c$, for $A$ is an open subset in $\mathbb{R}^N$.
\[rem:ori\] For $A>0$, even if $D_1(0)$ and $D_2(0)$ are disjoint, $D_1(t)$ and $D_2(t)$ may intersect. The basic reason is that the level set equation (\[eq:level\]) is not orientation free(If $u$ is a solution, there does not hold that $-u$ is also a solution for (\[eq:level\])).
In order to prove Theorem \[thm:fattening1\], we need the following lemma. This lemma gives the construction of an open evolution containing two disjoint components.
\[lem:sep\] Assume $D_1(t)$ and $D_2(t)$ being the open evolution of (\[eq:level\]) with $D_1(0)=U_1$ and $D_2(0)=U_2$. And $D(t)$ is denoted as the open evolution of (\[eq:level\]) with $D(0)=U_1\cup U_2$. If $D_1(t)\cap D_2(t)=\emptyset$ for $0\leq t\leq T$, then $D(t)=D_1(t)\cup D_2(t)$, $0\leq t\leq T$.
Under the condition $A=0$, $D_1(t)\cap D_2(t)=\emptyset$ holds automatically provided that $D_1(0)\cap D_2(0)=\emptyset$. But for $A>0$, it is not true. Therefore, we give the assumption $D_1(t)\cap D_2(t)=\emptyset$ for $0\leq t\leq T$.
First, we assume $D_1(t)\cap D_2(t)=\emptyset$ for $0\leq t\leq T$ and $\delta=:\min\limits_{0\leq t\leq T}\textrm{dist}(D_1(t),D_2(t))>0$. We define $$a_i(x)=\max\{\textrm{sd}(x,\partial D_i(0)),0\}, \ x\in\mathbb{R}^n,\ i=1,2.$$
![Proof of Lemma \[lem:sep\][]{data-label="fig:lemsep"}](separatetheorem.pdf){height="5cm"}
In the theory in [@G], there exist non-negative $\varphi_i(x,t)\in C_{c}(\mathbb{R}^N\times[0,T])$ being the solution of (\[eq:level\]) with $\varphi_i(x,0)=a_i(x)$. Then $D_i(t)=\{x\in\mathbb{R}^N\mid \varphi_i(x,t)>0\}$ and $\varphi_i=0$ hold outside of $D_i(t)$, $i=1,2$. Since $\textrm{supp}\varphi_1$ and $\textrm{supp}\varphi_2$ are seperated by $\delta$, it is easy to show that $\varphi(x,t)=:\max\{\varphi_1,\varphi_2\}(x,t)$ is also a viscosity solution of (\[eq:level\]). Then $D(t)=\{x\mid\varphi(x,t)>0\}=D_1(t)\cup D_2(t)$ with initial data $D(0)=U_1\cup U_2$, for $0\leq t\leq T$.
Next we prove the result only under the assumption $D_1(t)\cap D_2(t)=\emptyset$, $0\leq t\leq T$. Consider $\dis{D_i^j(t)=\{x\mid \varphi_i(x,t)>\frac{1}{j}\}}$.
We claim that $\min\limits_{0\leq t\leq T}\textrm{dist}(D_1^j(t),D_2^j(t))>0$, for all $j$. If $\min\limits_{0\leq t\leq T}\textrm{dist}(D_1^j(t),D_2^j(t))=0$, for some $j$, then there exist $t_0\in[0,T]$ and sequences $\{x_m\}\subset D_1^j(t_0)$, $\{y_m\}\subset D_2^j(t_0)$ such that $$|x_m-y_m|\rightarrow0,\ \ \varphi_1(x_m,t_0)>\frac{1}{j},\ \varphi_2(y_m,t_0)>\frac{1}{j}.$$ Then there exists $x$, such that $\lim\limits_{m\rightarrow\infty}x_m=\lim\limits_{m\rightarrow\infty}y_m=x$. Then $$\varphi_1(x,t_0)\geq\frac{1}{j}>0,\ \varphi_2(x,t_0)\geq\frac{1}{j}>0.$$ Consequently, $x\in D_1(t_0)\cap D_2(t_0)\neq\emptyset$, contradiction. Then we have $\min\limits_{0\leq t\leq T}\textrm{dist}(D_1^j(t),D_2^j(t))>0$, for all $j$. By the argument in the first step, there holds $D^j(t)=D_1^j(t)\cup D_2^j(t)$ is the open evolution with initial openset $\dis{\{x\mid \varphi_1(x,0)>\frac{1}{j}\}\cup\{x\mid \varphi_2(x,0)>\frac{1}{j}\}}$, for $0\leq t\leq T$.
Noting $\bigcup\limits_{j=1}^{\infty}D_1^j(0)\cup D_2^j(0)=U_1\cup U_2$ and using Theorem \[thm:mon\], $D(t)=\bigcup\limits_{j=1}^{\infty}D^j(t)=\bigcup\limits_{j=1}^{\infty}D_1^j(t)\cup D_2^j(t)=D_1(t)\cup D_2(t)$, for $0\leq t\leq T$.
\[thm:openevolutionmeancurvature\](Relation between evolution and mean curvature flow) Let $D(t)$, $E(t)$ be the open evolution and closed evolution, respectively. Assume that in an open region $U\times(t_1,t_2)\subset\mathbb{R}^N\times(0,T)$, $\partial D(t)$ and $\partial E(t)$ are the graph of continuous functions $v_1$, $v_2$. Precisely, $$\partial D(t)\cap U=\{x\in \mathbb{R}^N\mid x_N=v_1(x^{\prime},t), x^{\prime}\in U^{\prime}\}$$ and $$\partial E(t)\cap U=\{x\in \mathbb{R}^N\mid x_N=v_2(x^{\prime},t), x^{\prime}\in U^{\prime}\},$$ where $x^{\prime}=(x_1,\cdots,x_{N-1})$, $U^{\prime}=U\cap\{x_N=0\}$ and $v_1$, $v_2$ are continuous in $U^{\prime}\times(t_1,t_2)$. Then the function $v_1$($v_2$) is a viscosity supersolution(subsolution) of $$v_t=\left(\delta_{ij}-\frac{v_{x_i}v_{x_j}}{1+|\nabla v|^2}\right)v_{x_ix_j}+ A\sqrt{1+|\nabla v|^2}$$ or is a viscosity subsolution(supersolution) of $$v_t=\left(\delta_{ij}-\frac{v_{x_i}v_{x_j}}{1+|\nabla v|^2}\right)v_{x_ix_j}-A\sqrt{1+|\nabla v|^2}$$ where the signs of the last terms are determined by the direction of the normal velocity of $\partial D(t)\cap U$ and $\partial E(t)\cap U$.
We can use the similar method in [@ES] to prove this theorem. Here we omit it.
A Priori estimates
===================
In this section, we give an interior gradient estimate.
[**Graph equation**]{} Let $u(x,t)$ be some function on an open subset of $\mathbb{R}^n\times \mathbb{R}$, then the graph of $u(x,t)$ is a family of hypersurfaces in $\mathbb{R}^{n+1}$. If the family of hypersurfaces moves by $V=-\kappa+A$ if and only if $$u_t=\left(\delta_{ij}-\frac{u_{x_i}u_{x_j}}{1+|\nabla u|^2}\right)u_{x_ix_j}\pm A\sqrt{1+|\nabla u|^2},$$ where the signs of the last terms are determined by direction of the normal velocity $V$.
Under the case $A=0$, $$u_t=\dis{\left(\delta_{ij}-\frac{u_{x_i}u_{x_j}}{1+|\nabla u|^2}\right)u_{x_ix_j}},$$ The estimate for $|\nabla u|$ in entire space $\mathbb{R}^n$ is given by [@EH]. The interior estimate of gradient is first given by [@ES]. Here we give the estimate under the condition $A>0$. Since the proof for $n=1$ is not easier than $n\geq 2$, we prove it in $n$-dimensional setting.
\[thm:es\] For $u\in C^3(\Omega_{T})\cap C^0(\overline{\Omega}_T)$, $u$ satisfies $$\label{eq:graph}
u_t=\dis{\left(\delta_{ij}-\frac{u_{x_i}u_{x_j}}{1+|\nabla u|^2}\right)u_{x_ix_j}\pm A\sqrt{1+|\nabla u|^2}},$$ For the condition “$+$”(“$-$”), we assume $u<0$($u>0$) in $\Omega_T$, $u(0,T)=-v_0$($u(0,T)=v_0$). Then $$|\nabla u(0,T)|\leq (3+16v_0)e^{2K},$$ where $\dis{K=20v_0^2(4n+\frac{1}{T}+4A+\frac{A}{2v_0})}+2$, $\Omega_T=B_1(0)\times (0, 2T)$ and $$\delta_{ij}=\left\{
\begin{array}{lcl}
1,\ i=j\\
0,\ i\neq j
\end{array}
\right..$$
We only prove the condition “$+$”. For the condition, we can consider “$-u$” to get the result. Denote $w=\sqrt{1+|\nabla u|^2}$, $\nu^i=u_{x_i}/\sqrt{1+|\nabla u|^2}$, $g^{ij}=\delta_{ij}-\nu^i\nu^j$. We define the operator $L$ as $$Lh=g^{ij}h_{x_ix_j}-h_t+A\nu^kh_{x_k}.$$
We let $h=\eta(x,t,u(x,t))w$, where $\eta$ is a non-negative function and will be identified in future. By calculation, $$\begin{aligned}
Lh&=&g^{ij}(w_{x_ix_j}\eta+w_{x_i}(\eta)_{x_j}+(\eta)_{x_i}w_{x_j}+w(\eta)_{x_ix_j})\\
&-&(\eta)_tw-\eta w_t+A\nu^k(w_{x_k}\eta+(\eta)_{x_k}w)\\
&=&\eta Lw+wL\eta+2g^{ij}w_{x_i}(\eta)_{x_j}\\
&=&\eta Lw+wL\eta+2g^{ij}w_{x_i}\left(\frac{h_{x_j}-w_{x_j}\eta}{w}\right).\end{aligned}$$ Then $$Lh-2g^{ij}\frac{w_{x_i}}{w}h_{x_j}=\eta\left(Lw-2g^{ij}\frac{w_{x_i}w_{x_j}}{w}\right)+wL\eta.$$
We claim that $$Lw-2g^{ij}\frac{w_{x_i}w_{x_j}}{w}\geq 0.$$ Therefore, there holds $$\label{eq:grainter}
Lh-2g^{ij}\frac{w_{x_i}}{w}h_{x_j}\geq wL\eta.$$
We begin to prove the claim. Seeing $$w_{x_ix_j}=\nu^ku_{x_kx_ix_j}+\frac{1}{w}(u_{x_kx_i}u_{x_kx_j}-\nu^k\nu^lu_{x_kx_i}u_{x_lx_j}),$$ we have $$g^{ij}w_{x_ix_j}\geq\nu^kg^{ij}u_{x_kx_ix_j}=\nu^k((g^{ij}u_{x_ix_j})_{x_k}-g^{ij}_{x_k}u_{x_ix_j})$$ $$=\nu^k\left(u_{tx_k}-A\frac{u_{x_l}u_{x_lx_k}}{\sqrt{1+|\nabla u|^2}}\right)-\nu^kg_{x_k}^{ij}u_{x_ix_j}.$$ Combining $$g_{x_k}^{ij}=-\frac{1}{w}(\nu^ju_{x_ix_k}+\nu^iu_{x_jx_k})+\frac{2u_{x_i}u_{x_j}}{w^3}w_{x_k},$$ $$\begin{aligned}
\nu^kg^{ij}u_{x_kx_ix_j}
&=&w_t-A\nu^kw_{x_k}+\frac{\nu^ku_{x_ix_j}}{w}\left(\nu^ju_{x_ix_k}+\nu^iu_{x_jx_k}-\frac{2u_{x_i}u_{x_j}}{w^2}w_{x_k}\right)\\
&=&w_t-A\nu^kw_{x_k}+\frac{2}{w}g^{ij}w_{x_i}w_{x_j}.\end{aligned}$$ Therefore $$g^{ij}w_{x_ix_j}\geq w_t-A\nu^kw_{x_k}+\frac{2}{w}g^{ij}w_{x_i}w_{x_j}.$$ Then $$Lw\geq\frac{2}{w}g^{ij}w_{x_i}w_{x_j}.$$ We complete the proof of the claim.
Next we choose $\eta=f\circ\phi(x,t,u(x,t))$, $$\dis{\phi(x,t,z)=\left(\frac{z}{2v_0}+\frac{t}{T}(1-|x|^2)\right)^+}$$ and $$f(\phi)=e^{K\phi}-1.$$
When $\phi>0$, there holds $$\phi_z=\frac{1}{2v_0},\ \phi_t=\frac{1-|x|^2}{T},\ \phi_{x_i}=-\frac{2t}{T}x_i,\ \phi_{x_ix_j}=-\frac{2t}{T}\delta_{ij}.$$ Consequently, when $\phi>0$, $z<0$, $0<t<2T$, $$0\leq\phi\leq2,\ \sum\phi_{x_i}^2\leq\frac{4t^2}{T^2}\leq16.$$
By calculation, $$\begin{aligned}
L\eta&=&g^{ij}f^{\prime\prime}(\phi_{x_i}+\phi_{z}u_{x_i})(\phi_{x_j}+
\phi_{z}u_{x_j})+g^{ij}f^{\prime}(\phi_{x_ix_j}+\phi_zu_{x_ix_j})\\
&-&f^{\prime}(\phi_t+\phi_zu_t)+A\nu^kf^{\prime}(\phi_{x_k}+\phi_zu_{x_k})\\
&\geq&\frac{f^{\prime\prime}}{w^2}(\phi_{x_i}+\phi_zu_{x_i})^2+f^{\prime}(g^{ij}
\phi_{x_ix_j}-\phi_t+A\nu^k\phi_{x_k})+f^{\prime}\phi_zLu\end{aligned}$$ $$\begin{aligned}
&=&\frac{f^{\prime\prime}}{w^2}\left(-\frac{2t}{T}x_i+\frac{1}{2v_0}u_{x_i}\right)^2+f^{\prime}\left(-\frac{2t}{T}(n-\frac{|\nabla u|^2}{1+|\nabla u|^2})-\frac{1-|x|^2}{T}\right.\\
&-&\left. Ax_k\frac{2t}{T}\frac{u_{x_k}}{\sqrt{1+|\nabla u|^2}}\right)+f^{\prime}\phi_zLu.\end{aligned}$$ Combining $$Lu=g^{ij}u_{x_ix_j}-u_t+A\nu^ku_{x_k}=-\frac{A}{\sqrt{1+|\nabla u|^2}},$$ there holds $$\begin{aligned}
L\eta&\geq& \frac{f^{\prime\prime}}{w^2}\left(\frac{|\nabla u|^2}{8v_0^2}-8\right)+f^{\prime}\left(-4n-\frac{1}{T}-4A\right)-f^{\prime}\phi_z\frac{A}{\sqrt{1+|\nabla u|^2}}\\
&\geq& \frac{f^{\prime\prime}}{w^2}\left(\frac{|\nabla u|^2}{8v_0^2}-8\right)+
f^{\prime}\left(-4n-\frac{1}{T}-4A-\frac{A}{2v_0}\right)\\
&=&\frac{K^2e^{K\phi}}{w^2}\left(\frac{|\nabla u|^2}{8v_0^2}-8\right)+
Ke^{K\phi}\left(-4n-\frac{1}{T}-4A-\frac{A}{2v_0}\right).\end{aligned}$$
When $|\nabla u|\geq \max\{16v_0,2\}$, we have $$\frac{|\nabla u|^2}{16v^2_0}\geq8,\ \frac{|\nabla u|^2}{16}\geq\frac{1+|\nabla u|^2}{20}.$$
Then $$\begin{aligned}
L\eta&\geq&\frac{K^2e^{K\phi}}{w^2}\frac{|\nabla u|^2}{16v_0^2}+
Ke^{K\phi}\left(-4n-\frac{1}{T}-4A-\frac{A}{2v_0}\right)\\
&\geq&\frac{K^2e^{K\phi}}{20v_0^2}+Ke^{K\phi}(-4n-\frac{1}{T}-4A-\frac{A}{2v_0})\\
&=&Ke^{K\phi}\left(\frac{K}{20v_0^2}-4n-\frac{1}{T}-4A-\frac{A}{2v_0}\right)>0,\end{aligned}$$ when we choose $\dis{K=20v_0^2(4n+\frac{1}{T}+4A+\frac{A}{2v_0})+2}$, $\Omega_T=B_1(0)\times (0, 2T)$.
Therefore by (\[eq:grainter\]), there holds $$Lh-2g^{ij}\frac{w_{x_i}}{w}h_{x_j}\geq0\ \text{on}\ \{h>0\ \textrm{or}\ |\nabla u|>\max\{16v_0,2\}\}.$$ By maximum principle, $$\begin{aligned}
(e^{\frac{K}{2}}-1)w(0,T)&=&h(0,T)\leq\max\limits_{h=0\ \textrm{and} \ |\nabla u|=\max\{16v_0,2\}}h\\
&\leq&(e^{2K}-1)\max\{\sqrt{1+(16v_0)^2},\sqrt{5}\}.\end{aligned}$$ Consequently, $w(0,T)\leq e^{2K}(3+16v_0)$.
\[rem:es\] (1) In Theorem \[thm:es\], $\Omega_T$ can be replaced by $\Omega_T=B_R(x_0)\times(0,2T)$ and $v_0=u(x_0,T)$. Then the conclusion becomes $$|\nabla u(x_0,T)|\leq e^{2K}(3+16\frac{v_0}{R}),$$ where $\dis{K=20\frac{v_0^2}{R^2}\left(4n+\frac{R^2}{T}+\frac{4A}{R}+\frac{A}{2v_0}\right)}+2$. We can set $v(x,t)=\dis{\frac{u(Rx+x_0,R^2t)}{R}}$, then we can use Theorem \[thm:es\] for $v(x,t)$.
\(2) When $u$ is the solution of (\[eq:graph\]) for “+” without $u<0$, then we can set $$v=u-M-\epsilon$$ where $M=\sup\limits_{\overline{\Omega}_T} |u|$ and $\epsilon>0$. Using (1) in Remark \[rem:es\] to $v$, we can deduce $$|\nabla u(0,T)|\leq\left(3+16\frac{M-u(0,T)+\epsilon}{R}\right)e^{2\widetilde{K}_{\epsilon}},$$ where $\dis{\widetilde{K}_{\epsilon}=\frac{20(M-u(0,T)+\epsilon)^2}{R^2}\left(4n+\frac{R^2}{T}+\frac{4A}{R}
+\frac{A}{2(M+\epsilon-u(0,T))}\right)}$+2.
Tending $\epsilon\rightarrow0$, we have $$|\nabla u(0,T)|\leq \left(3+32\frac{M}{R}\right)e^{2\widetilde{K}},$$ where $\dis{\widetilde{K}=\frac{80M^2}{R^2}\left(4n+\frac{R^2}{T}+\frac{4A}{R}\right)
+\frac{20AM}{R^2}}+2$.
Then we use the (2) in Remark \[rem:es\] and the same method as [@AAG] to prove the next corollary.
\[cor:es\] For $s_1<s_2$, $\rho>0$ and $q\in\mathbb{R}^n$ we set $$\Omega=B_{\rho}(x_0)\times(s_1,s_2).$$ Suppose that $u\in C^3(\Omega)$ solves the equation (\[eq:graph\]) in $\Omega$ with $M=\sup\limits_{\overline{\Omega}}|u|<\infty$. For any $\epsilon>0$ there is a constant $C=C(M,\epsilon,n)$ such that $$|\nabla u|\leq C \ \textrm{on}\ \Omega_{\epsilon}=B_{\rho-\epsilon}(x_0)\times(s_1+\epsilon^2,s_2).$$
\[rem:hes\] (1) From Corollary \[cor:es\] and [@LSU], there exist $C_k(M,\epsilon,n)$ such that $$|\nabla^k u|\leq C_k,\ (x,t)\in B_(\rho-2\epsilon)\times(s_1+2\epsilon^2,s_2).$$ (2) Noting $C$ and $C_k$ are all independent on $s_2$, if the solution $u$ exists for all $t>s_1$, $s_2$ can be chosen as $\infty$.
Intersections and the Sturmian theorem
======================================
In this section, we want to introduce the intersection number argument. There are many applications for the intersection number principle. For example, in this paper:
(1). A type of derivative estimate in Theorem \[thm:grad\].
(2). The flow $\Gamma(t)$ evolving by $V=-\kappa+A$ with some initial curve does not intersect itself.(Proposition \[lem:alphad2\])
(3). The asymptotic behavior in Section 7.
Since the proof in $\mathbb{R}^2$ is not difficult than in higher dimension, some theorems and lemmas are proved in $\mathbb{R}^{n+1}$, $n\geq1$.
**Sturm’s classical result** The Sturmian theorem states that the number of zeros(counted with multiplicity) of a solution of linear parabolic equation of the type $$u_t=a(x,t)u_{xx}+b(x,t)u_x+c(x,t)u$$ doesn’t increase with time, provided that $u$ is defined on a rectangle $x_0\leq x\leq x_1$, $0<t<T$ and $u(x_j,t)\neq0$ for $j=0,1$, for all $t\in(0,T)$. This result also holds for the number of sign changing rather than the number of zeros of $u(\cdot,t).$
It is well known that the intersection number between two families of rotationally symmetric hypersurfaces $\Gamma_1(t)$ and $\Gamma_2(t)$ evolving by $V=-\kappa$ is non-increasing([@A2]). The definition of intersection number between two families of rotationally symmetric hypersurfaces is given following. But this result is not true under the condition $V=-\kappa+A$. Indeed seeing future, the intersection number between two families of rotationally symmetric hypersurfaces evolving by $V=-\kappa+A$ may increase. In this section we give some results about this.
[**Horizontal and vertical graph equation** ]{} If $\Gamma(t)$ is a family of rotationally symmetric hypersurfaces in $\mathbb{R}^{n+1}$, then parts of $\Gamma(t)$ may be represented either as horizontal graph, $r=u(x,t)$, or vertical graph, $x=v(r,t)$, where $(x,y_1,\cdots,y_n)\in\mathbb{R}^{n+1}$ and $r=\sqrt{y_1^2+y_2^2+\cdots+y_n^2}$.
If $\Gamma(t)$ is given as a horizontal graph, then $\Gamma(t)$ evolves by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$ and the direction of the normal velocity $V$ is chosen outward iff $u$ satisfies the horizontal graph equation $$\label{eq:1horizontal}
\frac{\partial u}{\partial t}=\frac{u_{xx}}{1+u_x^2}-\frac{n-1}{u}+A\sqrt{1+u_x^2}.$$ If $\Gamma(t)$ is given as a vertical graph, then $\Gamma(t)$ evolves by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$ iff $v$ satisfies the vertical graph equation $$\label{eq:1vertical+}
\frac{\partial v}{\partial t}=\frac{v_{rr}}{1+v_r^2}+\frac{n-1}{r}v_r+ A\sqrt{1+v_r^2},$$ or $$\label{eq:1vertical-}
\frac{\partial v}{\partial t}=\frac{v_{rr}}{1+v_r^2}+\frac{n-1}{r}v_r-A\sqrt{1+v_r^2},$$ where the signs of the last terms are determined by the direction of the normal velocity $V$(We choose “$+$($-$)” when the direction of $V$ is rightward(leftward)).
[**Intersection number for rotationally symmetric hypersurfaces**]{} For two rotationally symmetric hypersurfaces $\Gamma_1(t)$ and $\Gamma_2(t)$ are given by $\Gamma_1(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^{n}\mid r=u_1(x,t)\}$ and $\Gamma_2(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^{n}\mid r=u_2(x,t)\}$. The intersection number between $\Gamma_1(t)$ and $\Gamma_2(t)$ denoted by $\mathcal{Z}[\Gamma_1(t),\Gamma_2(t)]$ is defined by the number of intersections between $u_1(\cdot,t)$ and $u_2(\cdot,t)$.
\[thm:sl\]Two smooth families of smooth, closed, hypersurfaces given by $\Gamma_1(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=u_1(x,t),a_1(t)\leq x\leq b_1(t)\}$, $\Gamma_2(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=u_2(x,t),a_2(t)\leq x\leq b_2(t)\}$ evolve by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$, $0<t<T$. Then either $\Gamma_1\equiv\Gamma_2$ for all $t\in(0,T)$, or the number of intersections of $\Gamma_1(t)$ and $\Gamma_2(t)$ is finite for all $t\in(0,T)$. In the second case, if $a_1(t)$, $b_1(t)$, $a_2(t)$ and $b_2(t)$ are all different and their order remains unchanged for all $t\in(0,T)$, this number is nonincreasing in time, and decreases whenever $\Gamma_1(t)$ and $\Gamma_2(t)$ have a tangential intersection.
We only give the sketch of the proof. For example, if the order of $a_1$, $b_1$, $a_2$, $b_2$ is given by $a_1(t)<a_2(t)<b_1(t)<b_2(t)$, $0<t<T$, the intersections are only in the interval $[a_2(t),b_1(t)]$. Since $u_1(a_2(t),t)-u_2(a_2(t),t)\neq0$ and $u_1(b_1(t),t)-u_2(b_1(t),t)\neq0$, $0<t<T$, using Theorem D in [@A1], the intersection number between $u_1$ and $u_2$ is not increasing and decreases when tangentially intersecting in $[a_2(t),b_1(t)]$. Consequently, the intersection number between $\Gamma_1(t)$ and $\Gamma_2(t)$ is not increasing. We can prove the result for the other conditions with the same method.
\[thm:grad\] $\Gamma(t)=\{(x,y)\in\mathbb{R}^{n+1}\mid r=u(x,t),a_2(t)\leq x\leq b_2(t)\}$ is a smooth family of closed, smooth hypersurfaces in $\mathbb{R}^{n+1}$, $0<t<T$. If $\Gamma(t)$ evolves by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$, there is a function $\sigma$: $\mathbb{R}_+\times\mathbb{R}_+\rightarrow\mathbb{R}$ such that $$|u_x(x,t)|\leq \sigma(t,u(x,t))$$ holds for $0<t<T$, $a_2(t)<x<b_2(t)$. The function $\sigma$ only depends on $M=\max\limits_{a_2(0)<x<b_2(0)} u(x,0)$ and $T$.
Let $w_{0}(r)\in C^{\infty}((0,+\infty))$, $w^{\prime}_0(r)\geq0$ and $$x=w_0(r)=\left\{
\begin{array}{lcl}
0, \ 0\leq r<M+1\\
1, \ r>M+2
\end{array}\right..$$
![Proof of Theorem \[thm:grad\][]{data-label="fig:grad"}](intersectionnumber1.pdf){height="7.0cm"}
We let $w$ be the unique solution of the vertical equation (\[eq:1vertical-\]) with the boundary condition $$w_r(0,t)=0,\ t\geq0$$ and initial condition $$w(r,0)=w_0(r),\ r\geq0.$$ Differentiating (\[eq:1vertical-\]) in $r$, $$\label{eq:deriveeq}
p_t=a(r,t)p_{rr}+b(r,t)p_r+c(r,t)p,$$ where $p=w_r$, $a(r,t)=1/(1+w_r^2)$, $b(r,t)=-2w_rw_{rr}/(1+w_r^2)^2+(n-1)/r-Aw_r/\sqrt{1+w_r^2}$, $c(r,t)=-(n-1)/r^2$.
By the maximum principle, we have for all $r,t>0$, $w_r\geq 0$ and $\sup\limits_{r\geq0}w(r,t)$ is nonincreasing in time. It follows from classical estimate for parabolic equation that all derivative of $w$ are uniformly bounded for $r,t\geq0$.
We note that $w_r(r,0)>0$ for $M+1<r<M+2$. Using the property of Green’s function, for any $\delta$ satisfying $0<\delta<M+AT$, there exists $A_{\delta,T}>0$ such that $A_{\delta,T}$ decreases with respect to $\delta$ and $$\label{eq:derivativeblew}
p(r,t)\geq e^{-\frac{A_{\delta,T}}{t}},$$ for $\delta\leq r\leq M+AT$, $0<t<T$.
Since the strong maximum principle implies that $p(r,t)>0$ for $r>0$, the inverse of $x=w(r,t)$ exists, denoted by $r=v(x,t)$. Seeing the normal velocity of $x=w(r,t)$ is leftward, the normal velocity of $r=v(x,t)$ is upward. Then $v(x,t)$ satisfies the horizontal graph equation (\[eq:1horizontal\]) with the free boundary condition $$v(a(t),t)=0,\ v_x(a(t),t)=\infty,\ \lim\limits_{x\rightarrow b(t)}v(x,t)=\infty,\ \lim\limits_{x\rightarrow b(t)}v_x(x,t)=\infty,\ t>0.$$ Let $\Sigma(t)=\{(x,y)\in \mathbb{R}^{n+1}\mid r=v(x,t),a(t)\leq x< b(t)\}$ and $\Sigma_{\xi}(t)$ denote the translation of $\Sigma(t)$ given by $$x=w(r,t)+\xi.$$ $\Sigma_{\xi}(t)$ can be also represented by $r=v(x-\xi,t)$, $a(t)+\xi\leq x<b(t)+\xi$. Let $a_1(t)$ and $b_1(t)$ be the end point of $\Sigma_{\xi}(t)$, then $a_1(t)=\xi+a(t)$, $b_1(t)=\xi+b(t)$. Obviously, for $(x_0,t_0)\in (a_2(t_0),b_2(t_0))\times(0,T)$, there exists $\xi\in \mathbb{R}$ such that $$v(x_0-\xi,t_0)=u(x_0,t_0).$$ By the following Lemma \[lem:inters\], we can deduce that the graph of $u(x,t_0)$ intersects $v(x-\xi,t_0)$ only once.
![Proof of Theorem \[thm:grad\][]{data-label="fig:grad2"}](intersectionnumber2.pdf){height="7.0cm"}
Next we claim $$v_x(x_0-\xi,t_0)\geq u_x(x_0,t_0).$$ If not, $v_x(x_0-\xi,t_0)< u_x(x_0,t_0)$, then there exists $\delta>0$, such that $$u(x,t_0)>v(x-\xi,t_0),$$ for all $x\in(x_0,x_0+\delta)$. Since $\lim\limits_{x\rightarrow b_1(t)}v(x,t_0)=+\infty$, $\Sigma_{\xi}(t_0)$ intersects $\Gamma(t_0)$ at least twice. This yields a contradiction.
By maximum principle, it is easy to see $r=u(x,t)<M+At<M+AT$, $a_2(t)\leq x\leq b_2(t)$, $0<t<T$. Combining (\[eq:derivativeblew\]), there holds $$u_x(x_0,t_0)\leq\frac{1}{w_r(v(x_0-\xi,t_0),t_0)}\leq e^{\frac{A_{v(x_0-\xi,t_0),T}}{t_0}}=e^{\frac{A_{u(x_0,t_0),T}}{t_0}}:=\sigma(t_0,u(x_0,t_0)).$$
By considering the reflection $\widetilde{\Sigma}(0)=\{(x,y)\mid x=-w_0(r)\}$ and the equation (\[eq:1vertical+\]) with $w_r(0,t)=0,\ t\geq0$ and $w(r,0)=w_0(r),\ r\geq0$, the bound for $-u_x(x_0,t_0)$ can be got similarly.
\[lem:inters\] $\Sigma_{\xi}(t)$ and $\Gamma(t)$ is given by Theorem \[thm:grad\], then $\Sigma_{\xi}(t)$ intersects $\Gamma(t)$ at most once.
By the same argument as Theorem \[thm:sl\], the intersection number between $\Sigma_{\xi}(t)$ and $\Gamma(t)$ is not increasing provided that $a_1(t)$, $b_1(t)$, $a_2(t)$ and $b_2(t)$ are all different and that their order remains unchanged. So we only prove this result when the order of $a_1(t)$, $b_1(t)$, $a_2(t)$ and $b_2(t)$ changes.
[**Case 1.**]{} Assume $a_1(t)<a_2(t)<b_1(t)< b_2(t)$, $t<t_2$ and $a_1(t)<a_2(t)<b_2(t)< b_1(t)$, $t>t_2$. And for $t<t_2$, $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$. Then $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$, for $t>t_2$.
![Case 2 []{data-label="fig:22"}](2.pdf){width="5cm"}
![Case 2 []{data-label="fig:22"}](7.pdf){width="5cm"}
Since$\lim\limits_{x\rightarrow b_1(t_2)}v(x-\xi,t_2)=+\infty$ and $u(b_2(t_2),t_2)=0$, there exists a positive $\delta$ independent on $t$, such that $v(x-\xi,t_2)>u(x,t_2)$, $b_1(t_2)-\delta<x<b_1(t_2)$. By continuity, there exists $\epsilon$ such that $$\label{eq:1interlemma}
v(b_1(t_2)-\delta-\xi,t)>u(b_1(t_2)-\delta,t),\ t_2-\epsilon\leq t<t_2+\epsilon$$ and $$\label{eq:2interlemma}
v(x-\xi,t)>u(x,t),\ b_1(t_2)-\delta<x\leq b_2(t),\ t_2\leq t<t_2+\epsilon.$$ The assumptions in this case imply boundary condition $$u(a_2(t),t)-v(a_2(t)-\xi,t)< 0,\ t_2-\epsilon\leq t<t_2+\epsilon$$ and initial condition $$u(x,t_2-\epsilon)<v(x-\xi,t_2-\epsilon),\ a_2(t_2-\epsilon)\leq x\leq b_1(t_2)-\delta.$$ Combining the other boundary condition (\[eq:1interlemma\]), using maximum principle in domain $$\cup_ {t_2-\epsilon\leq t<t_2+\epsilon}\left(\left[a_2(t),b_1(t_2)-\delta\right]\times\{t\}\right),$$ there holds $$u(x,t)<v(x-\xi,t),\ a_2(t)\leq x\leq b_1(t_2)-\delta,\ t_2-\epsilon\leq t<t_2+\epsilon.$$ Seeing (\[eq:2interlemma\]), $u(x,t)<v(x-\xi,t)$, $a_2(t)\leq x\leq b_2(t)$, $t_2\leq t<t_2+\epsilon$. It means that $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$, for $t_2\leq t<t_2+\epsilon$. So by Theorem \[thm:sl\], $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$, for $t>t_2$.
[**Case 2.**]{} Assume $a_1(t)<a_2(t)$, $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$, $t<t_3$ and $a_1(t_3)=a_2(t_3)$.
![Case 3[]{data-label="fig:9"}](9.pdf){width="9cm"}
Since $\lim\limits_{x\rightarrow a_2(t)}u_x(x,t)=\infty$, there exist $\delta_1$ and $\epsilon$ such that $r=u(x,t)$ can be expressed as $x=h(r,t)$, $0\leq r\leq \delta_1$, $t_3-\epsilon<t<t_3+\epsilon$. The assumptions in this case imply that $$w(\delta_1,t)+\xi<h(\delta_1,t),\ t_3-\epsilon<t<t_3+\epsilon.$$ It is easy to see $w(r,t)+\xi$ and $h(r,t)$ satisfy the vertical graph equation $$\left\{
\begin{array}{lcl}
\dis{w_t=\frac{w_{rr}}{1+w_r^2}+\frac{n-1}{r}w_r-A\sqrt{1+w_r^2}, \ 0\leq r\leq\delta_1,\ t_3-\epsilon<t<t_3+\epsilon},\\
w_r(0,t)=0,\ t\geq0,\\
\end{array}
\right.$$ and $w(r,t_3-\epsilon)+\xi< h(r,t_3-\epsilon)$. By strong maximum principle, $w(r,t)+\xi<h(r,t)$, for $0\leq r<\delta_1,\ t_3-\epsilon<t<t_3+\epsilon$. Contradiction to $a_1(t_3)=a_2(t_3)$. It means that this case does not happen.
[**Case 3.**]{} Assume $a_2(t)<b_2(t)<a_1(t)<b_1(t)$, $t<t_6$ and $a_2(t)<a_1(t)<b_2(t)<b_1(t)$, $t>t_6$.
![Case 4[]{data-label="fig:62"}](5.pdf){width="5cm"}
![Case 4[]{data-label="fig:62"}](4.pdf){width="5cm"}
Obviously, $\Sigma_{\xi}(t)$ dosen’t intersect $\Gamma(t)$, $t<t_6$. For $\lim\limits_{x\rightarrow b_2(t)}u_x(x,t)=-\infty$ and $\lim\limits_{x\rightarrow a_1(t)}v_x(x-\xi,t)=\infty$, there exists $\epsilon$ such that $$u_x(x,t)-v_x(x-\xi,t)<0,\ a_1(t)\leq x\leq b_2(t),\ t_6<t<t_6+\epsilon.$$ Seeing $u(a_1(t),t)-v(a_1(t)-\xi,t)>0$ and $u(b_2(t),t)-v(b_2(t)-\xi,t)<0$, $u(x,t)$ intersects $v(x-\xi,t)$ only once in $[a_1(t),b_2(t)]$, $t_6<t<t_6+\epsilon$. Consequently, $\Sigma_{\xi}(t)$ intersects $\Gamma(t)$ only once, $t_6<t<t_6+\epsilon$. So by Theorem \[thm:sl\] we have $\Sigma_{\xi}(t)$ intersects $\Gamma(t)$ only once, $t>t_6$.
The other conditions can be proved similarly as the three cases above. We see that the intersection number increases only in Case 3.
Then we can conclude that
1\. if $a_1(0)<a_2(0)$, $\Sigma_{\xi}(t)$ does not intersect $\Gamma(t)$.
2\. if $a_2(0)<a_1(0)<b_2(0)$, $\Sigma_{\xi}(t)$ intersects $\Gamma(t)$ at most once.
3\. if $b_2(0)<a_1(0)$, $\Sigma_{\xi}(t)$ intersects $\Gamma(t)$ at most once.(Only in this case, the intersection number may increase)
We complete the proof.
\[rem:intersection1\] The intersection number between two closed, compact, rotationally symmetric hypersurfaces $\Gamma_1(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=u_1(x,t),a_1(t)\leq x\leq b_1(t)\}$, $\Gamma_2(t)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=u_2(x,t),a_2(t)\leq x\leq b_2(t)\}$ is denoted by $\mathcal{Z}(t):=\mathcal{Z}[\Gamma_1(t),\Gamma_2(t)]$. If $\Gamma_i(t)$ evolve by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$, seeing Theorem \[thm:sl\] and the proof in Lemma \[lem:inters\], we can similarly prove\
(a) $\mathcal{Z}(t)$ does not increase when $t$ satisfies $\mathcal{Z}(\Gamma_1(t),\Gamma_2(t))>0$.\
(b) If $\mathcal{Z}(t_0)=0$, then $\mathcal{Z}(t)\leq 1$, $t_0<t<T$.
Observing the proof of Case 3 in Lemma \[lem:inters\], it also holds that the intersection number will possibly increase once only for $a_1(0)>b_2(0)$ or $a_2(0)>b_1(0)$ in this remark. The results in this remark can be proved similarly as Lemma \[lem:inters\].
Observing the opinion in this remark similarly, since $\mathcal{Z}(0)\leq1$ in Lemma \[lem:inters\], there holds $\mathcal{Z}(t)\leq 1$ for $0<t<T$.
Using the intersection argument, we can prove the following theorem.
\[thm:gu\] Let $\Gamma(t)$, $t\in [0,T)$, be a family of smooth hypersurfaces evolving by $V=-\kappa+A$ in $\mathbb{R}^{n+1}$. If $\Gamma(0)$ is obtained by rotating the graph of a function around the $x$-axis, then so are the $\Gamma(t)$ for $t\in[0,T).$
For the proof of Theorem \[thm:gu\], we see that $\Gamma(t)$ is also rotationally symmetric because the equation is rotationally invariance. Since $\Gamma(0)$ is obtained by rotating the graph of a function around the $x$-axis, $\Gamma(0)$ can be written into $\Gamma(0)=\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=v_0(x)\}$ for some function $v_0(x)$. It means that all straight vertical line $x=c$ intersects $\Gamma(0)$ at most once. Using the same argument in Lemma \[lem:inters\], all $x=c$ intersects $\Gamma(t)$ at most once. Then $\Gamma(t)$ can be written into $\{(x,y)\in \mathbb{R}\times\mathbb{R}^n\mid r=u(x,t)\}$. We omit the details.
For the following argument, we only consider the results in $\mathbb{R}^2$. In our problem, the curve evolving by $V=-\kappa+A$ maybe intersect itself at $r=0$. To conquer this difficulty, we give the definition of the $\alpha$-domain first used by [@AAG].
\[def:alphad\] We say a domain is an $\alpha$-domain if
(1). Let $U\subset \mathbb{R}^{2}$ be an open set of the form $$U=\{(x,y)\in\mathbb{R}^2\mid r<u(x)\}.$$
(2). $I=\{x\in\mathbb{R}\mid u(x)>0 \}$ is a bounded, connected interval. Then there exist $a_1<a_2$ such that $\partial I=\{a_1,a_2\}$.
(3). $u$ is smooth on $I$;
(4). $\partial U$ intersects each cylinder $\partial C_{\rho}$ with $0<\rho\leq\alpha$ twice and these intersections are transverse, where $C_{\rho}=\{(x,y)\in\mathbb{R}^2\mid r<\rho\}$.
![$\alpha$-domain[]{data-label="fig:alphadom"}](alphadomain.pdf){height="5cm"}
From Figure \[fig:alphadom\], we observe that the boundary $\partial U$ of an $\alpha$-domain $U$ does not intersect itself at $y=0$. The condition (3) implies $\partial U$ is a smooth curve, except possibly at its endpoints $(a_1,0),(a_2,0)$. The condition (4) implies that there exist $\delta_1,\delta_2>0$ such that $$u(a_1+\delta_1)=u(a_2-\delta_2)=\alpha,$$ and $$u^{\prime}(x)=\left\{
\begin{array}{lcl}
>0,\ x\in(a_1,a_1+\delta_1],\\
<0,\ x\in[a_2-\delta_2,a_2).
\end{array}
\right.$$ Therefore, the inverse of $u|_{[a_1,a_1+\delta_1]}$ and $u|_{[a_2-\delta_2,a_2]}$ exist, denoted by $v_1$, $v_2:[0,\alpha]\rightarrow\mathbb{R}$. By the implicity theorem, they are smooth in $(0,\alpha]$. Moreover, $v_1^{\prime}(r)>0$, $v_2^{\prime}(r)<0$, $(0<r\leq\alpha)$ and $$\partial U\cap C_{\alpha}=\{(x,y)\in\mathbb{R}^2\mid0\leq r\leq\alpha,\ x=v_i(r),\ i=1,2\}.$$ The two components of $\partial U\cap C_{\alpha}$ are called the left and right caps of $\partial U$.
\[lem:alphad2\] Let $U$ be an $\alpha$-domain. Then there exists $t_U>0$ such that $D(t)$ denoted the open evolution with $D(0)=U$ is an $(\alpha+At)$-domain, $0<t<t_U$.
For proving the Lemma \[lem:alphad2\], we need the following lemma.
\[lem:in\] Let $(u,a,b)$ be the solution of $$\label{eq:eq1}
u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2}, \ x\in(a(t),b(t)), \ 0<t< T,$$ $$\label{eq:eq2}
u(a(t),t)=0,\ u(b(t),t)=0,\ 0\leq t< T,$$ $$\label{eq:eq3}
u_x(a(t),t)=+\infty,\ u_x(b(t),t)=-\infty,\ 0\leq t< T,$$ $$\label{eq:eq4}
u(x,0)=u_0(x),\ a(0)\leq x\leq b(0),$$ where $u_0\in C[a(0),b(0)]\cap C^1(a(0),b(0))$.
We denote $\gamma_1(t)$ consisting of the following three parts, $\gamma_{11}(t)=\{(x,y)\in \mathbb{R}^2\mid x=a(t), y<0\}$, $\gamma_{12}(t)=\{(x,y)\in \mathbb{R}^2\mid x=b(t), y<0\}$ and $\gamma_{13}(t)=\{(x,y)\in \mathbb{R}^2\mid y=u(x,t), a(t)\leq x\leq b(t)\}$. For all $C\in \mathbb{R}$, denote $\gamma_2(t)=\{(x,y)\in \mathbb{R}^2\mid y=C+At\}$.
Then the intersection number $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is not increasing in $t\in [0,T)$.
It is sufficient to show for $t_1\in(0,T)$, there exists $\epsilon>0$ such that $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is not increasing on $(t_1-\epsilon,t_1+\epsilon)$. For convenience, we denote $a(t_1)=x_1$ and $b(t_1)=x_2$.
Next we will prove this result by three cases separately.
First, if $C+At_1\leq0$. Since $u_x(x_1,t_1)=+\infty$ ,$u_x(x_2,t_1)=-\infty$ and $u(x_1,t_1)=u(x_2,t_1)=0$, there exist $\epsilon$, $\delta>0$ such that $$\label{eq:481interlemma}
u(x_1+\delta,t)>C+At,\ u(x_2-\delta,t)>C+At,\ t_1-\epsilon\leq t< t_1+\epsilon,$$ $$\label{eq:482interlemma}
u_x(x,t)>0,\ x\in(a(t),x_1+\delta),\ u_x(x,t)<0,\ x\in(x_2-\delta,b(t)),\ t_1-\epsilon\leq t< t_1+\epsilon,$$ and $$(x_1+\delta,x_2-\delta)\subset(a(t),b(t)),\ t_1-\epsilon\leq t< t_1+\epsilon.$$
**Case 1.** $C+At_1<0$.
Let $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]=h(t)+\mathcal{Z}[\gamma_{13}(t),\gamma_2(t)]$, where $h(t)$ is denoted as the intersection number between $\gamma_{2}(t)$ and half lines $\gamma_{11}(t)$, $\gamma_{12}(t)$.
In this condition, there exists $\epsilon$ such that $C+A(t_1+\epsilon)<0$. Then there holds $h(t)\equiv2$, $t_1-\epsilon\leq t< t_1+\epsilon$. (\[eq:481interlemma\]) and (\[eq:482interlemma\]) imply $\mathcal{Z}[\gamma_{13}(t),\gamma_2(t)]=\mathcal{Z}_{[x_1+\delta,x_2-\delta]}[\gamma_{13}(t),\gamma_2(t)]$, for $t\in(t_1-\epsilon,t_1+\epsilon)$. Where $\mathcal{Z}_{I}[\Gamma_{1}(t),\Gamma_2(t)]$ denotes the intersection number between $\Gamma_1(t)$ and $\Gamma_2(t)$ in set $I$. By (\[eq:481interlemma\]) again and Theorem D in [@A1], $\mathcal{Z}_{[x_1+\delta,x_2-\delta]}[\gamma_{13}(t),\gamma_2(t)]$ does not increase for $t_1-\epsilon\leq t< t_1+\epsilon$.
Therefore $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is non-increasing for $t\in(t_1-\epsilon,t_1+\epsilon)$.
**Case 2.** $C+At_1=0$.
For $t_1\leq t< t_1+\epsilon$, obviously, $h(t)=0$. And seeing $u(a(t),t)=u(b(t),t)=0$, (\[eq:481interlemma\]) and (\[eq:482interlemma\]), $C+At$ intersects $u(x,t)$ exactly twice in $[a(t),x_1+\delta)\cup(x_2-\delta,b(t)]$. Then $\mathcal{Z}_{[a(t),x_1+\delta)\cup(x_2-\delta,b(t)]}[\gamma_{13}(t),\gamma_2(t)]=2$, $t_1\leq t< t_1+\epsilon$. Therefore, $$h(t)+\mathcal{Z}_{[a(t),x_1+\delta)\cup(x_2-\delta,b(t)]}[\gamma_{13}(t),\gamma_2(t)]=2,\ t_1\leq t< t_1+\epsilon.$$
For $t_1-\epsilon< t\leq t_1$, obviously, $C+At<0$, then there holds $h(t)\equiv2$. (\[eq:481interlemma\]) and (\[eq:482interlemma\]) imply that $\mathcal{Z}_{[a(t),x_1+\delta)\cup(x_2-\delta,b(t)]}[\gamma_{13}(t),\gamma_2(t)]=0$, $t_1-\epsilon< t\leq t_1$. Then we have $$h(t)+\mathcal{Z}_{[a(t),x_1+\delta)\cup(x_2-\delta,b(t)]}[\gamma_{13}(t),\gamma_2(t)]=2,\ t_1-\epsilon< t< t_1+\epsilon.$$
On the other hand, by (\[eq:481interlemma\]) and Theorem D in [@A1], there holds $\mathcal{Z}_{[x_1+\delta, x_2-\delta]}[\gamma_{13}(t),\gamma_2(t)]$ is not increasing, $t_1-\epsilon<t< t_1+\epsilon$.
Therefore $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is non-increasing for $t\in(t_1-\epsilon,t_1+\epsilon)$.
![Proof of the case 3 in Lemma \[lem:in\][]{data-label="fig:lemalphadom2"}](lemma49.pdf){height="5cm"}
**Case 3.** $C+At_1>0$.
In this case, there exists $\epsilon$ such that $C+A(t_1-\epsilon)>0$. Then there hold $$h(t)\equiv0$$ and $$C+At>u(a(t),t)=u(b(t),t)=0,$$ $t\in(t_1-\epsilon,t_1+\epsilon)$. So by Theorem D in [@A1], $\mathcal{Z}[\gamma_{13}(t),\gamma_2(t)]$ is non-increasing and finite for $t\in(t_1-\epsilon,t_1+\epsilon)$. Consequently, $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is finite and non-increasing in $t\in(t_1-\epsilon,t_1+\epsilon)$.
Since $ U$ is an $\alpha$-domain, using Theorem \[thm:partialUmeancurvature\], $\partial D(t)=\{(x,y)\in\mathbb{R}^2\mid |y|=u(x,t), a(t)\leq x\leq b(t)\}$, where $(u,a,b)$ satisfies (\[eq:eq1\]), (\[eq:eq2\]), (\[eq:eq3\]). Moreover, there exists a maximal time $T_U>0$ such that $\partial D(t)$ is smooth, $0<t<T_U$.
Since $U$ is not contained in the cylinder $\overline{C_{\alpha}}$, there exists a small ball $B_{\epsilon}(P)\subset U\setminus\overline{C_{\alpha}}$. By (1) in Theorem \[thm:order\], $D(t)$ contains the ball $B_{\epsilon(t)}(P)$ for $0<t<\delta_1$. Where $\epsilon(t)$ satisfies $$\label{eq:ball2}
\epsilon^{\prime}(t)=A-\frac{1}{\epsilon(t)},\ 0<t<\delta,$$ with $\epsilon(0)=\epsilon$. Since $B_{\epsilon}(P)\cap\overline{C_{\alpha}}=\phi$, by (1b) in Theorem \[thm:conti\], there exists $t_1>0$, such that $B_{\epsilon(t)}(P)\cap \overline{C_{\alpha+At}}=\phi$, $0<t<t_1$.
For $0<\rho<\alpha+At_0$, $0<t_0\leq t_{U}^{\alpha}$, where $t_{U}^{\alpha}=\min\{\delta_1,t_1\}$, $y=\rho-At_0$ intersects $\gamma_1(0)$ exactly twice($\gamma_1(t)$ is constructed in Lemma \[lem:in\]). Lemma \[lem:in\] implies that $ y=\rho$ intersects $\gamma_1(t_0)$ at most twice. Consequently, $y=\rho$ intersects $y=u(x,t_0)$ at most twice, for $0<t_0< \min\{t_{U}^{\alpha},T_U\}$. On the other hand, there holds $B_{\epsilon(t_0)}(P)\subset D(t_0)\setminus\overline{C_{\alpha+At_0}}$, $0<t_0< \min\{t_{U}^{\alpha},T_U\}$, then $y=\rho$ intersects $y=u(x,t_0)$ at least twice, $0<t_0< \min\{t_{U}^{\alpha},T_U\}$. Therefore we have $\partial C_{\rho}$ intersects $\partial D(t_0)$ exactly twice, $0<t_0<\min\{t_{U}^{\alpha},T_U\}$.
Choosing $t_U=\min\{t_U^{\alpha},T_U\}$, $D(t)$ is an $(\alpha+At)$-domain, $0<t<t_U$. The proof is completed.
![Proof of Lemma \[lem:alphad2\][]{data-label="fig:4712"}](lemma4721.pdf){width="7cm"}
![Proof of Lemma \[lem:alphad2\][]{data-label="fig:4712"}](lemma4722.pdf){width="7cm"}
\[pro:sin\] For $t_U^{\alpha}$ and $T_U$ given in the proof Lemma \[lem:alphad2\], $t_U^{\alpha}\leq T_U$.
To prove the previous proposition we need the following lemma.
\[lem:sing1\] Assume that $D(t)=\{(x,y)\mid |y|<u(x,t), a(t)\leq x\leq b(t)\}$ is a $\rho$-domain, $0<t<T$. Let $w_1<w_2$ such that $$C_\rho\cap\partial D(t)=\{(x,y)\mid x=w_1(y,t),\ x=w_2(y,t)\}.$$ Then $$\lim\limits_{t\rightarrow T}w_1(y,t)=w_1(y,T)\ \ \ \ \ \textrm{and} \ \ \ \ \lim\limits_{t\rightarrow T}w_2(y,t)=w_2(y,T)$$ exist and these convergences are uniform for $|y|\leq\frac{\rho}{2}$. Moreover, $a(T)=:v_1(0,T)<v_1(r,T)$ and $b(T)=:v_2(0,T)>v_2(r,T)$, $0<r<\frac{\rho}{2}$, where $v_1(r,t)=w_1(y,t)$ and $v_2(r,t)=w_2(y,t)$.
$w_1(y,t)$ and $w_2(y,t)$ satisfy the equation (\[eq:graph\]), respectively for “$\mp$”. We only prove for $w_1(y,t)$. Since $w_1$ is uniformly bounded, Corollary \[cor:es\] and Remark \[rem:hes\] imply that derivatives $\frac{\partial ^j}{\partial y^j} w_1$, $j=1,2$, are uniformly bounded for $0\leq|y|\leq\frac{\rho}{2}$, $\frac{T}{2}\leq t<T$. Consequently, $\frac{\partial w_{1}}{\partial t}$ is bounded for $0\leq|y|\leq\frac{\rho}{2}$, $\frac{T}{2}\leq t<T$. So there exists $w_1(y,T)$ such that $w_1(y,t)$ converges to $w_1(y,T)$ uniformly for $0\leq|y|\leq\frac{\rho}{2}$, $\frac{T}{2}\leq t<T$.
Note there hold $$\frac{\partial v_1}{\partial r}(\frac{\rho}{2},t)>0,\ \frac{\partial v_1}{\partial r}(0,t)=0,\ 0<t<T$$ and $$\frac{\partial v_1}{\partial r}(r,0)>0,\ 0< r<\frac{\rho}{2}.$$ Since $p=\frac{\partial v_1}{\partial r}$ satisfies (\[eq:deriveeq\]), maximum principle implies $$\frac{\partial v_1}{\partial r}>0,\ 0<r<\frac{\rho}{2},\ 0<t\leq T.$$ Therefore $v_1(0,T)<v_1(r,T)$, for $0<r<\frac{\rho}{2}$.
If $T_U<t_U^{\alpha}$. By Lemma \[lem:alphad2\], there exists $\rho>0$ such that $D(t)$ is a $\rho$-domain, for $0<t<T_U$.
We divide $\partial D(t)$ into two parts: $\partial D(t)=(\partial D(t)\cap\{r< \rho/2\})\cup(\partial D(t)\cap\{r\geq \rho/2\})$.\
[**Step 1.**]{} $\partial D(t)\cap\{r< \rho/2\}$
Since $\partial D(t)$ is a $\rho$-domain, there exist $w_1<w_2$ such that $\partial D(t)\cap\{r<\rho\}=\{(x,y)\mid x=w_1(y,t), |y|<\rho\}\cup\{(x,y)\mid x=w_2(y,t), |y|<\rho\}$. By the same argument as in Lemma \[lem:sing1\], $\frac{\partial ^j}{\partial y^j} w_i$, $j=1,2$, $i=1,2$, are uniformly bounded for $0\leq|y|\leq\frac{\rho}{2}$, $\frac{T_U}{2}\leq t<T_U$. Therefore, the mean curvature of $\partial D(t)\cap\{r< \rho/2\}$ is bounded for $\frac{T_U}{2}\leq t<T_U$.\
[**Step 2.**]{} $\partial D(t)\cap\{r\geq\rho/2\}$
Recalling $\partial D(t)=\{(x,y)\mid |y|=u(x,t),a(t)\leq x\leq b(t)\}$, by Lemma \[lem:sing1\], there hold $a(T_U)<v_1(\rho/2,T_U)$ and $b(T_U)>v_2(\rho/2,T_U)$. Then for any $\epsilon$ small enough, $t$ is close to $T_U$ such that $$\label{eq:subset1}
(v_1(\rho/2,t),v_2(\rho/2,t))\subset (a(T_U)+\epsilon,b(T_U)-\epsilon).$$ Corollary \[cor:es\] and Remark \[rem:hes\] imply that $u_x$ and $u_{xx}$ are uniformly bounded for $x\in(a(T_U)+\epsilon,b(T_U)-\epsilon)$, $t$ close to $T_U$. (\[eq:subset1\]) implies that $u_x$ and $u_{xx}$ are uniformly bounded for $x\in(v_1(\rho/2,t),v_2(\rho/2,t))$, $t$ close to $T_U$. The curvature of $\partial D(t)\cap\{r\geq\rho/2\}$ is bounded for $t$ close to $T_U$.
Consequently, the curvature of $\partial D(t)$ is uniformly bounded as $t\uparrow T_U$. It contradicts to $\partial D(t)$ becoming singular at $T_U$.
\[rem:time\] In Lemma \[lem:alphad2\], $0<t<\min\{t_{U}^{\alpha},T_U\}$ can be replaced by $0<t<t_{U}^{\alpha}$. Seeing the choice of $t_{U}^{\alpha}$, if $U\subset W$, $t_{U}^{\alpha}\leq t_{W}^{\alpha}$.
[**Intersection number principle**]{} Lemma \[lem:inters\] and Lemma \[lem:alphad2\] show the possible intersection number between two curves evolving by $V=-\kappa+A$. Here we want to introduce a more general result about the intersection number. Consider the following problem which we call (Q): $$\left\{
\begin{array}{lcl}
\dis{u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2}},\ x\in(a(t),b(t)),\ 0<t< T,\\
u(a(t),t)=0,\ u(b(t),t)=0,\ 0\leq t< T,\\
u_x(a(t),t)=\tan\theta_-(t),\ u_x(b(t),t)=-\tan\theta_+(t),\ 0\leq t< T,\\
u(x,0)=u_0(x),\ a(0)\leq x\leq b(0),
\end{array}
\right.\tag{Q}$$ where $u_0\in C[a(0),b(0)]\cap C^1(a(0),b(0))$ and $\theta_{\pm}(t)$ are smooth functions with values in $[0,\pi/2]$. Let $$\gamma_1(t):=\left\{
\begin{array}{lcl}
\{(x,y)\mid y=\tan\theta_-(t)(x-a(t)),y<0\},\ \theta_-(t)<\pi/2\\
\{(x,y)\mid x=a(t),y<0\},\theta_-(t)=\pi/2,
\end{array}
\right.$$ $$\gamma_2(t):=\left\{
\begin{array}{lcl}
\{(x,y)\mid y=-\tan\theta_+(t)(x-b(t)),y<0\},\ \theta_+(t)<\pi/2\\
\{(x,y)\mid x=b(t),y<0\},\theta_+(t)=\pi/2,
\end{array}
\right.$$ and $$\gamma_3(t):=\{(x,y)\mid y=u(x,t),a(t)\leq x\leq b(t)\}.$$ The extension curve of $u(\cdot,t)$ is given by $$\gamma(t):=\gamma_1(t)\cup\gamma_2(t)\cup\gamma_3(t).$$
\[pro:intersection\] Let $u^{1}(x,t)$, $a^{1}(t)<x<b^{1}(t)$ be solution of (Q) for $\theta_{\pm}^1(t)\in[0,\pi/2)$, and $u^2(x,t)$, $a^{2}(t)<x<b^{2}(t)$ be solution of (Q) for $\theta_{\pm}^2(t)=\pi/2$, for $0\leq t<T$. Let $\gamma_i(t)$ be the extension curve of $u^i(x,t)$, respectively. Then $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ is non-increasing in $t\in[0,T)$ and is finite for each $t\in[0,T)$. Moreover, $\mathcal{Z}[\gamma_1(t),\gamma_2(t)]$ will drop when $\gamma_1(t)$ intersects $\gamma_2(t)$ tangentially.
For the proof of this proposition, it is similar as the proof of Lemma \[lem:in\] above or Proposition 2.4 in [@GMSW]. Here we omit it.
\[rem:1matanointersection\] (1). Proposition 2.4 in [@GMSW] only give the results under $\theta_{\pm}^i\in(0,\pi/2)$, $i=1,2$.
(2). For $\theta_{\pm}^i=\pi/2$, $i=1,2$, the results in Proposition \[pro:intersection\] are not true. Indeed, this condition is same as in Remark \[rem:intersection1\]\
(a). If $\mathcal{Z}(u^1(\cdot,0),u^2(\cdot,0))>0$, $\mathcal{Z}(u^1(\cdot,t),u^2(\cdot,t))$ will not increase for $0<t<T$ provided that $\mathcal{Z}(u^1(\cdot,t),u^2(\cdot,t))>0$, $0<t<T$.\
(b). If $\mathcal{Z}(u^1(\cdot,0),u^2(\cdot,0))=0$, $\mathcal{Z}(u^1(\cdot,t),u^2(\cdot,t))\leq 1$, $0<t<T$.
Proof of Theorem \[thm:exist\] and Theorem \[thm:fattening1\]
=============================================================
Denote $U=\{(x,y)\in \mathbb{R}^2\mid |y|<u_0(x),-b_0\leq x\leq b_0\}$, where $u_0$ is given by (\[eq:ineven\]). By assumption of $u_0$ in Section 1, we know that $U\cap\{x\geq0\}$ is an $\alpha$-domain with smooth boundary, for some $\alpha>0$.
We choose vector field $X\in C^1(\mathbb{R}^{2}\setminus\{(0,0)\}\rightarrow\mathbb{R}^{2})$ such that
\(i) At any $P\in \partial U$ not on the $x$-axis has $\langle X,\textbf{n}(P)\rangle<0$, $\textbf{n}$ is inward unit normal vector at $P$.
\(ii) Near the $(0,0)$, we set $X((x,y))=(0,-y/|y|)$ and set $X=(-1,0)$ near $(b,0)$, $X=(1,0)$ near $(-b,0)$.\
We note that $X$ has no definition at $(0,0)$.
Since $X\neq0 $ on $\partial U\setminus \{(0,0)\}$, there exists a neighbourhood $V\supset\partial U$ such that $|X|\geq \delta>0$ for some $\delta>0$ in $V\setminus \{(0,0)\}$.
\[pro:sigma2\] For $\rho$ small enough, there exists a smooth curve $\Sigma\subset V\setminus\{(0,0)\}$ with
\(i) $X(P)\notin T_P\Sigma$ at all $P\in\Sigma$,i.e., $\Sigma$ is transverse to the vector field $X$;
\(ii) $\Sigma=\partial U$ in $\{(x,y)\mid|y|\geq2\rho\}$;
\(iii) $\Sigma\cap\{(x,y)\mid|y|\leq\rho\}$ consists of discs $\Delta_{\pm c}=\{(\pm c,y)\mid|y|\leq\rho\}$ and pipe $B_d=\{(x,y)\mid-d \leq x\leq d,|y|=\rho\}$.
![Proof of Proposition \[pro:sigma2\][]{data-label="fig:sigma2"}](proposition55.pdf){height="5cm"}
Because $U\cap\{x\geq0\}$ is an $\alpha$-domain, there exist $\delta_j$, $\gamma_j$ and $0<\delta_j<\gamma_j$ such that $$u_0(\delta_j)=u_0(\gamma_j)=u_0(-\delta_j)=u_0(-\gamma_j)=\frac{\alpha}{2^j}$$ and $$\partial U\cap C_{\alpha}=\{(x,y)\mid x=\pm v(y), |y|<\alpha\}\cup\{(x,y)\mid x=\pm w(y),|y|<\alpha\},$$ where $v,w\in C^{\infty}((-\alpha,\alpha))$ and $0<v(y)<w(y)$ for $|y|<\alpha$.
We let $w_j\in C^{\infty}((-\alpha/2^{j-1},\alpha/2^{j-1}))$ be defined as following $$w_j(y)=\left\{
\begin{array}{lcl}
\gamma_{j+2},\ 0\leq\dis{|y|<\frac{\alpha}{2^{j+1}}}\\
w(y),\ \dis{\frac{\alpha}{2^{j}}<|y|<\frac{\alpha}{2^{j-1}}},
\end{array}
\right..$$
And $u_j\in C^{\infty}((-\delta_{j-1},\delta_{j-1}))$ is defined as following $$u_j(x)=\left\{
\begin{array}{lcl}
\dis{\frac{\alpha}{2^{j+1}}},\ x\in[0,\delta_{j+2}]\\
u_0(x),\ x\in[\delta_j,\delta_{j-1})
\end{array}
\right..$$
Let $\Sigma_j$ consist of three parts: $\{(x,y)\mid |y|=u_j(x),\ x\in(-\delta_j,\delta_j)\}$, $\{(x,y)\mid x=\pm w_j(y), |y|<\alpha/2^j\}$ and $\partial U\cap \{|y|\geq \alpha/2^{j}\}$. It is easy to see that for $j$ sufficient large, $\Sigma_j\subset V\setminus \{(0,0)\}$ satisfies (i), (ii), (iii) for $c=\gamma_{j+2}$, $\rho=\alpha/2^{j+1}$ and $d=\delta_{j+2}$.
Denote $\sigma(P,t):\Sigma\times(-\delta,\delta)\rightarrow V$($V$ is given at the begining of this section and $\Sigma$ is given by Proposition \[pro:sigma2\]) the flow generated by vector field $X$ in $\mathbb{R}^{2}$. Precisely, $\sigma(P,t)$ is defined as following: $$\left\{
\begin{array}{lcl}
\dis{\frac{d\sigma(P,t)}{dt}=X(\sigma(P,t))},\ P\in \Sigma,\\
\sigma(P,0)=P,\ \ \ \ \ P\in \Sigma.
\end{array}
\right.$$
Seeing (i) in Proposition \[pro:sigma2\], for any $C^{1}$ function $u:\Sigma\rightarrow\mathbb{R}$, “the image of $u$ under $\sigma$”—$\{\sigma(P,u(P))\mid P\in \Sigma\}$ is a $C^1$ curve. Conversely, for any curve $\Gamma\subset V$ being $C^1$ close to $\Sigma$, there exists a unique $C^1$ function $u:\Sigma\rightarrow\mathbb{R}$ such that $\Gamma=\{\sigma(P,u(P))\mid P\in \Sigma\}$. In other words, the map $\sigma(\cdot,t)$ defines a new coordinate from $\Sigma$ to $V$. Therefore, if $\Gamma(t)\subset V$$(0<t<T)$ is a smooth family of smooth curves and $C^1$ close to $\Sigma$, there exists a unique function $u \in C^\infty(\Sigma\times(0,t))$ such that $\Gamma(t)=\{\sigma(P,u(P,t))\mid P\in\Sigma\}$. Let $z$ be the local coordinate on an open subset of $\Sigma$. If $\Gamma(t)$ evolves by $V=-\kappa+A$, in this coordinate $u$ satisfies the following equation $$\label{eq:para1}
\frac{\partial u}{\partial t}=a(z,u,u_z)\frac{\partial^2u}{\partial z^2}+b(z,u,u_z).$$
Here $a$, $b$ are smooth functions of their arguments [@A2](Section 3). $a$ is always positive so that (\[eq:para1\]) is a parabolic equation.
For example, $\sigma(\cdot,t)$ is the flow defined as above. We can easily deduce that $$\sigma(P,t)=\left\{
\begin{array}{lcl}
(x,\rho-t),\ P\in B_d,\\
(-c+t,y),\ P\in \Delta_{-c},\\
(c-t,y),\ P\in \Delta_{c},
\end{array}
\right.$$ where we choose the local coordinates:\
(1). on $B_d$, $(x,\rho y)$ for $|y|=1$;\
(2). on $\Delta_{\pm c}$, $(\pm c,y)$.
Since $y=\pm1$ on $B_d$, $u$ only depends on $x$. Therefore on $B_d$, $a(x,u,u_x)=1/(1+u_x^2)$ and $b(x,u,u_x)=-A\sqrt{1+u_x^2}$. Then $u$ satisfies $$\label{eq:2dimhorieq}
u_t=\frac{u_{xx}}{1+u_x^2}-A\sqrt{1+u_x^2}.$$ On $\Delta_{\pm c}$, $u$ only depends on $y$. Then on $\Delta_{\pm c}$, $a(y,u,u_y)=1/(1+u_y^2)$, $b(y,u,u_y)=-A\sqrt{1+u_y^2}$. Therefore $u$ satisfies (\[eq:graph\]) for “$-$” and $n=1$.
In $\mathbb{R}^{n+1}$, $b$ obtained above may not be smooth. For example, $$u_t=\frac{u_{xx}}{1+u_x^2}+\frac{n-1}{\rho-u}-A\sqrt{1+u_x^2},$$ on $\{(x,y)\in\mathbb{R}\times\mathbb{R}^n\mid |y|=\rho, -d<x<d\}$. In this case, $b=\frac{n-1}{\rho-u}-A\sqrt{1+u_x^2}$. It is easy to see when $u=\rho$, $b$ is not smooth. This is the most different between 2-dimension and higher dimension.
\[lem:max\] For $v(x,t)$ being smooth function on $V\times(0,T)$, where $V$ is a compact set, we denote $m(t)$ as $$m(t)=\max\{v(x,t)\mid x\in V\}.$$ Then there exists $P_t\in V$ such that $v(P_t,t)=m(t)$ and $m^{\prime}(t)=v_t(P_t,t)$ for $t>0$.
It is a well known result. For example, the result can be found in [@M].
\[pro:uniq2\] Let $\Gamma_1$, $\Gamma_2$ be two families of curves with $\sigma^{-1}(\Gamma_j)$ the graph of $u_j(\cdot,t)$ for certain $u_j\in C(\Sigma\times[0,T))$. Assume $u_j$ are smooth on $\Sigma\times(0,T)$ and smooth on $\Sigma\setminus(\Delta_{\pm c}\cup B_d)\times[0,T)$. If $\Gamma_1(0)=\Gamma_2(0)$, there holds $\Gamma_1(t)=\Gamma_2(t)$, $0\leq t<T$.
Consider $v(P,t)=u_1(P,t)-u_2(P,t)$. From our assumptions, we have $v\in C(\Sigma\times[0,T))$ and that $v$ is smooth on $\Sigma\setminus(\Delta_{\pm c}\cup B_d)\times[0,T)$, as well as on $\Sigma\times(0,T)$. Moreover $v(P,0)\equiv0$. Define $m(t)=\max\{v(P,t)\mid P\in \Sigma\}$ and for each $0\leq t<T$ with $m(t)>0$. We want to show that $m^{\prime}(t)\leq Cm(t)$ for some constant $C$. Choose $P_t$ as in Lemma \[lem:max\] such that $m(t)=v(P_t,t)$ and $m^{\prime}(t)=v_t(P_t,t)$.
**Case 1.** $P_t\in B_d$, since $u_j$ satisfy the equation (\[eq:2dimhorieq\]), $v$ satisfies a parabolic equation $$v_t=a_1(x,t)v_{xx}+b_1(x,t)v_x,$$ where $a_1(x,t)$ and $b_1(x,t)$ is smooth, and $a_1(x,t)>0$. Since $v$ attains its maximum at $P_t$, $v_x(P_t,t)=0$ and $v_{xx}(P_t,t)\leq0$. Then $v_t(P_t,t)\leq0$. Considering Lemma \[lem:max\], $m^{\prime}(t)\leq0$.
**Case 2.** $P_t\in \Delta_{\pm c}$. We only consider $P_t\in \Delta_{-c}$. Then in the $y$-coordinates of $\Delta_{-c}$, $u_j$ satisfy the full graph equation, which is (\[eq:graph\]) for “$-$” and $n=1$. Therefore $v=u_1-u_2$ satisfies a parabolic equation $$v_t=a_2(y,t)v_{yy}+b_2(y,t)v_{y}.$$ Seeing $v_y(P_t,t)=0$ and $v_{yy}(P_t,t)\leq 0$, $m^{\prime}(t)\leq0$.
**Case 3.** $P_t\in \Sigma\setminus(\Delta_{\pm c}\cup B_d)$. Then we can choose coordinate $z$ on some neighbourhood of $P_t$ on $\Sigma$ and $u_j$ satisfy (\[eq:para\]). We may write this equation as $u_t=F(z,t,u, u_z, u_{zz})$. Then $v=u_1-u_2$ satisfies $$v_t=a_3(z,t)v_{zz}+b_3(z,t)v_{z}+c_3(z,t)v,$$ where $$c_3(z,t)=\int_{0}^1F_u(z,t,u^{\theta}, u_z^{\theta},u_{zz}^{\theta})d\theta,$$ where $u^{\theta}=(1-\theta)u_2+\theta u_1$.
By the assumption, outside of the disks $\Delta_{\pm c}$ and the pipe $B_d$, $u_i$ are smooth up to $t=0$, so the coefficient $c(z,t)$ is bounded, $0<t<T$, saying by $|c(z,t)|\leq M<\infty$. The constant $M$ may depend on the choice of local coordinate $z$. Noting $\Sigma$ is compact, by easy covering argument, we can choose $M$ independent of the choice of local coordinate. Since $v_z(P_t,t)=0$, $v_{zz}(P_t,t)\leq0$, $$v_t(P_t,t)\leq c(P_t,t)v(P_t,t)\leq Mv(P_t,t).$$ Consequently, $m^{\prime}(t)\leq Mm(t)$.
Combining these three cases, we have $m^{\prime}(t)\leq C m(t)$, for some constant $C>0$. Considering $m(0)=0$, $m(t)\leq 0$. Conversely, we can prove $M(t)=\min\{v(P,t)\mid P\in \Sigma\}\geq0$. Therefore $u_1\equiv u_2$. We complete the proof.
\[lem:closeas\] Then there exist $E_j$ closed such that $E_j^{\circ}$ are $\alpha/2^j$-domains and $E_j\downarrow \overline{U}$. Where $U$ is given at the beginning of the section and $E^{\circ}$ denotes the interior of the set $E$.
Since $U\cap \{x\geq0\}$ is an $\alpha$-domain, there exists unique $\delta_0>0$ such that $$u_0(\delta_0)=\alpha$$ and $$u_0^{\prime}(x)>0,\ 0<x<\delta_0.$$ For all $j\geq1$, there exists unique $\delta_j$, $0<\delta_j<\delta_0$ such that $$u_0(\delta_j)=\alpha/2^j.$$ We can contruct $v_j\in C^{\infty}((-b_0,b_0))$ being even such that $$v_j(x)=\left\{
\begin{array}{lcl}
\alpha/2^j,\ x\in (-\delta_j/2,\delta_j/2),\\
u_0(x), x\in [-b_0,-\delta_j]\cup[\delta_j,b_0],
\end{array}
\right.$$ $v_j(x)\geq u_0$, $x\in[-b_0,b_0]$ and $v^{\prime}_j(x)>0$, $x\in(\delta_j/2,\delta_j)$. It is easy to see $v_j\downarrow u_0$ uniformly in $[-b_0,b_0]$.
Let $E_j=\{(x,y)\mid|y|\leq v_j(x),\ -b_0\leq x\leq b_0\}$. Since $v_j\downarrow u_0$ uniformly in $[-b_0,b_0]$, $E_j\downarrow \overline{U}$. It is easy to check $E_j^{\circ}$ are $\alpha/2^j$-domain.
\[lem:closebou\] Let the same assumption in Theorem \[thm:exist\] be given. Then there exists $t_1>0$ such that, for all $t_2$ satisfying $0<t_2<t_1$, the second fundamental forms and derivatives of $\partial E_j(t)$ are uniformly bounded for $t_2\leq t\leq t_1$, where $E_j(t)$ denote the closed evolution of $V=-\kappa+A$ with $E_j(0)=E_j$.
Let $E_j(t)=\{(x,y)\mid|y|\leq v_j(x,t)\}$.
[**Step 1.**]{} For all $t_2$ satisfying $0<t_2<\delta$($\delta$ given by Theorem \[thm:exist\]), there exists a constant $c>0$ such that $$v_j(0,t)>c,\ t_2/2<t<\delta.$$
Let $U^{+}(t)$ denote the open evolution with $U^+(0)=U\cap\{x\geq0\}$. Using Theorem \[thm:partialUmeancurvature\], $U^{+}(t)$ is the domain surrounded by $\Lambda(t)$($\Lambda(t)$ is defined in Section 1). By (3) in Theorem \[thm:order\] and $U\cap\{x\geq0\}\subset E_j$, there holds $U^{+}(t)\subset E_j(t)$. By our assumption that $a_*(t)<0$, for $0<t\leq\delta$, there holds $(0,0)\in U^{+}(t)\subset E_j(t)$, $0<t<\delta$. For all $t_2$ satisfying $0<t_2<\delta$, there exists $c>0$ such that $v_j(0,t)> c$, $t_2/2\leq t\leq \delta$.
[**Step 2.**]{} Construction of four auxiliary balls.
Since $U\cap\{x\geq0\}$ is an $\alpha$-domain, there exist $\beta_2>\beta_1>0$ such that $u_0(\pm\beta_1)=u_0(\pm\beta_2)=\alpha$ and $u_0^{\prime}(x)<0$ for $x>\beta_2$, $u_0^{\prime}(x)>0$ for $0<x<\beta_1$. There exist $p>\beta_1$ and $0<q<\beta_2$ such that $\dis{u_0(\pm q)=u_0(\pm p)=\frac{\alpha}{2}}$. we consider the points $$Q=(-p,0),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P=(p,0),$$ $$Q^{\prime}=(-p,\alpha),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P^{\prime}=(p,\alpha).$$
![Proof of Lemma \[lem:closebou\][]{data-label="fig:uniformb"}](uniformb1.pdf){height="8cm"}
Since $P\in U$ and $P^{\prime}\in \overline{U}^{c}$, there exists $\epsilon$ such that $\overline{B_{\epsilon}(P)}\subset U$ and $\overline{B_{\epsilon}(P^{\prime})}\subset\overline{U}^c$. Consequently, $\overline{B_{\epsilon}(P)}\cup \overline{B_{\epsilon}(Q)}\subset E^{\circ}$ and $\overline{B_{\epsilon}(P^{\prime})}\cup \overline{B_{\epsilon}(Q^{\prime})}\subset E^{c}$. Then for $j$ large enough, $\overline{B_{\epsilon}(P)}\cup \overline{B_{\epsilon}(Q)}\subset E_j^{\circ}$ and $\overline{B_{\epsilon}(P^{\prime})}\cup \overline{B_{\epsilon}(Q^{\prime})}\subset E_j^{c}$. By (4) in Theorem \[thm:order\], $$\label{eq:q1}
\overline{B_{\epsilon(t)}(P)}\cup \overline{B_{\epsilon(t)}(Q)}\subset E_j(t)^{\circ},$$ $0<t<\delta_2$. By (1b) in Theorem \[thm:conti\], there exists $\delta_3>0$ such that $$\label{eq:q2}
\overline{B_{\epsilon(t)}(P^{\prime})}\cup \overline{B_{\epsilon(t)}(Q^{\prime})}\subset E_j(t)^c,$$ for $0< t<\delta_3$. Where $\epsilon(t)$ is the solution of (\[eq:ball2\]) with $\epsilon(0)=\epsilon$, $0<t<\delta_1$. Choose $\delta_2$ independent on $j$ such that $\epsilon(t)>\epsilon/2$, $0<t<\delta_2$.
![Proof of Lemma \[lem:closebou\][]{data-label="fig:uniformb2"}](uniformb2.pdf){height="7cm"}
[**Step 3.**]{} Divide $\partial E_j(t)$ into two parts by auxiliary balls.
Since for all $\rho<\alpha$, $C_{\rho}$ intersects $\partial E_j$ at most forth, by Proposition \[pro:intersection\], there exists $t_0>0$ such that $C_{\rho}$ intersects $\partial E_j(t)$ at most forth, $0<t<t_0$. By continuity, we can deduce that there exists $\delta_4$ such that for all $\rho<\alpha$, the equation $v_j(x,t)=\rho$ has most one root for $x>p$, for all $t<\delta_4$. By symmetry, it is so for $x<-p$.
Choosing $t_1=\min\{t_0,\delta_2,\delta_3,\delta_4\}$, Step 1 implies that $E_j(t)^{\circ}$ are all $c$-domains, $t_2/2<t<t_1$. Let $d<\min\{c,\epsilon/4\}$. By (\[eq:q1\]) in Step 2, we have $v_j(x,t)>d$, for $t_2/2<t<t_1$, $|x-p|<\sqrt{\epsilon^2(t)-d^2}$ or $|x+p|<\sqrt{\epsilon^2(t)-d^2}$. Seeing $\epsilon(t)>\epsilon/2$, there holds $$v_j(x,t)\geq d,\ \text{in}\ \Omega=(-p-\frac{\sqrt{3}}{4}\epsilon,p+\frac{\sqrt{3}}{4}\epsilon)\times(t_2/2,t_1).$$
For $x\leq-p$, by (\[eq:q2\]) in Step 2, $$v_j(x,t)<\alpha/2-\epsilon(t)<\alpha/2-\epsilon/2,\ x\leq-p,\ 0\leq t<t_1.$$ This is also true for $x\geq p$.
[**Step 4.**]{} The derivatives and second fundamental forms of $\partial E_j(t)$ are bounded in $\Omega^{\prime}=[-p,p]\times(t_2,t_1)$.
Since $v_j(x,t)\geq d$ in $\Omega=(-p-\frac{\sqrt{3}}{4}\epsilon,p+\frac{\sqrt{3}}{4}\epsilon)\times(t_2/2,t_1)$, Theorem \[thm:grad\] implies that $v_{jx}$ are uniformly bounded in $\Omega$. By Remark \[rem:hes\], $v_{jxx}$ are uniformly bounded in $\Omega^{\prime}$.
[**Step 5.**]{} The derivatives and second fundamental forms of $\partial E_j(t)$ are bounded for $x\leq -p$ and $x\geq p$, $t_2<t<t_1$.
We only consider for $x\leq-p$. For $0<t<t_1$ the part of $\partial E_j(t)$ on $x\leq-p$ can be represent by $x=w_j(y,t)$, for $|y|<\alpha/2$, $t\in (0,t_1)$. And $w_j$ satisfy the equation (\[eq:graph\]) in the condition “$-$” and $n=1$. Then Corollary \[cor:es\] and Remark \[rem:hes\] imply that all $\frac{\partial ^k}{\partial y^k}w_j(y,t)$, $k=1,2$, are uniformly bounded for $|y|\leq\alpha/2-\epsilon/2$, $t_2<t<t_1$ for any $t_2>0$. Then the derivatives and second fundamental forms of $\partial E_j(t)$ are uniformly bounded when $x\leq-p$, $t_2<t<t_1$.
The proof of this lemma is completed.
\[lem:opensy\] There exist $U_j$ being open and $U_j\cap\{x\geq0\}$ being an $\alpha$-domain such that $U_j\uparrow U$.
Since $U\cap\{x>0\}$ being $\alpha$-domain, for $j\geq1$, there exist $\delta_j$ satisfying $0<\delta_j<\delta_0$ such that $u_0(\delta_j)=\alpha/2^j$, where $\delta_0$ satisfies $u_0(\pm \delta_0)=\alpha$ and $u_0^{\prime}(x)>0$ for $0<x<\delta_0$. We set $u_j\in C^{\infty}((-b_0,b_0))$ and even satisfying $$u_j(x)=\left\{
\begin{array}{lcl}
0,\ x=0,\\
u_0(x),\ x\in[-b_0,-\delta_j]\cup[\delta_j,b_0],
\end{array}
\right.$$ and $u_j(x)\leq u_0$ for $x\in[-b_0,b_0]$, $u_j^{\prime}(x)>0$ for $x\in(0,\delta_j)$.
Let $U_j=\{(x,y)\mid |y|<u_j(x)\}$. Obviously $u_j\uparrow u_0$, then $U_j\uparrow U$. It is easy to check $U_j\cap\{x>0\}$ are $\alpha$-domain.
\[lem:openbou\] Let the same assumption in Theorem \[thm:exist\] be given. Then there exists $t_1>0$ such that for all $t_2$ satisfying $0<t_2<t_1$, the second fundamental forms and derivatives of $\partial U_j(t)$ is uniform bounded, $t_2<t<t_1$, where $U_j(t)$ is the open evolution of $V=-\kappa+A$ with $U_j(0)=U_j$.
Let $U(t)$ and $U^{+}(t)$ be the open evolution with $U(0)=U$ and $U^{+}(0)=U\cap\{x>0\}$. Seeing appendix, $\Lambda(t)=\partial U^{+}(t)$($\Lambda(t)$ is given in Section 1). Since $a_*(t)<0$, for $0<t<\delta$, $(0,0)\in\partial U^{+}(t)$.
By (1) in Theorem \[thm:order\] and $U\cap\{x>0\}\subset U$, we have $U^{+}(t)\subset U(t)$. Consequently, $(0,0)\in U(t)$, $0<t<\delta$. Then for $j$ large enough, $(0,0)\in U_j(t)$. The following parts can be proved similar as in Lemma \[lem:closebou\].
Seeing Lemma \[lem:closebou\] and \[lem:openbou\], $\partial U(t)$, $\partial E(t)$ are smooth curves and homeomorphic to the curve $\Sigma$ given by Proposition \[pro:sigma2\]. Consequently, $\partial U(t)$, $\partial E(t)$ satisfy the assumption of Proposition \[pro:uniq2\], $0\leq t<T_1$, for some $T_1$ satisfying $0<T_1<t_1$. Where $t_1$ is given by Lemma \[lem:closebou\] and \[lem:openbou\]. Then there holds $\partial U(t)=\partial E(t)$, $0<t<T_1$. If we let $\Gamma(t)=\partial E(t)\cap\{x\geq0\}$, $\Gamma(t)$ will be the unique solution of (\[eq:cur\]), (\[eq:Neum1\]) and (\[eq:initial1\]). The proof of Theorem \[thm:exist\] is completed.
Indeed, $\partial E(t)$ and $\partial U(t)$ are smooth on $\Sigma\setminus B_d\times[0,T_1)$ and $\Sigma\times(0,T_1)$. If we remove the assumption that $\Gamma_0$ is smooth at end point $(-b_0,0)$ and $(b_0,0)$, $\partial E(t)$ and $\partial U(t)$ will be smooth on $\Sigma\setminus( \Delta_{\pm c}\cup B_d)\times[0,T_1)$ and $\Sigma\times(0,T_1)$. Therefore the result is also true even if removing smoothness at end points.
It is sufficient to show that there is a ball $B$ such that $B\subset E(t)\setminus U(t)$, for some $t$.
[**Closed evolution $E(t)$.**]{} Since $E_j^{\circ}$(given by Lemma \[lem:closeas\]) are $\alpha/2^j$-domain with smooth boundary, by Lemma \[lem:alphad2\], there exists a positive time $t_1$, $t_1<\delta$($\delta$ is given in Theorem \[thm:fattening1\]) such that $E_j(t)^{\circ}$ are $(At+\alpha/2^j)$-domain for $0<t<t_1$. Combining $E_j(t)\downarrow E(t)$, we have $E(t)^{\circ}$ is an $At$-domain, $0<t<t_1$. Therefore $E(t)^{\circ}$ is an $At_1/2$-domain, $t_1/2<t<t_1$.
[**Open evolution $U(t)$.**]{} Denote $U^{\pm}(t)$ being the open evolutions with $U^{\pm}(0)=U\cap\{\pm x\geq 0\}$. Seeing appendix $\partial U^{+}(t)=\Lambda(t)$, $\partial U^{-}(t)=\{(-x,y)\mid(x,y)\in \Lambda(t)\}$, where $\Lambda(t)$ is given in Section 1. Thus the left end point of $U^+(t)$ and the right end point of $U^-(t)$ are ($a_*(t)$,0) and ($-a_*(t)$,0), respectively. By the assumption in this theorem $a_*(t)\geq 0$, $0\leq t<\delta$, it means that $-a_*(t)\leq a_*(t)$, $0\leq t<\delta$. Therefore, $U^+(t)\cap U^-(t)=\emptyset$, $0\leq t<\delta$. From Lemma \[lem:sep\], the inner evolution $U(t)$ satisfies $U(t)=U^{+}(t)\cup U^{-}(t)$, for $0\leq t<\delta$.
By (2a) in Theorem \[thm:conti\](the boundary of open evolution evolves continuously) and $a(t)\geq0$, there exists $\dis{\delta_1<\frac{At_1}{4}}$ such that $$\dis{B_{\delta_1}((0,\frac{At_1}{4}))}\cap U(t)=\emptyset,\ \dis{\frac{t_1}{2}<t<t_1}$$ and $$B_{\delta_1}((0,\frac{At_1}{4}))\subset E(t),\ \dis{\frac{t_1}{2}<t<t_1}.$$ Where $\dis{B_{\delta_1}((0,\frac{At_1}{4}))}$ is a ball centered at $(0,At_1/4)$ with radius $\delta_1$. Then $\dis{B_{\delta_1}((0,\frac{At_1}{4}))}\subset \Gamma(t)=E(t)\setminus U(t)$, for $\dis{\frac{t_1}{2}<t<t_1}$.
Formation of sigularity
=======================
In this section, we want to identify the singular formation of $\Gamma(t)$, when $\Gamma(t)$ becomes singular at $t=T<\infty$. Where $$\label{eq:singularT}
T=\sup\{t>0\mid \Gamma(s)\ \text{are}\ \text{smooth},0<s<t\}$$ and $\Gamma(t)$ is given by Theorem \[thm:exist\]. For convenience, we still consider $\Gamma(t)$ extended evenly. By Theorem \[thm:openevolutionmeancurvature\] and Theorem \[thm:gu\], $\Gamma(t)=\{(x,y)\in\mathbb{R}^2\mid|y|=u(x,t),-b(t)\leq x\leq b(t)\}$ and $(u,b)$ is the solution of the following free boundary problem $$ {
[lcl]{} , x(-b(t),b(t)), 0<t< T,\
u(-b(t),t)=0$, $u(b(t),t)=0$,\ $0t< T,\
u\_x(-b(t),t)=, u\_x(b(t),t)=-, 0t< T,\
u(x,0)=u\_0(x), -b\_0xb\_0.
. $$
Noting the choice of initial curve, $\Gamma(t)$ is not convex for $t$ near $0$. Therefore, $\Gamma(t)$ possible intersects itself at $y$-axis. Therefore, it is necessary to study the local minima of $u(\cdot,t)$.
As showed in section 4, the numbers of local maxima and local minima are a finite nonincreasing function of time. It follows that, after a while, the numbers of local maxima and local minima are constants. After discarding an initial section of the solution, we may even assume that $x\mapsto u(x,t)$ has $m$ local minima and $m+1$ local maxima. Let these minima and maxima be located at $\{\xi_j(t)\}_{1\leq j\leq m}$ and $\{\eta_j(t)\}_{0\leq j\leq m}$, respectively. And order the $\xi_j(t)$ and $\eta_j(t)$ so that $$\label{eq:ordermami}
-b(t)<\eta_0(t)<\xi_1(t)<\eta_1(t)<\cdots<\xi_m(t)<\eta_m(t)<b(t).$$ Since the number of critical points of $u(\cdot,t)$ drops whenever $u(\cdot,t)$ has degenerate critical point, the minima and maxima of $u(\cdot,t)$ are all nondegenerate. By the implicit function theorem the $\xi_j(t)$ and $\eta_j(t)$ are therefore smooth functions of time.
\[lem:convergeendmimax\] The limits $$\lim\limits_{t\rightarrow T}b(t)=b(T)$$ and $$\lim\limits_{t\rightarrow T}\xi_j(t)=\xi_j(T),\ \lim\limits_{t\rightarrow T}\eta_j(t)=\eta_j(T)$$ exist.
We prove this lemma by the method from [@AAG], first developed by [@CM]. But in our proof, there is a little difference, since the intersection number between two flows evolving by $V=-\kappa+A$ may increase. Therefore, the method in [@AAG] should be modified.
First, we prove $\lim\limits_{t\rightarrow T}b(t)$ exists. By the vertical equation $$w_t=\frac{w_{rr}}{1+w_r^2}+A\sqrt{1+w_r^2},$$ we can derive $b^{\prime}(t)=w_{rr}+A\leq A$ because of $w_{rr}(0)\leq0$. Then $b(t)-At$ is non-increasing. It is easy to see $b(t)-At$ is bounded for $t<T$. Therefore $\lim\limits_{t\rightarrow T}(b(t)-At)$ exists. Consequently, $\lim\limits_{t\rightarrow T}b(t)$ exists.
Next, we prove $\lim\limits_{t\rightarrow T}\xi_j(t)$ exists. We assume $$\limsup\limits_{t\rightarrow T}\xi_j(t)>\liminf\limits_{t\rightarrow T}\xi_j(t).$$ We can choose $x_0\in(\liminf\limits_{t\rightarrow T}\xi_j(t),\limsup\limits_{t\rightarrow T}\xi_j(t))$ and $x_0\neq 0$. Without loss of generality, we assume $-b(T)<x_0<0<b(T)$. Since $\xi_j(t)$ is continuous in $t$, there exists a sequence $t_m\rightarrow T$ such that $$\xi_j(t_m)=x_0 \ \textrm{and}\ u_x(x_0,t_m)=0.$$ We let $\widetilde{\Gamma}(t)$ be the reflection from $\Gamma(t)$ about $x=x_0$. Consequently, $\widetilde{a}(t):=2x_0-a(t)$ and $\widetilde{b}(t):=2x_0-b(t)$ are the end points of $\widetilde{\Gamma}(t)$. Obviously, $\widetilde{\Gamma}(t)$ evolves by $V=-\kappa+A$ and $\widetilde{a}(T)<-b(T)<x_0<\widetilde{b}(T)<b(T)$. For $t$ being sufficiently close to $T$, $\widetilde{a}(t)<a(t)<x_0<\widetilde{b}(t)<b(t)$, i.e., the order of $\widetilde{a}(t)$, $\widetilde{b}(t)$, $a(t)$, $b(t)$ dose not change. Using Theorem \[thm:sl\], since $\widetilde{\Gamma}(t_m)$ intersects $\Gamma(t_m)$ at $x_0$ tangentially, the intersection number between $\widetilde{\Gamma}(t)$ and $\Gamma(t)$ will drop infinite times, for $t$ close to $T$. But Theorem \[thm:sl\] shows that the intersection number between $\Gamma(t)$ and $\widetilde{\Gamma}(t)$ is finite(The choice of $x_0$ implies $\Gamma(t)$ is not identity to $\widetilde{\Gamma}(t)$). This yields a contradiction.
\[lem:singlepoint pinch\] If $\xi_{j}(T)<\eta_j(T)$, then for any compact interval $[c,d]\subset(\xi_j(T),\eta_j(T))$, there exists $t_1$ and $\delta>0$ such that $u(x,t)\geq\delta$ for $x\in[c,d]$, $t\in[t_1,T)$. (Similarly for $\eta_{j-1}(T)<\xi_j(T)$, $-b(T)<\eta_0(T)$, $\eta_m(T)<b(T)$).
Let $[a,b]\subset(\xi_j(T),\eta_j(T))$ be any compact interval, then there exists $t_1<T$ such that $[a,b]\subset(\xi_{j}(t),\eta_j(t))$ and $u_x(x,t)>0$, $x\in[a,b]$, $t\in(t_1,T)$. Letting $\theta=\arctan u_x$, $\theta$ satisfies $$\theta_t=\cos^2\theta\theta_{xx}+A\sin\theta\theta_x.$$ Since $u_x>0$, $x\in[a,b]$, $t\in(t_1,T)$, there holds $$\theta_t-\cos^2\theta\theta_{xx}-A\sin\theta\theta_x=0.$$
On the other hand, we let $\varphi(x,t)=\epsilon e^{-ct}\sin(\lambda(x-a))$, where $\lambda=\pi/(b-a)$, $c>A\lambda\pi+\lambda^2$, $0<\epsilon<\pi$. Since $\varphi_{xx}\leq0$, $x\in[a,b]$ and seeing $$\left|-A\lambda\frac{\sin(\epsilon e^{-ct}\sin(\lambda(x-a)))}{\sin(\lambda(x-a))}\cos(\lambda(x-a))\right|\leq A\lambda\pi,$$ there holds $$\begin{aligned}
\varphi_t&-&\cos^2\varphi\varphi_{xx}-A\sin\varphi\varphi_x\leq\varphi_t-\varphi_{xx}-A\sin\varphi\varphi_x\\
&=&
\epsilon e^{-ct}\sin(\lambda(x-a))\left(-c+\lambda^2-A\lambda\frac{\sin(\epsilon e^{-ct}\sin(\lambda(x-a)))}{\sin(\lambda(x-a))}\cos(\lambda(x-a))\right)\\&\leq&\epsilon e^{-ct}\sin(\lambda(x-a))(-A\lambda\pi-\lambda^2+\lambda^2+A\lambda\pi)=0,\end{aligned}$$ for $x\in[a,b],\ t\in(t_1,T).$ Since $u_x(x,t_1)$ is bounded from below for some positive constant in $[a,b]$, we can choose $\epsilon>0$ small enough such that $\varphi(x,t_1)\leq\theta(x,t_1)$. Seeing $$\varphi(a,t)=0<\theta(a,t),\ \varphi(b,t)=0<\theta(b,t),\ t\in(t_1,T).$$ By maximum principle, $$\theta(x,t)\geq\varphi(x,t),\ a<x<b,\ t_1<t<T.$$ Consequently, $$u_x\geq\arctan u_x=\theta\geq\epsilon e^{-ct}\sin(\lambda(x-a)),\ x\in[a,b],\ t\in(t_1,T).$$ $$u\geq\epsilon\frac{e^{-ct}}{\lambda}(1-\cos(\lambda(x-a))),\ x\in[a,b],\ t\in(t_1,T).$$ Then for all $[c,d]\subset(a,b)$, $u$ is uniformly bounded from below for $x\in[c,d],\ t\in[t_1,T)$.
\[lem:limitsurface\] $\lim\limits_{t\rightarrow T}u(x,t)=u(x,T)$ exists, and $u(x,t)$ converges uniformly to $u(x,T)$, for $x\in \mathbb{R}$, as $t\rightarrow T$. The function $u$ is smooth at $(x,t)\in \mathbb{R}\times(0,T]$ provided that $u(x,t)>0$. We interpret that $u(x,t)=0$ outside $(a(t),b(t))$.
By Lemma \[lem:singlepoint pinch\], for all $[c,d]\subset(\xi_{j-1}(T),\xi_{j}(T))$, $u(x,t)\geq \delta$, $x\in[c,d]$, $t\in[t_1,T)$. By Theorem \[thm:grad\], $u_x$ is uniformly bounded on $[c,d]\times[t_1,T)$, which implies $\frac{\partial^i}{\partial x^i}u(x,t)$, $i=1,2$ are bounded on any compact subinterval of $(c,d)$. On the other hand, from equation, $u_t(x,t)$ is uniformly bounded on such interval, so that $u(\cdot,t)$ converges uniformly on any such interval.
The same idea can be applied to the conditions in the intervals $(-b(T),\xi_1(T))$ and $(\xi_m(T),b(T))$. Siince outside of $[-b(T),b(T)]$,$u(x,T)$ is considered to be $0$, the result is true.
Except at $-b(T)$, $b(T)$ and $\xi_j(T)$s, $u(x,t)$ converges pointwise for every $x$ not equaling $-b(T)$, $b(T)$, $\xi_j(T)$, as $t\rightarrow T$. The convergence is uniform on any interval that does not contain any of the points.
Next we want to prove the functions $u(\cdot,t)$ are equicontinuous for $T/2<t<T$.
Assuming $x_1<x_2$, if $x_1$, $x_2$ are both not in the interval $(-b(T),b(T))$, the conclusion is obvious. Assume $x_1\in(-b(T),b(T))$.
Suppose that $|u(x_1,t)-u(x_2,t)|\geq\epsilon$. Then either $u(x_1,t)\geq\epsilon/2$ or $u(x_2,t)\geq\epsilon/2$ or both; we assume the first one. From Theorem \[thm:grad\], $|u_x|<\sigma(\epsilon/2,T/2)$ whenever $u(x,t)\geq\epsilon/2$, $T/2<t<T$. Thus, if $u(x,t)\geq\epsilon/2$ on $(x_1,x_2)$, $$x_2-x_1\geq\frac{|u(x_1,t)-u(x_2,t)|}{\sigma(\epsilon/2,T/2)}\geq\frac{\epsilon}{\sigma(\epsilon/2,T/2)}.$$ If $u(x,t)<\epsilon/2$ some where in the interval $(x_1,x_2)$, then there is a smallest $x_3$ satisfying $x_1<x_3$ at which $u(x_3,t)=\epsilon/2$. On the interval $(x_1,x_3)$, $u(x,t)\geq\epsilon/2$. Then $$x_2-x_1\geq x_3-x_1\geq\frac{u(x_1,t)-u(x_3,t)}{\sigma(\epsilon/2,T/2)}\geq\frac{\epsilon}{2\sigma(\epsilon/2,T/2)}.$$ So for every $\epsilon>0$, choose $\delta=\epsilon/(2\sigma(\epsilon/2,T/2))$ so that $$|u(x_1,t)-u(x_2,t)|<\epsilon,$$ $|x_1-x_2|<\delta$, for $T/2<t<T$.
Thus $u(x,t)$ is equicontinuous. Noting that $u(x,t)$ converges to $u(x,T)$ in $\mathbb{R}\setminus\{\xi_i(T),-b(T),b(T)\}$ and $\mathbb{R}\setminus\{\xi_i(T),-b(T),b(T)\}$ is dense in $\mathbb{R}$, the proof is completed.
\[lem:seprate\] Suppose that $u(\eta_0(T),T)>0$, then $-b(T)<\eta_0(T)$.
Since $u(\eta_0(T),T)>0$, there exists $\delta>0$ such that $\delta=\inf\limits_{0\leq t\leq T} u(\eta_0(t),t)$. We consider $$x=v(|y|,t),$$ being the inverse function of $|y|=u(x,t)$ for $x\in(a(t),\eta_0(t))$ and let $w(y,t)=v(|y|,t)$. $w(y,t)$ satisfies the equation (\[eq:graph\]) for the condition “$-$” and “$n=1$”, $|y|<\delta$, $0<t<T$. Clearly $w$ is uniformly bounded, so Corollary \[cor:es\] and Remark \[rem:hes\] imply that $\frac{\partial^k w}{\partial y^k}(y,t)$, $k=1,2$ are bounded for $|y|\leq\delta/2$, $T/2\leq t<T$. So the limit function $w(y,T)$ obtained by Lemma \[lem:limitsurface\] is smooth for $|y|\leq\delta/2$.
As the proof of Lemma \[lem:sing1\], using maximum principle, $v_r(r,T)>0$, $0<t<\delta/2$. Then $-b(T)=v(0,T)<v(\delta/2,T)<\eta_0(T)$.
Lemma \[lem:singlepoint pinch\] and Lemma \[lem:seprate\] imply that “width” and “height” become zero at same time. Therefore, $\Gamma(t)$ can not pinch at $y$-axis before shrinking. We prove the detail in the following theorem and corollary.
\[thm:formationofsingular\](Formation of singular)
1\. If $m=0$, $u(\eta_0(T),T)=0$ and $b(T)=0$. This implies that $\Gamma(t)$ shrinks to the origin $O$, as $t\rightarrow T$.
2\. If $m\geq1$, there is $j$ such that $u(\xi_j(T),T)=0$, $1\leq j\leq m$.
1\. First, we prove for $m=0$, i.e., $u(x,t)$ only has one maximum without local minimum. We prove this by contradiction.
[**Case 1.**]{} If $u(\eta_0(T),T)>0$, from Lemma \[lem:seprate\], $-b(T)<\eta_0(T)<b(T)$. $\Gamma(t)$ can be divided into three parts $\Delta_1(t)$, $\Delta_2(t)$ and $\Delta_3(t)$, for $t$ being very close to $T$, where $\Delta_1(t)$ and $\Delta_2(t)$ are the left and right caps of $\Gamma(t)$, $\Delta_3(t)$ is the middle part of $\Gamma(t)$ away form $x$-axis. It is easy to show the derivatives and second fundamental formations of $\Delta_1$, $\Delta_2$ and $\Delta_3$ are uniformly smooth for $t\rightarrow T$(We can similarly prove as Lemma \[lem:sing1\]), which contradicts to $\Gamma(t)$ becoming singular at $T$.
[**Case 2.**]{} If $b(T)> 0$, there holds $-b(T)<\eta_0(T)$ or $\eta_0(T)<b(T)$, assuming $-b(T)<\eta_0(T)$. By Lemma \[lem:singlepoint pinch\], for every $[c,d]\subset(-b(T),\eta_0(T))$, $u(x,t)\geq \delta>0$ in $[c,d]\times[t_1,T)$. Then $u(\eta_0(t),t)\geq\delta$, $t_1\leq t<T$. Consequently, $u(\eta_0(T),T)\geq\delta$. By the same argument as in Case 1, we get a contradiction. Here we complete the proof under the condition $m=0$.
2\. For $m\geq1$, if $u(\xi_j(T),T)>0$, for any $1\leq j\leq m$, we can divide $\Gamma(t)$ into three parts as above for $t$ being close to $T$. Then we can get contradiction similarly as in the condition $m=0$. So there is $j$ such that $u(\xi_j(T),T)=0$.
\[cor:2dsingular\] There is $t_1$ satisfying $0<t_1<T$ such that $u(x,t)$ loses all its local minima for $t\in[t_1,T)$. Moreover, $\Gamma(t)$ shrinks to a point, as $t\rightarrow T$.
Denote $h(t)=\max\limits_{a(t)<x<b(t)}u(x,t)$. By Lemma \[lem:in\], we can deduce that, for $t$ satisfying $t_2<t<T$ given, when $\rho<\min\{At_2,h(t)\}$, $y=\rho$ intersects $y=u(x,t)$ only twice.
If $u(x,t)$ does not lose its all local minima, the number of minima will not change denoted by $m\geq 1$. From Theorem \[thm:formationofsingular\], there exists $j$, $1\leq j\leq m$ such that $u(\xi_j(T),T)=0$. So we can choose $t_0$ satisfying $t_2<t_0<T$, there exists $\xi_j(t_0)$ such that $u(\xi_j(t_0),t_0)<At_2$. Obviously, $u(\xi_j(t_0),t_0)< h(t_0)$, then $u(\xi_j(t_0),t_0)<\min\{At_2,h(t_0)\}$. Consequently, $y=\rho=u(\xi_j(t_0),t_0)$ intersects $y=u(x,t_0)$ three times. Contradiction.
Therefore, there is $t_1$ such that $u(x,t)$ will lose its all local minima for $t\in[t_1,T)$. Seeing the proof in Theorem \[thm:formationofsingular\] for $m=0$, $u(\eta_0(T),T)=0$ and $b(T)=0$. It means that $\Gamma(t)$ shrinks to a point, as $t\rightarrow T$.
We note that all the proof in this section, we do not use the condition that $u(\cdot,t)$ is even. Therefore, the argument in this section can be used in any $x$-axisymmetric curve.
Asymptotic behaviors
=====================
In this section, we will prove Theorem \[thm:threecondition\] and Theorem \[thm:asym\]. For convenience, we still extend $\Gamma(t)$ by even, still denoted by $\Gamma(t)$ and let $$h(t)=\max\limits_{-b(t)\leq x\leq b(t)}u(x,t),\ \ l(t)=2b(t).$$ Denote $U(t)$ being the open set surrounded by $\Gamma(t)$.
All the proofs in this section are proved by intersection number principle introduced in section 4. For the proof of asymptotic behavior, there have so far been many methods. The intersection argument in proving asymptotic behavior is developed by Professor Hiroshi Matano. Saying roughly, if two functions $u(x,t)$ and $v(x)$ are satisfying the same parabolic equation, moreover, $u(x,t)$ intersects $v(x)$ at some fixed point tangentially, for any large $t$. Then there holds $u(x,t)\equiv v(x)$. Another important method in studying asymptotic behavior is by using Lyapunov function to prove $u(x,t)$ is independent on $t$. By the intersection argument, we can prove $u(x,t)$ is independent on $t$ without Lyapunov function.
The following lemma says that $l(t_0)$ being large enough deduce $h(t_0)$ being large. We prove it by using Proposition \[pro:intersection\]. Although the proof of Lemma \[lem:expanding\] is similar as in [@GMSW], for the reader’s convenience, we still give the proof for detail.
\[lem:expanding\] For any $\tau\in(0,T)$ and $M\in(0,A\tau/2)$, there exists $l_{M,\tau}>0$ such that, when $l(t_0)>l_{M,\tau}$ for some $t_0\in[\tau,T)$, it holds $h(t_0)>M$.
For given $\tau\in(0,T)$ and $M\in(0,A\tau/2)$, we choose $R_0$ such that $$R_0\geq\frac{2}{A}.$$ Let $R(t)$ be the solution of (\[eq:ball2\]) with $R(0)=R_0$. Since $R_0>1/A$, $R(t)$ is increased in $t$. Therefore $R^{\prime}(t)\geq A-1/R_0\geq A/2$. Integrating the inequality, there holds $$R(\tau)\geq R_0+A\tau/2\geq R_0+M.$$ So there exists $\tau_1\in(0,\tau]$ such that $$R(\tau_1)=R_0+M.$$
Now we let $$W(x,t):=\sqrt{R(t)^2-x^2}-R_0,\ \ x\in[\sigma_-(t),\sigma_+(t)],\ t\in(0,\tau_1],$$ where $\sigma_{-}(t)=-\sqrt{R(t)^2-R^2_0}$ and $\sigma_+(t)=\sqrt{R(t)^2-R^2_0}$. And we denote $$\theta_{\pm}(t)= \arctan\frac{\sqrt{R(t)^2-R^2_0}}{R_0}.$$ Obviously, $\pi/2>\theta_{\pm}(t)>0$.
We choose $l_{M,\tau}:=\sigma_+(\tau_1)-\sigma_-(\tau_1)=2\sqrt{R(\tau_1)^2-R^2_0}=2\sqrt{M^2+2R_0M}.$ We let $\gamma_1(t)$ and $\gamma_2(t)$ be the extension of $u(x,t)$ and $W(x,t)$ as in Proposition \[pro:intersection\]. Obviously, $(W(x,t),\sigma_{\pm}(t))$ is the solution of (Q) with $\theta_{\pm}(t)$(Proposition \[pro:intersection\]), so by Proposition \[pro:intersection\], we can deduce $$\mathcal{Z}(\gamma_1(t_0),\gamma_2(\tau_1))\leq \mathcal{Z}(\gamma_1(t_0-s),\gamma_2(\tau_1-s)),\ \textrm{for}\ s\in[0,\tau_1).$$ Since the extended curve $\gamma_2(\tau_1-s)$ converges to the $x$-axis, as $s\rightarrow\tau_1$, the right-hand side of the above inequality equals 2 for $s$ sufficiently close to $\tau_1$. Consequently, $$Z[\gamma_1(t_0),\gamma_2(\tau_1)]\leq2.$$
Assuming $l(t_0)>l_{M,\tau}$, for some $t_0\in[\tau,T)$, then $\sigma_{\pm}(\tau_1)$ satisfy $$-b(t_0)<\sigma_1(\tau_-)<\sigma_+(\tau_1)<b(t_0).$$ Hence $\gamma_1(t_0)$ intersects $\gamma_2(\tau_1)$ twice below the $x$-axis. So $u(x,t_0)>W(x,\tau_1)$ on the interval $[\sigma_-(\tau_1),\sigma_+(\tau_1)]$. Consequently, $h(t_0)>M$.
The following corollary gives that as long as $l(t)$ is unbounded, $\Gamma(t)$ will be expanding.
\[cor:expanding2\] Assume $T=\infty$ and there exists a sequence $s_m\rightarrow\infty$ such that $l(s_m)\rightarrow\infty$, as $m\rightarrow\infty$. Then $l(t)\rightarrow\infty$ and $h(t)\rightarrow\infty$, as $t\rightarrow\infty$.
We can use the same argument as in Lemma \[lem:expanding\], there exist $C>1/A$ and $m_0$ such that $u(x,s_{m_0})>\sqrt{(C+R_0)^2-x^2}-R_0$. Obviously, $\sqrt{(C+R_0)^2-x^2}-R_0>\sqrt{C^2-x^2}$, $-C\leq x\leq C$. Therefore $u(x,s_{m_0})\geq \sqrt{C^2-x^2}$, $-C\leq x\leq C$. By (b) in Remark \[rem:intersection1\], $u(x,s_{m_0}+t)\geq\sqrt{C(t)^2-x^2}$, $-C(t)\leq x\leq C(t)$, where $C(t)$ is the solution of (\[eq:ball2\]) with $C(0)=C$. Seeing the choice of $C$, we can deduce $C(t)\rightarrow\infty$, as $t\rightarrow\infty$. Then $h(t+s_{m_0})>C(t)\rightarrow\infty$ and $l(t+s_{m_0})>2C(t)\rightarrow\infty$, as $t\rightarrow\infty$.
The following lemma gives that as long as $h(t)$ is unbounded, $\Gamma(t)$ will be expanding.
\[lem:1expanding\] Assume $T=\infty$ and there exists a sequence $s_m\rightarrow\infty$ such that $h(s_m)\rightarrow\infty$, as $m\rightarrow\infty$. Then $l(t)\rightarrow\infty$ and $h(t)\rightarrow\infty$, as $t\rightarrow\infty$.
If $l(t)$ is unbounded, by Corollary \[cor:expanding2\], $h(t)\rightarrow\infty$ and $l(t)\rightarrow\infty$, $t\rightarrow\infty$. The result is true. Next we prove $l(t)$ is unbounded by contradiction.
Assume $l(t)$ is bounded.
[**Step 1.**]{} we are going to prove that $\lim\limits_{t\rightarrow\infty}b(t)$ exists. If $\liminf\limits_{t\rightarrow\infty}b(t)<\limsup\limits_{t\rightarrow\infty}b(t)$, we can choose $x_0$ such that $$\liminf\limits_{t\rightarrow\infty}b(t)<x_0<\limsup\limits_{t\rightarrow\infty}b(t).$$ We consider the function $u_1(x)=\sqrt{1/A^2-(x+1/A-x_0)^2}$. Obviously, $x_0$ is the right endpoint of $u_1(x)$ and $(u_1(x),x_0-2/A,x_0)$ is the solution of the problem (Q) with $\theta_{\pm}=\pi/2$. So $b(t)-x_0$ changes sign infinite many times as $t$ varying over $[0,\infty)$. There exists a sequence $p_m\rightarrow\infty$ such that $u(x,p_m)$ intersects $u_1(x)$ tangentially at $x_0$. Arguing as in Lemma \[lem:inters\], the intersection number between $u(x,t)$ with $u_1(x)$ drops at $b(p_m)=x_0$. Therefore, the intersection number between $u(x,t)$ and $u_1(x)$ drops infinite many times. This yields a contradiction. Then we let $\nu:=\lim\limits_{t\rightarrow\infty}b(t)$.
[**Step 2.**]{} We deduce the contradiction.
Since $h(s_m)\rightarrow\infty$, Lemma \[lem:alphad2\] implies that for $t_1=4/A^2$, there holds $y=\rho$ intersects $y=u(x,t)$ only twice, $\rho<At_1$, $t>t_1$. Here we choose $\rho_0=2/A$. Then there exists $w(y,t)>0$ such that $$C_{\rho_0}\cap\Gamma(t)=\{(x,y)\mid x=w(y,t)\ \text{or}\ x=-w(y,t)\}.$$ $w(y,t)$ satisfies (\[eq:graph\]) in the condition “$+$” and $n=1$, for $\{y\mid |y|<\rho_0\}\times(t_1,\infty)$. Since $w(0,t)=b(t)$ is bounded for $t>0$, by Corollary \[cor:es\] and Remark \[rem:hes\], $\frac{\partial^kw}{\partial y^k}(y,t)$, $k=1,2,3$ are uniformly bounded for $|y|\leq\rho_0/2$, $t>t_1+\epsilon^2$. From equation, $\frac{\partial^kw}{\partial t^k}w(y,t)$, $k=1,2$ are also bounded for $|y|<\rho_0/2$, $t>t_1+\epsilon^2$. So there exists $w_1(y,t)$, for any sequence satisfying $t_m\rightarrow \infty$ such that $w(\cdot,\cdot+t_m)$ converges to $w_1$ in $C^{2,1}([-\rho_0/2,\rho_0/2]\times[t_1+\epsilon^2,\infty))$ locally in time, as $m\rightarrow\infty$. Hence $w_1(y,t)$ also satisfies (\[eq:graph\]) with the condition “$+$” and $n=1$. Moreover, $w_1(0,t)=\nu$ and $\frac{\partial}{\partial y}w_1(0,t)=0$, $t>t_1+\epsilon^2$.
Next, we consider the function $w_2(y)=\nu-1/A+\sqrt{1/A^2-y^2}$. $w_2(y)$ satisfies (\[eq:graph\]) with the condition “$+$” and $n=1$. Moreover, $w_2(0)=\nu$ and $\frac{\partial}{\partial y}w_2(0)=0$. So $w_1(y,t)$ intersects $w_2(y)$ at $y=0$ tangentially for all $t>t_1+\epsilon^2$. By the same argument as in Lemma \[lem:inters\], there holds $w_1(y,t)\equiv w_2(y)$, $|y|\leq \rho_0/2$. Noting $\frac{\partial w_2}{\partial y}(1/A)=\infty$, $\frac{\partial w_1}{\partial y}(1/A,t)=\infty$. But $w_y(1/A,t)$ is bounded, as $t\rightarrow\infty$, by gradient interior estimate. This is a contradiction.(Indeed, $w(y,t)$ has definition for $y\in(-2/A,2/A)$, but the limit function $w_2(y)$ has definition only in $[-1/A,1/A]$.)
We complete the proof.
For $T<\infty$, seeing Corollary \[cor:2dsingular\], we get the conclusion.
For $T=\infty$, $h(t)$ is bounded or unbounded, as $t\rightarrow\infty$.
(1). $h(t)$ is unbounded. Lemma \[lem:1expanding\] yields that $l(t)\rightarrow\infty$, $h(t)\rightarrow\infty$, as $t\rightarrow\infty$.
(2). $h(t)$ is bounded.
By Corollary \[cor:expanding2\], $l(t)$ is also bounded. Next we want to prove $h(t)$ and $l(t)$ are bounded from below.
[**Step 1.**]{} We prove if there exists a sequence $s_m\rightarrow\infty$, as $m\rightarrow\infty$ such that $h(s_m)\rightarrow0$, as $m\rightarrow\infty$, then $l(s_m)\rightarrow0$, as $m\rightarrow\infty$.
By Lemma \[lem:expanding\] with $M=h(s_m)$, $$l(s_m)\leq l_{M,\tau}=2\sqrt{M^2+2R_0M}=2\sqrt{h(s_m)^2+2R_0h(s_m)}$$ Then we have $l(s_m)\rightarrow0$.
[**Step 2.**]{} $h(t)$ is bounded from below.
If there exists another sequence $t_m\rightarrow\infty$ such that $h(t_m)\rightarrow 0$, as $t_m\rightarrow\infty$, by Step 1, $l(t_m)\rightarrow 0$, as $t_m\rightarrow\infty$. Then there exists $t_{m_0}$ and $r<1/A$ such that $U(t_{m_0})\subset B_r((0,0))$, recalling $U(t)$ being the domain surrounded by $\Gamma(t)$. Then by comparison principle, we have $U(t+t_{m_0})\subset B_{r(t)}((0,0))$, where $r(t)$ is the solution of (\[eq:ball2\]) with $r(0)=r$. Obviously, $B_{r(t)}((0,0))$ shrinks to origin in finite time. Then it is also for $U(t)$. This contradicts to $T=\infty$.
Hence $h(t)$ is bounded from blew.
[**Step 3.**]{} Prove the result by contradiction. Assume there exists a sequence $s_m\rightarrow\infty$ such that $l(s_m)\rightarrow0$.
Since $h(t)$ is bounded from below, by Lemma \[lem:alphad2\], there exist $\rho_0$ and $t_1>0$ such that for all $\rho<\rho_0$, $y=\rho$ intersects $y=u(x,t)$ only twice for $t>t_1$. Then we let $w(y,t)>0$ such that $$C_{\rho_0}\cap\Gamma(t)=\{(x,y)\mid x=w(y,t)\ \text{or}\ x=-w(y,t)\}.$$ Arguing as the proof of Lemma \[lem:1expanding\], $\nu=\lim\limits_{t\rightarrow\infty}b(t)=0$, by $l(s_m)\rightarrow0$. And $w(\cdot,t)\rightarrow w_1$ in $C^{2,1}([0,\rho_0/2]\times[t_1+\epsilon^2,\infty))$ locally in time, as $m\rightarrow\infty$ and $w_1(y)=-1/A+\sqrt{1/A^2-y^2}\leq0$. But seeing $w(y,t)>0$, there holds $w_1(y)\geq0$ for $|y|<\rho_0/2$. Consequently, $w_1(y)\equiv0$, for $|y|<\rho_0/2$. Contradiction.
Therefore $h(t)$ and $l(t)$ are bounded from below.
The conclusion of the Shrinking case in Theorem \[thm:asym\] is obvious. We only need prove the case expanding and bounded.
In this case, since $h(t)$ and $l(t)$ tend to infinity, using the same argument as in the proof of Corollary \[cor:expanding2\], there exist $t_0$ and $C>1/A$ such that $B_C((0,0))\subset U(t_0)$. By comparison principle, $B_{C(t)}((0,0))\subset U(t_0+t)$. Therefore, $B_{C(t-t_0)}((0,0))\subset U(t)$, $t\geq t_0$, where $C(t)$ satisfies (\[eq:ball2\]) with $C(0)=C$.
On the other hand, seeing $U(0)$ being bounded, there exists $R>1/A$ such that $U(0)\subset B_R((0,0))$. Then $U(t)\subset B_{R(t)}((0,0))$, where $R(t)$ also satisfies (\[eq:ball2\]) with $R(0)=R$.
Denoting $R_1(t)=C(t-t_0)$ and $R_2(t)=R(t)$, $B_{R_1(t)}((0,0))\subset U(t)\subset B_{R_2(t)}((0,0))$, $t>t_0$. By the theory of ordinary equation, we can easily deduce that $\lim\limits_{t\rightarrow\infty}R_1(t)/t=\lim\limits_{t\rightarrow\infty}R_2(t)/t=A$. We complete the proof.
Since $h(t)$ is bounded from below, by Lemma \[lem:alphad2\], as before, there exist $t_1$, $\rho_0$ such that $$C_{\rho_0}\cap\Gamma(t)=\{(x,y)\mid x=w(y,t)\ \text{or}\ x=-w(y,t)\},\ t>t_1.$$
[**Step 1.**]{} Asymptotic behavior of $C_{\rho_0/2}\cap\Gamma(t)$.
Arguing as the proof of Lemma \[lem:1expanding\], there exist $\nu$ and $w_2(y)$ such that $\nu=\lim\limits_{t\rightarrow\infty}b(t)$ and $w(\cdot,t)\rightarrow w_2$ in $C^{2,1}([-\rho/2,\rho_0/2])$, as $t\rightarrow\infty$, where $w_2(y)=\nu-1/A+\sqrt{1/A^2-y^2}$.
[**Step 2.**]{} Asymptotic behavior of $\{(x,y)\mid|y|\geq\rho_0/2\}\cap\Gamma(t)$.
Noting that $w_2(\rho_0/4)<\nu=\lim\limits_{t\rightarrow\infty}b(t)$, then for all $\epsilon>0$, there is $t_2$ such that $(-w_2(\rho_0/4)-\epsilon,w_2(\rho_0/4)+\epsilon)\subset(-b(t),b(t))$, $t>t_2$. We consider $u(x,t)$ in the following set $[-w_2(\rho_0/4),w_2(\rho_0/4)]\times(t_2,\infty)$. Because $u(x,t)$ satisfies (\[eq:graph\]) under the condition $n=1$ and “$+$”, $\frac{\partial ^k }{\partial x^k}u$, $k=1,2,3$, are uniformly bounded in $(-w_2(\rho_0/4)-\epsilon/2,w_2(\rho_0/4)+\epsilon/2)\times(t_2+\epsilon^2,\infty)$. Therefore $\frac{\partial^i}{\partial t^i}u$, $i=1,2$, $\frac{\partial ^k }{\partial x^k}u$, $k=1,2,3$, are uniformly bounded in $[-w_2(\rho_0/4),w_2(\rho_0/4)]\times(t_2+\epsilon^2,\infty)$
Next we want to show $\lim\limits_{t\rightarrow\infty}u(0,t)$ exists. If $\limsup\limits_{t\rightarrow\infty}u(0,t)>\liminf\limits_{t\rightarrow\infty}u(0,t)$, we can choose $y_0$ such that $\limsup\limits_{t\rightarrow\infty}u(0,t)>y_0>\liminf\limits_{t\rightarrow\infty}u(0,t)$. We consider the function $u_2(x)=y_0-1/A+\sqrt{1/A^2-x^2}$. By the same argument in the proof of Lemma \[lem:1expanding\], we get the contradiction. Denote $\mu:=\lim\limits_{t\rightarrow\infty}u(0,t)$. We can show, as in the proof of Lemma \[lem:1expanding\], $u(\cdot,t)\rightarrow u_3$ in $C^{2,1}([-w_2(\rho_0/4),w_2(\rho_0/4)])$, as $t\rightarrow\infty$, where $u_3(x)=\mu-1/A+\sqrt{1/A^2-x^2}$.
[**Step 3.**]{} Identify $\nu$ and $\mu$.
Since the graph of $y=u_2(x)$ and $x=w_2(y)$ are identical with each other, for $\rho/4<y<\rho/2$, then $\nu=\mu=1/A$. Consequently, $u(x,t)$ converges to $\varphi(x)=\sqrt{1/A^2-x^2}$, $x\in \mathbb{R}$, as $t\rightarrow\infty$. Here we consider $u(x,t)$ and $\varphi(x)$ as 0 outside the domains of definition.(Indeed, seeing the proof, $\lim\limits_{t\rightarrow \infty}d_H(\Gamma(t),\partial B_{1/A}((0,0))=0$). We complete the proof.
Appendix
=========
In this section, we want to prove there exists unique smooth family of smooth hypersurfaces $\Gamma(t)$ satisfying $$\label{eq:hcur}
V=-\kappa+A,\ \text{on}\ \Gamma(t)\subset \mathbb{R}^{n+1},$$ where $\Gamma(0)=\partial U$ with $U$ is an $\alpha$-domain.
Seeing $\partial U$ is not necessary smooth, we also use the level set method and prove the interface evolution is not fattening.
\[def:alphad\] We say a domain being an $\alpha$-domain in $\mathbb{R}^{n+1}$ if
\(1) Let $U\subset \mathbb{R}^{n+1}$ be an open set of the form $$U=\{(x,y)\in\mathbb{R}\times\mathbb{R}^n\mid r<u(x)\}.$$
\(2) $I=\{x\in\mathbb{R}\mid u(x)>0 \}$ is a bounded, connected interval.
\(3) $u$ is smooth on $I$;
\(4) $\partial U$ intersects each cylinder $\partial C_{\rho}$ with $0<\rho\leq\alpha$ twice and these intersections are transverse, where $C_{\rho}=\{(x,y)\in\mathbb{R}\times\mathbb{R}^n\mid r<\rho\}$.
For $U$ being $\alpha$-domain, we choose smooth vector field $X:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}$ such that\
(i) At any point $P\in\partial U$ not on the $x$-axis has $\langle X(P), \textbf{n}(P)\rangle<0$, $\textbf{n}$ is inward unit normal vector at $P$.\
(ii) Near the two end points of $\partial U$, $X$ is constant vector with $X\equiv\pm e_0=(\pm1,0,\cdots,0).$
Since $X\neq0$ on the compact $\partial U$, there is an open neighbourhood $V\supset\partial U$ on which $|X|\geq\delta>0$ for some $\delta>0$.
![Vector field $X$[]{data-label="fig:vectorX"}](section5.pdf){height="4cm"}
\[pro:sigma1\]For small enough $\rho>0$ there exists a smooth hypersurface $\Sigma\subset V$ with\
(i) $X(P)\notin T_P\Sigma$ at all $P\in\Sigma$, i.e., $\Sigma$ is transverse to the vector field $X$.\
(ii) $\Sigma=\partial U$ in $\{(x,y)\in\mathbb{R}\times\mathbb{R}^{n}\mid |y|\geq2\rho\}$.\
(iii) $\Sigma\cap\{(x,y)\in\mathbb{R}\times\mathbb{R}^{n}\mid|y|\leq\rho\}$ consists of two flat disks $\Delta_a=\{(a,y)\in\mathbb{R}\times\mathbb{R}^{n}\mid|y|\leq\rho\}$ and $\Delta_b=\{(b,y)\in\mathbb{R}\times\mathbb{R}^{n}\mid|y|\leq\rho\}$ for some $a<b$.
Seeing Figure \[fig:sigma1\], this proposition can be proved as in Proposition \[pro:sigma2\].
![Proof of Proposition \[pro:sigma1\][]{data-label="fig:sigma1"}](proposition51.pdf){height="4cm"}
Let $\phi^{t}:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}(t\in\mathbb{R})$, $t\in(-\delta,\delta)$ be the flow generated by vector field $X$ on $\mathbb{R}^{n+1}$ determined by $$\left\{
\begin{array}{lcl}
\dis{\frac{d\phi^t(P)}{dt}=X(\phi^t)},\ P\in \Sigma,\\
\phi^0(P)=P,\ \ \ \ \ P\in \Sigma.
\end{array}
\right.$$
We denote $\sigma(P,s):=\phi^{s}(P)$. As in Section 5, suppose $\Gamma(t)\subset V$$(0<t<T)$ are smooth hypersurfaces with $\sigma^{-1}(\Gamma(t))$ being the graph $u(\cdot,t)$ for $u:\Sigma\times[0,T)\rightarrow\mathbb{R}$. Let $z_1,z_2,\cdots,z_n$ be local coordinates on an open subset of $\Sigma$. If $\Gamma(t)$ evolving by $V=-\kappa+A$, then in these coordinates $u$ satisfies the following parabolic equation $$\label{eq:para}
\frac{\partial u}{\partial t}=a_{ij}(z,u,\nabla u)\frac{\partial^2u}{\partial z_i\partial z_j}+b(z,u,\nabla u).$$
![The transportation from $\Sigma$ to $\Gamma$[]{data-label="fig:charact"}](charact.pdf){height="4cm"}
For example, on $\Delta_a$, by calculation, $\sigma(y_1,,y_2,\cdots,y_n,s)=(a-s,y_1,y_2,\cdots,y_n)$. Then $u$ satisfies the “$-$” condition of (\[eq:graph\]).
\[pro:uniq\] For $n\geq1$, let $\Gamma_1(t)$, $\Gamma_2(t)(0\leq t<T)$ be two families of hypersurface smooth and $\sigma^{-1}(\Gamma_j(t))$ be the graph of $u_j(\cdot,t)$ for certain $u_j\in C(\Sigma\times[0,T))$. Assume that the $u_j$ are smooth on $\Sigma\times(0,T)$ as well as on $\Sigma\setminus(\Delta_a\cup\Delta_b)\times[0,T)$. Then if the $\Gamma_j(t)$ evolve by $V=-\kappa+A$ and if $\Gamma_1(0)=\Gamma_2(0)$, then there holds $\Gamma_1(t)=\Gamma_2(t)$ for $0<t<T$.
We use the same method in [@AAG]. The proof is similar as in Proposition \[pro:uniq2\]. Here we omit it.
\[thm:partialUmeancurvature\] If $U$ is an $\alpha$-domain with smooth boundary, let $D(t)$ and $E(t)$ be the open and closed evolutions of $V=-\kappa+A$ with $D(0)=U$ and $E(0)=\overline{U}$. Then there exists $T>0$ such that $\partial D(t)$ and $\partial E(t)$ are smooth hypersurfaces for $0<t\leq T$ and $\partial D(t)=\partial E(t)$. Moreover, denoting $\Sigma(t)=\partial D(t)=\partial E(t)$, $\Sigma(t)$ can be written into $\Sigma(t)=\{(x,y)\in\mathbb{R}\times\mathbb{R}^n\mid |y|=u(x,t), a(t)\leq x\leq b(t)\}$ and $(u,a,b)$ is the solution of (Q) with $\theta_{\pm}=\pi/2$.
We only give the sketch of the proof. By approximate argument similarly in Lemma 6.2 and Lemma 6.4, $\partial D(t)$ and $\partial E(t)$ are smooth hypersurfaces and can be represented by $\sigma(P,u_j(P))$, for some $u_j$, $j=1,2$. Then we can use Proposition \[pro:uniq\] to prove $\partial D(t)=\partial E(t)$. Therefore $\Gamma(t)=\partial E(t)$ can be represented by $\Gamma(t)=\{(x,y)\in\mathbb{R}\times\mathbb{R}^n\mid |y|=u(x,t), a(t)\leq x\leq b(t)\}$. Using Theorem \[thm:openevolutionmeancurvature\], $(u,a,b)$ is the solution of (Q) with $\theta_{\pm}=\pi/2$.
[**Acknowledgment**]{}
The author expresses his hearty thanks to Professor Hiroshi Matano and Professor Yoshikazu Giga for their stimulating suggestions. The author learned the content about extended intersection number principle from Professor Hiroshi Matano. The author learned the techniques about viscosity solutions and formation of singularity in Section 6 contained in [@A2] from Professor Yoshikazu Giga. The author is grateful to the anonymous referee for valuable suggestion to improve the presentation of this paper.
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3
It has become clear that the intracellular nonlinear dynamics of calcium plays a crucial role in many biological processes [@Berridge98]. The nonlinearity of this problem is due to the fact that there exist calcium stores inside the cell which can be released via the opening of channels which themselves have calcium-dependent kinetics. Typically, these processes are modeled using a set of coupled equations for the calcium concentration (the diffusion equation with sources and sinks) and for the relevant channels; the latter is often described by a rate equation for the fraction of open channels per unit of area. More elaborate models take into account the discrete nature of these channels, their spatial clustering, and fluctuations in the process of their opening and closing [@KeizerSmith98; @Keizer98].
In this paper, we will propose and analyze a set of models which operate just with the channel dynamics alone. The justification for this is that the calcium field equilibrates quickly, with a diffusion time of perhaps 0.1s, as compared to the channel transition times, perhaps on the order of 1s for activation of a subunit to several seconds for its deactivation. One can then imagine solving for the quasi-stationary calcium concentration and thereafter using it to determine the conditional probabilities of channel opening or closing. In a subsequent paper[@FLT], we will show how this can be done in detail starting from a specific fully-coupled model (the DeYoung-Keizer-model [@DeYoung; @Keizer1]); here, we will make reasonable assumptions for these probabilities and study the resulting stochastic model in a one dimensional geometry.
For specificity, we will focus on systems that have 3 (inositol 1,4,5- trisphosphate) channels. Each of these channels consists of a number of subunits. Here we assume that $h$ subunits have to be activated for the channel to be open; experiments indicate that $h=3$ [@Bezprozvanny]. A subunit is activated when 3 ion is bound to its corresponding domain and is bound to its activating domain and [*not*]{} bound to its inhibiting site. The characteristic time of binding and unbinding of 3 is typically so fast (more than 20 times faster than other binding steps [@DeYoung]), that we can assume local balance of active/passive channels maintained at all times. Furthermore, we assume that the channels are spatially organized into clusters [@Parker1; @Parker2], with a fixed number of channels $N$ per cluster and a fixed inter-cluster distance.
Our model is as follows. We introduce two stochastic variables for each channel cluster: $n_i$, the number of activated subunits, and $m_i$, the number of inhibited subunits. At every time step, the number of activated subunits $n_i$ at site $i$ is changed due to three stochastic processes; activation of additional subunits by binding available to their activation domains, de-activation by unbinding from active subunits, and inhibition by binding available to their inhibition domains. We take these transition rates to depend on the number of open channels at site $i$, $c_i$, and on the number of open channels at the nearest neighboring sites $i\pm 1$, $c_{i\pm 1}$. Similarly, there will be binding and unbinding to the inhibitory domain, changing $m_i$. We denote by $p^\pm_{0(1)}$ the probability to activate/inhibit a subunit per number of open channels at the same site (0) or the neighboring site (1). To compute the actual probabilities, we need to multiply these by the number of open channels. Here, we use the simple expedient of taking this to equal $n_i^h/hN_s^{h-1}$ where the total number of subunits $N_s=hN$; this is easily shown to be the expected number of open channels for large enough $N$. This approach allows us to avoid keeping explicit account of each of the independent subunits. Also, we let $p^\pm_d$ be the deactivation and deinhibition probabilities which are $c$ independent.
Let us define the total probabilities $p^\pm=p_0^\pm+2p_1^\pm$ and the “diffusion constant” $\alpha=p_1^\pm/(p_0^\pm+2p_1^\pm)$. We also denote $C_i(t)=(1-2\alpha) c_i(t)+\alpha
c_{i-1}(t) + \alpha c_{i+1}(t)$, which mimics the amount of calcium at site $i$ due to open channels at sites $i,\ i\pm1$. Our model explicitly consists of the following coupled stochastic processes. $n_i$ is updated $$n_i(t+\Delta t)=n_i(t)+\Delta^+_n -\Delta^-_n - \delta_+$$ where $\Delta^+_n$ is a random integer number drawn from the binomial distribution $B(\Delta^+_n,N_s-n_i(t)-m_i(t),p^+C_i(t))$, $\Delta^-_n$ is drawn from $B(\Delta_n^-,n_i(t),p^-C_i(t))$, and $\delta^+_n$ is drawn from $B(\delta^+_n,n_i(t),p_d^+)$. The equation for $m_i$ reads $$m_i(t+\Delta t)=m_i(t)+\Delta^+_m - \delta^+_m$$ where $\Delta^+_m$ is drawn from $B(\Delta^+_m,N_s-m_i(t),p^-C_i(t))$, and $\delta^+_m$ is drawn from $B(\delta^+_m,m_i(t),p_d^-)$. We do not allow for transitions from the inhibited state to the activated state. In all these formulas, $B(x,y,p) \equiv {y \choose x}
p^x (1-p)^{y-x}$. Note that the probability that 3 is bound is included by rescaling the number of subunits.
As a first step, we consider a simplified version of the channel dynamics with the inhibition process excluded (all $p^-$=0), i.e. a subunit is activated whenever is attached to its activating site. Thus we take $m_i=0$, and arrive at the one-variable model for the number of activated subunits $n_i$. Let us first focus on fairly small $N_s$. Examples of the stochastic dynamics for several values of parameters are shown in Figure \[sim1\]. At small $\alpha$, an initial seed almost always ultimately dies giving rise to so-called abortive calcium waves. At larger values of $\alpha$ the region of activated channels typically expands at a finite rate. This transition mirrors what has been seen in many experimental systems [@Parker2].
As is well known for statistical models such as the contact process[@Harris], the critical value of $\alpha$ can be accurately determined by computing the distribution of survival times $\Pi(t)$ for the activation process started from a single active site. For $\alpha<\alpha_c$, the distribution falls exponentially at large $t$ as the wave of activation eventually dies out. On the contrary, at $\alpha>\alpha_c$, $\Pi(t)$ asymptotically reaches a constant value $\Pi_\infty$, since a non-zero fraction of runs produce ever-expanding active regions. At $\alpha=\alpha_c$, the distribution function exhibits a power-law asymptotic behavior with the slope determined by the universality class of the underlying stochastic process. Our data (not shown) indicate that $\alpha_c$ is inversely proportional to the number of subunits per site $N_s$. We have checked that our data is in the [*directed percolation*]{} (DP)[@DP] class. For example, in Fig. \[survival\_h3\] we show $\Pi (t)$ of a cluster of open channels at the critical value of $\alpha_c$ for $h=3$, $N_s=10$ and $\gamma=0.1$. The power-law dependence is consistent with DP prediction of $\Pi
(t) \propto t^{-0.159}$. This is perhaps not too surprising. According to the Janssen-Grassberger DP conjecture[@JanGras], any spatio-temporal stochastic process with short range interactions, fluctuating active phase and unique non-fluctuating (absorbing) state, single order parameter and no additional symmetries, should belong to the DP class. This result does open up the exciting possibility that intracellular calcium dynamics could be an experimental realization of the DP process.
Figure \[sim1\](c) shows the opposite limit where the dynamics becomes almost deterministic. If we take $N_s\to\infty$ and fix $pN_s/h\to P$, we can use a mean-field description in terms of the fraction of activated subunits $\rho_i=n_i/N_s$, $$\dot{\rho_i}=((1-2\alpha)\rho_i^h+\alpha\rho_{i-1}^h + \alpha\rho_{i+1}^h)
(1-\rho_i) -\gamma\rho_i.
\label{mf}$$ and where we rescaled time $t'=Pt/\Delta t$ and introduced $\gamma=p_d/P$. For all $h\geq 2$, if $\gamma<\gamma_{cr}$ Eq.(\[mf\]) the system possesses two stable uniform solutions, $\rho=0$ and $\rho=\rho_0$ and one unstable solution $\rho_u$, where $\rho_{0,u}$ are real roots of the algebraic equation $\rho^{h-1}(1-\rho)=\gamma$. The front is a solution connecting these two stable fixed points; it is easy to show that this front has a unique propagation velocity.
For small $\alpha$, the discreteness of our spatial lattice causes the front to become pinned, as the probability of activating subunits at the neighboring site $O(\alpha\rho_0^h)$ becomes smaller than the threshold value for excitation probability $O(\rho_u)$. The stationary front solution is described by the recurrence relation, $$(1-2\alpha)\rho_i^h+\alpha\rho_{i-1}^h + \alpha\rho_{i+1}^h=
\frac{\gamma\rho_i}{1-\rho_i}
\label{stfront}$$ The bifurcation line which separates pinned and moving fronts, can be found in the limit of small $\alpha$ by using the ideas of ref.[@mitkov]. Indeed, in this limit, the values of $\rho_i$ quickly (as $\alpha^i$) approach 0 and $\rho_0$ away from the front at $i\to\pm\infty$, respectively. We can thus replace $\rho_i$ by $\rho_0$ and $0$ everywhere to the left and to the right of the front position except for $\rho_\pm$ at the two sites nearest to the front, $i-1$ and $i+1$. Solving the resulting set of two algebraic equations up to $\alpha^2$, one can obtain the values of $\rho_\pm$. At any $\gamma$, there is a critical value of $\alpha_m$ at which the real solution $\rho_\pm$ vanishes. The family of these values $\alpha_m$ forms the bifurcation line for front pinning in $(\gamma,\alpha)$ plane. At large $\alpha$, discreteness of the mean field model (\[mf\]) becomes insignificant, and (\[mf\]) can be replaced by its continuum limit $$\partial_t\rho=(\rho^h-\alpha\partial_{x}^2\rho^h)(1-\rho) -\gamma\rho.
\label{cont}$$ which of course has no front pinning. Instead, $\alpha$ can be scaled out and there is specific value of $\gamma$ at which the system goes from forward to backward propagating fronts. Figure \[phdiag\] shows the phase diagram of the mean field equation (\[mf\]) for $h=3$. All the data (except possibly at the non-generic case $\gamma=0$) are consistent with expected[@mitkov] $(\alpha-\alpha_m)^{1/2}$ scaling.
How does one get from DP behavior to deterministic pinning/depinning? To investigate this issue, we have performed simulations for the front speed as a function $\alpha$ at various finite values of $N_s$; with the results given in Fig. \[vel\_h3\]. At large $N_s$, the velocity approaches the mean field prediction as long as $\alpha>\alpha_m$. Close to critical value $\alpha_m$, the velocity deviates from the mean-field dependence $V\propto(\alpha-\alpha_m)^{1/2}$ because of thermally activated “creep”; fluctuations let the front to overcome potential barriers associated with finite site separation, and lead to exponentially slow front propagation (see, e.g., [@ampt]). Directed percolation regime is not observed at large $N_s$ since the DP critical value $\alpha_c$ is less than $\alpha_m$. At smaller $N_s$, the relative magnitude of the fluctuations grows, and the DP threshold value $\alpha_c$ exceeds $\alpha_m$. Now, the front pinning is determined by fluctuations rather than discreteness, and the critical state exhibits the properties of directed percolation.
Now we return to the full two-variable stochastic model which describes both activation and inhibition. Since the probability of binding to the inhibition domain is typically much smaller than those for the activation domain, the inhibitor dynamics is slow. In the mean-field limit $N_s\to\infty$, this model is similar to the FitzHugh-Nagumo model often used to describe waves propagating in excitable systems. One therefore expects that for a certain range of binding/unbinding probabilities, the model gives rise to pulse propagation; that is, once the wave passes, the system goes into a state dominated by inhibition from which it slowly recovers as the inhibitory domains slowly unbind. This is indeed what we find for large enough $N_s$, as shown in Fig. \[sim2\](a). Behind the pair of outgoing pulses, the channels stay refractory for a certain time $O(1/p^-)$ and then return to the quiescent state.
However, we find that having only a modest number of channels $N$ leads to fluctuations which strongly affect the spatio-temporal behavior of the model. In fact, a new dynamical state is formed behind the outgoing fronts, a state which remains active at all subsequent times (see Fig.\[sim2\],b). This state is catalyzed by backfiring, i.e. the creation of oppositely propagating waves behind a moving front. In the deterministic limit of our model, this cannot occur as the system is completely refractory once the front has passed. At finite $N$ however, propagation of the front does not lead to the activation and subsequent inhibition of all the channels. Instead, a finite number of these remain inactivated, providing a supply of active elements that can still support wave propagation. There exist more complicated deterministic models[@zimmerman], such as one proposed for $CO$ oxidation on single crystal surfaces[@Ertl], which also appear to have pulse-induced backfiring. There, however, this effect is due to the loss of pulse stability which occurs due to the rather complex non-linear dynamics of the inhibitory field. Here, it is the fluctuations which allow for this phenomenon.
We have checked that this backfiring-induced state occurs as well in more realistic and more complex models which solve for the calcium concentration together with the channel dynamics. Again, the mechanism appears to be the lack of complete inhibition in the wake of the propagating pulse. Hence, our result that one should find this behavior in intracellular calcium dynamics is not an artifact of any of the simplifying assumptions used here. Also, this state persists when the model is studied in higher dimensions. A study of the exact nature of the transition to backfiring and a comparison of the deterministic versus stochastic pathways to its existence will be undertaken in future work[@FLT].
In summary, we proposed and studied a simple discrete model of calcium channel dynamics based on the assumption that calcium diffusion time is much smaller than the characteristic times of binding/unbinding. This model demonstrates familiar properties of deterministic reaction-diffusion systems in the limit $N\to\infty$ when fluctuations are small. For small $N$, we observed a transition to a directed percolation regime, in agreement with the general DP conjecture[@JanGras]. For the full model including inhibition, we found at small $N$ a novel persistent fluctuation driven state which emerges behind a front of outgoing activation; this occurs in a parameter regime where the corresponding deterministic system exhibits only single outgoing pulses.
The authors thank H. Hinrichsen, M.Or-Guil and I. Mitkov for helpful discussions. LST thanks Max Planck Institut für Physik komplexer Systeme, Dresden, Germany for hospitality. LST was supported in part by the Engineering Research Program of the Office of Basic Energy Sciences at the US Department of Energy under grants No. DE-FG03-95ER14516 and DE-FG03-96ER14592. HL was supported in part by US NSF under grant DMR98-5735. M.F. was supported in part by DFG grant Fa350/2-1.
TTo whom correspondence should be addressed at falcke@mpipks-dresden.mpg.de M. J. Berridge, M. D. Bootman, P. Lipp, [*Nature*]{} [**395**]{} 645 (1998) J. Keizer, G.D.Smith, Biophys.Chem. [**72**]{} 87 (1998) J. Keizer, G.D.Smith, S.Ponce-Dawson, J.E.Pearson, 1998, Biophys.J. [**75**]{} 595 (1998) M. Falcke, L. S. Tsimring, H. Levine, to be published. G. W. DeYoung, J. Keizer, [*Proc. Natl. Acad. Sci. USA*]{} [**89**]{} 9895 (1992) J. Keizer, G. W. DeYoung, [*J. Theor. Biol.*]{} [**166**]{} 431, (1994) I. Bezprozvanny, J. Watras, B. E. Ehrlich, [*Nature*]{}, [**351**]{} 751 (1991) X. P. Sun, N. Callamaras, J. S. Marchant, I. Parker, [*Journal of Physiology*]{} [**509.1**]{}, 67 (1998) N. Callamaras, J. S. Marchant, X. P. Sun, I. Parker, [*Journal of Physiology*]{} [**509.1**]{}, 81 (1998) E. T. Harris, [*Ann. Prob.*]{} [**2**]{} 969 (1974) S. R. Broadbent and J. M. Hammersley. [*Proc. Camb. Phil. Soc.*]{} [**53**]{}, 629 (1957). H. K. Janssen, [*Z. Phys.*]{} [**B 42**]{}, 151 (1981); P.Grassberger, [*Z.Phys.*]{} [**B 47**]{}, 365 (1982). I. Mitkov, K. Kladko, and J. E. Pearson, [**]{} [**81**]{}, 5453 (1999). I. S. Aranson, B. Malomed, L. M. Pismen, L. S. Tsimring, submitted to [**]{}. M. G. Zimmerman, S.O. Firle. M. A. Natiello, M. Hildebrand, M. Eiswirth, M. Bär, A. K. Bangia and I. G. Keverkides, [*Physica D*]{} [**110**]{}, 92 (1997). M. Bär, N. Gottschalk, M. Eiswirth and G. Ertl, [*J. Chem. Phys.*]{}, [**100**]{}, 1202 (1994).
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[**STRUCTURES OF GRAVITATIONAL VACUUM AND THEIR ROLE IN THE UNIVERSE**]{}
Vladimir Burdyuzha${}^{1}$, J.A.de Freitas Pacheco${}^{2}$, Grigory Vereshkov${}^{3}$
${}^{1}$ Astro-Space Center, Lebedev Physical Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, Moscow 117810, Russia\
${}^{2}$ Observatoire de la Cote d’Azur, bld.de l Observatoire, 06304 Nice Cedex 4, France\
${}^{3}$ Department of Physics, Rostov State University, Stachki 194,Rostov/Don 344104, Russia\
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The production of gravitational vacuum defects and their contribution in energy density of the Universe are discussed. These topological microstructures could be produced as the result of defect creation of the Universe from “nothing” as well as the result of the first relativistic phase transition. They must be isotropically distributed on background of the expanding Universe. After Universe inflation these microdefects smoothed, stretched and broke up. Parts of them have survived and now they are perceived as the structures of $\Lambda$-term (quintessence) and unclustered dark matter. It is shown that for phenomenological description of vacuum topological defects of different dimensions (worm-holes, micromembranes, microstrings and monopoles) the parametrizational noninvariant members of Wheeler -DeWitt equation can be used. The mathematical illustration of these processes may be the spontaneous breaking of local Lorentz-invariance of quasi-classical equations of gravity. In addition, 3-dimensional topological defects revalues $\Lambda$-term.
INTRODUCTION
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Previously cosmology of gravitational vacuum was not practically discussed although the influence of gravity on a vacuum was considered[@B1]. On the other hand cosmology of other vacua is very often discussed [@B2]. To be more exact we can say that we shall consider structures of a gravitational vacuum condensate. Among fundamental interactions gravity plays the central role as the determinant of the space-time structure and as the arena of physical reality [@B3]. The question is to find the internal structure of gravitational vacuum starting from the quantum regime. The quantum regime of gravity has not been satisfactorily explained although many approaches have been done [@B4].
We present some analogy between known vacuum structures and hypothetical structures of gravitational vacuum (as it is known condensates of the quark-gluon type consist of topological structures - instantons). The general representation about topological defects is: 3-dimensional topological structures ($D = 3$) - worm holes; 2-dimensional topological structures ($D = 2$) -membranes; 1-dimensional topological structures ($D = 1$) -strings; point defects (singularities) ($D = 0$) - gas of topological monopoles. The full theory of vacuum defects is absent although we understand that the presence of defects breaks the symmetry of a system. The difficult question is to know the loss of which symmetry in quantum theory of gravity applies to the presence of topological defects of gravitational vacuum. Here it is necessary to use the experience of vacuum physics.
Even now formulation of the problem of the microscopic description of gravitational properties of vacuum is absent. From general considerations we may propose that strong fluctuations of topology could take place on the Planck scale. Probably in these fluctuations stable structures averaging characteristics of which are constant or slowly changing in time may exist. We have taken into account that among possible parametrization-noninvariant potentials of Wheeler-DeWitt equation are ones topological properties of which can be used for macroscopic description of the gas of topological defects (worm-holes, micromembranes, microstrings and monopoles). In our opinion, these circumstances allow us to propose a hypothesis on which the future theory of quantum gravity and the problem of quantum topological structures and parametrization-noninvariant of Wheeler-DeWitt equation will be solved in combination. In the frame of this hypothesis based on mathematical observations and analogies we suggest the mathematical apparatus of systematic modelling of the superspace metric and the effective potential permitting us to choose structures which are concerned with a cosmological point of view.
Thus we discuss gravitational vacuum in which energy-momentum characteristics depends on the radius of a closed Universe. Our arguments are only heuristic:
1\. this is the simplest mathematical model;\
2. our model is synthesized with general property of Wheeler-DeWitt equation for a closed isotropic Universe. Gauge invariance of classic theory of gravity has an additional aspect - the parametrizational invariance on the relative choice of variables on which are put gauge conditions. In classic theory a parametrization and a gauge are single operations the division of which on separate steps is conditional. In quantum geometrodynamics (QGD) the situation is different - the basic equations of QGD are gauge invariant but parametrization is gauge noninvariant [@B5]. Physical consequences of the parametrizational noninvariance are not general. Authors [@B6] have suggested to reject this problem and to fix the choice of gauge variables by the concrete way;\
3. also a conjecture argument has arisen after the introduction in cosmology of quintessence: a gravitation vacuum condensate may be considered as a possible factor of the breaking of local vacuum Lorentz-invariance.\
Of course gravitational vacuum condensate (as well as and its structures) may arise after the first relativistic phase transition [@B7] but in this article we remain in the frame of Wheeler-DeWitt conceptions on which phase transitions are absent.
Generally speaking the nature of dependence of vacuum energy from time is good unknown and the mathematical simulation has discussion character. Although compensation mechanisms have probably taken place (vacuum energy may be decreased by jumps in the result of negative contributions during relativistic phase transitions [@B8]). Models of quintessence [@B9] appeal mainly to classical fields (scalar one as example). We shall show that in quasiclassical approach quintessence can be simulated using mathematical structures arising naturally in Wheeler-DeWitt quantum geometrodynamics.
BASIC STATEMENTS
Probably the Universe creation is a quantum geometrodynamical transition from “nothing” (the state of “nothing” has geometry of zero volume (a=0)) in the state of a closed 3-space of small sizes having some particles and fields. That is the closed isotropic Universe was created through tunneling process with a metric:
$$ds^{2} = N^{2} dt^{2} - a^{2}(t) dl^{2} \eqno(1)$$
$$dl^{2} = d\rho^{2} + sin^{2}\rho(d\Theta^{2} + sin^{2}\Theta d\Phi^{2}) \eqno(2)$$
here $N =\sqrt{g_{00}}$ is a gauge variable which is necessary to fix before the solution of Einstein equations and a tensor energy-momentum(TEM): $T^{\mu}_{\nu} = diag(\epsilon, -p,-p,-p)$ For simplisity we have restricted by the consideration only 3-space, although a geometrodynamical transition in the state of a closed space of large dimensions may be also possible. As known in the modern epoch, vacuum energy has overcome the curvature (it is a dominant component of the Universe) and now the Universe expands accelerationally: $\Omega_{\Lambda} \sim 0.7; \Omega_{m} \sim 0.3$ and no contradiction that the Universe was created closed. Besides vacuum energy (if it was positive) was decreasing by jumps because of negative contributions during its relativistic phase transitions [@B8]. On the first stage we work with the classic theory in which energy-carriers are described by the hydrodynamical TEM and which is locally Lorentz-invariant (generally covariant) and which has the standard $\Lambda$-term. For a closed Universe Einstein equations are:
$$R^{0}_{0} - \frac{1}{2}R \equiv 3(\frac{\dot{a}^{2}}{N^{2}a^{2}} +
\frac{1}{a^{2}}) = \ae (\epsilon_{s} + \Lambda_{0}) \eqno(3)$$
$$R^{k}_{i} - \frac{\delta^{k}_{i}}{2}R \equiv
\delta^{k}_{i} [\frac{1}{N^{2}}(2 \frac{\ddot{a}}{a} -
2 \frac{\dot{N} \dot{a}}{Na} + \frac{\dot{a}^{2}}{a^{2}}) +
\frac{1}{a^{2}}] = \ae \delta^{k}_{i}(-p_{s} + \Lambda_{0}) \eqno(4)$$
$$-R \equiv 6[\frac{1}{N^{2}}(\frac{\ddot{a}}{a} -
\frac{\dot{N} \dot {a}}{Na} +
\frac{\dot{a}^{2}}{a^{2}}) + \frac{1}{a^{2}}] =
\ae(\epsilon_{s} - 3p_{s} + 4 \Lambda_{0}) \eqno(5)$$
Quantum theory of a closed Universe - quantum geometrodynamics[@B10] is based on the Wheeler idea about a superspace. This idea includes the manifold of all possible geometries of 3-space, matter and field configurations in which the Universe wave function is defined. Before quantization of classical theory, it is necessary to impart the form of Lagrange and Hamilton theory with couples (we are able to quantify only Hamilton theory but the construction of Hamilton theory precedes the construction of Lagrange one). Also evidently that two variables must be in Lagrange formulation of the theory: dynamical variable $a(t)$ relating to an equation of motion and some Lagrange multiplier $ \lambda = \lambda(t)$ relating to an equation of couple. From a infinite multiplicity of variants of the inserting of a Lagrange multiplier we choose the concrete variant related with quantum geometrodynamics. In quantum geometrodynamics the task arises which has not classical analogy: it is necessary to formulate the procedure of operators ordering on a generalized momentum and a coordinate. It is assumed that the procedure of ordering must be based on the covariance principle of Wheeler-DeWitt equation in Wheeler superspace. The metric of superspace $ \gamma(a)$ together with a Lagrange multiplier are inserted for this conception.
The explicit form of function $ \gamma (a)$ is not fixed but this conception allows us to understand in which terms the problem of parametrizational invariance is formulated. The abovementioned program is carried out if: $N = \frac{\lambda a}{\gamma(a)}$ where $ \lambda$ is a Lagrange multiplier, $ \gamma(a)$ is the metric of a superspace. Then we have:
$$3\left(\frac{\gamma^{2}{\dot a}^{2}}{\lambda^{2} a^{4}} + \frac{1}{a^{2}}
\right) = \ae \left[ \epsilon_{s}(a) + \Lambda_{0} \right] \eqno(6)$$
$$\frac{\gamma^{2}}{\lambda^{2} a^{2}} \left( 2 \frac{\ddot {a}}{a} -
2 \frac{\dot {\lambda} \dot {a}}{\lambda a} + 2 \frac{\dot {\gamma} \dot {a}}
{\gamma a} - 3 \frac{\dot {a}^{2}}{a^{2}} \right) -
\frac{1}{a^{2}} = \frac{\ae}{3} \left[ \epsilon_{s}(a)+a \cdot
\frac{d \epsilon_{s}(a)}{da}+ \Lambda_{0} \right] \eqno(7)$$
The system of equations which is mathematically equivalent to these is obtained by the variational procedure from some effective action. The gravitational part of this action is the known expression written for a isotropic Universe.
$$S_g=\int \left(\frac{1}{2\ae} R+ \Lambda_{0} \right){\sqrt {-g}}d^4x \eqno(8)$$
Over this expression some standard operations are necessary to perform: transformation of it in a quadratic form on generalized velocity $\dot a$ by excluding the total derivative; integration on volume of a closed Universe $V=2\pi^2a^3$; inserting of parametrizational of time. The effective action of matter and radiation is added to the received result using Rubakov-Lapchinsky recept [@B11]. This recept is very simple: the energy density of matter $\epsilon_{s}(a)$ depending on radius of the Universe in an effective Lagrangian and as $ \Lambda$-term has the status of an effective potential energy. Therefore, for receiving of right expression it is necessary to do a replacement $\Lambda_{0}$ to $\Lambda_{0} + \epsilon_{s}(a)$. The final expression for effective action is:
$$S_{\gamma} \{a, \lambda\}= \int L_{\gamma} (a, \lambda) dt, \qquad
L_{\gamma} (a, \lambda) = \frac{6 \pi^{2}}{\ae} \frac{1}{\lambda}
\gamma(a) \dot{a}^{2} - \lambda \frac{aU(a)}{\gamma(a)} \eqno(9)$$
where
$$U(a) = \frac{6 \pi^{2}}{\ae} a - 2 \pi^{2} a^{3} [\epsilon_{s} (a) +\Lambda_{0}]$$
is a total effective potential energy accounting topology of a closed Universe, standard $\Lambda$-term and matter. The variation of action on $\lambda (t)$ gives equation:
$$\frac{\delta S_{\gamma} \{a, \lambda\}}{\delta \lambda}=
\frac{\partial L_{\gamma} (a, \lambda)}{\partial \lambda}=-\frac{2 \pi^{2}}{\ae \gamma}
\left(\frac{3\gamma^2{\dot a}^2 }{\lambda^2}+3a^2-\ae a^4[\epsilon_{s}(a)+
\Lambda_{0} \right) = 0 \eqno(10)$$
which is mathematically equivalent to the equation of the couple. The variation on dynamical variable $a(t)$ gives the Lagrange equation:
$$\frac{\delta S_{\gamma} \{a, \lambda\}}{\delta a}=-\frac{d}{dt}\frac {\partial
L_{\gamma} (a, \lambda)}{\partial \dot a}+\frac {\partial L_{\gamma} (a,\lambda)}{\partial a}=0 \eqno(11)$$
After some transformations of the Lagrange equation we have:
$$\frac{d}{dt} \frac{\partial L_{\gamma}(a, \lambda)}{\partial \dot{a}}
- \frac {\partial L{\gamma} (a, \lambda)}{\partial a} \equiv$$
$$\frac{6 \pi^{2} \lambda}{\ae \gamma} \left\{ \frac{\gamma^{2}}{\lambda^{2}}
\left( 2 \ddot {a} +2 \frac{\dot {\gamma} \dot{a}}
{\gamma} - 2 \frac{\dot{\lambda} \dot{a}}{\lambda} - 3 \frac{\dot{a}^{2}}{a} \right)
- a - \frac{\ae a^{3}}{3} (\epsilon_{s}(a) +a \frac{d \epsilon_{s}(a)}{da} +
\Lambda_{0}) + J \right\} = 0 \eqno(12)$$
where
$$J=\frac{2 \pi^{2} \lambda}{\ae \gamma}\cdot \frac{d\ln (a^{-3} \gamma
(a))}{da} \left( \frac{3 \gamma^{2} \dot {a}^{2}}{\lambda^{2}} + 3a^2-\ae
a^4[\epsilon_{s}(a)+\Lambda_{0}]\right)$$
These equations produce a total system of the Lagrange model. Of course, these equations must be considered in combination. It is easy to see that for $J=0$ the last equation is mathematically equivalent to the combination of Einstein’s equations which have been written before. Thus we have proved that this model gives us the Lagrange method of the description of an isotropic Universe, energy-carriers of which are setted by functions of the scale factor $\epsilon_s(a),\;p_s(a)$. In classic theory this result has a methodical character only (Einstein’s equations in the initial form are more convenient to work in classic theory). However we want to transfer these to the quantum geometrodynamics in which Hamilton formulation is necessary. Hamilton model can be built on basis of Lagrange one. Note, that introducing of function $\gamma(a)$ is the operation of parametrization. The index $\gamma$ shows that action and Lagrangian correspond to the definite parametrization. A Hamiltonian of our system is built on standard rules:
$$H\Phi_{s}=0 \eqno(13)$$
$$H = P \dot{a} - L = \lambda (\frac{\ae}{24 \pi^{2}} \frac{1}{\gamma} P^{2}+
aU(a)) \eqno(14)$$
where $P = \frac{\partial L}{\partial \dot{a}} = \frac{12 \pi^{2}}{\ae}
\frac{\gamma}{a} \dot{a}$ is a generalized momentum. Note that there are also the parametrization problems of Wheeler-DeWitt theory. The first problem of the commutation connection is given by the operator $p$ with accuracy to $\hat{p} =-i \hbar \frac{\partial}{\partial a} +f(a)$. The second problem is the ordering of operators in the Hamiltonian (this one is created for any nontrivial function $\gamma(a)$). The partial solution of these problems is proposed in the frame of hypothesis of covariant differentiation in a curved space. In the frame of this hypothesis Wheeler-DeWitt equation has the view:
$$-\frac{\ae \hbar^{2}}{24 \pi^{2}} \frac{1}{\sqrt{\gamma}} \frac{d}{da}
\frac{1}{\sqrt{\gamma}} \frac{d \Phi_{s}(a)}{da} + \frac{1}{\gamma}
[ \frac{6 \pi^{2}}{\ae} a^{2} - 2 \pi^{2} a^{4} (\epsilon_{s}(a) + \Lambda_{0})]
\Phi_{s} (a) = 0 \eqno(15)$$
Here we have introduced the quantum index (s) numerating quantum states of matter and vacuum. The wave function of the Universe satisfies the condition:
$$\int^{\infty}_{0} \sqrt{\gamma} \; da \Phi^{\star}_{s}(a) \Phi^{\star}_{s'}
(a)= \delta (s - s^{'}) \eqno(16)$$
where $\delta(s - s^{'})$ is the delta function or the discrete $\delta$ symbol in dependence on the concrete properties of Wheeler-DeWitt equation solutions. Unfortunately the hypothesis of covariant differentiation does not totally decide parametrizational problems of the theory. For a more evidential effect of the parametrizational invariance the multiplicative redefinition of wave function is necessary:
$\Phi_{s} (a) = \gamma^{1/4} \Psi_{s} (a)$
and Wheeler-DeWitt equation is rewritten in the form:
$$(\frac{\ae \hbar}{12 \pi^{2}})^{2} \frac{d^{2} \psi_{s}}{d a^{2}} +
[a^{2} - \frac{a^{4}}{3} (\ae \epsilon_{s} (a) + \ae \Lambda_{0} +
\ae \epsilon_{GVC} (a))] \Psi_{s} (a) = 0 \eqno(17)$$
where
$$\epsilon_{GVC}(a) = \frac{\ae \hbar^{2}}{192 \pi^{2}} \frac{1}{a^{4}}
(\mu^{''} - \frac{1}{4} (\mu^{'})^{2}) \eqno(18)$$
Here we meant ${'}$ as a derivative of parametric function $\mu(a) =
ln\gamma(a)$ on the scale factor. All parametrizational noninvariant effects are collected in the function $\epsilon_{GVC} (a)$, which we have named the density of energy of gravitational vacuum (gravitational vacuum condensate $(GVC)$). As it is easy to see that parametrizational noninvariant effects are clean quantum ones, $\epsilon_{GVC} \sim \hbar^{2}$. The parametrization noninvariant contributions have not the physical generalization if we do not know the physical nature of their creation. These contributions have arised from nonconservation of classical symmetry on quantum level. The experience of modern quantum field theory speaks that a vacuum state rebuilds when a symmetry is broken. For this reason we have named parametrizational noninvariant contributions as the density of GVC energy.
However general symmetric arguments do not have the clear physical connection to vacuum energy. In this situation the examples from QCD are useful. In quantum theory classical conform and chiral symmetries do not conserve resulting in the appearence of a quark-gluon condensate. Probably concrete vacuum topological structures exist in a gravitation vacuum and they are the consequence of the parametrizational noninvariance of quantum geometrodynamics. From general considerations it is evident that presence in space-time of topological defects makes it impossible for any continuous transformations of coordinates and time. That is, concrete properties of defects permitting us to determine the parametrization of gauge variables.
COSMOLOGICAL APPLICATION
In cosmology this means that properties of topological microscopic defects inside space on average are isotropic and homogeneous (isotropization in brane gas cosmology is also a natural consequence of the dynamics [@B12]. They are contained in the function $\mu(a)$. We propose that all topological quantum defects with $D \ge 1$ have the typical Planck size. On this reason breaking up defects with a change of their number in variable volume $V = a^{3} (t)$ must take place. From simple consideration the number of defects in this volume are:
$N_{D} \sim (\frac{a}{l_{pl}})^{D}, \;\;\; l_{pl} = (G \hbar)^{1/2} =
\frac{(\ae \hbar)^{1/2}}{\sqrt{8 \pi}},$ here $c = 1$.
In accordance with these representations we wait untill the energy density of the system of topological defects contains a constant part corresponding to worm-holes and also members of type $1/a^{3}; 1/a^{2}; 1/a$ corresponging to gas of point defects, micromembranes and microstrings. Besides, the function $\epsilon_ {GVC}(a)$ must contain additional members describing interactions of microdefects between each other. Accounted representations correspond to the next choice of function $\mu(a)$:
$$\mu(a) = c_{0} ln a + c_{1} a + \frac{1}{2} c_{2} a^{2} + \frac{1}{3}
c_{3} a^{3}, \;\;\; c_{i} = const \eqno(19)$$
After this it get easy:
$$\Lambda_{0} + \epsilon_{GVC} (a) = \Lambda_{0} - \frac{\ae \hbar^{2}}
{768 \pi^{2}}\; c^{2}_{3} + \frac{\ae \hbar^{2}}{192 \pi^{2}}
[- \frac{1}{2} \; c_{2} c_{3} \frac{1}{a} - (\frac{1}{4} c^{2}_{2} +
\frac{1}{2} c_{1} c_{3}) \frac{1}{a^{2}} +$$ $$+ (2 c_{3} - \frac{1}{2} c_{1} c_{2} - \frac{1}{2} c_{0} c_{3}) \frac{1}{a^{3}} +
(c_{2} - \frac{1}{4} c^{2}_{1} - \frac{1}{2} c_{0} c_{2}) \frac{1}{a^{4}} -
\frac{1}{2} c_{0} c_{1} \frac{1}{a^{5}} - (c_{0} + \frac{c^{2}_{0}}{4})
\frac{1}{a^{6}}] \eqno(20)$$
The last three members can be interpretated as energy of gravitational interaction of defects between each other but their discussion is not the case since quasiclassical dynamics is only right in a region of large a. Note also that 3-dimensional topological defects revalues $\Lambda$-term. Observed value of $\Lambda$-term is:
$$\Lambda = \Lambda_{0} - \frac{\ae \hbar^{2}}{768 \pi^{2}} c^{2}_{3} \eqno(21)$$
Probably the term $\frac{1}{a^{3}}$ may be like to dark matter(DM). This gives the limitation on parameters of function $\mu(a)$:
$$\frac{1}{3} \; l^{4}_{pl}\; [2 c_{3} (1 - \frac{c_{0}}{a}) - \frac{1}{2} c_{1}
c_{2}] = \ae M \eqno(22)$$
where $M$ is a mass in volume $a^{3}$. As known Wheeler-DeWitt quantum geometrodynamics is the extrapolation of quantum-theoretical conceptions on the scale of the Universe as the whole. The initial state of the Universe from QGD point of view was located in the region of small values of the scale factor in a minisuperspace. From classical point of view the initial state of the Universe is a structureless singular state. Here we must postulate the defect creation of the Universe if it was born from “nothing”. After the release of defects, probably in our Universe, the stage of quick expansion (inflation) has taken place. In the result defects are smoothed, stretched and broken up. Some defects have left and perceived now as $\Lambda$-term (quintessence) and unclustered DM. Note again, that topological defects of gravitational vacuum may also be produced after the first relativistic phase transition, but according to the ideas of Wheeler-DeWitt, in the frame of which we are, phase transitions are absent. More fully physics of topological defects arising during phase transitions is discussed by T.Kibble [@B13].
CONCLUSION
We have taken into account that among parametrizational noninvariant potentials of Wheeler-DeWitt equation are ones which have macroscopic properties suitable to macroscopic description of a gas of topological defects (worm-holes, micromembranes, microstrings and monopoles). This circumstance allows us to propose that in the future theory of quantum gravity, the problem of topological structures of gravitational vacuum and parametrizational noninvariance of Wheeler-DeWitt equation will be solved jointly (authors [@B14] even attempted to parametrize an equation of state of dark energy). Also we have shown that quasi-classical corrections (which are proportional $\sim \hbar^{2}$) are completely defined by a superspace metric (that is $\gamma(a)$). This means that these corrections in the theory of gravity are not entirely defined by physics of the 4-dim space-time. In the frame of quantum geometrodynamics a part of the corrections having an influence on the evolution of the Universe in 4-dim is defined by physics of a superspace (that is they come from another level). If we interpret these corrections as defects then this means that defects appear as the result of the interaction of universes in this superspace. The development of these ideas may be realized in the frame of tertiary quatization. Besides, it has been noted that the property of Lorentz invariance attributed to 4-manifold joints 1-dim time and 3-dim space. Here 3-dim defects (worm-holes) give the contribution in the Lorentz-invariant $\Lambda$-term). Quantum topological defects with D=0,1,2 give Lorentz-noninvariant contributions in vacuum TEM. Thus we emphasize that topological defects of gravitational vacuum are quantum structures produced at the Planck epoch of the evolution of the Universe (Lorentz invariance at the Planck scale must probably be modified [@B15]). Topological defects of the gravitational vacuum shall be included in the composition of $\Lambda$-term (quintessence) and unclustered dark matter. Probably, this data allows us to improve our understanding of the content of the Universe’s main components.
[15]{} S.Coleman and F.De Luccia, Phys.Rev.21, 3305 (1980). M.Bordag, Phys.Rev.D. 67, 065001 (2003); P.Jaikumar and A.Mazumbar, Phys.Rew.Lett 90, 191301 (2003); S.Kachru, R.Kallosh, A.Linde, S.Trivedi, hep-th/0301240 v.2. S.V.Babak and L.P.Grishchuk, gr-qc/0209006; A.Zee hep-th/0309032. G.Preparata, S.Rovelli, and S.-S.Xue, gr-qc/9806044; T.Banks, hep-th/0305206 v.1. M.J.Duncan, Preprint UMH-TII-916/96. S.W.Hawking and D.N.Page, Nucl.Phys.B264, 185 (1986). V.Burdyuzha, O.Lalakulich, Yu.Ponomarev, and G.Vereshkov, Phys.Rev.D55, 7340R (1997). V.Burdyuzha, O.Lalakulich, Yu.Ponomarev, and G.Vereshkov, Preprint of Yukawa Institute for Theoretical Physics (YITP-98-50) and gr-qc/9907101. R.R.Caldwell, R.Dave, P.J.Steinhardt, Phys.Rev.Lett. 80 1582 1998. B.DeWitt, Phys.Rev.160, 1113 (1967). V.G.Lapchinsky and V.A.Rubakov, Acta Phys.Polonica B10, 1041 (1979). S.Watson and R.Branderberger, Phys.Rev.D67, 043510 (2003). T.Kibble “Topological Defects and Non-Equilibrium Dynamics of Symmetry Breaking Phase Transitions” Eds.Yu.Bunkov.H.Godfrin, Kluwer Academic Publishers, The Netherlands (2000). P.S.Corasaniti and E.J.Copeland, Phys.Rev.D67, 063521 (2003). J.Magueijo and L.Smolin, Phys.Rev.D67, 044017 (2003).
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---
abstract: 'In spite of the recent surge of interest in quantile regression, joint estimation of linear quantile planes remains a great challenge in statistics and econometrics. We propose a novel parametrization that characterizes any collection of non-crossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar, vector and function valued parameters. Statistical models based on this parametrization inherit a fast computation of the likelihood function, enabling penalized likelihood or Bayesian approaches to model fitting. We introduce a complete Bayesian methodology by using Gaussian process prior distributions on the function valued parameters and develop a robust and efficient Markov chain Monte Carlo parameter estimation. The resulting method is shown to offer posterior consistency under mild tail and regularity conditions. We present several illustrative examples where the new method is compared against existing approaches and is found to offer better accuracy, coverage and model fit.'
author:
- |
Yun Yang and Surya Tokdar\
[*University of California, Berkeley*]{} and [*Duke University*]{}
bibliography:
- 'qr.bib'
title: Joint Estimation of Quantile Planes over Arbitrary Predictor Spaces
---
Introduction
============
Quantile regression [QR; @Koenker1978; @Koenker2005] has recently gained increased recognition as a robust alternative to standard least squares regression, with applications to ecology, economics, epidemiology and climate science research [@Burgette2011; @Elsner2008; @Dunham2002; @Abrevaya01]. By offering direct inference on the non-central parts of a response distribution, QR allows researchers to identify and quantify a wide range of regression heterogeneity where the predictors affect the quartiles or the tails of the response distribution differently than its mean or median. This is illustrated in Figure \[f:mcycle\](a), adapted from @Koenker2005vig, showing the estimated conditional quantile curves for the well-known motorcycle data [@Silverman1985] with “Acceleration” (head acceleration, in g) as the response and “Time” (time from impact, in ms) as the explanatory variable. The estimates do a much better job of capturing the complex relationship between the two variables than what could be inferred through a simple mean regression or from more modern nonparametric density regression techniques [@DeIorio2004; @Tokdar2010] as shown in Figures \[f:mcycle\](b)-(c).
-- -- --
-- -- --
The estimates in \[f:mcycle\](a) were generated by using the original linear quantile regression technique of @Koenker1978. For a response proportion $\tau \in (0, 1)$ let $Q_Y(\tau | X) = \inf\{a: P(Y \le a | X) \ge \tau\}$ denote the $\tau$-th conditional quantile of a response $Y$ given a predictor vector $X$. The linear quantile regression model postulates $$\label{eq1}
Q_Y(\tau | X) = \beta_0(\tau) + X^T\beta(\tau),$$ which is equivalent to saying $Y = \beta_0(\tau) + X^T\beta(\tau) + U$ with the error variable satisfying $Q_U(\tau|X) \equiv 0$. The model is linear in the model parameters $(\beta_0(\tau), \beta(\tau))$. The predictor vector $X$ may include non-linear and interaction terms of the original covariates. In the motorcycle data analysis, we used B-spline transforms (df = 15) of Time as predictors, with $\dim(X) = 15$. The model parameters are easily estimated by linear programming and the estimates are consistent, asymptotically Gaussian and robust against outliers. Current literature on quantile regression (QR) is both deep and diverse; see [@Koenker2005] for a comprehensive overview and [@Tokdar2012] for references to Bayesian approaches.
Most scientific applications of QR require inference over a dense grid of $\tau$ values, which is usually done by assimilating inference from single-$\tau$ model fits [e.g., @Elsner2008]. Such assimilations are often problematic. In Figure \[f:mcycle\](a) the estimated curves cross each other violating laws of probability; the waviness and the local optima of the curves change wildly across $\tau$ reflecting poor borrowing of information; all quantile curves nearly collapse to a single point at boundary, where uncertainty should have been high due to data scarcity. Post-hoc rearrangement of the estimated quantiles [@Chernozhukov2011] avoids the embarrassing issue of crossing (Figure \[f:mcycle\](d)), but the other two problems persist.
Joint estimation of the conditional quantile planes requires working with the linear specification simultaneously for all $\tau \in (0,1)$. These specifications together define a valid statistical model, parametrized by function valued parameters $\beta_0 : (0,1) \to {\mathbb{R}}$, $\beta: (0, 1) \to {\mathbb{R}}^p$, provided $$\label{eq2}
\beta_0(\tau_1) + x^T \beta(\tau_1) \ge \beta_0(\tau_2) + x^T \beta(\tau_2),~\mbox{for every pair}~\tau_1 > \tau_2~\mbox{and for every}~x \in {\mathcal{X}}$$ where ${\mathcal{X}}$ is a pre-specified domain for $X$. Such models and related methods are a minority in the current quantile regression literature, and existing approaches have severe shortcomings. The methods by [@He1997] and [@Bondell2010] impose serious restrictions on the shape of $\beta(\tau)$. The procedure by [@Dunson2005], based on substitution likelihood, does not scale to dense $\tau$ grids and the role of substitution likelihood in Bayesian estimation remains debated [@Monahan1992]. [@Tokdar2012] provide a complete, scalable solution for the univariate case, but their handling of multivariate $X$ through univariate, single index projection is unsatisfactory.
To date, the most comprehensive treatment is given by @Reich2011 who utilize monotonicity properties of Bernstein basis polynomials with non-negative coefficients to ensure non-crossing quantile planes in any dimension. Their use of truncated Gaussian prior distributions on the non-negative coefficients leads to an attractive Gibbs sampling based Bayesian model fitting. However, both the model and the computing algorithm of [@Reich2011] crucially depend on the predictor domain ${\mathcal{X}}$ being a hyper-rectangle in ${\mathbb{R}}^p$. This is a fairly major handicap that may lead to a poor fit for reasons explained below.
The specification of ${\mathcal{X}}$ is a critical model choice in QR. Without loss of generality, ${\mathcal{X}}$ can be chosen convex because holds over ${\mathcal{X}}$ if and only if it holds over the convex hull of ${\mathcal{X}}$. The convex hull of the observed predictor vectors presents the most obvious practical choice. In spite of convexity, such an ${\mathcal{X}}$ may have a fairly irregular shape and may occupy only a fraction of the volume of the encompassing hyper-rectangle. The B-spline transforms of Time in the motorcycle data analysis live on a tiny 1 dimensional manifold in ${\mathbb{R}}^{15}$. Quantiles planes that are required to be non-crossing over the larger hyper-rectangle will appear mostly parallel within the original ${\mathcal{X}}$, as can be seen in Figure \[f:mcycle\](e). Unfortunately, such narrow predictor convex hulls are unavoidable whenever non-linear effects are sought within the Koenker-Bassett program or the measured covariates are naturally correlated. These are also situations where assimilation techniques exhibit dramatic crossing problems and hence a sound statistical model is most needed for joint estimation.
For an arbitrary convex ${\mathcal{X}}$, the space of $\beta_0$, $\beta$ curves satisfying is highly non-regular and unsuitable for statistical modeling and investigation. For the case of $p = 1$, [@Tokdar2012] provides a much simpler representation parametrized by two monotonically increasing curves over $(0,1)$. A generalization of this to any $p \ge 1$ and any convex ${\mathcal{X}}$ of arbitrary shape is currently not available in the literature. In this paper we propose a novel theory that delivers the right modeling platform for joint quantile regression. Our theory covers any dimension $p$ and any bounded convex ${\mathcal{X}}$ of arbitrary shape. It provides a complete characterization of joint quantile regression in terms of a collection of scalars, vectors and curves all but one of which are entirely constraint-free. Even the one curve with a constraint has only a mild shape restriction on it; it is required to live in the space of all CDFs on $(0,1)$ with full support. Our reparametrization leads to an easy likelihood score calculation in the model parameters, making it ideally suited to develop practicable methods by using either penalized likelihood or Bayesian techniques.
We build upon this novel theory to introduce a semiparametric Bayesian methodology for joint quantile regression over any ${\mathcal{X}}$ and any $p$, where the curve valued model parameters are assigned Gaussian process and transformed Gaussian process priors within a hierarchical setting. Asymptotic frequentist properties of the method are studied in Section \[s:theory\] and we establish posterior consistency over a broad class of true data generating distributions with linear quantile curves. For parameter estimation, we propose a Monte Carlo technique that incorporates efficient model space discretization, adaptive Markov chain sampling [@Haario1999] and reduced rank approximation [@tokdar07; @Banerjee2008]. We provide empirical evidence (Section \[s:simu1\]-\[s:cs\]) that our Gaussian process method enjoys much better estimation accuracy and coverage than the method by @Reich2011, and our estimates are comparable to regularized versions of the classical single-$\tau$ estimates. We consider the developments here make a strong case for linear quantile regression to be used as a model based inferential method rather than just an exploratory tool!
A novel theory of joint quantile planes estimation {#s:necsuff}
==================================================
Characterizing non-crossing hyperplanes
---------------------------------------
We focus only on the case where the response distribution is non-atomic and admits a probability density function conditionally at every $X$, and hence is equivalent to requiring $\dot\beta_0(\tau) + x^T\dot\beta(\tau) > 0$ for all $\tau \in (0,1), x \in {\mathcal{X}}$. Our theory could be extended to atomic response distributions with known atoms. Assume $0$ is an interior point of ${\mathcal{X}}$. This can be achieved without any loss of generality by a simple translation of the predictors once a suitable interior point is found within the convex hull of the observed predictors; see Appendix \[A:centering\] for more details. Define a map $b \mapsto a(b, {\mathcal{X}})$ on ${\mathbb{R}}^p\cup \{\infty\}$ as $$a(b, {\mathcal{X}}) = \left\{\begin{array}{ll} \sup_{x \in {\mathcal{X}}} \{-x^Tb\}/\|b\| & b \ne 0,\\\infty & b = 0.\end{array}\right.$$ Note that for every $b \ne 0$ we have $a(b, {\mathcal{X}}) \in (0, \infty)$ because ${\mathcal{X}}$ is bounded with 0 as an interior point.
\[thm:char\] Let ${\mathcal{X}}$ be a bounded convex set in ${\mathbb{R}}^p$ with zero as an interior point and let $\beta_0(\tau)$ and $\beta(\tau) = (\beta_1(\tau), \cdots, \beta_p(\tau))^T$ be real, differentiable functions in $\tau \in (0, 1)$. Then $\dot\beta_0(\tau) + x^T\dot\beta(\tau) > 0$ for all $\tau \in (0, 1)$ at every $x \in {\mathcal{X}}$ if and only if $$\begin{aligned}
\dot\beta_0(\tau) > 0,~~\dot\beta(\tau) = \dot\beta_0(\tau) \frac{v(\tau)}{a(v(\tau), {\mathcal{X}}) \sqrt{1 + \|v(\tau)\|^2}},~~\tau \in (0,1), \label{c2}\end{aligned}$$ for some $p$-variate, real function $v(\tau) = (v_1(\tau), \cdots, v_p(\tau))^T$ in $\tau \in (0, 1)$.
Suppose holds. For any $\tau \in (0,1)$ either $v(\tau) = 0$ in which case $\dot\beta(\tau) = 0$ and so $\dot\beta_0(\tau) + x^T\dot\beta(\tau) > 0$ at every $x \in {\mathcal{X}}$. Otherwise, if $v(\tau) \ne 0$ then at any $x \in {\mathcal{X}}$. $$\begin{aligned}
\dot\beta_0(\tau) + x^T\dot\beta(\tau) & = \dot\beta_0(\tau)\left\{1 + \frac{x^T v(\tau)}{a(v(\tau), {\mathcal{X}}) \sqrt{1+ \|v(\tau)\|^2}} \right\}\\
& = \dot\beta_0(\tau) \left\{1 - \sqrt{\frac{\|v(\tau)\|^2}{1 + \|v(\tau)\|^2} } \frac{\{-x^T v(\tau) \}/ \|v(\tau)\|}{a(v(\tau), {\mathcal{X}})}\right\}\\
& \ge \dot\beta_0(\tau)\left\{1 - \sqrt{\frac{\|v(\tau)\|^2}{1 + \|v(\tau)\|^2} } \right\}\\
& > 0\end{aligned}$$
We must have $\dot\beta_0(\tau) > 0$ for all $\tau \in (0, 1)$ because ${\mathcal{X}}$ contains 0. For any $\tau \in (0, 1)$, if $\dot\beta(\tau) = 0$, set $v(\tau) = 0$. Otherwise, $\dot\beta_0(\tau) > \{-x^T\dot \beta(\tau)\}$ at every $x \in {\mathcal{X}}$ and hence $\dot\beta_0(\tau) > \|\dot\beta(\tau)\|a(\dot\beta(\tau), {\mathcal{X}})$. So the positive scalar $c(\tau) = [ [\dot\beta_0(\tau)/\{\|\dot\beta(\tau)\|a(\dot\beta(\tau), {\mathcal{X}})\}]^2 - 1]^{-1/2}$ satisfies $\|\dot\beta(\tau)\| a(\dot\beta(\tau), {\mathcal{X}}) / \dot\beta_0(\tau) = c(\tau) / \{1 + c(\tau)^2\}^{1/2}$. Set $$v(\tau) = c(\tau)\dot\beta(\tau) / \|\dot \beta(\tau)\|,~~\tau \in (0, 1).$$ A $v(\tau)$ constructed as above defines a real $p$-variate function on $\tau \in (0, 1)$ and satisfies .
An almost constraint-free parametrization of linear quantile regression
-----------------------------------------------------------------------
Theorem \[thm:char\] greatly reduces the monotonicity constraint on the quantile hyperplanes to that on a single function $\beta_0(\tau)$. Construction of a single monotone function is a relatively easy task, but some care is needed in handling the range $(\beta_0(0), \beta_0(1))$, which corresponds to the support of the conditional density of $Y$ given $X = 0$. We pursue a model for $\beta_0(\tau)$ based on a user specified or default “prior guess” $f_0(y)$ for this conditional density. In the special case where $(\beta_0(0), \beta_0(1))$ is a known finite interval, $f_0$ could be chosen with support equal to the same interval. In general $f_0$ should be chosen to have support $(-\infty, \infty)$, such as a standard normal density, or a Student-t density with a modest degrees of freedom if the response distribution is expected to have heavy tails. We focus only on this general case, although the model described below could be easily modified to $f_0$ supported on a bounded interval.
Let $f_0$ have support $(-\infty, \infty)$ and define its cumulative distribution function $F_0(y) = \int_{-\infty}^y f_0(z)dz$, quantile function $Q_0(\tau) = F_0^{-1}(\tau)$ and quantile density $q_0(\tau) = \dot Q_0(\tau)$. Let $\tau_0 = F_0(0)$; by the full support assumption, $0 < \tau_0 < 1$. We pursue a model for $\beta_0$ and $\beta$ as follows $$\begin{aligned}
& \beta_0(\tau_0) = \gamma_0,~~\beta(\tau_0) = \gamma \label{eq:tau0}\\
& \beta_0(\tau) - \beta_0(\tau_0) = \sigma \int_{\zeta(\tau_0)}^{\zeta(\tau)} q_0(u) du, ~~~ \tau \in (0,1) \label{eq:beta0}\\
& \beta(\tau) - \beta(\tau_0) = \sigma \int_{\zeta(\tau_0)}^{\zeta(\tau)} \frac{w(u)}{a(w(u), {\mathcal{X}})\sqrt{1 + \|w(u)\|^2}}q_0(u)du,~~~ \tau \in (0,1) \label{eq:beta}\end{aligned}$$ with model parameters $\gamma_0 \in {\mathbb{R}}$; $\gamma \in {\mathbb{R}}^p$; $\sigma > 0$; $w : (0, 1) \to {\mathbb{R}}^p$, an unconstrained $p$-variate function on $(0,1)$; and $\zeta : [0, 1] \to [0, 1]$, a differentiable, monotonically increasing bijection, i.e., a diffeomorphism, of $[0,1]$ onto itself. We write $(\beta_0, \beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$ to indicate $\beta_0, \beta$ defined as in -.
All model parameters, except the diffeomorphism $\zeta$, are essentially unconstrained. The function space of $\zeta$ is simply the space of cumulative distribution functions associated with all probability densities with support $[0,1]$. Such function spaces are easy to handle for statistical model fitting; a simple approach is presented in Section 3. Note that when $\zeta$ is the identity map of $(0,1)$ onto itself and $w(\tau) \equiv 0$, we get $\beta_0 = \sigma Q_0$, $\beta = 0$, and the resulting joint, linear quantile regression model simplifies to a standard homogeneous, linear model: $Y_i = \gamma_0 + X_i^T\gamma + \sigma \epsilon_i$ with $\epsilon_i \sim f_0$. This model indeed provides a complete representation of all $\beta_0$, $\beta$ satisfying the non-crossing condition , subject to a matching range criterion, as detailed below.
\[thm:model\] Let $\beta_0: (0,1) \to {\mathbb{R}}$, $\beta: (0,1) \to {\mathbb{R}}^p$ be differentiable with $(\beta_0(0), \beta_0(1)) = (-\infty, \infty)$ \[defined in the limit\]. Then holds if and only if $(\beta_0, \beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$ for some $\gamma_0 \in {\mathbb{R}}$, $\gamma \in {\mathbb{R}}^p$, $\sigma > 0$, $w : (0,1) \to {\mathbb{R}}^p$, and, $\zeta$, a diffeomorphism from $[0,1]$ onto itself $[0,1]$.
If $(\beta_0,\beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$ then, $$\dot\beta_0(\tau) = \sigma q_0(\zeta(\tau)) \dot\zeta(\tau), ~~\dot\beta(\tau) = \dot\beta_0(\tau) \frac{v(\tau) }{a(v(\tau), {\mathcal{X}}) \sqrt{1 + \|v(\tau)\|^2}},$$ with $v(\tau) = w(\zeta(\tau))$. Hence, by Theorem \[thm:char\], we only need establish that any real, differentiable function $\beta_0$ on $(0,1)$, with $\dot\beta_0(\tau) > 0$ for all $\tau \in (0,1)$ and $\beta_0(0) = \infty$, $\beta_0(1) = \infty$, can be constructed as in for some diffeomorphism $\zeta:[0,1]\to[0,1]$ and $\sigma > 0$. This is indeed true, since one could fix $\sigma > 0$ arbitrarily, and then take, $$\zeta(\tau) = F_0\left(\gamma_0 + \frac{\beta_0(\tau)-\gamma_0}\sigma \right),~~\tau \in (0,1),$$ which is differentiable and monotonically increasing in $(0,1)$ since $\dot\zeta(\tau) = f_0(\gamma_0 + (\beta_0(\tau) - \gamma_0)/\sigma) \dot\beta_0(\tau) > 0$ for all $\tau \in (0,1)$, and, $\zeta(0) = F_0(-\infty) = 0$, $\zeta(1) = F_0(\infty) = 1$.
When $\beta_0(0)$ or $\beta_0(1)$ is finite (or both), we can still write $\beta_0$ as in , but we need either $\zeta(0) > 0$ or $\zeta(1) < 1$ (or both). While such a $(\beta_0, \beta)$ does not strictly belong within our model space, they can be approximated arbitrarily well by a model element $\mathcal{T}(\gamma_0, \gamma, \sigma, v, \zeta)$, and consistently estimated from large samples (Lemma \[le:6\] and Section \[s:theory\]).
Likelihood evaluation {#ss:le}
---------------------
A salient feature of a valid specification of $Q_Y(\tau|x)$ for all $\tau \in (0,1)$ is that it uniquely defines the conditional response density $f_Y(y|x)$ over $x\in {\mathcal{X}}$, given by $$f_Y(y|x) = \left.\frac{1}{\frac{\partial}{\partial\tau} Q_Y(\tau | x)}\right|_{\tau = \tau_x(y)}$$ where $\tau_x(y)$ solves $Q_Y(\tau | x) = y$ in $\tau$ [@Tokdar2012]. Consequently, one can define a valid log-likelihood score $$\label{eq8}
\sum_{i}\log f_Y(y_i|x_i)=-\sum_i\log\big\{\dot{\beta}_0\big(\tau_{x_i}(y_i)\big)+
x_i^T\dot{\beta}\big(\tau_{x_i}(y_i)\big)\big\}.$$ in the model parameters based on observations $(x_i, y_i)$, $i = 1, \ldots, n$. From -, we could write $$\dot \beta_0(\tau) + x^T \dot \beta(\tau) = \sigma q_0(\zeta(\tau))\dot \zeta(\tau) \left\{1 + \frac{w(\zeta(\tau))}{a(w(\zeta(\tau), {\mathcal{X}})\sqrt{1+ \|w(\zeta(\tau))\|^2}}\right\}$$ and therefore a quick evaluation of the log-likelihood score is possible once we figure out $\tau_{x_i}(y_i)$ for each $i = 1, \ldots, n$, by solving $\tau = \int_{\tau_0}^\tau \{\dot\beta_0(u) + x_i^T \dot \beta(u)\}du$.
With enough resources, these numbers could be found up to any desired level of accuracy through standard numerical methods for integration and root finding. But for all practical needs, model fitting and inference could be restricted to a dense grid of $\tau \in \{t_1, \ldots, t_L\}$, for which finding $\tau_{x_i}(y_i)$ requires only a simple sequential search involving trapezoidal approximations to the integral of $\dot\beta_0(\tau) + x^T\dot\beta(\tau)$. Algorithm \[algo:loglik\] presents a pseudo-code for likelihood evaluation involving only simple matrix and vector multiplication. The code runs extremely fast when implemented in any low-level programming language with quick “for loops”. In our numerical studies we used a C implementation which offered 1000 likelihood evaluations in 2 seconds on an Intel(R) Core(TM) i7-3770 machine with $n = 1000$, $p = 7$ and a grid over $\tau$ with mesh size 0.01.
A practical issue with a discrete grid of $\tau$ is that it needs to cover the image of the data range mapped into the quantile space, while ensuring the grid length $L$ remains manageable. In our implementations we chose a data dependent grid as follows. We used equispaced grid points between $\tau = 0.01$ and $\tau = 0.99$ with an increment of $0.01$. Next, on the upper tail, we augmented the grid with new grid points $0.995$, $0.9975$, $\cdots$ until we covered $\tau = 1 - 1/(2n)$ where $n$ is the sampler size. Same augmentation strategy with geometrically reducing increment lengths were adopted on the lower tail to reach up to $\tau = 1/(2n)$.
Bayesian inference with hierarchical Gaussian process priors {#s:prior}
============================================================
Prior specification
-------------------
We adopt a Bayesian approach to parameter estimation with suitable prior distributions on the model parameters, including the function valued parameters $\zeta$ and $w = (w_1, \ldots, w_p)$. It is useful that $w_j$s are completely unrestricted, allowing us to handle them with Gaussian process prior distributions. For handling $\zeta$, we first introduce a constraint free version $w_0:(0,1) \to {\mathbb{R}}$ related to $\zeta$ through the “logistic transformation”: $$\zeta(\tau)=\frac{\int_0^\tau e^{w_0(u)}du}{\int_0^1 e^{w_0(u)}du}, \;\;\tau \in (0,1),
\label{eq:zeta0}$$ and use a Gaussian process prior on $w_0$; see [@Lenk88; @tokdar07] for similar uses in density estimation.
Recall that a Gaussian process $g = \{g(\tau): \tau \in (0,1)\}$ could be viewed as a random element of the Banach space of real valued functions on $(0,1)$ equipped with the supremum norm. Every Gaussian process $g$ is characterized by two functions, the mean function $m(\tau) = {\rm E} g(\tau)$ and the non-negative definite covariance function $c(\tau, \tau') = {\rm Cov}(g(\tau), g(\tau'))$, and we use the label $GP(m, c)$ to denote such a process. When $g \sim GP(m, c)$, for any finite set of points $\{\tau_1, \ldots, \tau_k\}$ the random vector $(g(\tau_1), \ldots, g(\tau_k))$ has a $k$-variate Gaussian distribution with mean $(m(\tau_1), \cdots, m(\tau_k))^T$ and $k\times k$ covariance matrix with elements $c(\tau_i, \tau_j)$.
Our prior specification can be expressed in the following hierarchical form: $$\begin{aligned}
w_j & \sim GP(0, \kappa_j^2 c^{\textit{\tiny SE}}(\cdot,\cdot|\lambda_j)),~~j = 0,\ldots,p \label{eq:w}\\
(\kappa_j^2, \lambda_j) & \sim \pi_k(\kappa_j^2)\pi_\lambda(\lambda_j), ~~ j = 0,\ldots, p\\
(\gamma_0, \gamma, \sigma^2) & \sim \pi(\gamma_0,\gamma,\sigma^2) \propto \frac1{\sigma^2},\end{aligned}$$ where $c^{\textit{\tiny SE}}(\tau, \tau'|\lambda^2) = \exp(-\lambda^2 (\tau - \tau')^2)$ is the so-called square exponential covariance function equipped with a rescaling parameter $\lambda$ . This particular choice of the covariance function is motivated by two facts. First, for any fixed $\lambda > 0$, the probability distribution $GP(0, c^{\textit{\tiny SE}}(\cdot,\cdot|\lambda))$ assigns 100% probability to the set of all continuous functions on $(0,1)$ and hence our prior specification does not [*a-priori*]{} rule out any valid specification of the joint linear QR model. Second, $\lambda$ plays the role of a bandwidth parameter for the sample paths generated from $GP(0, c^{\textit{\tiny SE}}(\cdot,\cdot|\lambda))$, with more wavy paths realized as $\lambda$ gets larger. In a seminal work, show that with a suitable prior distributions specified on $\lambda$, the resulting rescaled square-exponential Gaussian process prior offers adaptively efficient estimation in nonparametric mean regression and density estimation problems by automatically adjusting $\lambda$ to attain optimal smoothing.
For specifying $\pi_\lambda$, it is more insightful to fix a small $h > 0$ and consider the quantity $\rho_h(\lambda) = \exp(-h^2\lambda^2)$, which gives the correlation between $w_j(\tau)$ and $w_j(\tau + h)$ given $\lambda_j = \lambda$, and assign $\rho_h(\lambda)$ a $Be(a_\lambda,b_\lambda)$ prior. In our applications we use $h = 0.1$, $a_\lambda = 6$ and $b_\lambda = 4$, which assigns 95% mass to $\rho_{0.1}(\lambda) \in (0.3, 0.86)$. However, in our experience, the method shows little sensitivity to these choices. We take $\pi_\kappa$ to be $IG(a_\kappa, b_\kappa)$, the inverse gamma pdf with shape $a_\kappa$ and rate $b_\kappa$. The inverse gamma choice allows us to integrate out all $\kappa_j$ parameters at the time of model fitting. In our applications, we use $a_\kappa = b_\kappa = 3/2$, which is small enough to ensure a reasonably diffuse marginal prior on each $w_j$ while retaining a finite second moment.
Our choice of $\pi_\kappa$ and the right Haar prior on the location scale parameters $(\gamma_0, \gamma, \sigma^2)$ is partially motivated by our numerical experimentations in which we found these choices to lead to estimates and credible intervals most similar to the Koenker-Basette estimates and confidence intervals. Other reasonable choices could be made and we discuss in Section \[s:dis\] choices that offer useful shrinkage properties.
When no special information is available about the support of $Y$, we take $f_0$ to be a Student t-distribution with an unknown degrees of freedom parameter $\nu$ and assign $\nu/6$ a standard logistic prior distribution. The logistic prior is reasonably diffuse and helps the resulting method adapt well to a wide spectrum of tail behavior of the response distribution.
Model fitting via discretization and adaptive blocked Metropolis {#s:comp}
----------------------------------------------------------------
With likelihood evaluation discretized over a grid of $\tau$ values $\{t_1, \ldots, t_L\}$ as in Algorithm \[algo:loglik\], the curve valued parameters $w_j$, $1\le j \le p$ are needed to be tracked only over the specified grid, reducing each curve to a parameter vector of length $L$. The same applies to $w_0$ from which $\dot\zeta$ and $\zeta$ could be obtained on the grid by using the trapezoidal rule of integration. While it is theoretically possible to fit the model by running a Markov chain Monte Carlo over these parameters vector and the other model parameters, such a strategy is not entirely practicable. The parameter vector derived from any $w_j$ is conditionally an $L$ dimensional Gaussian variable given $\lambda_j$ and $\kappa_j$, and evaluating its log prior density requires factorizing or inverting a $L\times L$ covariance matrix which has an $O(L^3)$ computing complexity. Furthermore, a Markov chain sampler that operates on both these parameter vectors and the rescaling parameters $\lambda_j$s run into serious mixing problems.
To overcome these difficulties, we use two sets of further discretization. First, we replace $\pi_\lambda$ with a dense, discrete approximation covering the range $\rho_{0.1}(\lambda) \in (0.05, 0.95)$. Let $\pi^*_\lambda$ denote the approximating probability mass function with support points $\{\lambda^*_1, \ldots, \lambda^*_G\}$. We choose the support points to be more densely packed for smaller $\lambda$ values, the rational behind this and the exact manner in which the grid is chosen are discussed in Appendix \[a:support\].
Next, we fix a set of uniformly spaced knots $\{t^*_1, \ldots, t^*_m\} \subset[0,1]$, for some $m$ much smaller than $L$ and replace each $w_j$ curve with $$\tilde w_j(\tau) := E\{w_j(\tau) | w_j(t^*_1), \ldots, w_j(t^*_m)\}, \tau \in (0,1),
\label{eq:pp}$$ which provides an interpolation approximation to $w_j$ over $(0,1)$, passing through the points $(t^*_k, w_j(t^*_k))$, $k = 1, \ldots, m$, and determined entirely by the $m$-dimensional vector $W_{j*} = (w_j(t^*_1), \ldots, w_j(t^*_m))^T$, whose prior density evaluations require only $O(m^3)$ flops. Such interpolation based low rank approximations to Gaussian process priors are widely used in statistics and machine learning literature, see for example, .
Our treatment here, however, differs slightly from the above papers in that we carry out the conditional expectation in after marginalizing out both $\lambda_j$ and $\kappa_j$. Let $\tilde W_j$ denote the $L$-dimensional vector $(\tilde w_j(t_1), \ldots, \tilde w_j(t_L))^T$ that is needed for the likelihood evaluation. Then we can write, $$\tilde W_j = \sum_{g = 1}^G p_g(W_{j*})A_g W_{*j}$$ where $A_g$ denotes the $L\times m$ matrix $C_{o *}(\lambda_g)C_{**}(\lambda_g)^{-1}$ with $C_{o*}(\lambda_g) = ((c^{\textit{\tiny SE}}(t_l, t^*_k|\lambda_g)))_{l,k=1}^{L,m}$ and $C_{**}(\lambda_g) = ((c^{\textit{\tiny SE}}(t^*_l, ^*t_k|\lambda_g)))_{l,k=1}^{m}$, and $p_g(W_{j*}) \propto \pi^*_\lambda(\lambda_g)p(W_{j*} | \lambda_g)$ with $$p(W_{j*} | \lambda_g) \propto \pi^*_\lambda(\lambda_g)\left\{1 + \frac{W_{j*}^TC^{-1}_{**}(\lambda_g)W_{j*} }{2b_\kappa}\right\}^{-(a_\kappa + m/2)} \frac{\Gamma(a_\kappa + m/2)b_\kappa^{-m/2}}{\Gamma(a_\kappa)},$$ the multivariate t-density of $W_{j*}$ given $\lambda_j = \lambda_g$. Also notice that the marginal prior density of $W_{j*}$ is precisely $\sum_{g = 1}^G \pi^*_\lambda(\lambda_g) p(W_{j*}|\lambda_g)$.
With the help of the above sets discretization, our joint QR model is entirely determined by the $(m+1)(p+1)+2$ dimensional parameter vector $\theta = (W_{0*}^T, \ldots, W_{p*}^T, \gamma_0, \gamma^T, \sigma^2, \nu)^T$ and model fitting may be carried out by running a Markov chain sampler on $\theta$ followed by Monte Carlo approximations of posterior quantities. In our experience, an adaptive blocked Metropolis sampler has worked extremely well, offering fast mixing and reproducible results. For this sampler, we use $p+3$ block updates of $\theta$ per iteration of the sampler, where the first $p+1$ blocks are given by $(W_{j*}^T, \gamma_j)^T$, $j = 0,\ldots,p$ and the last two blocks are $(\gamma_0, \gamma^T)^T$ and $(\log \sigma^2, \log \nu)^T$. For each block, we perform a random walk Metropolis update governed by a multivariate Gaussian proposal distribution centered at the current realization of the block and with covariance that is slowly adapted to resemble, up to a scaler multiplication, the posterior covariance matrix of the block, where the scaler multiplier is also adapted slowly to achieve a pre-specified acceptance rate. We carry out these updates according to Algorithm 4 in @andrieu.thoms.
In our implementation, we precompute and save the matrix $A_g$ and a Cholesky factor $R_g$ of $C_{**}(\lambda_g)$ for every $g = 1, \ldots, G$ and plug them into the likelihood and prior density evaluations during Markov chain sampling. The precomputation step adds little overhead cost but results in a big jump in computing speed by drastically reducing the computing time for each Markov chain iteration.
Posterior consistency {#s:theory}
=====================
Frequentist justification of Bayesian methods are often presented in the form asymptotic properties of the posterior distribution. A basic desirable property is posterior consistency: the posterior mass assigned to any fixed neighborhood of the true data generating model element should converge to 1 in probability or almost surely as sample size goes to infinity. More refined evaluations of asymptotic properties emerge through posterior convergence rate calculations, where one considers a sequence of shrinking neighborhoods and calibrates the fastest rate of shrinkage for which the posterior mass assigned to these neighborhoods still converges to 1.
We restrict only to a study of [*weak*]{} posterior consistency of the Gaussian process based QR method developed in this paper. For a formal treatment, we consider a stochastic design setting where $X_i$s are drawn independently from a pdf $f_X$ on ${\mathcal{X}}$. Since any valid specification of the quantile planes $\{Q_Y(\tau|x)$: $\tau \in (0,1)$, $x \in {\mathcal{X}}\}$ uniquely corresponds to a specification of conditional response densities $\{f_Y(y|x) : y\in {\mathbb{R}}, x \in {\mathcal{X}}\}$, it also uniquely corresponds to a bivariate density function $f(x, y) = f_X(x)f_Y(y|x)$ under the stochastic design assumption. Hence our prior specification on the quantile planes induces a prior probability measure $\Pi$ on the space $\mathcal{F}$ of probability density functions on ${\mathcal{X}}\times {\mathbb{R}}$. If $f^*(x,y) = f_X(x)f^*_Y(y|x)$ is the true data generating element in this space, then the posterior is said to be weakly consistent at $f^*$ if $\Pi(U|(X_i,Y_i), i = 1, \ldots, n) \to 1$ almost surely for every weak neighborhood $U$ of $f^*$ in $\mathcal{F}$.
The celebrated Schwartz Theorem [@Schwartz1965] provides a fairly sharp sufficient condition for weak posterior consistency of $\Pi$ at $f^*$. Let $d_{KL}(p,q) := \int p \log (p/q)$ denote the Kullback-Leibler (KL) divergence. For any $f\in\mathcal{F}$ and $\epsilon > 0$, let $K_{\epsilon}(f)$ denote the $\epsilon$-KL neighborhood $\{g\in\mathcal{F}: d_{KL}(f,g)<\epsilon\}$. We say that $f^*$ is in the KL support of $\Pi$ if $\Pi(K_{\epsilon}(f^*))>0$ for all $\epsilon > 0$. @Schwartz1965 proved
\[thm:2\] The posterior is weakly consistent at $f^* \in \mathcal{F}$ if $f^*$ is in the KL support of $\Pi$.
We show that an $f^*$ with linear conditional quantiles $Q^*_Y(\tau|x) = \beta^*_0(\tau) + x^T\beta^*(\tau)$ belongs to the KL support of $\Pi$ under mild smoothness and tail conditions. Tail conditions are needed to ensure that $d_{KL}(f^*_Y(\cdot|x), f_Y(\cdot|x)) < \infty$, which holds when $f^*_Y(\cdot|x)$ has tails decaying faster than those of $f_Y(\cdot|x)$, with $f$ generated from $\Pi$. With our choice of $\Pi$, the tails of $f_Y(\cdot|x)$ are expected to be similar to those of $f_0$, and hence, a minimum requirement is that the tails of $f^*_Y(\cdot|x)$ decay faster than those of $f_0$. We make the notion of faster tail decay more precise with the following definitions.
Let $f$ be a probability density function on ${\mathbb{R}}$ with quantile function $Q$. Take $m = Q(\tau_0)$. All statements below are interpreted with respect a given $f_0$.
1. We say $f$ has a type I left tail if $Q(0) > -\infty$, and, for every $\sigma > 0$, $$\frac{\frac1\sigma f_0(m + \frac{Q(t) - m}\sigma)}{f(Q(t))} \to c_L(\sigma) \in (0, \infty), ~\mbox{as}~t \downarrow 0,
\label{eq:type1}$$ with, $c_L(\sigma) \to 0$ as $\sigma \downarrow 0$.
2. We say $f$ has a type II left tail if for every $\sigma > 0$, ${\frac1\sigma f_0(m + \frac{Q(t) - m}\sigma)}/{f(Q(t))} $ diverges to $\infty$ as $t \downarrow 0$ and, $$u_L(\sigma) := \inf\left\{t > 0: \frac{\frac1\sigma f_0(m + \frac{Q(t) - m}\sigma)}{f(Q(t))} \le 1\right\} > 0,
\label{eq:type2}$$ with, $u_L(\sigma) \to 0$ as $\sigma \downarrow 0$.
3. Same definitions apply to the right tail, with, $Q(1 - t)$ replacing $Q(t)$ in , , and $c_R$ and $u_R$ denoting the right tail counterparts of $c_L$ and $u_L$.
Recall that we have taken $f_0 = f_0(\cdot|\nu) = t_\nu$ with a prior on $\nu \in (0,\infty)$. Notice that an $f$ has a type I left tail with respect to any $t_\nu$, when ${\rm supp}(f)$ is bounded from below, which is same as saying $Q(0) > -\infty$, and, $f(y)$ is bounded away from zero near $Q(0)$. If $Q(0) > -\infty$ but $f(y) \to 0$ as $y \to Q(0)$ then $f$ has a type II left tail with respect to any $t_\nu$. If $Q(0) = -\infty$ and $f(y)$ decays to zero as $y \to -\infty$ at a polynomial or faster rate, then, $f$ has a type II left tail with respect to $t_\nu$ for all $\nu > 0$ sufficiently small. It is straightforward to see that $d_{KL}(f, f_0) < \infty$ whenever $f$ has tails that are type I or type II with respect to $f_0$.
It turns out that a type I or II tail condition on $f^*_Y(\cdot|0)$, coupled with some regularity conditions on $\beta_0,\beta$ are all that is needed to ensure consistency. Here is a precise statement.
\[th:2\] Suppose $\beta^*_0$, $\beta^*$ are differentiable on $(0,1)$. Also assume $\dot\beta^* / \dot\beta^*_0$ can be extended to a continuous function on $[0,1]$, and, there exists a $c_0 > 0$ such that $\dot\beta^*_0(t) + x^T\dot\beta^*(t) \ge c_0 \dot\beta^*_0(t)$ for all $t \in (0,1)$. Then $f^*$ belongs to the KL support of $\Pi$ whenever $f^*_Y(\cdot|0)$ has type I or II tails with respect to $t_\nu$ for all small enough $\nu > 0$.
A proof is given in Appendix \[a:proof4\]. The two regularity conditions on $(\dot\beta^*_0, \dot\beta^*)$ ensure that the conditional density functions do not exhibit pathological behaviors in the tails. Notice that the basic validity assumption $\dot\beta^*_0(t) + x^T\dot\beta^*(t) > 0$ for all $t \in (0,1)$ automatically guarantees that $\dot\beta^*(t)/\dot\beta^*_0(t)$ is bounded for all $t$. To see this, notice that ${\mathcal{X}}$ must contain an open ball of radius $r > 0$ around origin which is an interior point. So, for any $t \in (0,1)$ with $\dot\beta^*(t) \ne 0$, $u := -r\dot\beta^*(t)/\|\dot\beta^*(t)\| \in {\mathcal{X}}$, and hence, $0 \le \dot\beta^*_0(t) + u^T\dot\beta^*(t) = \dot\beta^*_0(t) - r \|\dot\beta^*(t)\|$, and hence, $\|\dot\beta^*(t) / \dot\beta^*_0(t)\| \le 1/ r$.
Numerical Experiments {#s:simu1}
=====================
A small experiment with a triangular ${\mathcal{X}}$.
-----------------------------------------------------
To illustrate why adjusting to the shape of ${\mathcal{X}}$ is important for joint QR estimation, we generated 200 synthetic observations from the model: $$\begin{aligned}
& X \sim \mbox{Uniform}({\mathcal{X}});~~{\mathcal{X}}= \{x = (x_1, x_2)^T \in {\mathbb{R}}^2: -1 \le x_1, x_2 \le 2, x_1 + x_2 \le 1\},\nonumber\\
& Q_Y(\tau|X) = \frac{1 - (X_1 + X_2)}{3}Q_N(\tau | 0,1) + \frac{2 + X_1 + X_2}{3}Q_N(\tau | 1, 0.2^2) \label{triQ}\end{aligned}$$ where $Q_N(\tau|\mu, \sigma^2)$ denotes the $\tau$-th quantile of the $N(\mu, \sigma^2)$ distribution. The hyperplanes on the right hand side of are correctly ordered on the triangular predictor space ${\mathcal{X}}$, but cross each other inside the smallest embedding rectangle $[-1,2]\times[-1,2]$, as seen on the left panel of Figure \[f:tri\]. This negatively impacts estimation by the @Reich2011 method (Figure \[f:tri\]), which cannot adapt to the triangular shape of the predictor space and is restricted to estimates that do not cross on the smallest rectangle enclosing all observed predictors. In contrast, our method, which works on the convex hull of the observed predictors, can retrieve the true parameter curves with a much higher accuracy.
Performance assessment: univariate $X$
--------------------------------------
For a thorough study of the frequentist performances of the proposed method, we simulated $100$ synthetic datasets each with $n = 1000$ observations from the model $$X \sim \mbox{Uniform}(-1,1);~~~Q_Y(\tau|X) = 3(\tau-\tfrac12) \log \frac{1}{\tau(1-\tau)} + 4(\tau-\tfrac12)^2 \log\frac1{\tau(1-\tau)}X,$$ and compared parameter estimation against the methods of @Reich2011 and @Koenker1978. Here $X$ is one dimensional, and the shape of ${\mathcal{X}}$ is a not an issue. However, the nearly quadratic $\beta_1(\tau)$ function is slightly challenging to estimate.
We used the default setting for the method by @Reich2011 with 5 basis functions. For each implementation, the Gibbs sampler was run for 10000 iterations and 200 samples from the second half of the chain were used for Monte Carlo. We also tried two other versions with 10 and 15 basis functions respectively. But increasing the number of basis functions resulted in a progressively poor performance, and thus we only report here the results from the 5 basis function setting. The `QuantReg` package in `R` was used to implement the classical method by @Koenker1978 and confidence intervals were constructed with 200 bootstrapped samples. For our Gaussian process method, we used 6 equispaced knots $\tau^*_k = (k-1)/5$, $k = 1,\ldots,6$. We ran the adaptive blocked Metropolis sampler for 10000 iterations with 10% burn-in and used 200 samples from the rest for Monte Carlo. Nearly identical results were obtained with 11 equispaced knots.
Figure \[f:simu1\] shows comparisons of pointwise mean absolute estimation errors of the three methods and also the coverage of the associated 95% confidence or credible bands, averaged across the 100 synthetic datasets. Our method offered lowest estimation errors over the entire range of $\tau$ values, and a consistently high coverage close to the nominal target of 95%. It is important to remember that the credible bands produced by our method are calibrated in a Bayesian way, and so 95% credible bands are not automatically guaranteed to offer 95% coverage.
Performance assessment: multivariate $X$
----------------------------------------
For assessing performance in the multivariate case, we ran another simulation study with synthetic data generated from the model: $$X \sim \mbox{Uniform}(\{x \in \mathbb{R}^7: \|x\| \le 1\});~~~Q_Y(\tau|X) = \beta_0(\tau) + X^T\beta(\tau)$$ with $\beta_0$ and $\beta$ specified by the equations $$\begin{aligned}
&\beta_0(0.5) = 0,~~~\beta(0.5) = \begin{pmatrix}0.96 & -0.38 & 0.05 & -0.22 & -0.80 & -0.80 & -5.97\end{pmatrix}^T,\\
& \dot\beta_0(\tau) = \frac{1}{\tau(1-\tau)};~~\dot\beta(\tau) = \frac{\dot\beta_0(\tau)v(\tau)}{\sqrt{1 + \|v(\tau)\|^2}},~~\tau \in (0,1),\end{aligned}$$ where $v_j(\tau) = \sum_{l = 0}^2 a_{lj} \cdot \phi(\tau; {l}/{2}, 1/{(3^2)})$, $1\le j \le 7$, with $\phi(\cdot|\mu,\sigma^2)$ denoting the $N(\mu,\sigma^2)$ density function and $$a = \begin{pmatrix} 0 & 0 & -3 & -2 & 0 & 5 & -1\\
-3 & 0 & 0 & 2 & 4 & 1 & 0\\
0 & -2 & 2 & 2 & -4& 0 & 0
\end{pmatrix}.$$ These specifications define a valid model by Theorem \[thm:char\] because $a(b, {\mathcal{X}}) = 1$ for any non-zero $b$ when ${\mathcal{X}}$ is the unit ball centered at zero. Also note that with these specifications, $Q_Y(\tau|0)$ is precisely the quantile function of the standard logistic distribution. For simulating an $(X,Y)$ from the model we set $X = U_1 Z/\|Z\|$ and $Y = Q_Y(U_2|X)$ where $U_1,U_2 \sim U(0,1)$ and $Z \sim N_7(0,I_7)$, drawn independently of each other. We evaluated each instance of $Q_Y(U_2|X)$ to a precision of $10^{-16}$ by numerically integrating $\dot\beta_0$ and $\dot\beta$ between $0.5$ and $U_2$ with the function in .
Figure \[f:simu7\] compares the estimation error and coverage of the three methods averaged across 100 datasets of size $n = 1000$ generated from the above simulation model. All three methods were set as in the previous example. Like before, our method again offered nearly lowest estimation errors and a consistently high coverage close to the nominal target of 95%, over the entire range of $\tau$ values.
Case studies {#s:cs}
============
Plasma concentration of beta-carotene
-------------------------------------
[@nierenberg] presents a study of the association of beta-carotene plasma concentrations with dietary intakes and drugs use for nonmelanoma skin cancer patients. The Statlib database (<http://lib.stat.cmu.edu/datasets/Plasma_Retinol>) hosts a subset of the data from 315 patients who had an elective surgical procedure during a three-year period to biopsy or remove a lesion of the lung, colon, breast, skin, ovary or uterus that was found to be non-cancerous. This dataset has been analyzed in the literature [@kai2011new] to assess how personal characteristics, smoking and dietary habits as well as dietary intake of beta-carotene affects concentration levels of beta-carotene in the plasma.
We analyzed the same data with our joint QR model with plasma beta-carotene concentration (ng/ml) as the response and 11 covariates consisting of age (years), sex (1=Male, 2=Female), smoking status (1=Never, 2=Former, 3=Current Smoker), Quetelet index or BMI (weight/(height$^2$)), vitamin use[^1] (1=No, 2=Yes, not often, 3=Yes, fairly often), and daily consumption of calories, fat (g), fiber (g), alcohol (number of drinks), cholesterol (mg) and dietary beta-carotene (mcg). These covariates gave rise to 13 predictors when the categorical variables (sex, smoking status and vitamin use) were coded with dummy indicators. Estimated intercept and slope curves, with 95% credible bands are shown in Figure \[f:plasma\].
The estimated intercept curve strongly suggests a longer right tail for the response distribution. The slope curve estimates indicate that being female, use of vitamin and consumption of fiber have reasonably strong positive effect on plasma concentration of beta-carotene, whereas, smoking and BMI have reasonably strong negative effect. Calories, fat, alcohol or cholesterol consumption appears to have little effect. Dietary intake of beta-carotene appears to have a positive effect, but the inference is not conclusive. The slope estimates in Figure \[f:plasma\] suggest more dramatic effects of some predictors on the upper quantiles, but the credible bands paint a more modest picture. However, credible bands for $\beta_j(0.9) - \beta_j(0.1)$ and $\beta_j(0.9) - \beta_j(0.5)$, constructed directly from the posterior draws, indeed suggest more enhanced positive and negative effects, respectively for heavy vitamin use and BMI, on the upper quantiles (Table \[t:plasma\]).
We also performed a ten fold validation study to assess how well our joint model captured the intricacies of the beta-carotene data. In each fold of the study, we randomly partitioned the 315 observations into training and test sets at roughly 2:1 ratio. We fitted our joint model on the training data and obtained estimates $\hat\beta_j$ of $\beta_j$, $j = 0,\ldots,p$ in the form of posterior means. These estimates were then used to evaluate the training and test data “check” loss at every $\tau \in \{0.1, \ldots, 0.9\}$ by averaging $\rho_\tau(Y_i - \hat\beta_0(\tau) - X_i^T\hat\beta(\tau))$ over, respectively, all training and all test set observations $(X_i, Y_i)$, where $\rho_\tau(r) = r\{\tau -I(r < 0)\}$. The same was done with Koenker-Bassette, [@Reich2011] and standard least squares estimates. The relative accuracy of a method at any $\tau$ was calculated as the reciprocal of its check loss at that $\tau$ relative to the least square method. Figure \[f:plasma-eff\] shows these relative accuracy measures for the three quantile regression methods, averaged across the 10 repetitions. Our joint QR method can be seen to offer the best test data accuracy across all $\tau$ values and maintain its advantage over least squares at the upper quantiles where the other two quantile regression methods appear to suffer a sharp loss of efficiency.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$j$ Predictor 95% CI for $\beta_j(0.9) - \beta_j(0.1)$ 95% CI for $\beta_j(0.9) - \beta_j(0.5)$
------------------------------------------------------------------------------------ ----------- ------------------------------------------ ------------------------------------------
1 Age $( -0.52 , 2.32 )$ $( -0.88 , 1.69 )$
2 Sex2 $( -43.34 , 54.12 )$ $( -33.53 , 55.31 )$
3 SmokStat2 $( -66.81 , 28.51 )$ $( -47.64 , 35.7 )$
4 SmokStat3 $( -102.75 , 26.64 )$ $( -95.05 , 19.97 )$
**5 & **Quetelet & $( -5.43 , 0.12 )$ & $\mathbf{( -4.93 , -0.14 )}$\
6 & VitUse1 & $( -21.05 , 54.43 )$ & $( -14.1 , 55.09 )$\
**7 & **VitUse2 & $\mathbf{( 12.27 , 177.02 )}$ & $\mathbf{( 10.57 , 153.73 )}$\
8 & Calories & $( -0.01 , 0.01 )$ & $( -0.01 , 0.02 )$\
9 & Fat & $( -0.75 , 0.33 )$ & $( -0.58 , 0.25 )$\
10 & Fiber & $( -3.7 , 3.14 )$ & $( -3.22 , 2.54 )$\
11 & Alcohol & $( -0.74 , 2.32 )$ & $( -0.72 , 1.51 )$\
12 & Cholesterol & $( -0.12 , 0.16 )$ & $( -0.12 , 0.09 )$\
13 & BetaDiet & $( 0 , 0.02 )$ & $( -0.01 , 0.01 )$********
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Evidence of more dramatic upper tail effects of certain predictors on plasma beta-carotene concentration. CI denotes posterior credible interval.[]{data-label="t:plasma"}
Survival analysis under right censoring
---------------------------------------
Joint estimation of quantile regression parameters could be particularly beneficial for survival analysis with censored response. A greater borrowing of information may help cover the information gaps left by censoring. A crossing-free estimation of the quantile functions means that the estimated survival curves are proper and interpretable. Also, a joint estimation offers an automatic way to quantify estimation uncertainty of the entire survival curves by simple inversions of estimated quantile functions. The probabilistic modeling framework of our joint quantile regression approach makes it particularly straightforward to handle right-censoring. The log-likelihood score calculation now changes to $$\begin{aligned}
\label{eq8c}
&\sum_{i} [(1 - c_i) \log f_Y(y_i|x_i) + c_i \log\{1 - F_Y(y_i|x_i)\}] \nonumber\\
&~~~~~~~~~~~~~~~=\sum_i \left[c_i \log\{1 - \tau_{x_i}(y_i)\}-(1 - c_i) \log\big\{\dot{\beta}_0\big(\tau_{x_i}(y_i)\big)+
x_i^T\dot{\beta}\big(\tau_{x_i}(y_i)\big)\big\}\right],\end{aligned}$$ where $c_i$ is the censoring status (1= right censored, 0 = observed). With this single change, the same prior specification and Markov chain Monte Carlo parameter estimation as detailed in Section \[s:prior\] remain applicable.
We illustrate these points with a reanalysis of the University of Massachusetts Aids Research Unit IMPACT Study data [UIS, @Hosmer1998 Table 1.3] in which we estimated the conditional quantiles of the logarithm of the time to return to drug use ($Y$) as linear functions of current treatment assignment (, 1 = Long course, 0 = Short course), number of prior drug treatments (), recent intravenous drug use (, 1 = Yes, 0 = No), Beck depression score (), a compliance factor measuring length of stay in the treatment relative to the course length (), race of the subject (, 1 = Non-white, 0 = White), age () and treatment site (). For model fitting, we used the 575 complete observations available in the data set of the R package . Return times were right censored for 111 of these subjects.
Figures \[f:uis-coef\]-\[f:uis-surv\] show parameter and survival curves (for 9 randomly chosen subjects) estimation with our joint quantile regression approach and also with the censored quantile regression approach described in @Koenker2008. The latter was implemented by using the function in R-package which uses a technique by @Portnoy2003. For joint estimation, we fixed the base probability density $f_0$ to be $N(0,1)$ instead of a $t_\nu$, since the tails of the distribution of log return time are expected to be fast decaying. The two sets of parameter estimates are comparable, except in the upper tails. The Portnoy method fails to produce an estimate beyond $\tau = 0.88$, and confidence intervals get extremely wide for $\tau$ close to this limit. In contrast, the credible bands from joint estimation are much more stable across the entire range of $\tau$. Estimated survival curves are remarkably similar, though for the Portnoy method, the issue of quantile crossing manifests in the form of estimated survival curves that are not strictly decreasing (e.g., subject \# 313).
Discussion {#s:dis}
==========
We have introduced a complete and practicable theoretical framework for simultaneous estimation of linear quantile planes in any dimension and over arbitrarily shaped convex predictor domains. Although we have pursued here a specific estimation procedure, our modeling platform is extremely broad and parameter estimation could be done in a variety of other manners. For example, one could choose to use spline based estimation of the basic functions $w_0, \ldots, w_p$ via penalized likelihood maximization or Bayesian averaging. Also, a variety of specifications could be used on the diffeomorphism parameter $\zeta$, e.g., one could model $\zeta$ as a mixture of beta cumulative distribution functions, or try estimating $\zeta$ directly by adding isotonic regression type constraints.
A number of interesting features could be added to the Bayesian parameter estimation method we have pursued here. An important consideration is shrinkage for large $p$. For moderately large $p$, any standard shrinkage prior could be used on $\gamma$, and the resulting posterior could be explored by the same Markov chain sampler as in Section \[s:comp\] as long as the prior density on $\gamma$ is available in an explicit form up to a normalizing constant. Shrinkage could also be applied on the curve valued parameters $w_j$, $j = 1, \ldots, p$, by choosing appropriate prior distributions on $(\kappa^2_1, \ldots, \kappa^2_p)$. An attractive choice is to replace the single gamma prior distribution we used in Section \[s:prior\] with a spike-slab type mixture of gamma distributions, e.g., $\kappa_j^{-2} \sim 0.5 Ga(a_\kappa, b_\kappa) + 0.5 Ga(a_\kappa, b_\kappa/100)$. Such a specification still allows integrating out $\kappa_j$ in and hence could be explored by the same Markov chain sampler as before.
The primary computational bottleneck of our method is that the likelihood evaluation involves a search over the grid of $\tau$ values for each observation. While our current implementation easily scales to thousands of observations, scaling it to even larger datasets will require further computing innovations. Fortunately, the likelihood evaluation is embarrassingly parallel in the observations and involves very simple arithmetic operations, and thus, it should be possible to obtain manyfold speed ups by the use of graphics processing units; such an implementation is currently underway.
Technical details
=================
This section presents a proof of Theorem \[th:2\], starting with a few fundamental results that allow comparing two probability density functions given information on their corresponding quantile density functions. We adopt the following notation in the remainder of this section: by a ‘probability function quartet’ we mean a four-tuple $(Q, q, F, f)$ of real valued functions where $Q:(0,1) \to {\mathbb{R}}$ is a non-atomic quantile function that admits a strictly positive derivative $q = \dot Q$, $F = Q^{-1}$ is the associated cumulative distribution function with probability density function $f = \dot F$. Recall the identities $f(y) = 1/q(F(y))$ and $q(t) = 1/f(Q(t))$.
Auxiliary results
-----------------
\[lem:aux2\] Let $(Q_1, q_1, F_1, f_1)$ and $(Q_2, q_2, F_2, f_2)$ be two probability function quartets and take $m_j = Q_j(\tau_0)$, $j = 1,2$. If there exist $0 < c_1 \le 1 \le c_2 < \infty$ such that $c_1 q_2(t) \le q_1(t) \le c_2 q_2(t)$, for all $t \in (0,1)$, then, $$f_2(y) = f_1(y + \Delta_1(y)) \cdot \Delta_2(y),~~\mbox{for all}~y \in {\rm supp}(f_2),$$ for two real valued functions $\Delta_1, \Delta_2$ satisfying $|\Delta_1(y)| \le \max(1 - c_1, c_2 - 1) |y - m_2| + |m_1 - m_2|$, and, $\Delta_2(y) \in [c_1, c_2]$.
By the assumption on $q_1, q_2$, for every $y \in {\rm supp}(f_2)$, $$c_1 y + b_1 \le Q_1F_2(y) \le c_2 y + b_2,$$ where $b_1 = m_1 - c_1m_2$, $b_2 = m_1 - c_2m_2$. Then, $\Delta_1(y) := Q_1F_2(y) - y$ satisfies, $\|\Delta_1(y)\| \le \max(1 - c_1, c_2 - 1)|y - m_2| + |m_1 - m_2|$. Since, $f_i(y) = 1/q_i(F_i(y))$, $i = 1,2$, we have, for any $y \in {\rm supp}(f_2)$, $f_1(Q_1(F_2(y)))q_1(F_2(y)) = 1$, and hence, $$f_2(y) = \frac{f_1(Q_1(F_2(y)))q_1(F_2(y))}{q_2(F_2(y))} = f_1(y + \Delta_1(y)) \cdot \frac{q_1(F_2(y))}{q_2(F_2(y))},$$ which proves the result.
\[lem:aux3\] Let $f^*$ satisfy the conditions of Theorem \[th:2\]. Given any $\delta, \sigma, c_1, c_2 > 0$, there exists an $\epsilon > 0$ such that , $$\sup_{x \in {\mathcal{X}}} \int_{[Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon|x)]} f^*_Y(y|x) \left|\log \{ f_0(y /\sigma + \Delta(y))/\sigma\}\right|dy < \delta$$ for every $\Delta: {\mathbb{R}}\to {\mathbb{R}}$ satisfying $|\Delta(y)| < c_1|y| + c_2$ for all $y \in {\mathbb{R}}$.
By the assumption on $f^*$, $q^*_Y(t|x) / q^*_Y(t|0) = 1 + x^T\dot\beta^*(t)/\dot\beta^*_0(t)$ is bounded away from zero and infinity. Hence, by Lemma \[lem:aux2\], there are constants $a, b > 0$ such that for every $x \in {\mathcal{X}}$, $Q^*_Y(F^*_Y(y|0)|x) = y + \Delta_{1,x}(y)$, with $|\Delta_{1,x}(y)| \le a|y| + b$ for all $y \in A_0 := {\rm supp}(f^*_Y(\cdot|0))$. Fix any $x \in {\mathcal{X}}$. By the change of variable $z = Q^*_Y(F^*_Y(y|x)|0)$, $$\begin{aligned}
\int f^*_Y(y|x) & \left|\log f_0\left(\frac{y}\sigma + \Delta(y)\right)\right|dy\\
& = \int_{A_0} f^*_Y(z|0)\left|\log f_0\left(\frac{Q^*_Y(F^*_Y(z|0)|x)}\sigma + \Delta(Q^*_Y((F^*_Y(z|0)|x))\right)\right|dz \\
& = \int_{A_0} f^*_Y(z|0) \left|\log f_0(z/\sigma + \Delta_2(z))\right|dz,\end{aligned}$$ with $|\Delta_2(z)| \le |\Delta_1(z)|/\sigma + |\Delta(z + \Delta_1(z))| \le a_1|z| + b_1$ where $a_1,b_1 > 0$ depend only on $\delta$, $\sigma$, $c_1$ and $c_2$. The tail assumption on $f^*_Y(\cdot|0)$ implies that the last integral is finite, proving the result!
Fix $\gamma_0 \in {\mathbb{R}}$, $\gamma \in {\mathbb{R}}^p$, $\sigma > 0$, $w : (0,1) \to {\mathbb{R}}^p$, and two differentiable, monotonically increasing functions $\zeta, {{\zeta}^\dagger} : [0,1] \to [0,1]$, with $[\zeta(0), \zeta(1)] \subset [{{\zeta}^\dagger}(0), {{\zeta}^\dagger}(1)]$. Let $(\beta_0, \beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$, $({{\beta}^\dagger}_0, {{\beta}^\dagger}) = \mathcal{T}({{\gamma}^\dagger}_0, {{\gamma}^\dagger}, \sigma, w, {{\zeta}^\dagger})$, where, $${{\gamma}^\dagger}_0 = \gamma_0 + \sigma \int_{\zeta(\tau_0)}^{{{\zeta}^\dagger}(\tau_0)} q_0(u) du,~~{{\gamma}^\dagger} = \gamma + \sigma\int_{\zeta(\tau_0)}^{{{\zeta}^\dagger}(\tau_0)} q_0(u) h(u) du,$$ with $h(\tau) := w(\tau)/\{a(w(\tau), {\mathcal{X}}) \sqrt{1 + \|w(\tau)\|^2}\}$, $\tau \in (0,1)$. Fix any $x \in {\mathcal{X}}$ and consider the probability function quartets $(Q_Y(\cdot|x), q_Y(\cdot|x), F_Y(\cdot|x), f_Y(\cdot|x))$, $({{Q}^\dagger}_Y(\cdot|x), {{q}^\dagger}_Y(\cdot|x), {{F}^\dagger}_Y(\cdot|x), {{f}^\dagger}_Y(\cdot|x))$ where $Q_Y(\tau|x) = \beta_0(t) +x^T\beta(\tau)$, ${{Q}^\dagger}_Y(\tau|x) = {{\beta}^\dagger}_0(\tau) + x^T{{\beta}^\dagger}(\tau)$. Then, $ {f_Y(y|x)}/{{{f}^\dagger}_Y(y|x)} = {{{\dot\zeta}^\dagger}({{F}^\dagger}_Y(y|x))}/{\dot\zeta(F_Y(y|x))}$ for all $y \in {\rm supp}(f_Y(\cdot|x))$. \[lem:aux\]
Let $\tau_1 = \zeta(\tau_0)$ and, define, $$q_0(\tau|x) = q_0(\tau) \{1 + x^T h(\tau)\},~~\tau \in (0,1), x \in {\mathcal{X}}.$$ Then $q_0(\tau|x)$ is strictly positive, and, hence, $$Q_0(\tau|x) := \int_{\tau_1}^{\tau} q_0(u|x) du,~~ \tau \in (0,1), x \in {\mathcal{X}},$$ defines valid quantile planes on ${\mathcal{X}}$. Denote the associated conditional distribution and density functions by $F_0(\cdot|x)$ and $f_0(\cdot|x)$. By definition of $(\beta_0, \beta)$, $q_Y(\tau|x) = \dot \beta_0(\tau) + x^T \dot\beta(\tau) = \sigma q_0(\zeta(\tau))\left\{1 + x^T h(\zeta(\tau))\right\}\dot\zeta(\tau) = \sigma q_0(\zeta(\tau)|x)\dot\zeta(\tau)$, and, hence, $$Q_Y(\tau|x) = \gamma_0 + x^T \gamma + \int_{\tau_0}^\tau q_Y(u|x)du = \gamma_0 + x^T \gamma + \sigma Q_0(\zeta(\tau)|x)\label{eq:a1}.$$ Similarly, ${{q}^\dagger}_Y(\tau|x) = \sigma q_0({{\zeta}^\dagger}(\tau)|x)\dot{{{\zeta}^\dagger}}(\tau)$, and, hence, $${{Q}^\dagger}_Y(\tau|x) = {{\gamma}^\dagger}_0 + x^T {{\gamma}^\dagger} + \int_{\tau_0}^\tau {{q}^\dagger}_Y(u|x)du = \gamma_0 + x^T \gamma + \sigma Q_0({{\zeta}^\dagger}(\tau)|x).$$ by the definitions of ${{\gamma}^\dagger}_0$, ${{\gamma}^\dagger}$. Inverting , we get, for every $x \in {\mathcal{X}}$, $$\zeta(\tau) = F_0\left(\frac{Q_Y(\tau|x) - \gamma_0 - x^T \gamma}{\sigma} \bigg| x \right), \tau \in (0,1).$$ Therefore, if $y \in (Q_Y(0|x), Q_Y(1|x))$, then, $$f_Y(y|x) = \frac{1}{q_Y(F_Y(y|x) | x)} = \frac{1}{\sigma q_0(\zeta(F_Y(y|x))|x) \dot \zeta(F_Y(y|x))} = \frac{f_0(\frac{y - \gamma_0 - x^T \gamma}{\sigma} | x)}{\sigma \dot \zeta(F_Y(y|x))}.$$ Similarly, ${{f}^\dagger}_Y(y|x) = f_0(\frac{y - \gamma_0 - x^T \gamma}{\sigma} | x) / \{\sigma \dot{{{\zeta}^\dagger}}({{F}^\dagger}_Y(y|x))$}, proving the result!
Approximating $f^*$ within assumed model space {#a:proof4}
----------------------------------------------
Let $\Pi_\nu$ denote the conditional prior distribution on $f$ under $\Pi$ given $\nu$. Theorem \[th:2\] is proved in two stages. Let $\nu_0 > 0$ such that the tails of $f^*_Y(\cdot|0)$ are of type I or II with respect to $f_0(\cdot|\nu)$ for every $0 < \nu \le \nu_0$. First we show that for any such $\nu$ and any given $\delta > 0$, there exists an ${{f}^\dagger} \in K_\delta(f^*)$ within our model space with nicely behaved underlying $w_j$ curves. Next we show $\Pi_\nu(f:\|\log ({{f}^\dagger} / f)\|_\infty < \delta) > 0$ which leads to the claim of Theorem \[th:2\]. The following lemma gives a precise statement of the first step.
\[le:6\] Let $f^*$ satisfy the conditions of Theorem \[th:2\]. For any small $\delta>0$ and $0 < \nu \le \nu_0$, there exists an ${{f}^\dagger} \in K_\delta(f^*)$ associated with $({{\beta}^\dagger}_0, {{\beta}^\dagger}) = \mathcal{T}({{\gamma}^\dagger}_0, {{\gamma}^\dagger}, {{\sigma}^\dagger}, {{w}^\dagger}, {{\zeta}^\dagger})$ where ${{w}^\dagger}: [0,1] \to {\mathbb{R}}^p$ is bounded continuous and ${{\zeta}^\dagger}: [0,1] \to [0,1]$ is a diffeomorphism with $\dot{{{\zeta}^\dagger}}(t) \in [e^{-B}, e^B]$ for all $t \in [0,1]$ for some finite $B > 0$.
Fix a $\nu \in (0, \nu_0)$ and a $\delta \in (0, \tau_0)$. All calculations below are carried out for this particular value of $\nu$ and we suppress $\nu$ from the notation $f_0(\cdot|\nu)$. Let $\gamma_0^* = \beta^*_0(\tau_0)$. Fix a $\sigma_L > 0$ such that $c_L(\sigma_L) \le 1/2$ if $f^*_Y(\cdot|0)$ has a type I left tail with respect to $f_0$, or, $u_L(\sigma_L) \log \{1/u_L(\sigma_L)\} \le \delta/2$ if the left tail is of type II. Similarly fix $\sigma_R$ and take $\sigma^* = \min\{\sigma_L, \sigma_R\}$. Define $\zeta^* : [0,1] \to [0,1]$ as $$\zeta^*_0(t) = F_0\left(\gamma^*_0 + \frac{\beta^*_0(t) - \gamma^*_0}{\sigma^*} \right), t \in [0,1],$$ which is differentiable and monotonically increasing, and, whose derivative can be written as, $$\dot\zeta^*_0(t) = \frac1{\sigma^*}f_0\left(\gamma^*_0 + \frac{\beta^*_0(t) - \gamma^*_0}{\sigma^*}\right)\dot\beta^*_0(t) = \frac{(1/\sigma^*)f_0\left(\gamma^*_0 + \frac{\beta^*_0(t) - \gamma^*_0}{\sigma^*}\right)}{f^*_Y(\beta^*_0(t)|0)},~t \in (0,1),$$ since $\beta^*_0(t) = Q_Y^*(t|0)$.
Because $\zeta^*$ has a continuously differentiable inverse on $[\zeta^*(0), \zeta^*(1)]$, the relation $h^*(\zeta^*(u)) = \dot \beta^*(u) / \dot \beta^*_0(u)$ defines a map $h^*: [\zeta^*(0), \zeta^*(1)] \to {\mathbb{R}}^p$ that is bounded and continuous by the assumption of Theorem \[th:2\], and hence, can be extended to a bounded continuous function $h^*:[0,1] \to {\mathbb{R}}^p$. Define $w^* :[0,1] \to {\mathbb{R}}^p$ as follows, essentially repeating the construction in the “Only if part” of the proof of Theorem \[thm:char\]. If $h^*(t) = 0$ then set $w^*(t) = 0$. Otherwise, take $c(t) = [ [1/\{\|h^*(t)\|a(h^*(\tau), {\mathcal{X}})\}]^2 - 1]^{-1/2}$ and set $w^*(t) = c(t) h^*(t) / \|h^*(t)\|$. By the assumption on $\dot\beta^*/\dot\beta^*_0$, $w^*$ is a bounded continuous function on $[0,1]$.
By construction $(\beta^*_0, \beta^*) = \mathcal{T}(\gamma^*_0, \gamma^*, \sigma^*, w^*, \zeta^*)$. However this parameter vector may not be in our model space since we may have either $[\zeta^*(0) , \zeta^*(1)] \ne [0,1]$ or $\|\dot \zeta^*\|_\infty = \infty$. We correct this by introducing a proper diffeomorphism ${{\zeta}^\dagger}$ on $[0,1]$ with $\dot{{{\zeta}^\dagger}}$ bounded away from 0 and infinity, such that ${{\zeta}^\dagger}(t) = \zeta^*(t)$ for $t \in [\delta_L, 1 - \delta_R]$ for suitably chosen small numbers $\delta_L, \delta_R > 0$. This is the crux of the approximation argument.
If the left tail is type I, then $\zeta^*(0) > 0$ and $\lim_{t \downarrow 0} \dot \zeta^*(t) = c_L(\sigma^*) \in (0,1/2]$. So one can fix $\delta_L > 0$ small enough such that $\zeta^*(\delta_L) > \delta_L$, $\dot\zeta^*(t) \in (c_L(\sigma^*)/2, 1]$ for all $t \in (0, \delta_L]$, and, $\delta_L \log [4 / \{c_L(\sigma^*)\delta_L\}] < \delta/2$. Otherwise, the left tail is type II, and in that case choose $\delta_L = u_L(\sigma^*)$, which automatically ensures $\dot \zeta^*(t) \ge 1$ for all $t \in (0,1)$ with $\dot\zeta^*(\delta_L) = 1$, and, $\delta_L\log(1/\delta_L) \le \delta/2$. Since $\zeta^*(0) \ge 0$, we also must have $\zeta^*(\delta_L) \ge \delta_L$. Fix $\delta_R$ by repeating the same steps with the right tail. Define ${{\zeta}^\dagger}: [0,1] \to {\mathbb{R}}$ as, $${{\zeta}^\dagger}(t) = \left\{ \begin{array}{ll} \zeta^*(t), & t \in [\delta_L,1 - \delta_R],\\[5pt] a_L t^2 + b_L t,& t \in [0, \delta_L) \\[5pt] 1 - a_R (1 - t)^2 - b_R (1 - t), & t \in (1 - \delta_R,1],\end{array}\right.$$ where, $$\begin{aligned}
& a_L = \frac{\delta_L \dot \zeta^*(\delta_L) - \zeta^*(\delta_L)}{ \delta_L^2},~~b_L = \frac{2 \zeta^*(\delta_L) - \delta_L \dot\zeta^*(\delta_L)}{\delta_L},\\
& a_R = \frac{\delta_R \dot \zeta^*(1 - \delta_R) - \{1 - \zeta^*(1 - \delta_R)\}}{ \delta_R^2},~~b_R = \frac{2 \{1 - \zeta^*(1 - \delta_R)\} - \delta_R \dot\zeta^*(1 - \delta_R)}{\delta_R}.\end{aligned}$$ By choice of $\delta_L$ and $\delta_R$, $a_L < 0$, $a_R < 0$ and $b_L \in [\dot\zeta^*(\delta_L), 2/\delta_L]$, $b_R \in [\dot\zeta^*(1 - \delta_R), 2 / \delta_R]$. It is straightforward to verify that ${{\zeta}^\dagger}$ defines a diffeomorphism from $[0,1]$ onto $[0,1]$, with ${{\dot \zeta}^\dagger}(t) \in [\dot\zeta^*(\delta_L), b_L]$ for all $t \in [0,\delta_L]$ and ${{\dot \zeta}^\dagger}(t) \in [\dot\zeta^*(1 - \delta_R), b_R]$ for all $t \in [1 - \delta_R, 1]$. Therefore there exists a $B > 0$ such that $\dot {{{\zeta}^\dagger}}(t) \in [e^{-B}, e^{B}]$ for all $t \in [0,1]$.
Take $({{\beta}^\dagger}_0, {{\beta}^\dagger})=\mathcal{T}(\gamma^*_0, \gamma^*, \sigma^*, w^*, {{\zeta}^\dagger})$ with valid conditional quantile planes ${{Q}^\dagger}_Y(\cdot|x)$ and associated cumulative distribution and probability density functions given by ${{F}^\dagger}_Y(\cdot|x)$ and ${{f}^\dagger}_Y(\cdot|x)$. By construction of ${{\zeta}^\dagger}$, ${{Q}^\dagger}_Y(\tau|x) = Q^*_Y(\tau|x)$ for all $\tau \in [\delta_L, 1 - \delta_R]$ and hence ${{F}^\dagger}_Y(y|x) = F^*_Y(y|x)$ for all $y \in [Q^*_Y(\delta_L|x), Q^*_Y(1 - \delta_R|x)]$. Hence, by Lemma \[lem:aux\], $$d_{KL}(f^*_Y(\cdot|x), {{f}^\dagger}_Y(\cdot|x)) = \int_{y \in Q^*_Y(\delta_L|x), Q^*_Y(1 - \delta_R|x)]^c} f^*_Y(y|x) \log \frac{{{\dot\zeta}^\dagger}({{F}^\dagger}_Y(y|x))}{\dot\zeta^*(F^*_Y(y|x))} dy.$$ Split the integral above into two integrals, one over $y < Q^*_Y(\delta_L|x)$ and the other over $y > Q^*_Y(1 - \delta_R|x)$. When $y < Q^*_Y(\delta_L|x)]$, both ${{F}^\dagger}_Y(y|x) < \delta_L$, and, $F^*_Y(y|x) < \delta_L$. Clearly, $\dot{{{\zeta}^\dagger}}({{F}^\dagger}_Y(y|x)) \le b_L \le 2 / \delta_L$. If left tail is type I then, $\dot \zeta^*(F^*_Y(y|x)) \ge c_L(\sigma^*) / 2$ and hence, $$\begin{aligned}
\int_{Q^*_Y(0|x)}^{Q^*_Y(\delta_L|x)} f^*_Y(y|x) \log \frac{{{\dot\zeta}^\dagger}({{F}^\dagger}_Y(y|x))}{\dot\zeta^*(F^*_Y(y|x))} dy & \le \left[\log \frac{4}{\delta_L c_L(\sigma^*)}\right]\int_{Q^*_Y(0|x)}^{Q^*_Y(\delta_L|x)} f^*_Y(y|x)dy\\
& = \delta_L \log \frac{4}{\delta_L c_L(\sigma^*)} \le \delta/2\end{aligned}$$ by the choice of $\delta_L$ for the type I left tail. On the other hand, if the left tail is type II, then $\dot\zeta^*(F^*_Y(y|x)) \ge 1$ and hence $$\int_{Q^*_Y(0|x)}^{Q^*_Y(\delta_L|x)} f^*_Y(y|x) \log \frac{{{\dot\zeta}^\dagger}({{F}^\dagger}_Y(y|x))}{\dot\zeta^*(F^*_Y(y|x))} dy \le \delta_L \log \frac{2}{\delta_L} \le \delta/2,$$ again by the choice of $\delta_L$ for this case. Same arguments apply to the integral over $y \in [Q^*_Y(1 - \delta_R|x), Q^*_Y(1|x)]$, and hence, for every $x \in {\mathcal{X}}$, $d_{KL}(f^*_Y(\cdot|x), {{f}^\dagger}_Y(\cdot|x)) \le \delta$. Therefore $d_{KL}(f^*, {{f}^\dagger}) = \int f_X(x) d_{KL}(f^*_Y(\cdot|x), {{f}^\dagger}_Y(\cdot|x))dx \le \delta$.
Proof of Theorem \[th:2\]
-------------------------
Since the prior on $\nu$ has full support, it suffices to show that given any $\delta > 0$ and $\nu < \nu_0$, the conditional prior $\Pi_\nu = \Pi(\cdot|\nu)$ assigns positive mass to the event $\{f:d_{KL}(f^*, f) < 3\delta\}$. Fix any $\delta > 0$, and $\nu < \nu_0$. By Lemma \[le:6\], there is a $({{\beta}^\dagger}_0, {{\beta}^\dagger}) = \mathcal{T}({{\gamma}^\dagger}_0, {{\gamma}^\dagger}, {{\sigma}^\dagger}, {{w}^\dagger}, {{\zeta}^\dagger})$ with the associated probability density function ${{f}^\dagger}$ satisfying $d_{KL}(f^*, {{f}^\dagger}) < \delta$, where, ${{w}^\dagger}: [0,1] \to {\mathbb{R}}^p$ is bounded continuous, and, ${{\zeta}^\dagger} :[0,1] \to [0,1]$ is a diffeomorphism with $\|\log \dot{{{\zeta}^\dagger}}\|_\infty < \infty$.
For any $\lambda > 0$, let $A_\lambda$ denote the set of $(\gamma_0, \gamma, \sigma, w, \zeta)$ such that $|\gamma_0 - {{\gamma}^\dagger}_0| < \lambda$, $\|\gamma - {{\gamma}^\dagger}\| < \lambda{{\sigma}^\dagger} / {\rm diam}({\mathcal{X}})$, $|\sigma /{{\sigma}^\dagger} - 1| < \lambda$, $w:[0,1] \to {\mathbb{R}}^p$ is continuous and $\sup_t \|w(t) - {{w}^\dagger}(t)\| < \lambda$, $\zeta: [0,1] \to [0,1]$ is a diffeomorphism and $\|\log \dot\zeta - \log {{\dot\zeta}^\dagger}\|_\infty < \lambda$. By construction and because of the full support properties of Gaussian processes , the conditional prior $\Pi_\nu$ assigns a positive mass to the set of $f$ associated with $(\beta_0, \beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$, $(\gamma_0, \gamma, \sigma, w, \zeta) \in A_\lambda$, for every $\lambda > 0$. So, it suffices to show that $\lambda > 0$ could be chosen small enough such that any $f$ associated with a $(\gamma_0, \gamma, \sigma, w, \zeta) \in A_\lambda$ satisfies $\int f^*_Y(y|x) \log \{{{f}^\dagger}_Y(y|x)/ f_Y(y|x)\} dy \le 2\delta$ for all $x \in {\mathcal{X}}$.
Let $b = \|\log\dot{{{\zeta}^\dagger}}\|_\infty + 1$. Since ${{w}^\dagger}$ is bounded on $[0,1]$, there exists a $B > 0$ such that ${{h}^\dagger}(t) := {{{w}^\dagger}(t) }/\{a({{w}^\dagger}(t), {\mathcal{X}})\sqrt{1 + \|{{w}^\dagger}(t)\|^2}\}$ satisfies $$1 + x^T{{h}^\dagger}(t) \in \left[e^{-B}, e^B\right],~\mbox{for all}~t \in [0,1].$$ Clearly, there exists a $\lambda_0 \in (0, 1/2)$ such that $\|\log\dot\zeta- \log\dot{{{\zeta}^\dagger}}\|_\infty < \lambda_0$ implies $\|\log\dot\zeta\|_\infty < 2b$, and, $\sup_t \|w(t) - {{w}^\dagger}(t)\| < \lambda_0$ implies $1 + x^T w(t) / \{a(w(t), {\mathcal{X}})\sqrt{1 + \|w(t)\|^2}\} \in [e^{-2B}, e^{2B}]$ for all $t \in [0,1]$. Take $c_1 = 1 + e^{2B}$, $c_2 = c_1\{|{{\gamma}^\dagger}_0| + \|{{\gamma}^\dagger}\| +1)\} + 1/2$, and, $\tilde c_1 = (1 + 2c_1)/{{\sigma}^\dagger}$, $\tilde c_2 = 2(c_2 + |{{\gamma}^\dagger}_0| + 1)/{{\sigma}^\dagger}$. By Lemma \[lem:aux3\] there exists an $0 < \epsilon < \delta / \max\{12b, 6(B+1)\}$ such that, $$\sup_{x \in {\mathcal{X}}} \int_{[Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon|x)]} f^*_Y(y|x) |\log f_0(y/{{\sigma}^\dagger} + \Delta(y))/{{\sigma}^\dagger}| dy < \delta/3.$$ for every $\Delta: {\mathbb{R}}\to {\mathbb{R}}$ satisfying $|\Delta(y)| < \tilde c_1 |y| + \tilde c_2$ for all $y \in {\mathbb{R}}$
Take any $(\gamma_0, \gamma, \sigma, w, \zeta) \in A_{\lambda_0}$ and let $(\beta_0, \beta) = \mathcal{T}(\gamma_0, \gamma, \sigma, w, \zeta)$, $(\beta^{e\dagger}_0, {\beta}^{e\dagger}) = \mathcal{T}(\gamma^{e\dagger}_0, \gamma^{e\dagger}, {{\sigma}^\dagger}, {{w}^\dagger}, e)$, and, $(\beta^e_0, \beta^e) = \mathcal{T}(\gamma^e_0, \gamma^e, \sigma, w, e)$, where $e$ denotes the identity function on $[0,1]$ onto itself and $$\begin{aligned}
\gamma^{e\dagger}_0 &= {{\gamma}^\dagger}_0 + {{\sigma}^\dagger} \int_{{{\zeta}^\dagger}(\tau_0)}^{\tau_0} q_0(u)du,~~\gamma^{e\dagger} = {{\gamma}^\dagger} + {{\sigma}^\dagger} \int_{{{\zeta}^\dagger}(\tau_0)}^{\tau_0} q_0(u){{h}^\dagger}(u) du,\\
\gamma^{e}_0 &= \gamma_0 + \sigma \int_{\zeta(\tau_0)}^{\tau_0} q_0(u)du,~~\gamma^{e} = \gamma + \sigma \int_{\zeta(\tau_0)}^{\tau_0} q_0(u) h(u) du.\\\end{aligned}$$ with $h(t) = w(t) / \{a(w(t), {\mathcal{X}})\sqrt{1 + \|w(t)\|^2}\} $, $t \in [0,1]$. The definitions of $\gamma_0, \gamma^e_0, \gamma^{e\dagger}_0, \gamma^{e\dagger}$ match the requirements of Lemma \[lem:aux\]. Let $(Q_Y(\cdot|x), q_Y(\cdot|X), F_Y(\cdot|x), f_Y(\cdot|x))$ denote the probability function quartet of $(\beta_0, \beta)$, and the same symbols with appropriate superscripts denote the same quantities associated with the other three pairs $(\beta^e_0, \beta^e)$, $({{\beta}^\dagger}_0, {{\beta}^\dagger})$ and $(\beta^{e\dagger}_0, \beta^{e\dagger})$.
Consider the following factorization in log-scale $$\log \frac{{{f}^\dagger}_Y(y|x)}{f_Y(y|x)} = \log \frac{{{f}^\dagger}_Y(y|x)}{f^{e\dagger}_Y(y|x)} + \log \frac{f^e_Y(y|x)}{f_Y(y|x)} + \log \frac{f^{e\dagger}_Y(y|x)}{f^{e}_Y(y|x)}.$$ By Lemma \[lem:aux\], $|\log \{f^{\dagger}_Y(y|x) / f^{e\dagger}_Y(y|x)\}| = |\log \dot{{{\zeta}^\dagger}}({{F}^\dagger}_Y(y|x))| \le 2b$, and, $|\log\{f^e_Y(y|x) / f_Y(y|x)\}| = |\log \dot\zeta(F_Y(y|x))|\le 2b$. Since $1 + x^T h(t) \in [e^{-2B}, e^{2B}]$ for all $t \in [0,1]$, we have, for any $x \in {\mathcal{X}}$, $q^e_Y(t|x) = q^e_Y(t|0) \cdot [e^{-2B}, e^{2B}]$ for all $t \in (0,1)$, and hence, by Lemma \[lem:aux2\], $f^e_Y(y|x) = f^e_Y(y + \Delta_{1,x}(y)|0) \Delta_{2,x}(y)$, with $|\Delta_{1,x}(y)| \le c_1 |y| + c_2$ for all $y \in {\mathbb{R}}$, and, $\|\log \Delta_{2,x}\|_\infty \le 2B$. But, $f^e_Y(y|0) = f_0((y - \gamma_0) / \sigma)/\sigma$ and so, $f^e_Y(y|x) = f_0(y/{{\sigma}^\dagger} + \tilde \Delta_{1,x}(y))\tilde \Delta_{2,x}(y)/{{\sigma}^\dagger}$ with $|\tilde \Delta_{1,x}(y)| \le \tilde c_1 |y| + \tilde c_2$, for all $y \in {\mathbb{R}}$, and, $\|\log \tilde \Delta_{2,x}\|_\infty \le B + 1$. The same calculations work for $f^{e\dagger}_Y$ because $({{\gamma}^\dagger}_0, {{\gamma}^\dagger}, {{\sigma}^\dagger}, {{w}^\dagger}, {{\zeta}^\dagger}) \in A_{\lambda_0}$. Therefore, $$\int_{[Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon | x)]^c} f^*_Y(y|x) \log \frac{{{f}^\dagger}_Y(y|x)}{f_Y(y|x)}dy < \delta,$$ for every $x \in {\mathcal{X}}$.
The map $(x, y) \mapsto \log {{f}^\dagger}_Y(y|x)$ is equicontinuous on $\{(x,y): x \in {\mathcal{X}}, y \in [Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon|x)]\}$, and hence there exist a $\kappa > 0$ such that $\log|{{f}^\dagger}_Y(y+z|x) / {{f}^\dagger}_Y(y|x)| < \delta/2$ for all $x \in {\mathcal{X}}$, $y \in [Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon|x)]$, $|z| < \kappa$. Fix a small $0 < \eta < \kappa/2$ such that $$\max(e^\eta - 1, 1 - e^{-\eta})\cdot\sup_{x\in{\mathcal{X}}} \max\{|Q^*_Y(\epsilon|x) - {{\gamma}^\dagger}_0 - x^T{{\gamma}^\dagger}|, |Q^*_Y(1 - \epsilon|x) - {{\gamma}^\dagger}_0 - x^T{{\gamma}^\dagger}|\} < \frac\kappa2.$$ By the equicontinuity of the maps $s \mapsto \log q_0(e^s)$ and $s \mapsto {{h}^\dagger}(e^s)$ on the interval $[\log \epsilon, \log (1 - \epsilon)]$, and the continuity of the transformation $v \mapsto v / \{a(v, {\mathcal{X}})\sqrt{1 + \|v\|^2}\}$, one can fix $0 < \lambda < \min(\lambda_0, \kappa/4)$ such that for any $(\gamma_0, \gamma, \sigma, w, \zeta) \in A_\lambda$, $$\frac{q_Y(t|x)}{{{q}^\dagger}_Y(t|x)} = \frac\sigma{{{\sigma}^\dagger}} \times \frac{q_0(\zeta(t))}{q_0({{\zeta}^\dagger}(t))} \times \frac{1 + x^Th(\zeta(t))}{1 + x^T{{h}^\dagger}({{\zeta}^\dagger}(t))} \times \frac{\dot \zeta(t)}{\dot{{{\zeta}^\dagger}}(t)} \in [e^{-\eta}, e^{\eta}],$$ for every $t \in [\epsilon,1-\epsilon]$ and $x \in {\mathcal{X}}$. Consequently, by Lemma \[lem:aux2\] for every $x \in {\mathcal{X}}$ and $y \in [Q^*_Y(\epsilon|x), Q^*_Y(1 - \epsilon|x)]$, $|\log \{f_Y(y|x) / {{f}^\dagger}_Y(y|x)\}| < \delta$. This proves the result.
Computational details
=====================
( `// Could be parallelized in l`)[$l = 1:m$]{}[set $v_l = \omega(\zeta(t_l))$ calculate $a_{{\mathcal{X}}} = \max_{1 \le i \le n} \{-a_{i}\} / \sqrt{\|v_l\|}$ ]{} set $\ell\ell = 0$ ( `// Could be parallelized in i`)[$i = 1:n$]{}[ calculate $Q_0 = \gamma_0 + \gamma^T x_i$ ]{}
Centering the predictors {#A:centering}
------------------------
A preprocessing step of our method is to center the observed predictors $\{x_1, \ldots, x_n\}$ around an interior point of their convex hull (Figure \[fig:centering\]). While the sample mean vector automatically gives an interior point, it may lie too close to the hull boundary and lead to poorer model fit. A better strategy is to use the mean of the extreme points of the data cloud, but finding the extreme points becomes computationally intensive for $p > 2$. Instead, we employ a fast algorithm that recursively identifies $p + 1$ points $x^*_1, \ldots, x^*_{p+1}$, from the data cloud that are close to the boundary and far away from each other.
Consider a Gaussian process $f$ on ${\mathbb{R}}^p$ with covariance function $C(x, x') = \exp\{-\|\Delta^{-1}(x - x')\|^2\}$, where $\Delta$ is the $p\times p$ diagonal matrix with $j$-th element equaling the observed range of the $j$-th predictor, $j = 1, \ldots,p$. Take $x^*_1 = x_1$ and recursively select $x^*_j$ as the $x \in \{x_1, \ldots, x_n\}$ with maximum ${\rm Var}(f(x) | x^*_1, \ldots, x^*_{j-1})$, $j = 2, \ldots, p+1$. This recursive selection can be carried out extremely fast, with computational complexity of the order $(p+1)n\log n$ flops, by carrying out a rank-$(p+1)$ incomplete, pivoted Cholesky decomposition of the $n\times n$ non-negative definite matrix $K = ((C(x_i, x_j)))$, for example, by using the [inchol]{} function of the [R]{} package [kernlab]{}. Such implementations depend on the order in which the $x_i$s are stored. To encourage selection close from the boundary, we prearrange the $x_i$s in decreasing order of their Mahalanobis distance $\|S^{-1}(x_i - \bar x)\|$ from mean $\bar x$, where $S$ denotes the sample covariance.
Choosing $\lambda$ grid points {#a:support}
------------------------------
In choosing the grid points $\lambda_g$, $g = 1, \ldots, G$, for $\lambda_j$, it is important to ensure that the conditional prior distributions $N(0, \kappa_j^2 C_{**}(\lambda_g))$ remain sufficiently overlapped for neighboring $\lambda_g$ values, since otherwise, the grid based discretization of the prior on $\lambda$ may lead to poor mixing of the Markov chain sampler. If overlap is measured by the Kullback-Leibler divergence $d(\lambda, \lambda') := d_{KL}(N(0, \kappa_j^2 C_{**}(\lambda)), N(0, \kappa_j^2 C_{**}(\lambda')))$, which does not depend on $\kappa_j$, it is easy to see that one must use a non-uniform grid of $\lambda$ values since for a given $\Delta > 0$, $d(\lambda, \lambda + \Delta)$ is much larger for a small $\lambda$ than a large one. To choose this non-uniform grid, we set $\lambda_1$ to be the smallest value in the predetermined range, one that gives $\rho_{0.1}(\lambda_1) = 0.99$, and then increment $\lambda$ recursively so that $d(\lambda_{g-1}, \lambda_{g}) = 1$, $g = 2, 3, \ldots$, until the whole range is covered.
[^1]: relabeled for better clarity as ‘$3~-$ the original label’
|
---
abstract: 'Cosmic-ray proton and helium spectra have been measured with the balloon-borne Cosmic Ray Energetics And Mass experiment flown for 42 days in Antarctica in the 20042005 austral summer season. High-energy cosmic-ray data were collected at an average altitude of $\sim$38.5 km with an average atmospheric overburden of $\sim$3.9 g cm$^{-2}$. Individual elements are clearly separated with a charge resolution of $\sim$0.15 $e$ (in charge units) and $\sim$0.2 $e$ for protons and helium nuclei, respectively. The measured spectra at the top of the atmosphere are represented by power laws with a spectral index of $-$2.66 $\pm$ 0.02 for protons from 2.5 TeV to 250 TeV and 2.58 $\pm$ 0.02 for helium nuclei from 630 GeV nucleon$^{-1}$ to 63 TeV nucleon$^{-1}$. They are harder than previous measurements at a few tens of GeV nucleon$^{-1}$. The helium flux is higher than that expected from the extrapolation of the power law fitted to the lower-energy data. The relative abundance of protons to helium nuclei is 9.1 $\pm$ 0.5 for the range from 2.5 TeV nucleon$^{-1}$ to 63 TeV nucleon$^{-1}$. This ratio is considerably smaller than the previous measurements at a few tens of GeV nucleon$^{-1}$.'
author:
- 'Y. S. Yoon, H.S. Ahn, P. S. Allison, M. G. Bagliesi, J. J. Beatty, G. Bigongiari, P. J. Boyle, J. T. Childers, N. B. Conklin, S. Coutu, M. A. DuVernois, O. Ganel, J.H. Han, J.A. Jeon, K.C. Kim, M.H. Lee, L. Lutz, P. Maestro, A. Malinine, P.S. Marrocchesi, S. A. Minnick, S. I. Mognet, S. Nam, S. Nutter, I. H. Park, N.H. Park, E.S. Seo , R. Sina, S. Swordy, S. P. Wakely, J. Wu, J. Yang, R. Zei, S.Y. Zinn'
title: |
Cosmic-Ray Proton and Helium Spectra\
from the First CREAM Flight
---
Introduction
============
Cosmic rays are the product of energetic processes in the universe, and their interactions with matter and fields are the source of much of the diffuse gamma-ray, X-ray, and radio emissions that are observed. Therefore, the origin of cosmic rays and how they propagate have a major impact on our understanding of the universe. Supernova shock waves could provide the power required to sustain the galactic cosmic-ray intensity, but details of the acceleration mechanism are not completely understood. The shock acceleration mechanism is believed to be a prevalent process in astrophysical plasmas on all scales throughout the universe. It has been shown to work in the heliosphere, e.g., at planetary bow shocks, at interplanetary shocks in the solar wind, and at the solar wind termination shock.
It is a characteristic of diffusive shock acceleration that the resulting particle energy spectrum is much the same for a wide range of shock properties. This energy spectrum, when corrected for leakage from the Galaxy, is consistent with the observed spectrum of Galactic cosmic rays. In the most commonly used form of the theory, the characteristic limiting energy is about $Z$ $\times$ $10^{14}$ eV, where $Z$ is the particle charge [@Lagage1983]. The observed composition should begin to change beyond about $10^{14}$ eV, the limiting energy for protons, and the Fe spectrum would start to steepen at an energy 26 times higher. In this scenario, protons would be the most dominant element at low energies, but heavier elements would become relatively more abundant at higher energies, at least up to the acceleration limit for iron.
Compelling evidence that supernova remnants (SNRs) are common sites for shock acceleration of electrons comes from observations of non-thermal synchrotron radiation from several shell-type remnants [@Koyama1995; @Allen1997; @LeBohec2000]. Non-thermal X-ray spectra indicate the presence of very high energy electrons which, at least in the case of SN 1006, have energies $>$2 $\times 10^{14}$ eV [@Koyama1995]. These electrons were likely accelerated at the remnant because at this energy electrons cannot travel far from their origin before they are attenuated by synchrotron losses. There are other sources of particle acceleration that may also contribute to the cosmic-ray beam [@Dermer2001]. Recent *Chandra* X-ray observations of *Tycho’s* SNR have shown hot stellar debris keeping pace with an outward-moving shock wave indicated by high-energy electrons.
Semi-direct evidence for the acceleration of cosmic-ray protons could come in the form of gamma rays from pion decay [@Ellison2005]. Indeed, the observation of TeV gamma rays, possibly of $\pi^{0}$-origin, from the SNR RX J1713.7$-$3946 [@Enomoto2002; @Aharonian2007] may have revealed the first specific site where protons are accelerated to energies typical of the main cosmic-ray component. Their hadronic origin is yet to be confirmed, but the CANGAROO collaboration has shown that the energy spectrum of gamma-ray emission from SNR RX J1713.7$-$3946 matches that expected if the gamma rays are the decay products of neutral pions generated in *p-p* collisions. Although the proton scenario is favored because of the spectral shape, gamma rays may originate from either electrons or protons. A complete understanding of gamma-ray emission processes may need a broadband approach [@Aharonian2006], using all the available measurements in different wavelength regions. Direct measurements of nuclear particle composition changes would provide strong corroborating evidence that shocks associated with shell-type SNRs provide the acceleration sites for cosmic rays.
Shock acceleration is the generally accepted explanation for the characteristic power-law feature of cosmic-ray energy spectra, although ground-based measurements have shown that the all-particle spectrum extends far beyond the highest energy thought possible for supernova shock acceleration. These measurements have also shown that the energy spectrum above $10^{16}$ eV is somewhat steeper than the spectrum below $10^{14}$ eV, which lends credence to the possibility of a different source. Of course, the “knee” structure might be related to energy-dependent leakage effects during the propagation process [@Ptuskin1993; @Swordy1995] or to other effects, such as reacceleration in the galactic wind [@Voelk2003] and acceleration in pulsars [@Bednarek2002]. Whether and how the spectral “knee” is related to the mechanisms of acceleration, propagation, and confinement are among the major current questions in particle astrophysics.
CREAM Experiment
================
The Cosmic Ray Energetics And Mass (CREAM) experiment [@Seo2008] was designed and constructed to extend balloon and space-based direct measurements of cosmic-ray elemental spectra to the highest energy possible in a series of balloon flights. The detailed energy dependence of elemental spectra at very high energies, where the rigidity-dependent supernova acceleration limit could be reflected in composition change, provides a key to understanding the acceleration and propagation of cosmic rays. We report in this paper the proton and helium spectra as well as their ratios observed from the maiden flight of the CREAM payload in Antarctica. Results from the CREAM experiment such as B/C ratio and heavy elemental spectra are discussed in elsewhere [@Ahn2008TRD; @Ahn2009CREAM2; @Ahn2010CREAM12].
CREAM Flight 20042005
---------------------
The first Long Duration Balloon (LDB) flight of the CREAM payload was launched from McMurdo Station, Antarctica on December 16, 2004. It subsequently circumnavigated the South Pole three times for a record-breaking duration of 42 days; the flight was terminated on January 27, 2005. The instrument float altitude remained between 37 and 40 km through most of the flight. The corresponding atmospheric overburden was 3.9 $\pm$ 0.4 g cm$^{-2}$. The diurnal altitude variation due to the Sun angle change was very small, $<$1 km, near the pole, i.e., at high latitude, which increased as the balloon spiraled out to lower latitudes [@Seo2008]. The temperature of the various instrument boxes stayed within the required operational range with daily variation of a few $^{\circ}$C, consistent with the Sun angle. A total of 60 GB of data including $\sim$4 $\times$ 10$^{7}$ science events were collected.
The science instrument was supported by the command and data module developed by the NASA Wallops Flight Facility (WFF) [@Thompson2008]. This is in contrast to typical LDB payloads which utilize the support instrumentation package provided by the Columbia Scientific Balloon Facility. CREAM was the first LDB mission to transmit all the prime science and housekeeping data (up to 85 kbps) in near real-time through the Tracking and Data Relay Satellite System (TDRSS) via a high-gain antenna, in addition to having an onboard data archive. To fit the data into this bandwidth, science event records excluded information from channels that had levels consistent with their pedestal value. This “data sparsification” reduced the average high-energy shower event record size by nearly 95$\%$. The science instrument was controlled from a science operation center at the University of Maryland throughout the flight after line-of-sight operations ended at the launch site. Primary command uplink was via TDRSS, with Iridium serving as backup whenever the primary link was unavailable due to schedule or traversing zones of exclusion. The nearly continuous availability of command uplink and data downlink allowed rapid response to changing conditions on the payload (e.g., altitude-dependent effects) throughout the flight. More details about flight operations and the data acquisition system are discussed elsewhere [@Yoon2005Op; @Zinn2005].
CREAM Instrument
----------------
The instrument was designed to meet the challenging and conflicting requirements to have a large enough geometry factor to collect adequate statistics for the low flux of high-energy particles, and yet stay within the weight limit for near-space flights [@Ahn2007CREAMInst]. It was comprised of a suite of particle detectors to determine the charge and energy of the very high energy particles. As shown schematically in Figure \[fig:inst\], the detector configuration included a timing charge detector (TCD), a transition radiation detector (TRD) with a Cerenkov detector (CD), a silicon charge detector (SCD), hodoscopes (HDS), and a tungsten/scintillating fiber calorimeter. Starting from the top, the TCD consists of two crossed layers of four 5 mm thick and 1.2 m long plastic scintillators [@Ahn2009TCD]. It defines the 2.2 m$^{2}$ sr trigger geometry and determines charge based on the fact that the incident particle enters the TCD before developing a shower in the calorimeter, and the backscattered albedo particles arrive several nanoseconds later. A layer of scintillating fibers, S3, located between the carbon target and the tungsten calorimeter provides a reference time.
\[t\]
The TRD determines the Lorentz factor for $Z$ $\geqslant$ 3 nuclei by measuring transition X-rays using thin-wall gas tubes. Transition radiation is produced when a relativistic particle traverses an inhomogeneous medium, in particular the boundary between materials of different dielectric properties. The TRD consists of a foam radiator and 16 layers of proportional tubes filled with a mixture of xenon (95$\%$) and methane (5$\%$) gas [@Ahn2008TRD]. The CD between the two TRD sections provides low-energy particle rejection at the flight site, Antarctica, where the geomagnetic cutoff is low. It also provides additional charge identification.
The SCD is comprised of 380 $\mu$m thick Si sensors [@Park2007]. It is segmented into pixels, each about 2.12 cm$^{2}$ in area to minimize multiple hits in a segment due to backscattered particles. The targets are comprised of blocks of densified graphite cemented in carbon/epoxy composite cradles. The vertical thickness of the carbon targets is about 0.5 interaction lengths. They force hadronic interactions in the calorimeter, which measures the shower energy and provides tracking information to determine which segment(s) of the charge detectors to use for the charge measurement [@Seo1996].
The calorimeter consists of 20 tungsten layers interleaved with scintillating fiber ribbon layers which are alternately oriented in the $x$- and $y$-directions. Each tungsten layer is 1 radiation length thick to sample the shower every radiation length. Each layer consists of fifty 1 cm wide and 0.5 mm thick fiber ribbons to measure the longitudinal and lateral distributions of the shower. The light signal from each ribbon is collected by means of an acrylic light-mixer coupled to a bundle of clear fibers. This is split into three sub-bundles, each feeding a pixel of a hybrid photo diode (HPD). In this way the wide dynamic range of the calorimeter is divided into three sub-ranges (low, mid, high) with different gains, chosen to match the dynamic range of the front-end electronics [@Lee2006]. Tracking for showers is accomplished by extrapolating each shower axis back to the charge detectors. The HDS S0/S1 and S2, comprised of 2 mm thick and 2 mm wide scintillating fibers, provide additional tracking information above the tungsten stack [@Yoon2005HDS; @Marrocchesi2004]. The tracking uncertainty is smaller than the pixel size of the SCD [@Ahn2001].
Tracking for non-interacting particles is achieved in the TRD with better accuracy (1 mm resolution with 67 cm lever arm, 0.0015 radians). The TRD and calorimeter have different systematic biases in determining particle energy. The use of both instruments allows in-flight cross-calibration of the two techniques and, consequently, provides a powerful method for measuring cosmic-ray energies [@Maestro2007]. Details of the detectors and their performance are discussed elsewhere [@Lee2006; @Park2004; @Ahn2007CREAMInst].
Data Analysis
=============
The main trigger conditions for science events were (1) significant energy deposit in the calorimeter for high-energy particles or (2) large pulse height, $Z$ $>$ 2, in the TCD for heavy nuclei. The former requires each of six consecutive layers in the calorimeter to have at least one ribbon recording a deposit of more than 45 MeV. The high-energy shower events that meet this calorimeter trigger condition were used in this analysis.
Event Selection {#sec:evtsel}
---------------
The ribbon with the highest energy deposit and the neighboring ribbons on both sides were used to determine the position in each layer of maximum energy deposits. The shower axis was reconstructed by a least-squares fit of a straight line through a combination of these hit positions in the $XZ$ and $YZ$ planes [@Ahn2007CREAMHeavy]. Hits not along the straight line were excluded from the fit. The resulting trajectory resolution is $\sim$1 cm when projected to the SCD. The reconstructed trajectories were required to traverse the SCD active area and the bottom of the calorimeter active area.
At this stage non-interacting particles are removed, but some events have their first hadronic interaction in the calorimeter layers instead of the carbon targets. These late interacting events could result in an underestimation of deposited energy, or misidentification of charge due to large uncertainties in the trajectory reconstruction. Since their longitudinal shower profiles are different, events with small energy deposit in the top few layers of the calorimeter were removed to ensure that the selected events had their first interactions either in the carbon targets or in the top of the calorimeter.
Charge Determination
--------------------
In order to determine the incident particle charge, the reconstructed shower axis from the calorimeter was extrapolated to the SCD and a 7 $\times$ 7 pixel area, about 10 $\times$ 10 cm$^{2}$, centered on the extrapolated position, was scanned to seek for the highest pixel signal. The scanned area was optimized to sustain the charge identification efficiency of 99$\%$ in all energy bins, accounting for dead and noisy SCD and calorimeter channels ($\sim$15$\%$ and 13$\%$, respectively), and determined to be a 7 $\times$ 7 pixel area. That highest pixel signal was then corrected for the particle path length (calculated from the reconstructed incidence angle) in the sensor. The signal reflects the ionization energy loss per unit path length ($dE/dx$) of an incident particle in the SCD. The energy loss is proportional to $Z^{2}$. According to Monte Carlo (MC) simulations and beam tests, the expected contamination from secondary particles back scattering from the calorimeter is $<$3$\%$ when this tracking-based selection method is used [@ParkN2007]. The resulting SCD signal distribution is shown in Figure \[fig:scdz\]. Events with $Z$ $<$ 1.7 were selected as protons, while events with 1.7 $\leqslant$ $Z$ $<$ 2.7 were selected as helium nuclei. The charge resolutions are estimated as $\sim$0.15 $e$ and $\sim$0.2 $e$ for protons and helium nuclei, respectively. The proton and helium losses due to $dE/dx$ Landau tails were corrected by charge selection efficiencies, which will be discussed in Section \[sec:absoluteflux\]. The proton events in the helium range were removed as a background in the helium selection, and the helium events in the proton range were removed as a background in the proton selection, which will be discussed in Section \[sec:background\]. Unstable SCD channels identified by their large root-mean-square pedestal variations throughout the flight were excluded from the analysis. Including dead or noisy channels, $\sim$15$\%$ of the total 2,912 SCD channels were masked.
\[t\]
Energy Measurement
------------------
An ionization calorimeter is the only practical way to measure the energy of protons and helium nuclei above $\sim$1 TeV, but calorimeters with full containment of hadronic showers are too massive to be incorporated into space-based or balloon-borne experiments [@Ganel1999]. A thin calorimeter offers a practical approach but the calorimeter calibration requires the use of accelerator beam particles having known energy. The CREAM calorimeter was calibrated before the flight with electron beams at the European Organization for Nuclear Research (CERN). Each of the 1000 fiber ribbons was exposed to 150 GeV electrons. The responses from the 50 ribbons in a given layer are equalized by moving the detector in steps of 1 cm vertically or 1 cm horizontally, so the electron beam is centered each time on the center of a different ribbon in each X or Y layer.
The calorimeter was designed to measure the energy deposit from showers initiated by nuclei with energies up to 10$^{15}$ eV and higher. Its sampling fraction for isotropically incident TeV proton showers initiated in the graphite targets is about 0.13$\%$ of the parent’s energy in the active media. With electron test beam energies of 150 GeV or less, only 8$-$10 layers around the shower maximum register enough scintillation to allow calibration. To address this, the calibration scan was carried out in three sets of runs by exposing the calorimeter from the bottom with additional targets along the beam line as described in @Ahn2007CREAMInst.
The energy deposit expected along the shower core in each layer was calculated using MC simulations of electron showers. Conversion factors from analog-to-digital conversion (ADC) signals to MeV were obtained from the ratio of MC simulation of the energy deposited in each ribbon to the measured ADC signal from the calibration beam test. The MC simulations were based on GEANT/FLUKA 3.21 [@Brun1984; @Fasso1993]. The ADC signals were corrected for the HPD quantum efficiency and gain difference from the different HPD high-voltage settings between the beam test and the flight.
Inter-calibration between the low- and mid- energy ranges, and between the mid- and high- energy ranges were carried out with flight data by comparing the signals from two ranges of the same ribbon generated by the same shower. None of the proton and helium event candidates saturated in the middle range, so the high range optical division was not needed for this analysis. More details about the calibration can be found in @Yoon2005BT [@Yoon2007Calib] and @Ahn2006BT.
The calorimeter, HDS, and SCD were also exposed to nuclear fragments (A/Z = 2) of a 158 GeV nucleon$^{-1}$ Indium beam at CERN [@Yoon2007Calib; @Ahn2006BT; @Marrocchesi2004; @Park2004]. The energy response was linear up to the maximum beam energy of $\sim$9 TeV. Above the available accelerator beam energy, MC simulations indicate that the calorimeter response is quite linear in the CREAM measurement energy range. Simulations also indicate that the calorimeter energy resolution is nearly energy independent [@Ahn2001]. Nevertheless, our energy deconvolution included corrections for the small energy dependence of the energy resolution due to shower leakage [@Ahn2009CREAM2].
Spectral Deconvolution
----------------------
Entries in the deposited energy bins were deconvolved into incident energy bins using matrix relations. The counts, $N_{inc,i}$, in incident energy bin $i$ were estimated from the measured counts, $N_{dep,j}$, in deposited energy bin $j$ by the relation [@Buckley1994; @Ahn2006asr] $$N_{inc,\, i} = \sum_{j} P_{ij} N_{dep,\, j},$$ where matrix element $P_{ij}$ is a probability that the events in the deposited energy bin $j$ are from incident energy bin $i$. The matrix element $P_{ij}$ was estimated from the response matrix generated by MC simulation results obtained separately for protons and helium nuclei. The response matrix and corresponding deconvolution matrix were generated and tested by varying the indices between $-$2.5 and $-$2.8. We verified that the flux deconvolution process was not sensitive to the assumed spectral index used, within that range, to generate the matrix elements.
The MC simulations for helium and heavy nuclei used FRITIOF/RQMD/DPMJET-II [@Kim1999; @Wang2001] interfaced to the GEANT/FLUKA 3.21 hadronic simulation package. FRITIOF [@Andersson1993] is based upon semiclassical considerations of string dynamics for high-energy hadronic collisions. The relativistic quantum molecular dynamics (RQMD) model was adopted for simulations of heavy ions for energies in the center-of-mass frame less than 5 GeV nucleon$^{-1}$. RQMD is a semiclassical microscopic approach which combines classical propagation with stochastic interactions [@Sorge1995]. DPMJET-II [@Ranft1995; @Ferrari1996] was based on the dual parton model, a framework for hadronhadron interactions and production in hadronnucleus and nucleusnucleus collisions at high energies.
Background Corrections {#sec:background}
----------------------
The primary background is comprised of events with misidentified charge, which result mainly from secondary particles generated by interactions above the SCD or from particles back-scattered from the calorimeter. This is the case for the protons; however, there is an additional cause of misidentified events for helium nuclei: the proton $dE/dx$ Landau tail. Misidentified event counts of protons and helium nuclei were estimated from the MC simulations with a power-law input spectrum. Due to the Landau tails, back-scattered and secondary particles, 5.1$\%$ of measured protons were misidentified helium nuclei and 6.8$\%$ of measured helium nuclei were misidentified protons, as shown in Table \[tbl:effi\]. About 0.2$\%$ of incident carbon nuclei were identified as protons, and 2.8$\%$ of incident carbon nuclei were misidentified as helium nuclei, using the energy spectra of individual cosmic-rays compiled by @Wiebel-Sooth1998. Less than 1$\%$ of trigger and reconstructed protons and helium events are from secondary particles. Additional background comes from the events that are not within the geometry, but which satisfy the trigger and reconstruction conditions; they are either entering the instrument acceptance from outside the SCD area or exiting the side of the calorimeter instead of the bottom. According to MC simulations, this is about 3.6$\%$ and 4.0$\%$ of the selected events for protons and helium nuclei, respectively. The total background was 9$\%$ for protons and 11$\%$ for helium nuclei.
Absolute Flux {#sec:absoluteflux}
-------------
The measured spectra are corrected for the instrument acceptance as shown below to obtain the absolute flux F: $$F = \frac{dN}{dE} \frac {1}{GF ~ \varepsilon ~ T ~ \eta},
\label{eq:flux}$$ where $dN$ is the number of events in an energy bin, $dE$ is the energy bin size, $GF$ is the geometry factor, $\varepsilon$ is the efficiency (defined below), $T$ is the live time, and $\eta$ is the survival fraction after accounting for atmospheric attenuation. The geometry factor was calculated to be 0.43 m$^{2}$ sr using an MC simulations by requiring the extrapolated calorimeter trajectory of the incident particle to traverse the SCD active area and the bottom of the calorimeter. Out of 42 days of the flight, the stable period was about 24 days when no commands were sent, e.g., for instrument tuning, power-cycle, or high-voltage adjustments. After the dead-time correction, the live time, $T$, of 1,099,760 s was used for this analysis.
*Efficiency.* The efficiency, $\varepsilon$ in Equation (\[eq:flux\]) includes efficiencies from all analysis steps, including trigger condition, event reconstruction, charge identification, and removing events with late interactions: $$\varepsilon = \varepsilon_{trig} ~ \varepsilon_{rec} ~
\varepsilon_{sel} ~ \varepsilon_{charge}\,.$$ The trigger efficiency, $\varepsilon_{trig}$, was obtained from the fraction of events satisfying the trigger condition among all events within the geometry, i.e., passing through the bottom of the calorimeter and the SCD active area, using MC simulations. This is energy dependent at low energies where the trigger is not fully efficient. Above 3 TeV, it is nearly constant around 76$\%$ for protons and 91$\%$ for helium nuclei, respectively. The reconstruction efficiency, $\varepsilon_{rec}$, was taken to be the ratio of events satisfying the reconstruction and trigger conditions to events satisfying only the trigger condition. The reconstruction efficiency was 98$\%$ for protons and 99$\%$ for helium nuclei, respectively, based on MC simulations. The event selection efficiency, $\varepsilon_{sel}$, was estimated with the MC simulations after removing events with late interactions and was 90$\%$ protons and 96$\%$ for helium nuclei. The charge efficiency, $\varepsilon_{charge}$, takes into account lost events due to the noisy or dead SCD channels, interactions above SCD and misidentified charges. It was calculated to be 77$\%$ for protons and 67$\%$ for helium nuclei, respectively, using MC simulations. The efficiencies are summarized in Table \[tbl:effi\].
[lcc]{} Trigger efficiency & 76 $\pm$ 2 & 91 $\pm$ 1\
Reconstruction efficiency & 98 $\pm$ 1 & 99 $\pm$ 1\
Late interaction events efficiency & 90 $\pm$ 1 & 96 $\pm$ 1\
Charge selection efficiency & 77 $\pm$ 2 & 67 $\pm$ 2\
Background from reconstruction & 3.6 $\pm$ 0.1 & 4.0 $\pm$ 0.2\
Background from misidentified charge & 5.1 $\pm$ 0.2 & 6.8 $\pm$ 0.2\
The trigger efficiency for proton and helium nuclei cannot be estimated with the flight data, since we do not know how many un-triggered events occurred. However, the event selection efficiency, $\varepsilon_{sel}$, and charge efficiency, $\varepsilon_{charge}$, were estimated in a limited way using flight data for combined protons and helium nuclei events. It is not as accurate as individual MC simulations because the composition (abundance) of the incident particles is unknown. When the abundance ratio of protons and helium nuclei was assumed to be 1:1 and the abundance of heavy nuclei above helium nuclei was ignored, the combined efficiencies were 68$\%$ from the flight data and 67$\%$ for the MC simulations.
*Interactions in air.* The attenuation loss due to the atmospheric overburden, 3.9 $\pm$ 0.4 g cm$^{-2}$, was corrected for survival fractions of protons and helium nuclei. This air depth was measured by pressure sensors during the flight. Interaction cross sections have been measured in many fixed target experiments, and cross sections are known up to a few tens of GeV [@Hagen1977; @Webber1990; @Papini1996]. We used the cross section formula from @Hagen1977 to calculate interaction lengths and survival fractions for protons and helium nuclei. The mean incident angle of 35$^{\circ}$, estimated from the flight data, was used to estimate the losses. The survival fraction, $\eta$, used to characterize atmospheric attenuation was determined to be 95$\%$ for protons and 91$\%$ for helium nuclei, respectively.
The ratio of secondary to primary protons and helium nuclei in the atmosphere above GeV energies has been reported [@Kawamura1989; @Abe2003]. @Papini1996 calculated that the secondary to primary proton ratio at an air depth of 3 g cm$^{-2}$ was less than 1$\%$ above 40 GeV, and the secondary to primary helium nuclei ratio was less than 2$\%$ at 10 GeV nucleon$^{-1}$. Our MC simulations showed that the fraction of secondary protons and helium nuclei produced from carbon and iron nuclei interactions in the air was less than 1$\%$ at 10 TeV.
*Energy-bin representation.* For the number of events ($dN$) in each energy bin with upper- and lower-energy limits, $E_{j+1}$ and $E_{j}$, respectively ($dE = E_{j+1} - E_{j}$), the differential flux is $dN/dE$ at $E_{m}$, where $E_{m}$ can be taken as the arithmetic mean of $E_{j}$ and $E_{j+1}$ in logarithmic range or else using a suitably weighted average of $E_{j}$ and $E_{j+1}$. We also investigated an alternative procedure to determine $E_{m}$, as suggested by @Lafferty1995: $$f(E_{m}) = \frac{1}{E_{j+1} - E_{j}} \int_{E_{j}}^{E_{j+1}} f(E) dE.$$ For a power-law spectrum, $f(E) = A E^{-\gamma}$, $E_{m}$ can be calculated as, $$E_{m} = \Big( \frac{E_{j+1}^{1-\gamma}-E_{j}^{1-\gamma}}{(E_{j+1}-E_{j})(1-\gamma)} \Big)^{-1/\gamma}.$$ In this analysis, $E_{m}$ was used and the difference between $E_{m}$ and the center of the bin in logarithmic range is less than 1$\%$.
Uncertainties {#sec:uncertainties}
-------------
The statistical uncertainty in each energy bin was estimated by the relation $\delta N_{inc,i} = \delta (\sum_{j} P_{ij} N_{dep,j})$, considering 68.3$\%$ the Poisson confidence interval determined by @Feldman1998. The uncertainties were estimated by propagating uncertainties of measured entries in each bin and uncertainties of deconvolution components, $P_{ij}$, from MC simulations, while in the paper reported by @Ahn2010CREAM12, uncertainties were estimated by propagating uncertainties from measured entries with $P_{ij}$. This estimation gives more conservative results than the reported results.
Several sources of systematic uncertainties were identified. The systematic uncertainties for efficiencies and backgrounds were estimated within each energy range to account for the energy-dependent effects determined using MC simulations. They are summarized in Table \[tbl:effi\]. Efficiency uncertainties were about 12$\%$ and background uncertainties were about 5$\%$.
The geometry factor uncertainty was 2$\%$ for both protons and helium nuclei; it was estimated with MC simulations. The precision of estimated live-time fraction was about 3.3$\%$ and the accuracy of estimated dead time due to timeouts in TCD readout, which delayed processing, was about 2.6$\%$. The overall uncertainties for the estimated live time were 4$\%$ for both protons and helium nuclei. The systematic uncertainties for the survival fractions in the atmosphere were calculated analytically. The $p$$p$ cross section difference between 10 TeV and 100 TeV is about 28$\%$, according to the most recent reference from the Particle Data Group [@PDG2008]. Using a conservative estimate of 30$\%$ for cross section uncertainties, the estimated uncertainties of survival fractions were 2$\%$ and 3$\%$ for protons and helium nuclei, respectively. The range of incident angle was from 0$^{\circ}$ to 66$^{\circ}$. The uncertainty in correcting for atmospheric losses introduced by using an assumed mean incident angle was at the level of 1$\%$ for protons and 1.6$\%$ for helium nuclei. The energy calibration accuracy was found to be 1$\%$. The systematic uncertainties of the measured number in each energy bin, considering the 1$\%$ energy calibration accuracy, were 3$\%$ for both protons and helium nuclei. To estimate uncertainties in the spectral deconvolution, the unfolding procedure was repeated by varying input spectral indices. The difference of the proton fluxes varying the input spectral indices between 2.64 and 2.68 was less than 1$\%$. Similarly, the helium flux difference was also less than 1$\%$ for input spectra between indices 2.56 and 2.60. The overall systematic uncertainties were found to be 9$\%$ for both protons and helium nuclei. These systematic uncertainties are energy independent. They do not change the spectral shape, but they might shift the normalization of the spectra up or down.
Results
=======
The measured proton fluxes from 2.5 TeV to 250 TeV and helium fluxes from 630 GeV nucleon$^{-1}$ to 63 TeV nucleon$^{-1}$ at the top of the atmosphere are given in Tables \[tbl:proton\] and \[tbl:helium\], while previously reported results in the paper by @Ahn2010CREAM12 are presented in a plot. The statistical uncertainties were re-estimated, as discussed in Section \[sec:uncertainties\]. The CREAM proton and helium spectra are each consistent with a single power law over the measured range. The best-fit parameters for the spectra for protons and helium nuclei are represented by $$\frac{d\Phi}{dE} \,=\, \Phi_{0} E^{-\beta} ~~~ (\textmd{m}^{2}\,\textmd{sr\,s\,GeV\,nucleon}^{-1})^{-1}.$$ The best-fit parameters for the spectra for protons and helium nuclei are given by [@Ahn2010CREAM12]: $$\Phi_{0,p} \,=\, ( 7.8 \pm 1.9 ) \times 10^{3}
~~(\textmd{m}^{2}\,\textmd{sr\,s})^{-1} (\textmd{GeV nucleon}^{-1})^{1.66},$$ $$\beta_{p} \,=\, 2.66 \pm 0.02,$$ and $$\Phi_{0,He} \,=\, ( 4.2 \pm 0.8 ) \times 10^{2}
~~(\textmd{m}^{2}\,\textmd{sr\,s})^{-1}\,(\textmd{GeV\,nucleon}^{-1})^{1.58},$$ $$\beta_{He} \,=\, 2.58 \pm 0.02.$$
The spectral indices for proton and helium nuclei were calculated both with the least squares fit and maximum likelihood method. The results from both methods were consistent. Uncertainties for the spectral indices were estimated with the maximum likelihood method.
[cc]{}
$2.5 {\times 10}^{3} - 4.0 {\times 10}^{3}$ & $ ( 3.72 \pm 0.10 ) {\times 10}^{-6}$\
$4.0 {\times 10}^{3} - 6.3 {\times 10}^{3}$ & $ ( 1.10 \pm 0.04 ) {\times 10}^{-6}$\
$6.3 {\times 10}^{3} - 1.0 {\times 10}^{4}$ & $ ( 3.19 \pm 0.19 ) {\times 10}^{-7}$\
$1.0 {\times 10}^{4} - 1.6 {\times 10}^{4}$ & $ ( 9.47 \pm 0.80 ) {\times 10}^{-8}$\
$1.6 {\times 10}^{4} - 2.5 {\times 10}^{4}$ & $ ( 2.80 \pm 0.35 ) {\times 10}^{-8}$\
$2.5 {\times 10}^{4} - 4.0 {\times 10}^{4}$ & $ ( 8.1 \pm 1.5 ) {\times 10}^{-9}$\
$4.0 {\times 10}^{4} - 6.3 {\times 10}^{4}$ & $ ( 2.2 \pm 0.6 ) {\times 10}^{-9}$\
$6.3 {\times 10}^{4} - 1.0 {\times 10}^{5}$ & $ ( 6.1 ^{+2.6}_{-2.2} ) {\times 10}^{-10}$\
$1.0 {\times 10}^{5} - 1.6 {\times 10}^{5}$ & $ ( 1.8 ^{+1.2}_{-0.9} ) {\times 10}^{-10}$\
$1.6 {\times 10}^{5} - 2.5 {\times 10}^{5}$ & $ ( 4.2 ^{+5.4}_{-3.4} ) {\times 10}^{-11}$\
[cc]{}
$6.3 {\times 10}^{2} - 1.0 {\times 10}^{3}$ & $ ( 1.42 \pm 0.04 ) {\times 10}^{-5}$\
$1.0 {\times 10}^{3} - 1.6 {\times 10}^{3}$ & $ ( 4.35 \pm 0.16 ) {\times 10}^{-6}$\
$1.6 {\times 10}^{3} - 2.5 {\times 10}^{3}$ & $ ( 1.31 \pm 0.07 ) {\times 10}^{-6}$\
$2.5 {\times 10}^{3} - 4.0 {\times 10}^{3}$ & $ ( 3.83 \pm 0.31 ) {\times 10}^{-7}$\
$4.0 {\times 10}^{3} - 6.3 {\times 10}^{3}$ & $ ( 1.27 \pm 0.14 ) {\times 10}^{-7}$\
$6.3 {\times 10}^{3} - 1.0 {\times 10}^{4}$ & $ ( 4.19 \pm 0.64 ) {\times 10}^{-8}$\
$1.0 {\times 10}^{4} - 1.6 {\times 10}^{4}$ & $ ( 1.15 \pm 0.27 ) {\times 10}^{-8}$\
$1.6 {\times 10}^{4} - 2.5 {\times 10}^{4}$ & $ ( 3.4 ^{+1.1}_{-1.0} ) {\times 10}^{-9}$\
$2.5 {\times 10}^{4} - 4.0 {\times 10}^{4}$ & $ ( 8.2 ^{+4.9}_{-3.8} ) {\times 10}^{-10}$\
$4.0 {\times 10}^{4} - 6.3 {\times 10}^{4}$ & $ ( 2.9 ^{+2.4}_{-1.5} ) {\times 10}^{-10}$\
The CREAM proton spectrum is harder than previous measurements at lower energies such as AMS [@AMS2002], 2.78 $\pm$ 0.009 at 10200 GV and BESS [@Haino2004], 2.732 $\pm$ 0.011 from 30 GeV to a few hundred GeV. Likewise, the CREAM helium spectrum is harder than AMS, 2.740 $\pm$ 0.01 at 20$-$200 GV and BESS, 2.699 $\pm$ 0.040 from 20 GeV nucleon$^{-1}$ to a few hundred GeV nucleon$^{-1}$.
Figure \[fig:cream1hhe275\] compares our measured spectra with previous measurements: AMS, BESS, CAPRICE98 [@Boezio2003], ATIC-2 [@Panov2009], JACEE [@Asakimori1998] and RUNJOB [@Derbina2005]. The error bars shown in the figures represent the statistical uncertainties. The CREAM results are consistent with JACEE where its measurement energy range overlaps with CREAM but indicate higher fluxes, particularly for helium, with respect to RUNJOB. The proton and helium fluxes are both higher than that expected by extrapolating the power law fitted to the lower-energy measurements, which verifies that our TeV spectra are harder than the lower-energy spectra. At 20 TeV nucleon$^{-1}$ the helium flux measured by CREAM is about 4$\sigma$ higher than the flux expected from a power-law extrapolation of the AMS helium flux and spectral index.
\[t\]
The proton to helium ratio as a function of energy provides insight into whether the proton and helium spectra have the same spectral index. This has long been a tantalizing question, mainly because of the limited energy range individual experiments could cover. The ratio from the first CREAM flight provides a much needed higher energy, low-statistical uncertainty, measurement. The ratio is compared with previous measurements in Figure \[fig:cream1ratio\]: ATIC-2, CAPRICE94 [@Boezio1999], CAPRICE98, JACEE [@Asakimori1993he], LEAP [@Seo1991], and RUNJOB. The CREAM ratios are consistent with JACEE where its measurement energy range overlaps. The measured CREAM ratio at the top of the atmosphere is on average 9.1 $\pm$ 0.5 for the range from 2.5 TeV nucleon$^{-1}$ to 63 TeV nucleon$^{-1}$, which is significantly lower than the ratio of $\sim$20 obtained from the lower-energy measurements.
\[t\]
Discussion and Conclusion
=========================
The energy spectra of primary cosmic rays are known with good precision up to energies around 10$^{11}$ eV, where magnetic spectrometers have been able to carry out such measurements. Above this energy the composition and energy spectra are not accurately known, although there have been some pioneering measurements [@Mueller1991; @Asakimori1998; @Apanasenko2001]. The collecting power of CREAM is about a factor of two larger than that of ATIC for protons and helium nuclei and, considering the much larger geometry factor of the TRD, about a factor of 10 larger for heavier nuclei. TRACER has a larger geometry factor than CREAM, but a smaller dynamic charge range ($Z$ = 826) was reported for its 10-day Antarctic flight. Although its dynamic charge range was improved to $Z$ =326 for its $\sim$4 day flight from Sweden to Canada in 2006, it is still insensitive to protons and helium nuclei.
The CREAM payload maintained a high altitude, corresponding to an atmospheric overburden of 3.9 g cm$^{-2}$ for vertically incident particles. That implies about 6.8 g cm$^{-2}$ at the maximum acceptance angle for this analysis, which is smallest among comparable experiments. For example, the average vertical depth for RUNJOB was more than twice that of CREAM, due to its low flight altitude. Considering the RUNJOB acceptance of particles at large zenith angles, its effective atmospheric depth was as large as 50 g cm$^{-2}$. For that depth, large corrections are required to account for the fact that 41$\%$ of protons would have interacted before reaching the detector.
The CREAM calorimeter is much deeper than either that of JACEE or RUNJOB, so it provides better energy measurements. CREAM also has excellent charge resolution, sufficient to clearly identify individual nuclei, whereas JACEE and RUNJOB reported elemental groups. Our observation did not confirm a softer spectrum of protons above 2 TeV reported by @Grigorov1970 or a bend around 40 TeV [@Asakimori1993p]. An increase in the flux of helium relative to protons could be interpreted as evidence for two different types of sources for protons and helium nuclei as proposed by @Biermann1993. The observed harder spectra compared to prior low-energy measurements may require a significant modification of conventional acceleration and propagation models, with significant impact for the interpretation of other experimental observations.
The CREAM experiment was planned for Ultra Long Duration Balloon (ULDB) flights lasting about 100 days with super-pressure balloons. While waiting for development of these exceptionally long flights, the CREAM instrument has flown five times on LDB flights in Antarctica. It should be noted that a 7 million cubic foot ($\sim$0.2 million cubic meters) super-pressure balloon was flown successfully for 54 days during the 2008-2009 austral summer season. As ULDB flights become available for large science payloads, long-duration exposures can be achieved faster and more efficiently with reduced payload refurbishment and launch efforts. Whatever the flight duration, data from each flight reduces the statistical uncertainties and extends the reach of measurements to energies higher than previously possible.
This work was supported in the U.S. by NASA grants NNX07AN54H, NNX08AC11G, NNX08AC15G, NNX08AC16G and their predecessor grants, in Italy by INFN, in Korea by the Creative Research Initiatives of MEST/NRF. The authors wish to acknowledge NASA/WFF for provision and operation of flight support systems; CERN for provision of excellent accelerator beams; and CSBF, National Science Foundation’s Office of Polar Programs, and Raytheon Polar Services Company for outstanding support of launch, flight, and recovery operations in Antarctica.
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|
---
abstract: 'For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein–Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so called “monodromy transform” approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases.'
author:
- |
G. A. Alekseev$^{1,}$[^1] and J. B. Griffiths$^{2,}$[^2]\
\
$^1$Steklov Mathematical Institute, Gubkina 8, Moscow 117966, GSP-1, Moscow, Russia.\
$^2$Department of Mathematical Sciences, Loughborough University, Loughborough,\
Leics. LE11 3TU, U.K.\
title: 'Infinite hierarchies of exact solutions of the Einstein and Einstein–Maxwell equations for interacting waves and inhomogeneous cosmologies'
---
Introduction {#introduction .unnumbered}
------------
A number of solution-generating techniques are known which provide tools for the construction of vacuum and electrovacuum solutions of Einstein’s equations for space-times with symmetries. These methods are based on the integrability of the symmetry reduced Einstein equations (viz. the Ernst equations). However, most of them were primarily designed to construct exact stationary axisymmetric solutions for which an additional regularity condition should be satisfied on the axis. This condition does not apply to interacting waves or cosmological models as considered here.
Apart from the completely linearizable subcase of Einstein–Rosen vacuum gravitational waves, the only techniques which provide nontrivial tools for the construction of solutions for the dynamical case are the vacuum Belinskii–Zakharov inverse-scattering method [@BelZak78], the so called “monodromy transform” approach [@Alek85; @Alek87; @Alek99], and the group-theoretical approach recently developed by Hauser and Ernst [@HauErn99]. In particular, the methods of [@BelZak78] enable the construction of soliton perturbations of homogeneous cosmological models and some specific solutions for wave interaction regions. For example, the Khan–Penrose [@KhaPen71] or Nutku–Halil [@NutHal77] solutions for the interaction region for colliding impulsive gravitational waves on a Minkowski background formally turn out to be two-soliton solutions on a symmetric Kasner background.
Here we consider the monodromy transform approach and the linear singular integral equations which arise in this context as an alternative form of the reduced Einstein equations. We present a new method for the solution of these equations which gives rise to infinite hierarchies of exact solutions. Among many other solutions, these include the particular cases mentioned above together with other soliton solutions on the symmetric Kasner background and their non-soliton extensions.
Integral equation form of reduced Einstein equations {#integral-equation-form-of-reduced-einstein-equations .unnumbered}
----------------------------------------------------
According to methods developed in [@Alek85; @Alek87; @Alek99], any solution of the Ernst equations can be constructed from the solution of the linear singular integral equation $${1\over \pi i}\int_L
{[\lambda ]_\zeta\over\zeta-\tau}
{\cal H}(\tau,\zeta)
{\bphi}(\xi,\eta,\zeta)\, d\zeta =-\k(\tau)
\label{IntEqs}$$ considered here for the hyperbolic case only. The parameters $\xi$, $\eta$ are two real null space-time coordinates, e.g. $(\xi,\eta)=(x+t,x-t)$. These coordinates span some local region in the neighbourhood of some initial regular space-time point $P_0$: $\xi=\xi_0$, $\eta=\eta_0$, in which local solutions of the reduced Einstein equations are considered.
The integration in (\[IntEqs\]) is performed along the path $L$ on the spectral plane $w$ which consists of two disconnected parts $L_{\scriptscriptstyle+}$ and $L_{\scriptscriptstyle-}$. In the hyperbolic case, these are chosen as the segments of the real axis in the $w$-plane, which go from $w=\xi_0$ to $w=\xi$, and from $w=\eta_0$ to $w=\eta$ respectively.(We choose $\xi_0\ne \eta_0$ and take $\xi$ and $\eta$ sufficiently close to $\xi_0$ and $\eta_0$ that the segments $L_{\scriptscriptstyle\pm}$ do not overlap.)
The integral in (\[IntEqs\]) splits into two, one of which possesses a singular kernel of Cauchy type and should be understood as a Cauchy principal value integral. The integration parameter $\zeta$ and a parameter $\tau$ span both of the contours $L_{\scriptscriptstyle+}$ and $L_{\scriptscriptstyle-}$. Sometimes it will be convenient to introduce suffices: $\zeta_{\scriptscriptstyle+},\tau_{\scriptscriptstyle+}\in
L_{\scriptscriptstyle+}$ and $\zeta_{\scriptscriptstyle-},\tau_{\scriptscriptstyle-}\in
L_{\scriptscriptstyle-}$.
In the integrand in (\[IntEqs\]), $[\lambda]_\zeta={1\over
2}(\lambda_{\rm left}-\lambda_{\rm right})$. This represents the jump on the contour, i.e. half of the difference between left and right limit values at the point $\zeta\in
L_{\scriptscriptstyle+}$ or $\zeta\in L_{\scriptscriptstyle-}$ of some “standard” function $\lambda(\xi,\eta,w)$. This function is a product of two functions $\lambda(\xi,\eta,w)=\lambda_{\scriptscriptstyle+}(\xi,w)
\lambda_{\scriptscriptstyle-}(\eta,w)$ given by $$\label{lampm}
\lambda_{\scriptscriptstyle+}=\sqrt{w-\xi\over w-\xi_0},
\qquad \lambda_{\scriptscriptstyle-}=\sqrt{w-\eta\over w-\eta_0},$$ with the additional conditions $\lambda_{\scriptscriptstyle+}|_{w=\infty}
=\lambda_{\scriptscriptstyle-}|_{w=\infty}=1$. Each of these functions is an analytic function on the whole spectral plane $w$ apart from the cut $L_{\scriptscriptstyle+}$ or $L_{\scriptscriptstyle-}$ respectively, whose endpoints are the branching points of the corresponding function.
In the equations (\[IntEqs\]), the three-dimensional complex vector function $\bphi(\xi,\eta,\zeta)$ is unknown, and the right hand side $\k(\tau)$ is a three-dimensional complex vector function of the spectral parameter which may be taken to be $$\k(w)=\{1,{\bf u}(w),\v(w)\},
\label{kw}$$ where ${\bf u}(w)$ and $\v(w)$ are arbitrary functions. The kernel of the integral in (\[IntEqs\]) is a scalar function ${\cal H}(\tau,\zeta)$ given by $$\begin{aligned}
&&{\cal H}(\tau,\zeta) =1+i (\zeta-\beta_0) ({\bf
u}(\tau)-{\bf u}^\dagger(\zeta)) +\alpha_0^2{\bf u}(\tau){\bf
u}^\dagger(\zeta) \nonumber \\
&&\qquad\qquad\qquad -4 (\zeta-\xi_0)(\zeta-\eta_0)\v(\tau)\v^\dagger(\zeta)
\label{calH} \end{aligned}$$ where the dagger denotes complex conjugation: e.g. ${\bf u}^\dagger(w)\equiv \overline{{\bf u}(\overline{w})}$. The additional constants in (\[calH\]) are $\alpha_0=(\xi_0-\eta_0)/2$ and $\beta_0=(\xi_0+\eta_0)/2$.
It is important to emphasize that the integral equations (\[IntEqs\]), and hence the functions ${\bf u}(w)$, $\v(w)$ and $\bphi(\xi,\eta,w)$, only need to be evaluated on the two cuts $L_{\scriptscriptstyle+}$ and $L_{\scriptscriptstyle-}$ in the spectral plane. Thus all the above vector and scalar functions of the spectral parameter are actually determined by pairs of functions which represent their values on these contours. For convenience we shall denote the values of these functions on $L_{\scriptscriptstyle\pm}$ by the corresponding suffices: $$\{\u(w),\v(w)\}=
\left\{\matrix{\{\u_{\scriptscriptstyle+}(w),
\v_{\scriptscriptstyle+}(w)\},\qquad w\in
L_{\scriptscriptstyle+}\cr
\{\u_{\scriptscriptstyle-}(w),
\v_{\scriptscriptstyle-}(w)\},\qquad w\in
L_{\scriptscriptstyle-}}\right.
\label{defuvpm}$$ Thus, in (\[IntEqs\]) written in a more explicit form, we actually have two unknown vector functions $\bphi_{\scriptscriptstyle\pm}$. For any of these suffixed functions we can use also an alternative definition, for example, $$\bphi(\xi,\eta,\tau_{\scriptscriptstyle\pm})\equiv
\bphi_{\scriptscriptstyle\pm}(\xi,\eta,\tau).$$
Using this notation, it is convenient to split the integral in (\[IntEqs\]) into separate integrals over $L_{\scriptscriptstyle+}$ and $L_{\scriptscriptstyle-}$ and to consider separately the cases $\tau=\tau_{\scriptscriptstyle+}\in
L_{\scriptscriptstyle+}$ and $\tau=\tau_{\scriptscriptstyle-}\in
L_{\scriptscriptstyle-}$. It is also convenient to denote the four scalar kernels which appear in the integrands of (\[IntEqs\]) in the form $$\begin{aligned}
{\cal H}(\tau_{\scriptscriptstyle+},\zeta_{\scriptscriptstyle+})\equiv
{\cal H}_{\scriptscriptstyle++}(\tau,\zeta), && \qquad {\cal
H}(\tau_{\scriptscriptstyle+},\zeta_{\scriptscriptstyle-})\equiv
{\cal H}_{\scriptscriptstyle+-}(\tau,\zeta), \nonumber \\
{\cal H}(\tau_{\scriptscriptstyle-},\zeta_{\scriptscriptstyle+})\equiv
{\cal H}_{\scriptscriptstyle-+}(\tau,\zeta), && \qquad {\cal
H}(\tau_{\scriptscriptstyle-},\zeta_{\scriptscriptstyle-})\equiv
{\cal H}_{\scriptscriptstyle--}(\tau,\zeta)
\nonumber
% \label{calHpm}
\end{aligned}$$ where the functions ${\cal H}_{\scriptscriptstyle++}(\tau,\zeta)$, ${\cal H}_{\scriptscriptstyle+-}(\tau,\zeta)$, ${\cal H}_{\scriptscriptstyle-+}(\tau,\zeta)$ and ${\cal H}_{\scriptscriptstyle--}(\tau,\zeta)$ can be determined explicitly in terms of the four functions ${\bf u}_{\scriptscriptstyle\pm}(w)$ and $\v_{\scriptscriptstyle\pm}(w)$ using (\[calH\]).
To conclude our description of the structure of the master integral equations, we recall that the four functions ${\bf u}_{\scriptscriptstyle\pm}(w)$ and $\v_{\scriptscriptstyle\pm}(w)$ appearing in (\[kw\]) and (\[defuvpm\]) play a significant role in the entire construction. They determine completely the coefficients of the integral equations in the electrovacuum case. In the vacuum case there are only two such functions ${\bf u}_{\scriptscriptstyle\pm}(w)$, as $\v_{\scriptscriptstyle\pm}(w)\equiv0$. As shown in [@Alek87], they characterize unambiguously every individual solution of the Ernst equations. Moreover, the singular integral equations (\[IntEqs\]) possess a unique solution for any given choice of analytical functions ${\bf u}_{\scriptscriptstyle\pm}(w)$ and $\v_{\scriptscriptstyle\pm}(w)$.
We recall now also, that the general local solution of the hyperbolic Ernst equations can be expressed by quadratures in terms of the solution of (\[IntEqs\]) $$\begin{aligned}
&&{\cal E} =-1
-{2\over\pi} \int_L [\lambda]_\zeta \big[1-i (\zeta-\beta_0) {\bf
u}^\dagger(\zeta)\big] \bphi^{[u]}(\xi,\eta,\zeta)\,d\zeta \nonumber
\\ &&\Phi ={2\over\pi} \int_L [\lambda]_\zeta \big[1-i
(\zeta-\beta_0) {\bf u}^\dagger(\zeta)\big]
\bphi^{[v]}(\xi,\eta,\zeta)\,d\zeta
\label{Potentials}\end{aligned}$$ where $\bphi^{[u]}$ and $\bphi^{[v]}$, in some association with the definition (\[kw\]), denote respectively the second and third components of the vector solutions $\bphi$ of the master integral equation (\[IntEqs\]), corresponding to a given choice of the monodromy data functions ${\bf u}_{\scriptscriptstyle\pm}(w)$ and $\v_{\scriptscriptstyle\pm}(w)$. In a more explicit form, each of the the integrals in (\[Potentials\]) should be split into two integrals evaluated over $L_{\scriptscriptstyle+}$ and $L_{\scriptscriptstyle-}$.
New hierarchies of solutions {#new-hierarchies-of-solutions .unnumbered}
-----------------------------
Here we will construct a class of hyperbolic solutions that is determined by the rational monodromy data $$\label{uvpm}
\u_{\scriptscriptstyle\pm}(w) ={U_{\scriptscriptstyle\pm}(w)\over
Q_{\scriptscriptstyle\pm}(w)},\qquad
\v_{\scriptscriptstyle\pm}(w) ={V_{\scriptscriptstyle\pm}(w)\over
Q_{\scriptscriptstyle\pm}(w)}
\label{rational}$$ where $U_{\scriptscriptstyle+}(w)$, $V_{\scriptscriptstyle+}(w)$, $Q_{\scriptscriptstyle+}(w)$ and $U_{\scriptscriptstyle-}(w)$, $V_{\scriptscriptstyle-}(w)$, $Q_{\scriptscriptstyle-}(w)$ are arbitrary complex polynomials, provided $\u_+(w)$, $\v_+(w)$ and $\u_-(w)$, $\v_-(w)$ do not have poles on $L_+$ and $L_-$ respectively.
For what follows, it is convenient to calculate some auxiliary polynomials of two variables – we introduce the four polynomials $P_{\scriptscriptstyle\pm\pm}(\tau,\zeta)$ defined by the relations $${\cal H}_{\scriptscriptstyle\pm\dot\pm}(\tau,\zeta)
={P_{\scriptscriptstyle\pm\dot\pm}(\tau,\zeta)\over
Q_\pm(\tau) Q_{\dot\pm}^\dagger(\zeta)},
\label{Hpmpm}$$ and four polynomials $R_{\scriptscriptstyle\pm\pm}(\tau,\zeta)$ defined from them by $$R_{\scriptscriptstyle\pm\dot\pm}(\tau,\zeta)=
{P_{\scriptscriptstyle\pm\dot\pm}(\tau,\zeta)-
P_{\scriptscriptstyle\pm\dot\pm}(\zeta,\zeta)\over \zeta-\tau}.
%=\sum\limits_{k=0}^{N_{\scriptscriptstyle\pm}}
%R_{\scriptscriptstyle\pm\dot\pm k}
%(\zeta) \tau^k
\label{Rpmpm}$$ In these definitions there are two sets of suffices, denoted as dotted and undotted, which should each be taken to be the same. Finally, it is convenient to introduce a redefinition of the unknown functions$$\begin{aligned}
&&\bphi_{\scriptscriptstyle+}(\zeta)=
-{ \lambda_{\scriptscriptstyle-}^{-1}(\zeta)
Q_{\scriptscriptstyle+}^\dagger(\zeta)
\over P_{\scriptscriptstyle++}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle+}(\zeta), \nonumber \\
&&\bphi_{\scriptscriptstyle-}(\zeta)=
-{ \lambda_{\scriptscriptstyle+}^{-1}(\zeta)
Q_{\scriptscriptstyle-}^\dagger(\zeta)
\over P_{\scriptscriptstyle--}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle-}(\zeta).
\label{Phitilde}\end{aligned}$$ Hereafter we do not show explicitly the arguments $\xi$ and $\eta$ of $\bphi_\pm$ and $\lambda$ or the suffices $\pm$ at the points $\zeta$ and $\tau$, unless it is necessary.
A direct substitution of (\[uvpm\]) into equations (\[IntEqs\]) with the use of (\[Hpmpm\])–(\[Phitilde\]) leads to the following convenient form of linear equations with polynomial right hand sides $$\begin{aligned}
\label{PolyEqs}
{1\over \pi i} \int\limits_{\xi_0}^{\xi}
{[\lambda_{\scriptscriptstyle+}]_\zeta
\over\zeta-\tau_{\scriptscriptstyle+}}
\widetilde{\bphi}_{\scriptscriptstyle+}(\zeta)\,d\zeta
&&=-{1\over \pi i} \int\limits_{\xi_0}^{\xi}
[\lambda_{\scriptscriptstyle+}]_\zeta
{R_{\scriptscriptstyle++}(\tau_{\scriptscriptstyle+},\zeta)\over
P_{\scriptscriptstyle++}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle+}(\zeta)\,d\zeta\nonumber
\\
&&\qquad-{1\over \pi i}
\int\limits_{\eta_0}^{\eta}
[\lambda_{\scriptscriptstyle-}]_\zeta
{R_{\scriptscriptstyle+-}(\tau_{\scriptscriptstyle+},\zeta)
\over P_{\scriptscriptstyle--}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle-}(\zeta)\,d\zeta
+\pmatrix{Q_{\scriptscriptstyle+}(\tau_{\scriptscriptstyle+})\cr
U_{\scriptscriptstyle+}(\tau_{\scriptscriptstyle+})\cr
V_{\scriptscriptstyle+}(\tau_{\scriptscriptstyle+})}, \nonumber
\\ &&{} \label{lineqs} \\
{1\over \pi i}
\int\limits_{\eta_0}^{\eta}{[\lambda_{\scriptscriptstyle-}]_\zeta
\over\zeta-\tau_{\scriptscriptstyle-}}
\widetilde{\bphi}_{\scriptscriptstyle-}(\zeta)\,d\zeta
&&=-{1\over \pi i} \int\limits_{\eta_0}^{\eta}
[\lambda_{\scriptscriptstyle-}]_\zeta
{R_{\scriptscriptstyle--}(\tau_{\scriptscriptstyle-},\zeta)\over
P_{\scriptscriptstyle--}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle-}(\zeta)\,d\zeta\nonumber
\\
&&\qquad-{1\over \pi i}
\int\limits_{\xi_0}^{\xi} [\lambda_{\scriptscriptstyle+}]_\zeta
{R_{\scriptscriptstyle-+}
(\tau_{\scriptscriptstyle-},\zeta)\over
P_{\scriptscriptstyle++}(\zeta,\zeta)}
\widetilde{\bphi}_{\scriptscriptstyle+}(\zeta)\,d\zeta
+\pmatrix{Q_{\scriptscriptstyle-}(\tau_{\scriptscriptstyle-})\cr
U_{\scriptscriptstyle-}(\tau_{\scriptscriptstyle-})\cr
V_{\scriptscriptstyle-}(\tau_{\scriptscriptstyle-})} \nonumber
\end{aligned}$$ if we impose constraints on the coefficients of the rational functions (\[uvpm\]) such that $$P_{\scriptscriptstyle+-}(\zeta,\zeta)
=P_{\scriptscriptstyle-+}(\zeta,\zeta)=0.
\label{ansatz1}$$ This leads to a large class of explicit solutions $\widetilde{\bphi}_{\scriptscriptstyle\pm}(\xi,\eta,\tau)$ of (\[lineqs\]) that are regular on the cuts $L_\pm$. However, the solution of the Ernst equations needs the solutions $\bphi_{\scriptscriptstyle\pm}(\xi,\eta,\tau)$ of (\[IntEqs\]) to be regular on the cuts $L_\pm$. Fortunately, all additional singularities (poles) of $\bphi_{\scriptscriptstyle+}(\xi,\eta,\tau)$ on $L_+$ and $\bphi_{\scriptscriptstyle-}(\xi,\eta,\tau)$ on $L_-$, which arise from the denominators in (\[Phitilde\]), can be avoided by the additional restrictions that $\u_+(\eta_0)=-i/\alpha_0$ and $\u_-(\xi_0)=i/\alpha_0$. We therefore specify $$\begin{aligned}
&&\u_+(w)=-{i\over\alpha_0}+(w-\eta_0) {C_+(w)\over Q_+(w)} \nonumber\\
&&\u_-(w)={i\over\alpha_0}+(w-\xi_0) {C_-(w)\over Q_-(w)}
\label{upm}
\end{aligned}$$ where $C_+(w)$, $C_-(w)$, $Q_+(w)$ and $Q_-(w)$ are arbitrary polynomials. With these, the ansatz (\[ansatz1\]) leads to the constraint $C_-(w)=B(w)C_+^\dagger(w)-4 i A(w)V_+^\dagger(w)$ and, for the polynomials in (\[rational\]), the general solution of (\[ansatz1\]) reads $$\begin{aligned}
&&U_+(w)= -\displaystyle{i\over\alpha_0}Q_+(w)+(w-\eta_0) C_+(w) \nonumber\\
&&U_-(w)= B(w)\left(\displaystyle{i\over\alpha_0}Q_+^\dagger(w)+(w-\beta_0)
C_+^\dagger(w)\right)
-4 i (w-\xi_0) A(w) V_+^\dagger(w) \nonumber\\
&&V_-(w)=A(w)\left(Q_+^\dagger(w)-i\alpha_0^2 C_+^\dagger(w)\right)
\nonumber\\
&&Q_-(w)=B(w)\left(Q_+^\dagger(w)-i\alpha_0^2 C_+^\dagger(w)\right)
\label{evacpoly}
\end{aligned}$$ where the polynomials $A(w)$, $B(w)$, $C_+(w)$, $V_+(w)$ and $Q_+(w)$ can be chosen arbitrarily, provided the corresponding functions $\u_\pm(w)$, $\v_\pm(w)$ have no poles on the cuts $L_+$ and $L_-$ respectively. The vacuum case, which occurs when $A(w)=V_+(w)=0$ and $B(w)=1$, yields simpler expressions which involve just two arbitrary polynomials $C_+(w)$ and $Q_+(w)$.
Returning to (\[PolyEqs\]), we note that the integral operators in the left hand sides can be inverted using the Poincaré–Bertrand formula [@Gakhov] for singular integrals $${1\over \pi i}
\int_L {[\lambda]_\zeta\over\zeta-\tau}
{\bphi}(\zeta)\,d\zeta = f(\tau) \quad\Leftrightarrow\quad
\bphi(\tau)={1\over \pi i}
\int_L {[\lambda^{-1}]_\zeta\over\zeta-\tau} f(\zeta)\,d\zeta.
\label{PB}$$ This can be applied to the integrals over $L_{\scriptscriptstyle+}$ (using $\lambda_{\scriptscriptstyle+}$), or over $L_{\scriptscriptstyle-}$ (using $\lambda_{\scriptscriptstyle-}$).
Since the right hand sides of (\[PolyEqs\]) are polynomials in $\tau$, the inversion (\[PB\]) leads to the solution in the form $$\label{sums}
\widetilde{\bphi}_{\scriptscriptstyle\pm}(\tau)
=\sum\limits_{k=0}^{N_{\scriptscriptstyle\pm}}
\pmatrix {\widetilde{q}_{k{\scriptscriptstyle\pm}}\cr
\widetilde{u}_{k{\scriptscriptstyle\pm}}\cr
\widetilde{v}_{k{\scriptscriptstyle\pm}}}
Z_{k{\scriptscriptstyle\pm}}(\tau)$$ where $N_{\scriptscriptstyle+}$ and $N_{\scriptscriptstyle-}$ are the maxima of the degrees of the polynomials $U_{\scriptscriptstyle+}$, $V_{\scriptscriptstyle+}$, $Q_{\scriptscriptstyle+}$ and $U_{\scriptscriptstyle-}$, $V_{\scriptscriptstyle-}$, $Q_{\scriptscriptstyle-}$ respectively, $\widetilde{u}_{k{\scriptscriptstyle\pm}}$, $\widetilde{v}_{k{\scriptscriptstyle\pm}}$, $\widetilde{q}_{k{\scriptscriptstyle\pm}}$ are unknown $\tau$-independent functions of $\xi$ and $\eta$, and $Z_{k{\scriptscriptstyle\pm}}(\tau)$ are “standard” functions given by $$Z_{k{\scriptscriptstyle\pm}}(\tau)=\displaystyle{1\over \pi i}
\int_{L_\pm}{[\lambda_{\scriptscriptstyle\pm}^{-1}]_\zeta\over\zeta-
\tau}\,
\zeta^k\,d\zeta.$$ All these functions (integrals) can be evaluated as the residues of their integrands at $\zeta=\infty$ are polynomials in $\tau$ of degree $k$.
We note now, that the vector integral equations (\[IntEqs\]) decouple into three pairs of equations – one pair for each of the three components of $\widetilde{\bphi}_{\scriptscriptstyle+}$ and the corresponding component of $\widetilde{\bphi}_{\scriptscriptstyle-}$. All these pairs of equations possess the same kernels but different right hand sides. Therefore, substituting the expressions (\[sums\]) into (\[PolyEqs\]) and using (\[Rpmpm\]) with (\[ansatz1\]), we get three decoupled algebraic systems, each of order $(N+2)\times (N+2)$ where $N=N_{\scriptscriptstyle+}+N_{\scriptscriptstyle-}$ and for the sets of unknowns $\widetilde{q}_{k{\scriptscriptstyle\pm}}$, $\widetilde{u}_{k{\scriptscriptstyle\pm}}$, $\widetilde{v}_{k{\scriptscriptstyle\pm}}$ respectively. However, in view of (\[Potentials\]), we need the solutions of two of these systems only: $$\matrix{
\sum\limits_{{\scriptscriptstyle B}=0}^{N+1}
{\cal D}_{\scriptscriptstyle A B}
\widetilde{u}_{\scriptscriptstyle B}=u_{\scriptscriptstyle A},\cr
\sum\limits_{{\scriptscriptstyle B}=0}^{N+1}
{\cal D}_{\scriptscriptstyle A B}
\widetilde{v}_{\scriptscriptstyle B}
=v_{\scriptscriptstyle A}}\qquad
{\cal D}=\pmatrix {D_{\scriptscriptstyle++} &
D_{\scriptscriptstyle+-}\cr
D_{\scriptscriptstyle-+} & D_{\scriptscriptstyle--}}
\label{AlgSys}$$ where the indices $A,B=0,1,\ldots N+1$. The column vectors $u_{\scriptscriptstyle A}$, $v_{\scriptscriptstyle A}$ (shown below as rows) are composed of the coefficients of the polynomials $U_{\scriptscriptstyle\pm }(\zeta)$ and $V_{\scriptscriptstyle\pm}(\zeta)$: $$\begin{aligned}
&&u_{\scriptscriptstyle A}
=\{u_{0{\scriptscriptstyle+}},u_{1{\scriptscriptstyle+}},
\ldots,u_{N_{\scriptscriptstyle+}},u_{0{\scriptscriptstyle-}},
u_{1{\scriptscriptstyle-}},\ldots,u_{N_{\scriptscriptstyle-}}
\} \nonumber \\
&&v_{\scriptscriptstyle
A}=\{v_{0{\scriptscriptstyle+}},v_{1{\scriptscriptstyle+}},
\ldots,v_{N_{\scriptscriptstyle+}},v_{0{\scriptscriptstyle-}},
v_{1{\scriptscriptstyle-}},\ldots,v_{N_{\scriptscriptstyle-}}
\}.
\end{aligned}$$ Similarly, we combine the coefficients $\widetilde{u}_{k{\scriptscriptstyle\pm}}$, $\widetilde{v}_{k{\scriptscriptstyle\pm}}$ in (\[sums\]) to form the column vectors (rows) $$\begin{aligned}
&&\widetilde{u}_{\scriptscriptstyle A}(\xi,\eta)
=\{\widetilde{u}_{0{\scriptscriptstyle+}},
\widetilde{u}_{1{\scriptscriptstyle+}},
\ldots,\widetilde{u}_{N_{\scriptscriptstyle+}},
\widetilde{u}_{0{\scriptscriptstyle-}},
\widetilde{u}_{1{\scriptscriptstyle-}},\ldots,
\widetilde{u}_{N_{\scriptscriptstyle-}}\} \nonumber\\
&&\widetilde{v}_{\scriptscriptstyle A}(\xi,\eta)
=\{\widetilde{v}_{0{\scriptscriptstyle+}},\widetilde{v}_{1{
\scriptscriptstyle+}},
\ldots,\widetilde{v}_{N_{\scriptscriptstyle+}},
\widetilde{v}_{0{\scriptscriptstyle-}},\widetilde{v}_{1{
\scriptscriptstyle-}},
\ldots,\widetilde{v}_{N_{\scriptscriptstyle-}}\}\end{aligned}$$
The matrix $\Vert{\cal D}\Vert$ consists of the blocks $D_{\scriptscriptstyle++}$, $D_{\scriptscriptstyle+-}$, $D_{\scriptscriptstyle-+}$, $D_{\scriptscriptstyle--}$ of orders $(N_{\scriptscriptstyle+}+1)\times
(N_{\scriptscriptstyle+}+1)$, $(N_{\scriptscriptstyle+}+1)\times
(N_{\scriptscriptstyle-}+1)$, $(N_{\scriptscriptstyle-}+1)\times
(N_{\scriptscriptstyle+}+1)$ and $(N_{\scriptscriptstyle-}+1)\times
(N_{\scriptscriptstyle-}+1)$ respectively. Their components are determined by the integrals: $$\begin{aligned}
&& (D_{\scriptscriptstyle++})_{kl}(\xi)=\delta_{kl}+{1\over \pi i}
\int\limits_{\xi_0}^{\xi} [\lambda_{\scriptscriptstyle+}]_\zeta
{(R_{\scriptscriptstyle++})_k(\zeta)\over
P_{\scriptscriptstyle++}(\zeta,\zeta)}
Z_{l{\scriptscriptstyle+}}(\zeta)
\,d\zeta\nonumber\\
&&(D_{\scriptscriptstyle+-})_{kl}(\eta)={1\over \pi
i}\int\limits_{\eta_0}^{\eta}
[\lambda_{\scriptscriptstyle-}]_\zeta
{(R_{\scriptscriptstyle+-})_k(\zeta)\over
P_{\scriptscriptstyle--}(\zeta,\zeta)}
Z_{l{\scriptscriptstyle-}}(\zeta)
\,d\zeta\nonumber\\
&&(D_{\scriptscriptstyle-+})_{kl}(\xi)={1\over \pi
i}\int\limits_{\xi_0}^{\xi}
[\lambda_{\scriptscriptstyle+}]_\zeta
{(R_{\scriptscriptstyle-+})_k(\zeta)\over
P_{\scriptscriptstyle++}(\zeta,\zeta)}
Z_{l{\scriptscriptstyle+}}(\zeta)
\,d\zeta \\
&&(D_{\scriptscriptstyle--})_{kl}(\eta)=\delta_{kl}+{1\over \pi i}
\int\limits_{\eta_0}^{\eta} [\lambda_{\scriptscriptstyle-}]_\zeta
{(R_{\scriptscriptstyle--})_k(\zeta)\over
P_{\scriptscriptstyle--}(\zeta,\zeta)}
Z_{l{\scriptscriptstyle-}}(\zeta)
\,d\zeta\nonumber
\end{aligned}$$ where $(R_{\scriptscriptstyle\pm\pm})_k$ are the coefficients in the expansions $R_{\scriptscriptstyle+\pm}(\tau,\zeta)
=\sum_{k=0}^{N_{\scriptscriptstyle+}}
(R_{\scriptscriptstyle+\pm})_k(\zeta) \tau^k$ and $R_{\scriptscriptstyle-\pm}(\tau,\zeta)
=\sum_{k=0}^{N_{\scriptscriptstyle-}}
(R_{\scriptscriptstyle-\pm})_k(\zeta)\tau^k$.
To calculate the final expressions for the Ernst potentials, we need to evaluate the additional sets of integrals $$\begin{aligned}
J_{k{\scriptscriptstyle+}}(\xi) ={1\over\pi i} \int\limits_{\xi_0}^{\xi}
[\lambda_{\scriptscriptstyle+}]_\zeta
{Q_{\scriptscriptstyle+}^\dagger(\zeta)-i(\zeta-\beta_0)
U_{\scriptscriptstyle+}^\dagger (\zeta)\over
P_{\scriptscriptstyle++}(\zeta,\zeta)}
Z_{k{\scriptscriptstyle+}}(\zeta)
\,d\zeta\nonumber\\
J_{k{\scriptscriptstyle-}}(\eta)
={1\over\pi i}\int\limits_{\eta_0}^{\eta}
[\lambda_{\scriptscriptstyle-}]_\zeta
{Q_{\scriptscriptstyle-}^\dagger(\zeta)
-i(\zeta-\beta_0)
U_{\scriptscriptstyle-}^\dagger (\zeta)\over
P_{\scriptscriptstyle--}(\zeta,\zeta)}
Z_{k{\scriptscriptstyle-}}(\zeta) \,d\zeta, \nonumber\end{aligned}$$ and to combine them into one row vector $$J_{\scriptscriptstyle A}
=\{J_{0{\scriptscriptstyle+}},J_{1{\scriptscriptstyle+}},
\ldots,J_{N_{\scriptscriptstyle+}},J_{0{\scriptscriptstyle-}},
J_{1{\scriptscriptstyle-}},\ldots,
J_{N_{\scriptscriptstyle-}}\}.$$ Let us also define two additional $(N+2)\times (N+2)$ matrices $${\cal G}_{\scriptscriptstyle AB} ={\cal D}_{\scriptscriptstyle AB}
-2iu_{\scriptscriptstyle A} J_{\scriptscriptstyle B},\qquad
{\cal F}_{\scriptscriptstyle AB} ={\cal D}_{\scriptscriptstyle AB}-2 i
v_{\scriptscriptstyle A} J_{\scriptscriptstyle B}.$$ All integrals determining the components of the matrices ${\cal
G}_{\scriptscriptstyle AB}$, ${\cal F}_{\scriptscriptstyle AB}$ and ${\cal D}_{\scriptscriptstyle AB}$ can be evaluated in terms of the residues of their integrands at the zeros of $P_{\scriptscriptstyle++}(w,w)$ and $P_{\scriptscriptstyle--}(w,w)$ and at $w=\infty$. We then have $${\cal E}= -{\det\Vert{\cal G}_{\scriptscriptstyle AB}
\Vert \over \det\Vert{\cal D}_{\scriptscriptstyle AB}\Vert},
\qquad
\Phi ={\det\Vert{\cal F}_{\scriptscriptstyle AB}
\Vert \over \det\Vert{\cal D}_{\scriptscriptstyle AB}\Vert},
\label{pots}$$ which are the final expressions for the Ernst potentials. These solutions generally possess essentially nonlinear properties. They are not trivial time-dependent analogues of any stationary axisymmetric solutions with regular axis of symmetry which have different structures of monodromy data. The expressions (\[pots\]) generally are not rational functions of $\xi$, $\eta$.
When evaluating explicit examples, it may be noted that solutions with a diagonal metric occur when ${\bf u}_{\scriptscriptstyle\pm}^\dagger =-{\bf u}_{\scriptscriptstyle\pm}$. The plane symmetric (type D) Kasner metric with ${\cal E}=-\alpha/\alpha_0$ is obtained using the constants ${\bf u}_{\scriptscriptstyle+}=-i/\alpha_0$, ${\bf u}_{\scriptscriptstyle-}=i/\alpha_0$ and ${\bf v}_{\scriptscriptstyle\pm}=0$. The Khan–Penrose solution [@KhaPen71] for colliding plane impulsive gravitational waves is obtained with $\v_+(w)=\v_-(w)=0$ and $${\bf u}_+(w) =i k_+{w-a_+\over w-b_+},\qquad
{\bf u}_-(w) =ik_-{w-a_-\over w-b_-}$$ when the constants $a_\pm$, $b_\pm$ and $k_\pm$ are real. The nondiagonal Nutku–Halil solution [@NutHal77] for non-colinear impulsive waves is obtained from the same expression using complex constants. This explicitly demonstrates that the above method is applicable to both the linear and nonlinear cases.
This work was partly supported by the EPSRC and by the grants 99-01-01150 and 99-02-18415 from the RFBR.
[99]{}
V. A. Belinskii and V. E. Zakharov, Sov. Phys. JETP [**48**]{} (1978) 985
G. A. Alekseev, Dokl. Akad. Nauk SSSR, [**283**]{}, 577 (1985); Sov. Phys. Dokl. [**30**]{}, 565 (1985).
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G. A. Alekseev, gr-qc/9911045, gr-qc/9912109.
I. Hauser and F. J. Ernst, gr-qc/9903104, gr-qc/0002049.
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[^1]: Email address: [G.A.Alekseev@mi.ras.ru]{}
[^2]: E–mail: [J.B.Griffiths@Lboro.ac.uk]{}
|
---
abstract: 'We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.'
address:
- 'Department of Mathematics, University of California, Riverside, CA 92521, USA'
- 'College of Mathematics and Computer Sciences, Hunan Normal University, Changsha, 410081, China. '
author:
- Wee Liang Gan
- Liping Li
title: Coinduction functor in representation stability theory
---
Introduction
============
Conventions
-----------
Let $k$ be a commutative ring. By a category, we shall always mean a small category. If ${{\mathscr{C}}}$ is a category, a ${{\mathscr{C}}}$-module over $k$ is a functor from ${{\mathscr{C}}}$ to the category of $k$-modules. We shall refer to ${{\mathscr{C}}}$-modules over $k$ simply as ${{\mathscr{C}}}$-modules. A morphism of ${{\mathscr{C}}}$-modules is a natural transformation of functors. We shall write ${{\mathscr{C}}}\operatorname{-Mod}$ for the category of ${{\mathscr{C}}}$-modules, and ${{\mathscr{C}}}\operatorname{-mod}$ for the category of finitely generated ${{\mathscr{C}}}$-modules.
If ${{\mathscr{C}}}$ is a category and $m,n \in\operatorname{Ob}({{\mathscr{C}}})$, we shall write $k{{\mathscr{C}}}(m,n)$ for the free $k$-module on the set ${{\mathscr{C}}}(m,n)$. Let $k{{\mathscr{C}}}= \bigoplus_{m,n\in\operatorname{Ob}({{\mathscr{C}}})} k{{\mathscr{C}}}(m,n)$ and denote by $e_n \in {{\mathscr{C}}}(n,n)$ the identity morphism of $n\in \operatorname{Ob}({{\mathscr{C}}})$. We have a natural algebra structure on $k{{\mathscr{C}}}$ where the product of $\alpha\in {{\mathscr{C}}}(r,n)$ and $\beta\in {{\mathscr{C}}}(m,l)$ is defined to be the composition $\alpha\beta$ if $r=l$; it is defined to be $0$ if $r\neq l$. A $k{{\mathscr{C}}}$-module $V$ is said to be *graded* if $V=\bigoplus_{n\in \operatorname{Ob}({{\mathscr{C}}})} e_n V$. If $V$ is a ${{\mathscr{C}}}$-module, then $\bigoplus_{n\in \operatorname{Ob}(C)} V(n)$ has a natural structure of a graded $k{{\mathscr{C}}}$-module which we shall also denote by $V$. Conversely, any graded $k{{\mathscr{C}}}$-module $V$ defines a natural ${{\mathscr{C}}}$-module denoted again by $V$, with $V(n)=e_n V$. Thus, we shall not distinguish between the notion of ${{\mathscr{C}}}$-modules and the notion of graded $k{{\mathscr{C}}}$-modules.
Main results
------------
Let ${{\mathbb{Z}}}_+$ be the set of non-negative integers. For any $n\in{{\mathbb{Z}}}_+$, we write $[n]$ for the set $\{1,\ldots, n\}$; in particular, $[0]=\emptyset$. Let $G$ be a finite group.
Let ${{\mathrm{FI}}}_G$ be the category whose set of objects is ${{\mathbb{Z}}}_+$, and whose morphisms from $m$ to $n$ are all pairs $(f,c)$ where $f:[m]\to [n]$ is an injective map and $c:[m]\to G$ is an arbitrary map. The composition of morphisms $(f_1,c_1)\in {{\mathrm{FI}}}_G(m,l)$ and $(f_2,c_2)\in {{\mathrm{FI}}}_G(l,n)$ is defined by $(f_2,c_2)(f_1,c_1) = (f_3,c_3)$ where $$f_3(t) = f_2(f_1(t)), \quad c_3(t) = c_2(f_1(t))c_1(t), \quad \mbox{ for all } t\in [m].$$ We write ${{\mathrm{FI}}}$ for ${{\mathrm{FI}}}_G$ when $G$ is trivial.
Let ${{\mathbb{F}}}$ be a finite field.
Let ${{\mathrm{VI}}}$ be the category whose set of objects is ${{\mathbb{Z}}}_+$, and whose morphisms from $m$ to $n$ are all injective linear maps from ${{\mathbb{F}}}^m$ to ${{\mathbb{F}}}^n$. The composition of morphisms is defined to be the composition of maps.
Suppose ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. There is a natural monoidal structure $\odot$ on ${{\mathscr{C}}}$ defined on objects by $m \odot n = m+n$. Define a functor $\iota : {{\mathscr{C}}}\to {{\mathscr{C}}}$ by $$\label{iota}
\iota(n) = 1\odot n, \quad \iota(\alpha) = e_1\odot \alpha, \quad \mbox{ for each } n\in \operatorname{Ob}({{\mathscr{C}}}),\; \alpha\in \operatorname{Mor}({{\mathscr{C}}}).$$ The functor $\iota$ is faithful and gives rise to a restriction functor: $${{\mathsf{S}}}: {{\mathscr{C}}}\operatorname{-Mod}\longrightarrow {{\mathscr{C}}}\operatorname{-Mod}, \quad V \mapsto V\circ \iota.$$ From the usual tensor-hom adjunction, one can define a right adjoint functor ${{\mathsf{Q}}}$ to ${{\mathsf{S}}}$ called the coinduction functor. The main results of this paper are the following.
\[main result on q 1\] Suppose that $k$ is a commutative ring, and ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$. Let $m \in \operatorname{Ob}({{\mathscr{C}}})$. Then ${{\mathsf{Q}}}(k {{\mathscr{C}}}e_m)$ is isomorphic to $k{{\mathscr{C}}}e_m\oplus k{{\mathscr{C}}}e_{m+1}$.
The proof of Theorem \[main result on q 1\] will be given in Section \[FI\_G section\].
Denote by $q$ the number of elements of ${{\mathbb{F}}}$.
\[main result on q 2\] Suppose that $k$ is a commutative ring, and ${{\mathscr{C}}}$ is the category ${{\mathrm{VI}}}$. Let $m\in \operatorname{Ob}({{\mathscr{C}}})$. If $q$ is a unit of $k$, then ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ contains a direct summand isomorphic to $k{{\mathscr{C}}}e_{m+1}$.
To prove Theorem \[main result on q 2\], we construct a non-obvious surjective homomorphism from ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ to $k{{\mathscr{C}}}e_{m+1}$. The details will be given in Section \[VI section\].
Applications
------------
Let us indicate quickly the use of the coinduction functor in this paper.
Suppose ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. It is known that ${{\mathsf{S}}}(V)$ is projective if $V$ is a finitely generated projective ${{\mathscr{C}}}$-module. Hence, by the Eckman-Shapiro lemma, one has $\operatorname{Ext}^1_{{\mathscr{C}}}({{\mathsf{S}}}(V),W) = \operatorname{Ext}^1_{{\mathscr{C}}}(V, {{\mathsf{Q}}}(W))$ for any ${{\mathscr{C}}}$-modules $V$ and $W$. It follows that ${{\mathsf{Q}}}(W)$ is injective if $W$ is injective.
Suppose now that $k$ is a field of characteristic 0. In this case, ${{\mathscr{C}}}$ is locally Noetherian. By reducing to finite categories, one can show that $k{{\mathscr{C}}}e_0$ is an injective ${{\mathscr{C}}}$-module. It follows by induction, using Theorems \[main result on q 1\] and \[main result on q 2\], that $k{{\mathscr{C}}}e_n$ is injective for all $n\in\operatorname{Ob}({{\mathscr{C}}})$. Hence, we deduce the following result.
\[main-result-1\] Suppose that $k$ is a field of characteristic 0, and ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. Let $V$ be a finitely generated projective ${{\mathscr{C}}}$-module. Then $V$ is an injective ${{\mathscr{C}}}$-module.
We shall show, in Section \[inj\], that if $k$ is a field of positive characteristic and ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}$, the projective ${{\mathscr{C}}}$-module $k{{\mathscr{C}}}e_0$ is not injective.
\[infinite global dimension\] Suppose that $k$ is a field of characteristic 0, and ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. Let $V$ be a finitely generated ${{\mathscr{C}}}$-module. Then $V$ has a finite projective resolution in the category ${{\mathscr{C}}}\operatorname{-mod}$ if and only if $V$ is a projective ${{\mathscr{C}}}$-module.
If $F$ is a nonzero finitely generated torsion-free ${{\mathscr{C}}}$-module, then there is a smallest $a\in \operatorname{Ob}({{\mathscr{C}}})$ such that $F(a)\neq 0$, and it is easy to see that ${{\mathsf{S}}}^a(F)$ contains $k{{\mathscr{C}}}e_0$ as a ${{\mathscr{C}}}$-submodule. Since $k{{\mathscr{C}}}e_0$ is injective, it is a direct summand of ${{\mathsf{S}}}^a(F)$. By the adjunction of ${{\mathsf{S}}}$ and ${{\mathsf{Q}}}$, it follows that there exists a nonzero homomorphism from $F$ to ${{\mathsf{Q}}}^a (k{{\mathscr{C}}}e_0)$. Thus, when ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$, it follows from Theorem \[main result on q 1\] that every nonzero finitely generated torsion-free ${{\mathscr{C}}}$-module has a nonzero homomorphism to a finitely generated projective ${{\mathscr{C}}}$-module. From this, it is not difficult to deduce the following result.
\[main-result-2\] Suppose that $k$ is a field of characteristic 0, and ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}_G$. Then one has the following.
\(i) Any finitely generated injective ${{\mathscr{C}}}$-module is a direct sum of a finite dimensional injective ${{\mathscr{C}}}$-module and a finitely generated projective ${{\mathscr{C}}}$-module.
\(ii) Any finitely generated ${{\mathscr{C}}}$-module has a finite injective resolution in the category of finitely generated ${{\mathscr{C}}}$-modules.
The proofs of Theorems \[main-result-1\] and \[main-result-2\] will be given in Section \[applications of coinduction\]. We do not know if Theorem \[main-result-2\] holds for the category ${{\mathrm{VI}}}$.
Theorems \[main-result-1\] and \[main-result-2\] were first proved for the category ${{\mathrm{FI}}}$ by Sam and Snowden in [@SS-GL]. Our proofs are new and independent of their results.
In [@Kuhn], Kuhn studied the category of ${{\mathscr{C}}}$-modules where ${{\mathscr{C}}}$ is the category of finite dimensional ${{\mathbb{F}}}$-vector spaces with all linear maps as morphisms. He showed that if $k$ is a field and $q$ is a unit in $k$, the category of ${{\mathscr{C}}}$-modules is equivalent to the product over all $n\in {{\mathbb{Z}}}_+$ of the categories of $\mathrm{GL}_n({{\mathbb{F}}})$-modules; see [@Kuhn Theorem 1.1]. As a consequence, if $k$ is a field of characteristic 0, the category of ${{\mathscr{C}}}$-modules is semisimple; see [@Kuhn Corollary 1.3]. In contrast, the categories ${{\mathrm{FI}}}_G\operatorname{-Mod}$ and ${{\mathrm{VI}}}\operatorname{-Mod}$ do not have a similar decomposition, and Corollary \[infinite global dimension\] implies that the categories ${{\mathrm{FI}}}_G\operatorname{-mod}$ and ${{\mathrm{VI}}}\operatorname{-mod}$ have infinite global dimension when characteristic of $k$ is 0.
Representation stability {#rep stability}
------------------------
An upshot of Theorem \[main-result-2\] is an alternative proof of a key theorem of Church, Ellenberg, and Farb in their theory of representation stability. We shall discuss this in the more general situation of wreath product groups.
Suppose that $k$ is a splitting field for $G$ of characteristic 0. Denote by $\operatorname{Irr}(G)$ the set of isomorphism classes of simple $kG$-modules, and $\chi_1 \in \operatorname{Irr}(G)$ the trivial class. For each $n\in {{\mathbb{Z}}}_+$, the isomorphism classes of simple $k G\wr S_n$-modules are parametrized by partition-valued functions ${{\underline{\lambda}}}$ on $\operatorname{Irr}(G)$ such that $|{{\underline{\lambda}}}|=n$, where $$|{{\underline{\lambda}}}| = \sum_{\chi\in\operatorname{Irr}(G)} |{{\underline{\lambda}}}(\chi)|.$$
Suppose ${{\underline{\lambda}}}$ is any partition-valued function on $\operatorname{Irr}(G)$ and $n$ is any integer $\geqslant |{{\underline{\lambda}}}|+a$, where $a$ is the biggest part of ${{\underline{\lambda}}}(\chi_1)$. Following [@SS-G-maps], we define the partition-valued function ${{\underline{\lambda}}}[n]$ on $\operatorname{Irr}(G)$ by $${{\underline{\lambda}}}[n](\chi) = \left\{ \begin{array}{ll}
(n-|{{\underline{\lambda}}}|, {{\underline{\lambda}}}(\chi_1)) & \mbox{ if } \chi=\chi_1,\\
{{\underline{\lambda}}}(\chi) & \mbox{ if } \chi\neq \chi_1.
\end{array} \right.$$ Let $L({{\underline{\lambda}}})_n$ be a simple $k G\wr S_n$-module belonging to the isomorphism class corresponding to ${{\underline{\lambda}}}[n]$.
Suppose $V_n$ is a sequence of $k G\wr S_n$-modules equipped with linear maps $\phi_n : V_n \to V_{n+1}$. We say that $\{V_n\}$ is a *consistent* sequence if the following diagram commutes for each $g\in G \wr S_n$: $$\xymatrix{ V_n \ar[r]^{\phi_n} \ar[d]_{g} & V_{n+1} \ar[d]^{g} \\ V_n \ar[r]_{\phi_n} & V_{n+1} }$$ Here, $g$ acts on $V_{n+1}$ by its image under the standard inclusion $G\wr S_n \hookrightarrow G\wr S_{n+1}$.
A consistent sequence $\{V_n\}$ of $kG\wr S_n$-modules is *representation stable* if there exists $N>0$ such that for each $n\geqslant N$, the following three conditions hold:
- [**Injectivity:**]{} The map $\phi_n : V_n \longrightarrow V_{n+1}$ is injective.
- [**Surjectivity:**]{} The span of the $G\wr S_{n+1}$-orbit of $\phi_n(V_n)$ is all of $V_{n+1}$.
- [**Multiplicities:**]{} There is a decomposition $$V_n = \bigoplus_{{{\underline{\lambda}}}} L({{\underline{\lambda}}})_n ^{\oplus c({{\underline{\lambda}}})}$$ where the multiplicities $0\leqslant c({{\underline{\lambda}}}) \leqslant \infty$ do not depend on $n$.
Our terminology of representation stable follows [@Farb]. In [@CF] and [@CEF], this is called *uniformly* representation stable.
Suppose ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. Let $(\iota_n, c_n)\in \operatorname{Hom}_{{\mathscr{C}}}(n,n+1)$ be the morphism where $\iota_n : [n] \hookrightarrow [n+1]$ is the standard inclusion and $c_n:[n] \to G$ is the constant map whose image is the identity element of $G$. If $V$ is a ${{\mathscr{C}}}$-module, then $\{ V(n) \}$ is a consistent sequence of $kG \wr S_n$-modules where the maps $\phi_n : V(n)\to V(n+1)$ are induced by the morphisms $(\iota_n, c_n)$. The following theorem was first proved in [@CEF Theorem 1.13] when $G$ is trivial.
\[rep stable thm\] Suppose that $k$ is a splitting field for $G$ of characteristic 0. Let $V$ be a ${{\mathrm{FI}}}_G$-module. Then $V$ is finitely generated if and only if $\{V(n)\}$ is a representation stable sequence of $k G\wr S_n$-modules with $\dim_k V(n)<\infty$ for each $n$.
It is easy to see that $V$ is finitely generated if and only if condition (RS2) holds and $\dim_k V(n)< \infty$ for each $n$; see [@GL Proposition 5.2]. Suppose that $V$ is a finitely generated ${{\mathscr{C}}}$-module. Then condition (RS1) is a simple consequence of the fact that $V$ is Noetherian; see [@GL Proposition 5.1]. The real task is to prove that condition (RS3) holds. But by Theorem \[main-result-2\], it suffices to verify condition (RS3) for finitely generated projective ${{\mathscr{C}}}$-modules, and this is easily accomplished by Pieri’s formula. We shall give the details of the proof of Theorem \[rep stable thm\] in Section \[last section\].
Bibliographical remarks
-----------------------
Church and Farb introduced the notion of representation stability for various families of groups in [@CF]. The connection of representation stability for sequences of $S_n$-representations to ${{\mathrm{FI}}}$-modules was subsequently made by Church, Ellenberg, and Farb in [@CEF]; their paper contains many interesting examples of representation stable sequences in algebra, geometry and topology. It was also proved in [@CEF Theorem 1.3] that ${{\mathrm{FI}}}$ is locally Noetherian over any field $k$ of characteristic 0; this result was later extended to an arbitrary Noetherian ring $k$ by Church, Ellenberg, Farb, and Nagpal in [@CEFN Theorem A].
In [@Wilson], Wilson defined and studied ${{\mathrm{FI}}}_{\mathcal{W}}$-modules associated to the Weyl groups of type B/C and type D. In particular, she gave a proof of the analogue of Theorem \[rep stable thm\] in these cases along the same lines as the proof in [@CEF]. Using [@CEFN Theorem A], she proved that ${{\mathrm{FI}}}_{\mathcal{W}}$ is locally Noetherian over any Noetherian ring $k$. In the type B/C case, the category ${{\mathrm{FI}}}_{\mathcal{W}}$ is same as the category ${{\mathrm{FI}}}_G$ for the group $G$ of order 2.
From a different point of view, Snowden [@Snowden Theorem 2.3] also proved that ${{\mathrm{FI}}}$ is locally Noetherian over any field $k$ of characteristic 0. In fact, he proved this for any twisted commutative algebra finitely generated in order 1. Theorems \[main-result-1\] and \[main-result-2\] for the category ${{\mathrm{FI}}}$ were proved by Sam and Snowden in [@SS-GL Corollary 4.2.5], [@SS-GL Theorem 4.3.1], and [@SS-GL Theorem 4.3.4]. The strategy of their proofs is to pass to the Serre quotient of the category of finitely generated ${{\mathrm{FI}}}$-modules by the subcategory of finite dimensional ${{\mathrm{FI}}}$-modules. Using their results, Sam and Snowden deduced a formula for the character polynomial of a finitely generated ${{\mathrm{FI}}}$-module over a field of characteristic 0.
In [@GL], we gave a simple proof that ${{\mathrm{FI}}}_G$ and ${{\mathrm{VI}}}$ are locally Noetherian over any field $k$ of characteristic 0 (which is sufficient for the present paper); a similar proof for ${{\mathrm{FI}}}$ and ${{\mathrm{VI}}}$ was also obtained by Putman (unpublished). At about the same time, using Gröbner basis methods, Putman and Sam [@PS Theorem A] proved that ${{\mathrm{VI}}}$ is locally Noetherian when ${{\mathbb{F}}}$ is any finite ring and $k$ is any Noetherian ring, and Sam and Snowden [@SS-Grobner Theorem 10.1.2] proved that ${{\mathrm{FI}}}_G$ is locally Noetherian for any Noetherian ring $k$. In [@SS-G-maps Theorem 1.2.4], Sam and Snowden proved that if $k$ is a field in which the order of $G$ is invertible, then representations of ${{\mathrm{FI}}}_G$ are in fact equivalent to representations of ${{\mathrm{FI}}}\times \mathrm{FB}^{a}$, where $\mathrm{FB}$ is the groupoid of finite sets and $a$ is the number of non-trivial irreducible representations of $G$. Using this, they proved a wreath-product version of Murnaghan’s stability theorem [@SS-G-maps Theorem 5.2.1].
It was first observed by Church, Ellenberg, Farb, and Nagpal in [@CEFN Proposition 2.12] that the functor ${{\mathsf{S}}}$ for the category ${{\mathrm{FI}}}$ has the property that ${{\mathsf{S}}}(V)$ is projective whenever $V$ is a finitely generated projective ${{\mathrm{FI}}}$-module. A more precise version of this property plays a crucial role in their paper. In [@GL-Koszulity Section 5], we showed that the functor ${{\mathsf{S}}}$ for many other categories have this property, including ${{\mathrm{FI}}}_G$ and ${{\mathrm{VI}}}$. To the best of our knowledge, no one has studied the coinduction functor ${{\mathsf{Q}}}$, even in the case of ${{\mathrm{FI}}}$.
Preliminaries
=============
Notations and terminology {#notations and terminology}
-------------------------
Recall that an EI category is a category in which every endomorphism is an isomorphism (see [@Dieck]). Throughout this paper, we shall denote by ${{\mathscr{C}}}$ an EI category satisfying the following conditions:
- $\operatorname{Ob}({{\mathscr{C}}})={{\mathbb{Z}}}_+$;
- ${{\mathscr{C}}}(m,n)$ is an empty set if $m>n$;
- ${{\mathscr{C}}}(m,n)$ is a nonempty finite set if $m\leqslant n$;
- for $m\leqslant l\leqslant n$, the composition map ${{\mathscr{C}}}(l,n) \times {{\mathscr{C}}}(m,l) \to {{\mathscr{C}}}(m,n)$ is surjective.
Suppose $n\in \operatorname{Ob}({{\mathscr{C}}})$. We shall use the following notations:
- $G_n$ denotes the group ${{\mathscr{C}}}(n,n)$;
- ${{\mathscr{C}}}_n$ denotes the full subcategory of ${{\mathscr{C}}}$ with $\operatorname{Ob}({{\mathscr{C}}}_n)=\{0, 1, \ldots, n\}$.
Suppose that $k$ is a commutative ring and $V$ is a ${{\mathscr{C}}}$-module. For $n\in \operatorname{Ob}({{\mathscr{C}}})$ and $v\in V(n)$, we say that $v$ has degree $n$, and write $\deg(v)$ for the degree of $v$. By a set of generators of $V$, we shall always mean a subset $S$ of $\bigcup_{n\in \operatorname{Ob}({{\mathscr{C}}})} V(n)$ such that the only submodule of $V$ containing $S$ is $V$ itself. We say that $V$ is generated in degrees $\leqslant n$ if $V$ has a set $S$ of generators such that the degree of each element of $S$ is at most $n$. We say that $V$ is finitely generated if $V$ has a finite set of generators. We say that $V$ is Noetherian if every submodule of $V$ is finitely generated. We say that ${{\mathscr{C}}}$ is locally Noetherian if every finitely generated ${{\mathscr{C}}}$-module is Noetherian.
Baer’s criterion
----------------
We omit the proof of the following lemma, which is standard (see [@Weibel page 39]).
Suppose that $k$ is a commutative ring, and $V\in {{\mathscr{C}}}\operatorname{-Mod}$. Suppose that for all $n\in \operatorname{Ob}({{\mathscr{C}}})$ and for all ${{\mathscr{C}}}$-submodule $U$ of $k{{\mathscr{C}}}e_n$, every homomorphism $U \to V$ can be extended to a homomorphism $k{{\mathscr{C}}}e_n \to V$. Then $V$ is injective in ${{\mathscr{C}}}\operatorname{-Mod}$.
The following corollary is immediate.
\[injectivity in C-mod\] Suppose that $k$ is a commutative ring and ${{\mathscr{C}}}$ is locally Noetherian. Let $V\in {{\mathscr{C}}}\operatorname{-mod}$. Then $V$ is injective in ${{\mathscr{C}}}\operatorname{-Mod}$ if and only if $V$ is injective in ${{\mathscr{C}}}\operatorname{-mod}$.
Projective resolutions
----------------------
Suppose that $k$ is a commutative ring. For any $n\in \operatorname{Ob}({{\mathscr{C}}})$, the ${{\mathscr{C}}}$-module $k{{\mathscr{C}}}e_n$ is clearly projective. Suppose $V$ is a ${{\mathscr{C}}}$-module and $S$ is a set of generators of $V$. Then there is a surjective homomorphism $\bigoplus_{s\in S} k{{\mathscr{C}}}e_{\deg(s)} \to V$ whose restriction to the direct summand corresponding to $s$ is $\alpha\mapsto \alpha s$. Hence, any ${{\mathscr{C}}}$-module $V$ has a projective resolution $$\cdots \to P^{-2} \to P^{-1} \to P^{0} \to V \to 0$$ such that each $P^{-i}$ is a direct sum of projective ${{\mathscr{C}}}$-modules of the form $k{{\mathscr{C}}}e_n$ where $n\in \operatorname{Ob}({{\mathscr{C}}})$.
Restriction to ${{\mathscr{C}}}_n$
----------------------------------
Suppose that $k$ is a commutative ring. Let $n\in \operatorname{Ob}({{\mathscr{C}}})$. Recall that ${{\mathscr{C}}}_n$ denotes the full subcategory of ${{\mathscr{C}}}$ with $\operatorname{Ob}({{\mathscr{C}}}_n)=\{0, 1, \ldots, n\}$; see Subsection \[notations and terminology\]. Denote by $\jmath: {{\mathscr{C}}}_n \hookrightarrow {{\mathscr{C}}}$ the inclusion functor. We have the pullback functor $$\jmath^*: {{\mathscr{C}}}\operatorname{-Mod}\longrightarrow {{\mathscr{C}}}_n\operatorname{-Mod}, \quad V \mapsto V\circ \jmath.$$ We also have the pushforward functor $$\jmath_*: {{\mathscr{C}}}_n\operatorname{-Mod}\longrightarrow {{\mathscr{C}}}\operatorname{-Mod}$$ which regards a ${{\mathscr{C}}}_n$-module as a ${{\mathscr{C}}}$-module in the obvious way. The pushforward functor $\jmath_*$ is a right adjoint functor to the pullback functor $\jmath^*$.
\[eckmann-shapiro1\] Suppose that $k$ is a commutative ring. Let $n\in\operatorname{Ob}({{\mathscr{C}}})$ and denote by $\jmath: {{\mathscr{C}}}_n \hookrightarrow {{\mathscr{C}}}$ the inclusion functor. For any $V\in {{\mathscr{C}}}\operatorname{-Mod}$ and $W\in {{\mathscr{C}}}_n\operatorname{-Mod}$, one has $$\operatorname{Ext}_{{{\mathscr{C}}}_n}^i(\jmath^*(V), W) = \operatorname{Ext}_{{{\mathscr{C}}}}^i(V,\jmath_*(W))$$ for all $i\geqslant 1$.
Observe that $\jmath^*(k{{\mathscr{C}}}e_m)$ is a projective ${{\mathscr{C}}}_n$-module for all $m\in \operatorname{Ob}({{\mathscr{C}}})$. Thus, the required result follows from the Eckmann-Shapiro lemma.
\[reduction to C\_n\] Suppose that $k$ is a commutative ring and ${{\mathscr{C}}}$ is locally Noetherian. Let $V, W \in {{\mathscr{C}}}\operatorname{-mod}$. Then there exists $N \in \operatorname{Ob}({{\mathscr{C}}})$ such that for all $n\geqslant N$, one has $$\operatorname{Ext}_{{{\mathscr{C}}}}^1(V, W) = \operatorname{Ext}_{{{\mathscr{C}}}_n}^1(\jmath^*(V),\jmath^*(W)),$$ where $\jmath: {{\mathscr{C}}}_n \hookrightarrow {{\mathscr{C}}}$ is the inclusion functor.
Since ${{\mathscr{C}}}$ is locally Noetherian, there exists a projective resolution $$\cdots \to P^{-2} \to P^{-1} \to P^{0} \to V \to 0$$ such that each $P^{-i}$ is a finitely generated projective ${{\mathscr{C}}}$-module. Thus, there exists $N\in \operatorname{Ob}({{\mathscr{C}}})$ such that $P^{-2}$ and $P^{-1}$ are both generated in degrees $\leqslant N$.
Suppose $n\geqslant N$. Let $U$ be the submodule $\bigoplus_{m>n} W(m)$ of $W$. We have a short exact sequence $$0 \longrightarrow U \longrightarrow W \longrightarrow \jmath_*( \jmath^*(W) )\longrightarrow 0,$$ and hence a long exact sequence $$\begin{gathered}
\cdots \to \operatorname{Ext}^1_{{{\mathscr{C}}}} (V,U) \to \operatorname{Ext}^1_{{{\mathscr{C}}}} (V, W) \to \operatorname{Ext}^1_{{{\mathscr{C}}}} (V, \jmath_*( \jmath^*(W) ))
\to \operatorname{Ext}^2_{{{\mathscr{C}}}}(V,W) \to \cdots .\end{gathered}$$ But, for $i=1, 2$, one has $\operatorname{Hom}_{{{\mathscr{C}}}}(P^{-i}, U)=0$ and so $\operatorname{Ext}^i_{{{\mathscr{C}}}} (V,U)=0$. It follows that $$\operatorname{Ext}^1_{{{\mathscr{C}}}} (V, W) = \operatorname{Ext}^1_{{{\mathscr{C}}}} (V, \jmath_*( \jmath^*(W) )) = \operatorname{Ext}^1_{{{\mathscr{C}}}_n}(\jmath^*(V), \jmath^*(W)),$$ using Lemma \[eckmann-shapiro1\].
Injective resolutions of finite dimensional modules
---------------------------------------------------
Suppose that $k$ is a field. We denote by ${{\mathsf{D}}}$ the standard duality functor $\operatorname{Hom}_k ( - , k)$ between the categories ${{\mathscr{C}}}_n\operatorname{-mod}$ and ${{\mathscr{C}}}_n^{\operatorname{op}}\operatorname{-mod}$. Any finite dimensional injective ${{\mathscr{C}}}_n$-module is isomorphic to ${{\mathsf{D}}}(P)$ for some finite dimensional projective ${{\mathscr{C}}}_n^{\operatorname{op}}$-module $P$.
\[finite dimensional injective 1\] Suppose $k$ is a field. If $W$ is a finite dimensional injective ${{\mathscr{C}}}_n$-module for some $n\in \operatorname{Ob}({{\mathscr{C}}})$, then $\jmath_*(W)$ is a finite dimensional injective ${{\mathscr{C}}}$-module, where $\jmath: {{\mathscr{C}}}_n\hookrightarrow {{\mathscr{C}}}$.
It is obvious that $\jmath_*(W)$ is finite dimensional. The injectivity of $\jmath_*(W)$ follows from Lemma \[eckmann-shapiro1\].
\[inj res of finite dim module\] Suppose $k$ is a field of characteristic 0. Then every finite dimensional ${{\mathscr{C}}}$-module $V$ has a finite injective resolution $$0 \to V \to I^0 \to I^1 \to \cdots I^m \to 0$$ where $I^r$ is finite dimensional for each $r\in\{0, 1,\ldots, m\}$.
Choose $n\in\operatorname{Ob}({{\mathscr{C}}})$ such that $V = \jmath_*(\jmath^*(V))$ where $\jmath: {{\mathscr{C}}}_n \hookrightarrow {{\mathscr{C}}}$. It is easy to see that any finite dimensional ${{\mathscr{C}}}_n^{\operatorname{op}}$-module has a finite projective resolution in the category ${{\mathscr{C}}}_n^{\operatorname{op}}\operatorname{-mod}$. Using the functor ${{\mathsf{D}}}$, we deduce that $\jmath^*(V)$ has a finite injective resolution in the category ${{\mathscr{C}}}_n\operatorname{-mod}$. By Lemma \[finite dimensional injective 1\], applying the functor $\jmath_*$ to this resolution gives a resolution of $V$ of the required form.
Suppose $k$ is a field, and $V$ is a finite dimensional injective ${{\mathscr{C}}}$-module. It is easy to see that if $n\in \operatorname{Ob}({{\mathscr{C}}})$ and $V=\jmath_*(\jmath^*(V))$, where $\jmath: {{\mathscr{C}}}_n \hookrightarrow {{\mathscr{C}}}$ is the inclusion functor, then $\jmath^*(V)$ is an injective ${{\mathscr{C}}}_n$-module.
Injectivity of $k{{\mathscr{C}}}e_0$ {#inj}
====================================
Field of characteristic 0
-------------------------
We say that the category ${{\mathscr{C}}}$ satisfies the *transitivity condition* if for all $n\in \operatorname{Ob}({{\mathscr{C}}})$, the action of $G_{n+1}$ on ${{\mathscr{C}}}(n,n+1)$ is transitive. (Recall that $G_n$ denotes the group ${{\mathscr{C}}}(n,n)$; see Subsection \[notations and terminology\].)
\[kC\_n e\_0 is injective\] Suppose that $k$ is a field of characteristic 0. If ${{\mathscr{C}}}$ satisfies the transitivity condition, and $0$ is an initial object of ${{\mathscr{C}}}$, then $k{{\mathscr{C}}}_n e_0$ is an injective ${{\mathscr{C}}}_n$-module for all $n\in \operatorname{Ob}({{\mathscr{C}}})$.
It suffices to prove that $\operatorname{Ext}^1_{{{\mathscr{C}}}_n} (k G_m, k{{\mathscr{C}}}_n e_0) = 0$ for all $m\in \operatorname{Ob}({{\mathscr{C}}}_n)$. This is clear when $m=n$ for $k G_n$ is a projective ${{\mathscr{C}}}_n$-module.
Suppose $m<n$. Let $P=k{{\mathscr{C}}}_n e_m$ and let $U$ be the submodule $\bigoplus_{l>m} P(l)$ of $P$. We have a short exact sequence $$0\longrightarrow U \longrightarrow P \longrightarrow k G_m \longrightarrow 0,$$ and hence a long exact sequence $$\begin{gathered}
0 \to \operatorname{Hom}_{{{\mathscr{C}}}_n}(k G_m, k{{\mathscr{C}}}_n e_0) \to \operatorname{Hom}_{{{\mathscr{C}}}_n} (P, k{{\mathscr{C}}}_n e_0) \to \operatorname{Hom}_{{{\mathscr{C}}}_n} (U, k{{\mathscr{C}}}_n e_0) \\
\to \operatorname{Ext}^1_{{{\mathscr{C}}}_n} (k G_m, k{{\mathscr{C}}}_n e_0) \to \operatorname{Ext}^1_{{{\mathscr{C}}}_n} (P, k{{\mathscr{C}}}_n e_0) \to \cdots .\end{gathered}$$ Since $m<n$, one has $\operatorname{Hom}_{{{\mathscr{C}}}_n}(k G_m, k{{\mathscr{C}}}_n e_0)=0$. Since $P$ is projective, $\operatorname{Ext}^1_{{{\mathscr{C}}}_n} (P, k{{\mathscr{C}}}_n e_0) =0$. Note that $$\dim_k \operatorname{Hom}_{k{{\mathscr{C}}}_n} (P, k{{\mathscr{C}}}_n e_0) = \dim_k k{{\mathscr{C}}}_n (0,m) = 1.$$ Since $U$ is generated by $U(m+1)$, and $G_{m+1}$ acts transitively on $U(m+1)$, one has $$\dim_k \operatorname{Hom}_{k{{\mathscr{C}}}_n} (U, k{{\mathscr{C}}}_n e_0) \leqslant \dim_k \operatorname{Hom}_{k G_{m+1}} (U(m+1), k{{\mathscr{C}}}_n (0,m+1)) \leqslant 1.$$ Hence, we must have $\operatorname{Ext}^1_{{{\mathscr{C}}}_n} (k G_m, k{{\mathscr{C}}}_n e_0) = 0$
\[kCe\_0 is injective\] Suppose that $k$ is a field of characteristic 0. If ${{\mathscr{C}}}$ is locally Noetherian, satisfies the transitivity condition, and $0$ is an initial object of ${{\mathscr{C}}}$, then $k{{\mathscr{C}}}e_0$ is an injective ${{\mathscr{C}}}$-module.
By Corollary \[injectivity in C-mod\], it suffices to show that $k{{\mathscr{C}}}e_0$ is injective in ${{\mathscr{C}}}\operatorname{-mod}$. By Lemma \[reduction to C\_n\], this follows from injectivity of $k{{\mathscr{C}}}_n e_0$ which is proved in Lemma \[kC\_n e\_0 is injective\].
Field of characteristic $p$
---------------------------
Let us show that Corollary \[kCe\_0 is injective\] is false when $k$ is a field of characteristic $p>0$ and ${{\mathscr{C}}}$ is the category ${{\mathrm{FI}}}$.
For any $n\geqslant p$, we have a right action of the symmetric group $S_p$ on ${{\mathscr{C}}}(p,n)$. Let $U(n)$ be the subspace of $S_p$-invariant elements in $k{{\mathscr{C}}}(p,n)$, and let $U=\bigoplus_{n\geqslant p} U(n)$. Then $U$ is a submodule of $k{{\mathscr{C}}}e_p$. Now let ${{\mathscr{J}}}(n)$ be the set of $S_p$-orbits in ${{\mathscr{C}}}(p,n)$. For each orbit $J\in{{\mathscr{J}}}(n)$, let $\xi_J\in U(n)$ be the sum of all the $p!$ elements in $J$. Then the collection of $\xi_J$ for $J\in{{\mathscr{J}}}(n)$ is a basis for $U(n)$. There is a homomorphism $f: U \to k{{\mathscr{C}}}e_0$ such that $f(\xi_J)$ is the unique element of ${{\mathscr{C}}}(0,n)$ if $J\in {{\mathscr{J}}}(n)$. We define a submodule $W$ of $k{{\mathscr{C}}}e_0 \oplus k{{\mathscr{C}}}e_p$ as follows. For any $n\geqslant p$, let $W(n)$ be the set of all elements in $k{{\mathscr{C}}}(0,n)\oplus U(n)$ of the form $f(\xi) + \xi$ for $\xi \in U(n)$.
Let $V=(k{{\mathscr{C}}}e_0 \oplus k{{\mathscr{C}}}e_p)/W$, and let $\pi: k{{\mathscr{C}}}e_0 \oplus k{{\mathscr{C}}}e_p \to V$ be the canonical projection. Denote by $i: k{{\mathscr{C}}}e_0 \to V$ the restriction of $\pi$ to $k{{\mathscr{C}}}e_0$. It is clear that $i$ is a monomorphism. We claim that the short exact sequence $$0 \longrightarrow k{{\mathscr{C}}}e_0 \stackrel{i}{\longrightarrow} V \longrightarrow V/i(k{{\mathscr{C}}}e_0) \longrightarrow 0$$ does not split. Indeed, any homomorphism $k{{\mathscr{C}}}e_0 \oplus k{{\mathscr{C}}}e_p\to k{{\mathscr{C}}}e_0$ whose restriction to $k{{\mathscr{C}}}e_0$ is the identity map cannot vanish identically on $W$, for any homomorphism $k{{\mathscr{C}}}e_p \to k{{\mathscr{C}}}e_0$ must vanish identically on $U$.
Restriction and coinduction along genetic functors
==================================================
Throughout this section, $k$ denotes any commutative ring.
Restriction and coinduction
---------------------------
A coinduction functor can be defined whenever one has a subring of a ring, and it is a right adjoint functor to the restriction functor; we refer the reader to [@Benson Section 2.8] for a clear exposition on the definition and basic properties of the coinduction functor in such a general setting. In this section, we begin our study of the coinduction functor in our special setting, where the pair of subring and the ring are isomorphic.
Let $\iota : {{\mathscr{C}}}\to {{\mathscr{C}}}$ be a faithful functor such that $\iota(n)=n+1$ for all $n\in\operatorname{Ob}({{\mathscr{C}}})$. We define the restriction functor ${{\mathsf{S}}}: {{\mathscr{C}}}\operatorname{-Mod}\longrightarrow {{\mathscr{C}}}\operatorname{-Mod}$ by ${{\mathsf{S}}}(V)=V\circ \iota$ for all $V\in{{\mathscr{C}}}\operatorname{-Mod}$; thus, ${{\mathsf{S}}}(V)(n)=V(n+1)$.
Suppose $V\in{{\mathscr{C}}}\operatorname{-Mod}$. We define ${{\mathsf{Q}}}(V)\in{{\mathscr{C}}}\operatorname{-Mod}$ by $${{\mathsf{Q}}}(V)(n) = \operatorname{Hom}_{{\mathscr{C}}}({{\mathsf{S}}}(k{{\mathscr{C}}}e_n), V) \quad \mbox{ for each } n\in \operatorname{Ob}({{\mathscr{C}}}).$$ We call ${{\mathsf{Q}}}: {{\mathscr{C}}}\operatorname{-Mod}\longrightarrow {{\mathscr{C}}}\operatorname{-Mod}$ the *coinduction* functor.
Observe that any $\alpha \in {{\mathscr{C}}}(m,n)$ defines a ${{\mathscr{C}}}$-module homomorphism $${{\mathsf{S}}}(k{{\mathscr{C}}}e_n) \longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_m), \quad \gamma \mapsto \gamma\alpha.$$ The ${{\mathscr{C}}}$-module structure on ${{\mathsf{Q}}}(V)$ is defined in the natural way as follows: if $\alpha\in {{\mathscr{C}}}(m,n)$ and $\varrho\in {{\mathsf{Q}}}(V)(m)$, then $\alpha(\varrho)\in {{\mathsf{Q}}}(V)(n)$ is the ${{\mathscr{C}}}$-module homomorphism $${{\mathsf{S}}}(k{{\mathscr{C}}}e_n) \longrightarrow V, \quad \gamma \mapsto \varrho(\gamma\alpha).$$
\[Q is right adjoint\] The functor ${{\mathsf{Q}}}$ is right adjoint to the functor ${{\mathsf{S}}}$.
Let $$M = \bigoplus_{\substack{m\geqslant 0,\\n\geqslant 1}} k{{\mathscr{C}}}(m,n).$$ We have a $k{{\mathscr{C}}}$-bimodule structure on $M$ defined by $$\alpha \cdot \gamma = \iota(\alpha)\gamma,\qquad \gamma \cdot \alpha = \gamma\alpha,$$ for $\alpha \in k{{\mathscr{C}}}$ and $\gamma \in M$. By the tensor-hom adjunction, one has $$\operatorname{Hom}_{k{{\mathscr{C}}}} (M\otimes_{k{{\mathscr{C}}}} V, W) = \operatorname{Hom}_{k{{\mathscr{C}}}} (V, \operatorname{Hom}_{k{{\mathscr{C}}}} (M, W))$$ for any $V, W\in {{\mathscr{C}}}\operatorname{-Mod}$.
The $k{{\mathscr{C}}}$-module homomorphism $$M\otimes_{k{{\mathscr{C}}}} V \longrightarrow {{\mathsf{S}}}(V), \quad \gamma \otimes v \mapsto \gamma v$$ has an inverse defined on ${{\mathsf{S}}}(V)(n)$ by $v\mapsto e_{n+1}\otimes v$, for each $n\in \operatorname{Ob}({{\mathscr{C}}})$. Hence, $M\otimes_{k{{\mathscr{C}}}} V$ is isomorphic to ${{\mathsf{S}}}(V)$.
On the other hand, there is a $k{{\mathscr{C}}}$-module direct sum decomposition $$M = \bigoplus_{m\geqslant 0} {{\mathsf{S}}}(k{{\mathscr{C}}}e_m),$$ so $$\operatorname{Hom}_{k{{\mathscr{C}}}} (M, W) = \prod_{m\geqslant 0} \operatorname{Hom}_{{{\mathscr{C}}}} ({{\mathsf{S}}}(k{{\mathscr{C}}}e_m), W).$$ But since $V$ is a *graded* $k{{\mathscr{C}}}$-module, the image of any $k{{\mathscr{C}}}$-module homomorphism from $V$ to $\operatorname{Hom}_{k{{\mathscr{C}}}}(M,W)$ lies in ${{\mathsf{Q}}}(W)$. It follows that $$\operatorname{Hom}_{k{{\mathscr{C}}}} (V, \operatorname{Hom}_{k{{\mathscr{C}}}} (M, W)) = \operatorname{Hom}_{{\mathscr{C}}}(V, {{\mathsf{Q}}}(W)).$$ Thus, $\operatorname{Hom}_{{\mathscr{C}}}({{\mathsf{S}}}(V), W) = \operatorname{Hom}_{{\mathscr{C}}}( V,{{\mathsf{Q}}}(W))$.
Following [@GL-Koszulity], we call $\iota$ a *genetic* functor if, for each $n\in \operatorname{Ob}({{\mathscr{C}}})$, the ${{\mathscr{C}}}$-module ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$ is projective and generated in degrees $\leqslant n$.
\[eckmann-shapiro2\] Suppose $\iota$ is a genetic functor. Then for any $V, W\in {{\mathscr{C}}}\operatorname{-Mod}$, one has $$\operatorname{Ext}^i_{{{\mathscr{C}}}} ({{\mathsf{S}}}(V), W) = \operatorname{Ext}^i_{{{\mathscr{C}}}} (V,{{\mathsf{Q}}}(W))$$ for all $i\geqslant 1$.
This is immediate from Lemma \[Q is right adjoint\] and the Eckmann-Shapiro lemma.
The condition in Lemma \[eckmann-shapiro2\] that $\iota$ is a genetic functor can be weakened. Indeed, we only need to use the property that for each $n\in \operatorname{Ob}({{\mathscr{C}}})$, the ${{\mathscr{C}}}$-module ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$ is projective.
Genetic functors for ${{\mathrm{FI}}}_G$ and ${{\mathrm{VI}}}$
--------------------------------------------------------------
Suppose ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. There is a natural monoidal structure $\odot$ on ${{\mathscr{C}}}$ such that $m\odot n=m+n$ for all $m, n \in \operatorname{Ob}({{\mathscr{C}}})$. Let us recall this monoidal structure.
[**Case 1**]{}: Suppose ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. For any $(f_1, c_1)\in {{\mathscr{C}}}(m_1,n_1)$ and $(f_2, c_2)\in {{\mathscr{C}}}(m_2,n_2)$, we define $(f_1, c_1) \odot (f_2, c_2)$ to be the morphism $(f,c)\in {{\mathscr{C}}}(m_1+m_2, n_1+n_2)$ where $$\begin{gathered}
f(t) = \left\{ \begin{array}{ll}
f_1(t) & \mbox{ if } t\leqslant m_1,\\
f_2(t-m_1)+n_1 & \mbox{ if } t> m_1.
\end{array}\right.\end{gathered}$$ and $$\begin{gathered}
c(t) = \left\{ \begin{array}{ll}
c_1(t) & \mbox{ if } r\leqslant m_1,\\
c_2(t-m_1) & \mbox{ if } t> m_1.
\end{array}\right.\end{gathered}$$
[**Case 2**]{}: Suppose ${{\mathscr{C}}}$ is ${{\mathrm{VI}}}$. For any $f_1\in {{\mathscr{C}}}(m_1, n_1)$ and $f_2 \in {{\mathscr{C}}}(m_2,n_2)$, we define $f_1 \odot f_2 \in {{\mathscr{C}}}(m_1+m_2, n_1+ n_2)$ by $$f_1 \odot f_2 = f_1 \oplus f_2 : {{\mathbb{F}}}^{m_1}\oplus {{\mathbb{F}}}^{m_2} \longrightarrow {{\mathbb{F}}}^{n_1}\oplus {{\mathbb{F}}}^{n_2}.$$
In both cases, we let $\iota: {{\mathscr{C}}}\to {{\mathscr{C}}}$ be the functor defined by (\[iota\]). It is clear that $\iota$ is faithful. In the next two sections, we shall briefly recall the proof that $\iota$ is a genetic functor, and examine the structure of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ for each $m\in \operatorname{Ob}({{\mathscr{C}}})$.
Structure of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ when ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$ {#FI_G section}
================================================================================================
Throughout this section, $k$ denotes any commutative ring.
Structure of ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$ {#structure of S for FI_G}
--------------------------------------------------
Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. Let $n\in \operatorname{Ob}({{\mathscr{C}}})$. We now recall the structure of ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$.
Denote by $e$ the identity element of $G$, and define the morphisms $$(f_n,, c_n) \in {{\mathscr{C}}}(n,n+1) \quad\mbox{ and }\quad (f_{n,r,g}, c_{n,r,g}) \in {{\mathscr{C}}}(n, n) \quad \mbox{ for } r\in [n],\, g\in G,$$ by $$\begin{aligned}
f_n(t) &= t+1,\\
c_n(t) &= e,\\
f_{n,r,g}(t) &= \left\{ \begin{array}{ll}
t+1 & \mbox{ if }t<r,\\
1 & \mbox{ if } t=r,\\
t & \mbox{ if }t>r,
\end{array} \right. \\
c_{n,r,g}(t) &= \left\{ \begin{array}{ll}
e & \mbox{ if }t \neq r,\\
g & \mbox{ if } t=r,
\end{array} \right.\end{aligned}$$ for $t\in [n]$.
Now, for any $l \in \operatorname{Ob}({{\mathscr{C}}})$, $r\in [n]$, $g\in G$, define the maps $$\begin{gathered}
\Phi_{n,0} : {{\mathscr{C}}}(n, l) \longrightarrow {{\mathscr{C}}}(n,l+1), \quad \alpha\mapsto \iota(\alpha)\circ (f_n,c_n);\\
\Phi_{n,r,g} : {{\mathscr{C}}}(n-1,l) \longrightarrow {{\mathscr{C}}}(n,l+1), \quad \alpha\mapsto \iota(\alpha)\circ (f_{n,r,g},c_{n,r,g}).\end{gathered}$$ We may extend these maps linearly to ${{\mathscr{C}}}$-module homomorphisms $$\Phi_{n,0} : k{{\mathscr{C}}}e_n \longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n) \quad \mbox{ and } \quad \Phi_{n,r,g} : k{{\mathscr{C}}}e_{n-1}\longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n).$$ Let $$\label{phi for FI_G}
\Phi_n : k{{\mathscr{C}}}e_n \oplus \left( \bigoplus_{r\in [n],\, g\in G} k{{\mathscr{C}}}e_{n-1} \right) \longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$$ be the ${{\mathscr{C}}}$-module homomorphism whose restriction to $k{{\mathscr{C}}}e_n$ is $\Phi_{n,0}$ and whose restriction to the direct summand $k{{\mathscr{C}}}e_{n-1}$ indexed by $r\in [n]$, $g\in G$ is $\Phi_{n,r,g}$. It is straightforward to verify that $\Phi_n$ is an isomorphism (see [@GL-Koszulity Section 5]). Thus, $\iota$ is a genetic functor.
Preliminary discussion of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ {#preliminary discussion 1}
---------------------------------------------------------------
We retain the notations of subsection \[structure of S for FI\_G\]. Let $m, n \in \operatorname{Ob}({{\mathscr{C}}})$. By the isomorphism (\[phi for FI\_G\]), one has the identification $${{\mathsf{Q}}}(k {{\mathscr{C}}}e_m)(n) = \operatorname{Hom}_{{{\mathscr{C}}}}( k{{\mathscr{C}}}e_n, k{{\mathscr{C}}}e_m ) \oplus \left( \bigoplus_{r\in [n],\, g\in G} \operatorname{Hom}_{{{\mathscr{C}}}} ( k{{\mathscr{C}}}e_{n-1}, k{{\mathscr{C}}}e_m) \right).$$ Denote by $$\begin{aligned}
\Psi_{n,0}: k{{\mathscr{C}}}(m,n) &\longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n),\\
\Psi_{n,r,g} : k{{\mathscr{C}}}(m,n-1) &\longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)\end{aligned}$$ the linear maps where $\Psi_{n,0}$ is the natural bijection of $k{{\mathscr{C}}}(m,n)$ with the direct summand $\operatorname{Hom}_{{{\mathscr{C}}}} (k{{\mathscr{C}}}e_n, k{{\mathscr{C}}}e_m)$ of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$, and $\Psi_{n,r,g}$ is the natural bijection of $k{{\mathscr{C}}}(m,n-1)$ with the direct summand $\operatorname{Hom}_{{{\mathscr{C}}}}(k{{\mathscr{C}}}e_{n-1}, k{{\mathscr{C}}}e_m)$ of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$ indexed by $r\in [n]$, $g\in G$. We have a linear bijection $$\Psi_n: k{{\mathscr{C}}}(m,n) \oplus \left( \bigoplus_{r\in [n],\, g\in G} k{{\mathscr{C}}}(m,n-1) \right) \longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$$ whose restriction to $k{{\mathscr{C}}}(m,n)$ is $\Psi_{n,0}$ and whose restriction to the direct summand $k{{\mathscr{C}}}(m,n-1)$ indexed by $r\in [n]$, $g\in G$ is $\Psi_{n,r,g}$.
The next lemma describes the ${{\mathscr{C}}}$-module structure of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ in terms of the identifications $\Psi_n$ for $n\in \operatorname{Ob}({{\mathscr{C}}})$. We shall use the following notations. For any $l \geqslant 1$ and $r\in [l]$, let $\partial_r : [l]\setminus\{r\} \to [l-1]$ be the unique nondecreasing bijection. If $\alpha = (f,c) \in {{\mathscr{C}}}(n,l)$, $r\in [l]\setminus {{\mathrm{Im}}}(f)$, and $s\in [n]$, we let $$\begin{aligned}
\partial_r \alpha &= (\partial_r\circ f, c) \in {{\mathscr{C}}}(n,l-1), \\
\alpha_s &= (\partial_{f(s)}\circ f \circ \partial_s^{-1} , c\circ \partial_s^{-1} )\in {{\mathscr{C}}}(n-1,l-1),\end{aligned}$$ where $\partial_s^{-1} : [n-1]\to [n]\setminus \{ s \}$ is the inverse map of $\partial_s$.
\[preliminary structure 1\] Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$.
\(i) Let $\alpha =(f,c) \in {{\mathscr{C}}}(n,l)$ and $\beta \in {{\mathscr{C}}}(m,n)$. Then $$\alpha \Psi_{n,0}(\beta) = \Psi_{l,0}(\alpha\beta) + \sum_{\substack{r\in [l]\setminus {{\mathrm{Im}}}(f),\\ g\in G } } \Psi_{l, r, g} (\partial_r \alpha \beta).$$ (ii) Let $\alpha = (f,c) \in {{\mathscr{C}}}(n,l)$, $\beta \in {{\mathscr{C}}}(m,n-1)$, $s \in [n]$, and $h \in G$. Then $$\alpha \Psi_{n,s,h}(\beta) = \Psi_{l, f(s), h\cdot c(s)^{-1}} (\alpha_s \beta).$$
\(i) Suppose $\gamma \in {{\mathscr{C}}}(l,i)$. Then $$\begin{aligned}
\alpha \Psi_{n,0}(\beta) (\Phi_{l,0}(\gamma) ) &= \Psi_{n,0}(\beta) ( \iota(\gamma)\circ (f_l,c_l)\circ \alpha )\\
&= \Psi_{n,0}(\beta) ( \iota(\gamma)\circ \iota(\alpha) \circ (f_n,c_n) )\\
&= \Psi_{n,0}(\beta) ( \Phi_{ n, 0} (\gamma \alpha) )\\
&= \gamma\alpha\beta\\
&= \Psi_{l,0}(\alpha\beta) (\Phi_{l,0}(\gamma)).\end{aligned}$$
Suppose $\gamma \in {{\mathscr{C}}}(l-1,i)$, $r\in [l]\setminus {{\mathrm{Im}}}(f)$, and $g\in G$. Then $$\begin{aligned}
\alpha \Psi_{n,0}(\beta) (\Phi_{l,r,g}(\gamma) ) &= \Psi_{n,0}(\beta) ( \iota(\gamma)\circ (f_{l,r,g},c_{l,r,g})\circ \alpha )\\
&= \Psi_{n,0}(\beta) ( \iota(\gamma)\circ \iota(\partial_r \alpha ) \circ (f_n,c_n) )\\
&= \Psi_{n,0}(\beta) (\Phi_{n,0}(\gamma \partial_r \alpha) ) \\
&= \gamma \partial_r \alpha \beta \\
&= \Psi_{l, r, g} (\partial_r \alpha \beta) (\Phi_{l,r,g}(\gamma) ).\end{aligned}$$
Suppose $\gamma \in {{\mathscr{C}}}(l-1,i)$, $r\in {{\mathrm{Im}}}(f)$, and $g\in G$. Then $$\begin{aligned}
\alpha \Psi_{n,0}(\beta) (\Phi_{l,r,g}(\gamma) ) &= \Psi_{n,0}(\beta) ( \iota(\gamma)\circ (f_{l,r,g},c_{l,r,g})\circ \alpha )\\
&= 0.\end{aligned}$$
\(ii) Suppose $\gamma \in {{\mathscr{C}}}(l,i)$. Then $$\begin{aligned}
\alpha \Psi_{n,s,h}(\beta) (\Phi_{l,0}(\gamma) ) &= \Psi_{n,s,h}(\beta)( \iota(\gamma)\circ (f_l,c_l)\circ \alpha ) \\
&= \Psi_{n,s,h}(\beta)( \iota(\gamma)\circ \iota(\alpha) \circ (f_n,c_n) )\\
&= 0.\end{aligned}$$
Suppose $\gamma \in {{\mathscr{C}}}(l-1,i)$, $r\in [l]$, and $g\in G$.
If $r = f(s)$ and $g \cdot c(s) = h$, then $$\begin{aligned}
\alpha \Psi_{n,s,h}(\beta) (\Phi_{l,r,g}(\gamma) ) &= \Psi_{n,s,h}(\beta)( \iota(\gamma)\circ (f_{l,r,g},c_{l,r,g})\circ \alpha )\\
&= \Psi_{n,s,h}(\beta)( \iota(\gamma)\circ \iota( \alpha_s ) \circ (f_{n,s,h}, c_{n,s,h} ) ) \\
&= \Psi_{n,s,h}(\beta)(\Phi_{n,s,h}(\gamma \alpha_s) )\\
&= \gamma \alpha_s\beta \\
&= \Psi_{l, f(s), h\cdot c(s)^{-1}} (\alpha_s \beta) ( \Phi_{l,r,g}(\gamma) )\end{aligned}$$
If $r \neq f(s)$ or $g \cdot c(s) \neq h$, then $$\begin{aligned}
\alpha \Psi_{n,s,h}(\beta) (\Phi_{l,r,g}(\gamma) ) &= \Psi_{n,s,h}(\beta)( \iota(\gamma)\circ (f_{l,r,g},c_{l,r,g})\circ \alpha )\\
&= 0.\end{aligned}$$
Proof of Theorem \[main result on q 1\]
---------------------------------------
We retain the notations of subsection \[preliminary discussion 1\].
Let ${{\mathscr{C}}}$ be ${{\mathrm{FI}}}_G$. Let $m\in \operatorname{Ob}({{\mathscr{C}}})$. We need to prove that ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ is isomorphic to $k{{\mathscr{C}}}e_m \oplus k{{\mathscr{C}}}e_{m+1}$.
For each $n\in \operatorname{Ob}({{\mathscr{C}}})$, let $U(n)$ be the image of $\displaystyle\bigoplus_{r\in [n],\, g\in G} k{{\mathscr{C}}}(m,n-1)$ in ${{\mathsf{Q}}}(k {{\mathscr{C}}}e_m)(n)$ under $\Psi_n$, and let $U = \bigoplus_{n\in \operatorname{Ob}({{\mathscr{C}}})} U(n)$. By Lemma \[preliminary structure 1\], $U$ is a ${{\mathscr{C}}}$-submodule of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ and there is a short exact sequence $$0 \longrightarrow U \longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m) \longrightarrow k{{\mathscr{C}}}e_m \longrightarrow 0.$$ Since $k{{\mathscr{C}}}e_m$ is projective, this short exact sequence splits. It suffices to show that $U$ is isomorphic to $k{{\mathscr{C}}}e_{m+1}$.
For each $n\in \operatorname{Ob}({{\mathscr{C}}})$, define a linear map $\Theta_n: U(n) \to k{{\mathscr{C}}}(m+1, n)$ by $$\Theta_n ( \Psi_{n,s,h} (\beta) ) = (f_{n,s,h}, c_{n,s,h})^{-1} \iota(\beta)$$ for all $s\in [n]$, $h\in G$, and $\beta \in {{\mathscr{C}}}(m,n-1)$. Let $\Theta: U \to k{{\mathscr{C}}}e_{m+1}$ be the linear map whose restriction to $U(n)$ is $\Theta_n$. It is easy to see that $\Theta$ is bijective.
We show now that $\Theta$ is a ${{\mathscr{C}}}$-module homomorphism. Suppose $\alpha =(f,c)\in {{\mathscr{C}}}(n,l)$, $\beta\in {{\mathscr{C}}}(m,n-1)$, $s\in [n]$, and $h\in G$. Observe that $$\iota(\alpha_s) \circ (f_{n,s,h}, c_{n,s,h}) = (f_{l, f(s), h\cdot c(s)^{-1}}, c_{l, f(s), h\cdot c(s)^{-1}}) \circ \alpha.$$ Hence, $$\begin{aligned}
\Theta_l (\alpha \Psi_{n,s,h}(\beta)) &= \Theta_l(\Psi_{l, f(s), h\cdot c(s)^{-1}} (\alpha_s \beta)) \\
&= (f_{l, f(s), h\cdot c(s)^{-1}}, c_{l, f(s), h\cdot c(s)^{-1}} )^{-1} \iota(\alpha_s \beta)\\
&= \alpha(f_{n,s,h}, c_{n,s,h})^{-1} \iota(\beta) \\
&= \alpha \Theta_n ( \Psi_{n,s,h} (\beta) ).\end{aligned}$$
Structure of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ when ${{\mathscr{C}}}$ is ${{\mathrm{VI}}}$ {#VI section}
==============================================================================================
Throughout this section, $k$ denotes any commutative ring.
Structure of ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$ {#structure of S for VI}
--------------------------------------------------
Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{VI}}}$. Let $n\in \operatorname{Ob}({{\mathscr{C}}})$. We now recall the structure of ${{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$.
We write elements $\alpha\in {{\mathscr{C}}}(n,l)$ as a $l\times n$-matrix. We write elements $u\in {{\mathbb{F}}}^n$ as a column vector and $u^t$ for its transpose. Let ${{\mathrm{P}}}({{\mathbb{F}}}^n)$ be the set of one dimensional vector subspaces of ${{\mathbb{F}}}^n$. For any $u\in {{\mathbb{F}}}^n$ and $\ell\in {{\mathrm{P}}}({{\mathbb{F}}}^n)$, we write $u^t(\ell)\neq 0$ if $u^t v \neq 0$ for any nonzero vector $v$ in $\ell$. For each $\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$, we choose and fix a $(n-1)\times n$-matrix $\varpi_\ell : {{\mathbb{F}}}^n \to {{\mathbb{F}}}^{n-1}$ whose kernel is $\ell$. We shall denote identity matrices by $I$.
Now, for any $u\in {{\mathbb{F}}}^n$ and $\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$ such that $u^t(\ell)\neq 0$, define the maps $$\begin{gathered}
\Phi_{n,u,0} : {{\mathscr{C}}}(n, l) \longrightarrow {{\mathscr{C}}}(n,l+1), \quad \alpha\mapsto
\left( \begin{matrix} 1& 0 \\ 0 & \alpha \end{matrix} \right) \left( \begin{matrix} u^t \\ I \end{matrix} \right) ; \\
\Phi_{n,u,\ell} : {{\mathscr{C}}}(n-1,l) \longrightarrow {{\mathscr{C}}}(n,l+1), \quad \alpha\mapsto
\left( \begin{matrix} 1& 0 \\ 0 & \alpha \end{matrix} \right) \left( \begin{matrix} u^t \\ \varpi_\ell \end{matrix} \right) .\end{gathered}$$ We may extend these maps linearly to ${{\mathscr{C}}}$-module homomorphisms $$\Phi_{n,u,0} : k{{\mathscr{C}}}e_n \longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n) \quad \mbox{ and } \quad \Phi_{n,u,\ell} : k{{\mathscr{C}}}e_{n-1}\longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n),$$ Let $$\label{phi for VI}
\Phi_n : \left( \bigoplus_{u\in {{\mathbb{F}}}^n} k{{\mathscr{C}}}e_n \right) \oplus \left( \bigoplus_{u\in {{\mathbb{F}}}^n} \bigoplus_{\substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ u^t(\ell)\neq 0 }} k{{\mathscr{C}}}e_{n-1} \right) \longrightarrow {{\mathsf{S}}}(k{{\mathscr{C}}}e_n)$$ be the ${{\mathscr{C}}}$-module homomorphism whose restriction to the direct summand $k{{\mathscr{C}}}e_n$ indexed by $u\in {{\mathbb{F}}}^n$ is $\Phi_{n,u,0}$ and whose restriction to the direct summand $k{{\mathscr{C}}}e_{n-1}$ indexed by $u\in {{\mathbb{F}}}^n$, $\ell\in {{\mathrm{P}}}({{\mathbb{F}}}^n)$ is $\Phi_{n,u,\ell}$. It is straightforward to verify that $\Phi_n$ is an isomorphism (see [@GL-Koszulity Section 5]). Thus, $\iota$ is a genetic functor.
Preliminary discussion of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ {#preliminary discussion 2}
---------------------------------------------------------------
We retain the notations of subsection \[structure of S for VI\]. Let $m, n \in \operatorname{Ob}({{\mathscr{C}}})$. By the isomorphism (\[phi for VI\]), one has the identification $${{\mathsf{Q}}}(k {{\mathscr{C}}}e_m)(n) = \left( \bigoplus_{u\in {{\mathbb{F}}}^n} \operatorname{Hom}_{{{\mathscr{C}}}}( k{{\mathscr{C}}}e_n, k{{\mathscr{C}}}e_m ) \right) \oplus \left( \bigoplus_{u\in {{\mathbb{F}}}^n} \bigoplus_{\substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ u^t(\ell)\neq 0 }} \operatorname{Hom}_{{{\mathscr{C}}}} ( k{{\mathscr{C}}}e_{n-1}, k{{\mathscr{C}}}e_m) \right).$$ Denote by $$\begin{aligned}
\Psi_{n,u,0}: k{{\mathscr{C}}}(m,n) &\longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n),\\
\Psi_{n,u,\ell} : k{{\mathscr{C}}}(m,n-1) &\longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)\end{aligned}$$ the linear maps where $\Psi_{n,u,0}$ is the natural bijection of $k{{\mathscr{C}}}(m,n)$ with the direct summand $\operatorname{Hom}_{{{\mathscr{C}}}} (k{{\mathscr{C}}}e_n, k{{\mathscr{C}}}e_m)$ of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$ indexed by $u\in {{\mathbb{F}}}^n$, and $\Psi_{n,u,\ell}$ is the natural bijection of $k{{\mathscr{C}}}(m,n-1)$ with the direct summand $\operatorname{Hom}_{{{\mathscr{C}}}}(k{{\mathscr{C}}}e_{n-1}, k{{\mathscr{C}}}e_m)$ of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$ indexed by $u\in {{\mathbb{F}}}^n$, $\ell\in {{\mathrm{P}}}({{\mathbb{F}}}^n)$. We have a linear bijection $$\Psi_n:\left( \bigoplus_{u\in {{\mathbb{F}}}^n} k{{\mathscr{C}}}(m,n) \right) \oplus \left( \bigoplus_{u\in {{\mathbb{F}}}^n} \bigoplus_{\substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ u^t(\ell)\neq 0 }} k{{\mathscr{C}}}(m, n-1) \right)\longrightarrow {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$$ whose restriction to the direct summand $k{{\mathscr{C}}}(m,n)$ indexed by $u\in{{\mathbb{F}}}^n$ is $\Psi_{n,u,0}$ and whose restriction to the direct summand $k{{\mathscr{C}}}(m,n-1)$ indexed by $u\in {{\mathbb{F}}}^n$, $\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$ is $\Psi_{n,u,\ell}$.
The next lemma describes the ${{\mathscr{C}}}$-module structure of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ in terms of the identifications $\Psi_n$ for $n\in \operatorname{Ob}({{\mathscr{C}}})$. We shall use the following notation. For any $\alpha\in {{\mathscr{C}}}(n,l)$ and $\wp\in {{\mathrm{P}}}({{\mathbb{F}}}^n)$, let $\alpha_{\wp} \in {{\mathscr{C}}}(n-1,l-1)$ be the unique linear map such that the following diagram commutes: $$\xymatrix{ {{\mathbb{F}}}^n \ar[r]^{\alpha} \ar[d]_{\varpi_{\wp}} & {{\mathbb{F}}}^l \ar[d]^{\varpi_{\alpha(\wp)} } \\
{{\mathbb{F}}}^{n-1} \ar[r]_{\alpha_{\wp}} & {{\mathbb{F}}}^{l-1} }$$
\[preliminary structure 2\] Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{VI}}}$.
\(i) Let $\alpha\in {{\mathscr{C}}}(n,l)$, $\beta\in {{\mathscr{C}}}(m,n)$, and $v\in {{\mathbb{F}}}^n$. Then $$\alpha \Psi_{n,v,0}(\beta) = \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \Psi_{l,u,0}(\alpha\beta) + \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l) \\ u^t(\ell)\neq 0 \\ \ell \nsubseteq {{\mathrm{Im}}}(\alpha) } } \Psi_{l,u,\ell} ( \varpi_\ell \alpha \beta ) .$$
\(ii) Let $\alpha\in {{\mathscr{C}}}(n,l)$, $\beta\in {{\mathscr{C}}}(m,n-1)$, $v\in {{\mathbb{F}}}^n$, and $\wp\in {{\mathrm{P}}}({{\mathbb{F}}}^n)$. Suppose $v^t(\wp) \neq 0$. Then $$\alpha \Psi_{n,v,\wp} (\beta) = \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l)\\ \ell = \alpha(\wp) } } \Psi_{l,u,\ell}( \alpha_{\wp} \beta ) .$$
\(i) Suppose $\gamma \in {{\mathscr{C}}}(l,i)$ and $u\in {{\mathbb{F}}}^l$. One has $$\label{preliminary structure 2 equation 1}
\Phi_{l,u,0}(\gamma)\alpha
= \left( \begin{matrix} 1& 0 \\ 0 & \gamma \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ \alpha \end{matrix}\right)
= \left( \begin{matrix} 1& 0 \\ 0 & \gamma \alpha \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ I \end{matrix} \right).$$ Thus, $$\alpha \Psi_{n,v,0}(\beta) (\Phi_{l,u,0}(\gamma)) = \Psi_{n,v,0}(\beta) (\Phi_{l,u,0}(\gamma)\alpha)
= \left\{ \begin{array}{ll}
0 & \mbox{ if } u^t\alpha \neq v^t, \\
\gamma\alpha\beta & \mbox{ if } u^t\alpha = v^t.
\end{array} \right.$$ In particular, when $u^t\alpha=v^t$, one has $$\alpha \Psi_{n,v,0}(\beta) (\Phi_{l,u,0}(\gamma)) = \Psi_{l,u,0}(\alpha\beta)(\Phi_{l,u,0}(\gamma)).$$
Now suppose $\gamma\in {{\mathscr{C}}}(l-1,i)$, $u\in {{\mathbb{F}}}^l$, $\ell\in {{\mathrm{P}}}({{\mathbb{F}}}^l)$, and $u^t(\ell)\neq 0$. One has $$\Phi_{l,u,\ell}(\gamma)\alpha = \left( \begin{matrix} 1& 0 \\ 0 & \gamma \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ \varpi_\ell \alpha \end{matrix}\right)
= \left( \begin{matrix} 1& 0 \\ 0 & \gamma \varpi_\ell \alpha \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ I \end{matrix} \right).$$ Thus, $$\begin{aligned}
\alpha \Psi_{n,v,0}(\beta) (\Phi_{l,u,\ell}(\gamma)) &= \Psi_{n,v,0}(\beta) (\Phi_{l,u,\ell}(\gamma)\alpha) \\
&= \left\{ \begin{array}{ll}
0 & \mbox{ if } u^t\alpha \neq v^t \mbox{ or } \ell \subseteq {{\mathrm{Im}}}(\alpha), \\
\gamma \varpi_\ell \alpha\beta & \mbox{ if } u^t\alpha = v^t \mbox{ and } \ell \nsubseteq {{\mathrm{Im}}}(\alpha).
\end{array} \right.\end{aligned}$$ In particular, when $ u^t\alpha = v^t$ and $\ell \nsubseteq {{\mathrm{Im}}}(\alpha)$, one has $$\alpha \Psi_{n,v,0}(\beta) (\Phi_{l,u,\ell}(\gamma)) = \Psi_{l,u,\ell} ( \varpi_\ell \alpha \beta ) (\Phi_{l,u,\ell}(\gamma)).$$
\(ii) Suppose $\gamma \in {{\mathscr{C}}}(l,i)$ and $u\in {{\mathbb{F}}}^l$. From (\[preliminary structure 2 equation 1\]), one has $$\alpha \Psi_{n,v,\wp} (\beta)(\Phi_{l,u,0}(\gamma)) = \Psi_{n,v,\wp} (\beta) (\Phi_{l,u,0}(\gamma)\alpha) =0.$$
Now suppose $\gamma\in {{\mathscr{C}}}(l-1,i)$, $u\in {{\mathbb{F}}}^l$, $\ell\in {{\mathrm{P}}}({{\mathbb{F}}}^l)$, and $u^t(\ell)\neq 0$. One has $$\Phi_{l,u,\ell}(\gamma)\alpha = \left( \begin{matrix} 1& 0 \\ 0 & \gamma \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ \varpi_\ell \alpha \end{matrix}\right).$$ We can write $\left( \begin{matrix} u^t \alpha \\ \varpi_\ell \alpha \end{matrix}\right)$ in the form $ \left( \begin{matrix} 1& 0 \\ 0 & * \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ \varpi_{\wp} \end{matrix}\right)$ if and only if $\ell = \alpha(\wp)$. If $\ell = \alpha(\wp)$, then $$\Phi_{l,u,\ell}(\gamma)\alpha = \left( \begin{matrix} 1& 0 \\ 0 & \gamma \alpha_{\wp} \end{matrix} \right) \left( \begin{matrix} u^t \alpha \\ \varpi_{\wp} \end{matrix}\right).$$ Thus, $$\begin{aligned}
\alpha\Psi_{n,v,\wp} (\beta) \Phi_{l,u,\ell}(\gamma) &= \Psi_{n,v,\wp} (\beta) (\Phi_{l,u,\ell}(\gamma)\alpha) \\
&= \left\{ \begin{array}{ll}
0 & \mbox{ if } u^t\alpha \neq v^t \mbox{ or } \ell \neq \alpha(\wp), \\
\gamma \alpha_{\wp} \beta & \mbox{ if } u^t\alpha = v^t \mbox{ and } \ell = \alpha(\wp).
\end{array} \right.\end{aligned}$$ In particular, when $u^t\alpha = v^t$ and $ \ell = \alpha(\wp)$, one has $$\alpha\Psi_{n,v,\wp} (\beta) (\Phi_{l,u,\ell}(\gamma) ) = \Psi_{l,u,\ell}( \alpha_{\wp} \beta ) (\Phi_{l,u,\ell}(\gamma)).$$
Proof of Theorem \[main result on q 2\]
---------------------------------------
We retain the notations of subsection \[preliminary discussion 2\].
Let ${{\mathscr{C}}}$ be ${{\mathrm{VI}}}$. Let $m\in \operatorname{Ob}({{\mathscr{C}}})$. We need to prove that ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ contains a direct summand isomorphic to $k{{\mathscr{C}}}e_{m+1}$.
Since $k{{\mathscr{C}}}e_{m+1}$ is a projective ${{\mathscr{C}}}$-module, it suffices to construct a surjective homomorphism $$\pi : {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m) \longrightarrow k{{\mathscr{C}}}e_{m+1}.$$ For each $n\in \operatorname{Ob}({{\mathscr{C}}})$, we define a linear map $\pi_n : {{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n) \longrightarrow k{{\mathscr{C}}}(m+1, n)$ as follows:
1. If $\beta\in {{\mathscr{C}}}(m,n)$ and $v\in {{\mathbb{F}}}^n$, let $$\pi_n (\Psi_{n,v,0}(\beta) ) = -q^{-n} \sum_{ \substack{ \wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ v^t(\wp)\neq 0 \\ \wp \nsubseteq {{\mathrm{Im}}}(\beta) } } \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_{\wp} \beta \end{matrix} \right).$$
2. If $\beta\in {{\mathscr{C}}}(m,n-1)$, $v\in {{\mathbb{F}}}^n$, $\wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$, and $v^t(\wp)\neq 0$, let $$\pi_n (\Psi_{n,v, \wp}(\beta) )= q^{-n} \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \beta \end{matrix} \right).$$
Let $\pi$ be the linear map whose restriction to ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)(n)$ is $\pi_n$. We claim that: $\pi$ is surjective, and $\pi$ is a ${{\mathscr{C}}}$-module homomorphism.
To show that $\pi$ is surjective, consider any $\gamma \in {{\mathscr{C}}}(m+1, n)$. We want to show that $\gamma$ is in the image of $\pi_n$. To this end, we write the $m+1$ columns of $\gamma$ as $\gamma_1, \ldots, \gamma_{m+1} \in {{\mathbb{F}}}^n$. Let $\wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$ be the span of $\gamma_1$; so one has $\varpi_{\wp} \gamma_1=0$. Since $\gamma: {{\mathbb{F}}}^{m+1} \to {{\mathbb{F}}}^n$ is injective, its transpose $\gamma^t : {{\mathbb{F}}}^n \to {{\mathbb{F}}}^{m+1}$ is surjective. Hence, there exists $v\in {{\mathbb{F}}}^n$ such that $$v^t \gamma_1 = 1, \quad v^t \gamma_2 = \cdots v^t \gamma_{m+1} = 0.$$ Choose such a $v$. Then one has $$\left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right) \gamma = \left( \begin{matrix} 1 & 0 \\ 0 & \beta \end{matrix} \right)$$ for some $\beta\in {{\mathscr{C}}}(m,n-1)$. Therefore, $$\pi_n( q^n \Psi_{n,v, \wp}(\beta) ) = \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \beta \end{matrix} \right) = \gamma.$$
It remains to check that $\pi$ is a ${{\mathscr{C}}}$-module homomorphism. Let $\alpha\in {{\mathscr{C}}}(n,l)$.
Observe that if $v\in {{\mathbb{F}}}^n$, then $$\# \{ u\in {{\mathbb{F}}}^l \mid u^t\alpha = v^t \} = q^{l-n}.$$
Suppose that $\beta\in {{\mathscr{C}}}(m,n)$ and $v\in {{\mathbb{F}}}^n$. One has:
$$\begin{aligned}
& \pi_l ( \alpha \Psi_{n,v,0}(\beta) ) \\
=& \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \pi_l( \Psi_{l,u,0}(\alpha\beta) ) + \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l) \\ u^t(\ell)\neq 0 \\ \ell \nsubseteq {{\mathrm{Im}}}(\alpha) } } \pi_l( \Psi_{l,u,\ell} ( \varpi_\ell \alpha \beta ) ) \\
=& - q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{ \ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l) \\ u^t (\ell)\neq 0 \\ \ell \nsubseteq {{\mathrm{Im}}}(\alpha \beta) } } \left( \begin{matrix} u^t \\ \varpi_\ell \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_\ell \alpha \beta \end{matrix} \right) \\
&\quad + q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l) \\ u^t (\ell)\neq 0 \\ \ell \nsubseteq {{\mathrm{Im}}}(\alpha) } } \left( \begin{matrix} u^t \\ \varpi_\ell \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_\ell \alpha \beta \end{matrix} \right) \\
=& - q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{ \ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l) \\ u^t (\ell)\neq 0 \\ \ell \subseteq {{\mathrm{Im}}}(\alpha) \setminus {{\mathrm{Im}}}(\alpha \beta) } } \left( \begin{matrix} u^t \\ \varpi_\ell \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_\ell \alpha \beta \end{matrix} \right) \\
=& - q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{ \wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ v^t (\wp)\neq 0 \\ \wp \nsubseteq {{\mathrm{Im}}}(\beta) } } \left( \begin{matrix} u^t \\ \varpi_{\alpha(\wp)} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_{\alpha(\wp)} \alpha \beta \end{matrix} \right) \\
=& - q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{ \wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ v^t (\wp)\neq 0 \\ \wp \nsubseteq {{\mathrm{Im}}}(\beta) } } \left( \begin{matrix} u^t \\ \varpi_{\alpha(\wp)} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \alpha_{\wp} \varpi_{\wp} \beta \end{matrix} \right).\end{aligned}$$
Observe that when $u^t\alpha = v^t$, one has $$\left( \begin{matrix} u^t \\ \varpi_{\alpha(\wp)} \end{matrix} \right) \alpha = \left( \begin{matrix} v^t \\ \alpha_{\wp} \varpi_{\wp} \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & \alpha_{\wp} \end{matrix} \right) \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)$$ which implies $$\label{key result equation}
\left( \begin{matrix} u^t \\ \varpi_{\alpha(\wp)} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \alpha_{\wp} \end{matrix} \right) = \alpha \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1}.$$ Hence, continuing our calculation from above, $$\begin{aligned}
& \pi_l ( \alpha \Psi_{n,v,0}(\beta) ) \\
=& - q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{ \wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ v^t (\wp)\neq 0 \\ \wp \nsubseteq {{\mathrm{Im}}}(\beta) } } \alpha \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_{\wp} \beta \end{matrix} \right) \\
=& - q^{-l} \cdot q^{l-n} \sum_{ \substack{ \wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n) \\ v^t (\wp)\neq 0 \\ \wp \nsubseteq {{\mathrm{Im}}}(\beta) } } \alpha \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \varpi_{\wp} \beta \end{matrix} \right) \\
=& \alpha \pi_n ( \Psi_{n,v,0}(\beta) ) .\end{aligned}$$
Now suppose that $\beta \in {{\mathscr{C}}}(m, n-1)$, $v\in {{\mathbb{F}}}^n$, $\wp \in {{\mathrm{P}}}({{\mathbb{F}}}^n)$, and $v^t(\wp)\neq 0$. One has:
$$\begin{aligned}
& \pi_l ( \alpha \Psi_{n,v,\wp}(\beta) ) \\
=& \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l)\\ \ell = \alpha(\wp) } } \pi_l ( \Psi_{l,u,\ell}( \alpha_{\wp} \beta ) ) \\
=& q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l)\\ \ell = \alpha(\wp) } } \left( \begin{matrix} u^t \\ \varpi_\ell \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \alpha_{\wp} \beta \end{matrix} \right) \\
=& q^{-l} \sum_{\substack{u \in {{\mathbb{F}}}^l \\ u^t \alpha = v^t}} \sum_{ \substack{\ell \in {{\mathrm{P}}}({{\mathbb{F}}}^l)\\ \ell = \alpha(\wp) } } \alpha \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \beta \end{matrix} \right) \quad \mbox{ using (\ref{key result equation}) } \\
=& q^{-l} \cdot q^{l-n} \cdot \alpha \left( \begin{matrix} v^t \\ \varpi_{\wp} \end{matrix} \right)^{-1} \left( \begin{matrix} 1 & 0 \\ 0 & \beta \end{matrix} \right) \\
=& \alpha \pi_n (\Psi_{n,v, \wp}(\beta)) .\end{aligned}$$
This completes the verification that $\pi$ is a ${{\mathscr{C}}}$-module homomorphism.
Applications of coinduction functor {#applications of coinduction}
===================================
Throughout this section, we assume that $k$ is a field of characteristic 0.
Proofs of Theorem \[main-result-1\] and Corollary \[infinite global dimension\]
-------------------------------------------------------------------------------
Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$ or ${{\mathrm{VI}}}$. Recall that ${{\mathscr{C}}}$ is locally Noetherian by [@GL].
It suffices to prove that $k{{\mathscr{C}}}e_n$ is injective for each $n\in \operatorname{Ob}({{\mathscr{C}}})$. We prove this by induction on $n$.
By Corollary \[kCe\_0 is injective\], $k{{\mathscr{C}}}e_n$ is injective when $n=0$. Now suppose that $k{{\mathscr{C}}}e_n$ is injective when $n=m$ for some $m\in \operatorname{Ob}({{\mathscr{C}}})$. By Lemma \[eckmann-shapiro2\], ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ is injective. But $k{{\mathscr{C}}}e_{m+1}$ is a direct summand of ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ by Theorems \[main result on q 1\] and \[main result on q 2\]. It follows that $k{{\mathscr{C}}}e_n$ is injective for $n=m+1$.
To carry out the inductive argument in the above proof, we need to know that ${{\mathsf{Q}}}(k{{\mathscr{C}}}e_m)$ contains a direct summand isomorphic to $k{{\mathscr{C}}}e_{m+1}$. This was verified for the categories ${{\mathrm{FI}}}_G$ and ${{\mathrm{VI}}}$ by the explicit computations in Sections \[FI\_G section\] and \[VI section\]; we do not know if a conceptual proof can be given based on certain underlying combinatorial properties of the category ${{\mathscr{C}}}$.
Suppose $V$ is a finitely generated ${{\mathscr{C}}}$-module and $$0 \to P^{-r} \to \cdots \to P^{-1} \to P^0 \to V \to 0$$ is an exact sequence where $P^0, \ldots, P^{-r}$ are finitely generated projective ${{\mathscr{C}}}$-modules. Then there are short exact sequences $0 \to Q^{i-1} \to P^i \to Q^i \to 0$ for $i=0, \ldots, -(r-1)$ where $Q^0, \ldots, Q^{-r}$ are finitely generated ${{\mathscr{C}}}$-modules such that $Q^0=V$ and $Q^{-r}=P^{-r}$. It follows from Theorem \[main-result-1\] that these short exact sequences split; in particular, $V$ is a direct summand of $P^0$.
Torsion-free modules
--------------------
Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$.
A ${{\mathscr{C}}}$-module $F$ is *torsion-free* if $\operatorname{Hom}_{{\mathscr{C}}}(T, F)=0$ for all finite dimensional ${{\mathscr{C}}}$-modules $T$.
\[torsion-pair\] Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. Let $V$ be a finitely generated ${{\mathscr{C}}}$-module. Then there exists a short exact sequence $$0 \longrightarrow T \longrightarrow V \longrightarrow F \longrightarrow 0$$ such that $T$ is a finite dimensional ${{\mathscr{C}}}$-module and $F$ is a torsion-free ${{\mathscr{C}}}$-module.
Since $V$ is Noetherian, there exists a maximal finite dimensional submodule $T$ of $V$. It is plain that $V/T$ is torsion-free.
\[nonzero hom to projective\] Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. Let $F$ be a finitely generated torsion-free ${{\mathscr{C}}}$-module. If $F\neq 0$, then there exists $n\in \operatorname{Ob}({{\mathscr{C}}})$ such that $\operatorname{Hom}_{{\mathscr{C}}}(F, k{{\mathscr{C}}}e_n)\neq 0$.
Since $F\neq 0$, there exists a smallest $a\in \operatorname{Ob}({{\mathscr{C}}})$ such that $F(a)\neq 0$. Thus, ${{\mathsf{S}}}^a(F)(0) \neq 0$. Choose a nonzero element $s$ of ${{\mathsf{S}}}^a(F)(0)$ and let $$f: k{{\mathscr{C}}}e_0 \longrightarrow {{\mathsf{S}}}^a(F), \quad \alpha \mapsto \alpha s.$$ Since $F$ is torsion-free, the homomorphism $f$ is injective. But $k{{\mathscr{C}}}e_0$ is an injective ${{\mathscr{C}}}$-module by Theorem \[main-result-1\]. Thus, there exists a nonzero homomorphism from ${{\mathsf{S}}}^a(F)$ to $k{{\mathscr{C}}}e_0$. It follows from Lemma \[eckmann-shapiro2\] that $$\operatorname{Hom}_{{\mathscr{C}}}(F, {{\mathsf{Q}}}^a(k{{\mathscr{C}}}e_0)) = \operatorname{Hom}_{{\mathscr{C}}}({{\mathsf{S}}}^a(F), k{{\mathscr{C}}}e_0) \neq 0.$$ But by Theorem \[main result on q 1\], ${{\mathsf{Q}}}^a(k{{\mathscr{C}}}e_0)$ is isomorphic to $k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r}$ for some $n_1,\ldots, n_r\in \operatorname{Ob}({{\mathscr{C}}})$. Hence, the result follows.
Suppose $V$ is a finitely generated ${{\mathscr{C}}}$-module. Then there exists $l \in \operatorname{Ob}({{\mathscr{C}}})$ such that $V$ is generated in degrees $\leqslant l$; clearly, one has $\operatorname{Hom}_{{\mathscr{C}}}( V, k{{\mathscr{C}}}e_n) = 0$ for all $n>l$.
For any finitely generated ${{\mathscr{C}}}$-module $V$, let $$\kappa(V) = \sum_{n \in \operatorname{Ob}({{\mathscr{C}}})} \kappa (V, n),$$ where $\kappa(V, n) = \dim_k \operatorname{Hom}_{{{\mathscr{C}}}} (V, k{{\mathscr{C}}}e_n)$ for each $n\in \operatorname{Ob}({{\mathscr{C}}})$.
\[injection to projective\] Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$. Let $F$ be a finitely generated torsion-free ${{\mathscr{C}}}$-module. Then there exists an injective homomorphism from $F$ to $k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r}$ for some $n_1,\ldots, n_r\in \operatorname{Ob}({{\mathscr{C}}})$.
If $\kappa(F)=0$, then $F=0$ by Lemma \[nonzero hom to projective\]. We shall prove the proposition by induction on $\kappa(F)$. Suppose $\kappa(F)>0$. Then there exists a nonzero homomorphism $f: F \to k{{\mathscr{C}}}e_m$ for some $m\in \operatorname{Ob}({{\mathscr{C}}})$. Let $E=\operatorname{Ker}(f)$ and $W={{\mathrm{Im}}}(f)$. We have a short exact sequence $$\label{ses for induction}
0 \longrightarrow E \longrightarrow F \longrightarrow W \longrightarrow 0.$$ By Theorem \[main-result-1\], the ${{\mathscr{C}}}$-module $k{{\mathscr{C}}}e_n$ is injective, so $$\kappa(E,n) + \kappa(W,n) = \kappa (F, n) \quad \mbox{ for each } n\in \operatorname{Ob}({{\mathscr{C}}}).$$ Therefore, $\kappa(E)+ \kappa(W) = \kappa(F)$. But $\kappa(W,m)>0$, so $\kappa(E)<\kappa(F)$. Observe that $E$ is a finitely generated torsion-free ${{\mathscr{C}}}$-module. By induction hypothesis, there exists an injective homomorphism from $E$ to $k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r}$ for some $n_1,\ldots, n_r\in \operatorname{Ob}({{\mathscr{C}}})$. Since we also have $W \hookrightarrow k{{\mathscr{C}}}e_m$ and the ${{\mathscr{C}}}$-modules $k{{\mathscr{C}}}e_1, \ldots, k{{\mathscr{C}}}e_r$, $k{{\mathscr{C}}}e_m$ are injective, it follows from (\[ses for induction\]) by a standard argument that there exists an injective homomorphism from $F$ to $k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r}\oplus k{{\mathscr{C}}}e_m$.
Proof of Theorem \[main-result-2\]
----------------------------------
Suppose that ${{\mathscr{C}}}$ is ${{\mathrm{FI}}}_G$.
\(i) Suppose that $V$ is a finitely generated injective ${{\mathscr{C}}}$-module. By Lemma \[torsion-pair\], there is a short exact sequence $0 \to T \to V \to F \to 0$ where $T$ is a finite dimensional ${{\mathscr{C}}}$-module and $F$ is a torsion-free ${{\mathscr{C}}}$-module. For any ${{\mathscr{C}}}$-module $U$, there is a long exact sequence $$\cdots \to \operatorname{Hom}_{{\mathscr{C}}}(U, F) \to \operatorname{Ext}^1_{{\mathscr{C}}}(U,T) \to \operatorname{Ext}^1_{{\mathscr{C}}}(U, V) \to \cdots .$$ Since $V$ is injective, one has $\operatorname{Ext}^1_{{\mathscr{C}}}(U,V)=0$. Since $F$ is torsion-free, one has $\operatorname{Ext}^1_{{\mathscr{C}}}(U, T)=0$ whenever $U$ is finite dimensional.
Choose $l\in \operatorname{Ob}({{\mathscr{C}}})$ such that $T(m)=0$ for all $m>l$. Suppose $W$ is a finitely generated ${{\mathscr{C}}}$-module. By Lemma \[reduction to C\_n\], we can choose $n>l$ such that: $\operatorname{Ext}^1_{{{\mathscr{C}}}}(W, T) = \operatorname{Ext}^1_{{{\mathscr{C}}}_n} (\jmath^*(W), \jmath^*(T))$ where $\jmath: {{\mathscr{C}}}_n\hookrightarrow {{\mathscr{C}}}$ denotes the inclusion functor. Let $U= \jmath_*(\jmath^*(W))$. Observe that $\jmath^*(U) = \jmath^*(W)$ and $\jmath_*(\jmath^*(T))=T$. Thus, by Lemma \[eckmann-shapiro1\], one has $$\operatorname{Ext}^1_{{{\mathscr{C}}}_n} (\jmath^*(W), \jmath^*(T)) = \operatorname{Ext}^1_{{{\mathscr{C}}}} (U, T) = 0.$$ Therefore, $\operatorname{Ext}^1_{{{\mathscr{C}}}}(W, T) = 0$. It follows from Corollary \[injectivity in C-mod\] that $T$ is an injective ${{\mathscr{C}}}$-module.
We deduce that $V$ is isomorphic to $T\oplus F$, so $F$ is an injective ${{\mathscr{C}}}$-module. By Proposition \[injection to projective\], it follows that $F$ is a direct summand of a projective ${{\mathscr{C}}}$-module. Therefore, $F$ is a projective ${{\mathscr{C}}}$-module.
\(ii) Suppose that $V$ is a finitely generated ${{\mathscr{C}}}$-module generated in degrees $\leqslant l$. We shall prove the result by induction on $l$.
We may assume, without loss of generality, that $V$ has no projective direct summands. By Lemma \[torsion-pair\], there is a short exact sequence $0 \to T \to V \to F \to 0$ where $T$ is a finite dimensional ${{\mathscr{C}}}$-module and $F$ is a torsion-free ${{\mathscr{C}}}$-module. By Proposition \[injection to projective\], there is an injective homomorphism $$f: F \to k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r} \quad
\mbox{ for some } n_1,\ldots, n_r\in \operatorname{Ob}({{\mathscr{C}}}).$$
Observe that $F$ is generated in degrees $\leqslant l$ and has no projective direct summands. Since there is no nonzero homomorphism from $F$ to $k{{\mathscr{C}}}e_n$ for any $n>l$, we can assume $n_1, \ldots, n_r \leqslant l$. We claim that there is also no nonzero homomorphism from $F$ to $k{{\mathscr{C}}}e_l$. Indeed, the image of any homomorphism from $F$ to $k{{\mathscr{C}}}e_l$ is a submodule of $k{{\mathscr{C}}}e_l$ generated in degrees $\leqslant l$, but any such submodule of $k{{\mathscr{C}}}e_l$ is a direct summand of $k{{\mathscr{C}}}e_l$ and hence a projective ${{\mathscr{C}}}$-module. Since $F$ has no projective direct summands, it follows that any homomorphism from $F$ to $k{{\mathscr{C}}}e_l$ must be 0. If $l=0$, this implies that $F=0$, so $V=T$, and we are done by Lemma \[inj res of finite dim module\].
Suppose $l>0$. By the above observations, we can assume that $n_1, \ldots, n_r \leqslant l-1$. Let $W$ be the cokernel of $f$. Since $W$ is generated in degrees $\leqslant l-1$, it follows by induction hypothesis that $W$ has a finite injective resolution $$0 \to W \to J^1 \to \cdots \to J^a \to 0$$ in the category ${{\mathscr{C}}}\operatorname{-mod}$. From this, we obtain a finite injective resolution $$0 \to F \stackrel{f}{\to} J^0 \to J^1 \to \cdots \to J^a \to 0$$ where $J^0 = k{{\mathscr{C}}}e_{n_1} \oplus \cdots \oplus k{{\mathscr{C}}}e_{n_r}$. Recall that by Lemma \[inj res of finite dim module\], the ${{\mathscr{C}}}$-module $T$ has a finite injective resolution in ${{\mathscr{C}}}\operatorname{-mod}$. We conclude by the horseshoe lemma (see [@Weibel page 37]) that $V$ has a finite injective resolution in ${{\mathscr{C}}}\operatorname{-mod}$.
Homological approach to representation stability {#last section}
================================================
Throughout this section, we assume that $k$ is a splitting field for $G$ of characteristic 0.
Simple modules of wreath product groups
---------------------------------------
Let ${{\mathscr{C}}}$ be ${{\mathrm{FI}}}_G$. Then $G_n = G \wr S_n$.
For each $m, l \in \operatorname{Ob}({{\mathscr{C}}})$, we consider $G_m \times G_l$ as a subgroup of $G_{m+l}$ via $$G_m \times G_l \hookrightarrow G_{m+l}, \quad \big( (f_1,c_1), (f_2,c_2) \big) \mapsto (f_1,c_1)\odot (f_2,c_2).$$ If $X$ is a $kG_m$-module, and $Y$ is a $kG_l$-module, we set $$X \circledast Y = kG_{m+l} \otimes_{k(G_m\times G_l)} (X\otimes_k Y).$$ If $A$ is a $k G$-module, and $E$ is a $k S_n$-module, we write $A\wr E$ for the $k G\wr S_n$-module $A^{\otimes n} \otimes_k E$.
We denote by $\operatorname{Irr}(G)=\{\chi_1, \ldots, \chi_r\}$ the set of isomorphism classes of simple $k G$-modules; in particular, let $\chi_1 \in \operatorname{Irr}(G)$ be the trivial class. For each $\chi\in\operatorname{Irr}(G)$, let $A(\chi)$ be a simple $k G$-module belonging to the isomorphism class $\chi \in \operatorname{Irr}(G)$. Recall that the isomorphism classes of simple $k S_n$-modules are parametrized by the partitions of $n$. For each partition $\lambda$ of $n$, let $E(\lambda)$ be a simple $k S_n$-module whose isomorphism class corresponds to $\lambda$.
If ${{\underline{\lambda}}}$ is a partition-valued function on $\operatorname{Irr}(G)$, we let $$\widetilde{L}({{\underline{\lambda}}}) = \Big( A(\chi_1) \wr E({{\underline{\lambda}}}(\chi_1)) \Big) \circledast \cdots \circledast \Big( A(\chi_r) \wr E({{\underline{\lambda}}}(\chi_r)) \Big).$$ The following result on the classification of simple $k G\wr S_n$-modules is well-known (see [@Macdonald]).
Suppose that $k$ is a splitting field for $G$ of characteristic 0. Then the set of $\widetilde{L}({{\underline{\lambda}}})$ for all partition-valued functions ${{\underline{\lambda}}}$ on $\operatorname{Irr}(G)$ with $|{{\underline{\lambda}}}|=n$ is a complete set of non-isomorphic simple $k G \wr S_n$-modules.
Proof of Theorem \[rep stable thm\]
-----------------------------------
The homological properties of the category ${{\mathscr{C}}}\operatorname{-mod}$ proved in the previous section allow one to give a quick proof that finite generation implies condition (RS3).
For any partition-valued function ${{\underline{\lambda}}}$ on $\operatorname{Irr}(G)$, and integer $n\geqslant |{{\underline{\lambda}}}|+a$ where $a$ is the biggest part of ${{\underline{\lambda}}}(\chi_1)$, let $$L({{\underline{\lambda}}})_n = \widetilde{L}({{\underline{\lambda}}}[n]).$$
As explained in subsection \[rep stability\], it only remains to verify that condition (RS3) holds for finitely generated projective ${{\mathscr{C}}}$-modules. But by [@Dieck Theorem 11.18], any finitely generated projective ${{\mathscr{C}}}$-module is a direct sum of ${{\mathscr{C}}}$-modules of the form $k {{\mathscr{C}}}e_m \otimes_{k G_m} L({{\underline{\lambda}}})_m$ for some $m\in \operatorname{Ob}({{\mathscr{C}}})$ and partition-valued function ${{\underline{\lambda}}}$ on $\operatorname{Irr}(G)$.
Let $n\geqslant 2m$. Denote by $\mu$ the trivial partition of $n-m$. One has $$k{{\mathscr{C}}}(m,n) = k G_m \circledast \Big(A(\chi_1) \wr E(\mu) \Big).$$ Therefore, $$\begin{gathered}
k{{\mathscr{C}}}(m,n) \otimes_{k G_m} L({{\underline{\lambda}}})_m
= L({{\underline{\lambda}}})_m \circledast \Big(A(\chi_1) \wr E(\mu) \Big)\\
= \Big( A(\chi_1) \wr E({{\underline{\lambda}}}[m](\chi_1)) \Big) \circledast \Big(A(\chi_1) \wr E(\mu) \Big) \circledast \left( {\mathop{\mathpalette\b@gCircledast\relax}}_{i=2}^r A(\chi_i) \wr E({{\underline{\lambda}}}(\chi_i)) \right).\end{gathered}$$ The rest of the proof is same as [@Hemmer Lemma 2.3]. By Pieri’s formula, $$\Big( A(\chi_1) \wr E({{\underline{\lambda}}}[m](\chi_1)) \Big) \circledast \Big(A(\chi_1) \wr E(\mu) \Big) \\
= \bigoplus_{\nu \in P(n)} A(\chi_1) \wr E(\nu),$$ where $P(n)$ denotes the set of all partitions $\nu$ whose Young diagram can be obtained from the Young diagram of ${{\underline{\lambda}}}[m](\chi_1)$ by adding $n-m$ boxes with no two in the same column. Let $\hbar: P(n) \to P(n+1)$ be the map which assigns to $\nu\in P(n)$ the partition $\hbar(\nu)\in P(n+1)$ whose Young diagram is obtained from the Young diagram of $\nu$ by adding a box in the first row. It is plain that $\hbar$ is injective. Since $n\geqslant 2m$, the map $\hbar$ must also be surjective. The result follows.
[99]{}
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B. Farb, Representation stability, to appear in Proceedings of ICM 2014, arXiv:1404.4065.
W.L. Gan, L. Li, Noetherian property of infinite EI categories, arXiv:1407.8235.
W.L. Gan, L. Li, Koszulity of directed categories in representation stability theory, arXiv:1411.5308.
D. Hemmer, Stable decompositions for some symmetric group characters arising in braid group cohomology, J. Combin. Theory Ser. A 118 (2011), no. 3, 1136-1139.
N. Kuhn, Generic representation theory of finite fields in nondescribing characteristic, Adv. Math. 272 (2015), 598-610, arXiv:1405.0318.
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---
abstract: 'We consider the holographic duality between type-A higher-spin gravity in $AdS_4$ and the free $U(N)$ vector model. In the bulk, linearized solutions can be translated into twistor functions via the Penrose transform. We propose a holographic dual to this transform, which translates between twistor functions and CFT sources and operators. We present a twistorial expression for the partition function, which makes global higher-spin symmetry manifest, and appears to automatically include all necessary contact terms. In this picture, twistor space provides a fully nonlocal, gauge-invariant description underlying both bulk and boundary spacetime pictures. While the bulk theory is handled at the linear level, our formula for the partition function includes the effects of bulk interactions. Thus, the CFT is used to solve the bulk, with twistors as a language common to both. A key ingredient in our result is the study of ordinary spacetime symmetries within the fundamental representation of higher-spin algebra. The object that makes these “square root” spacetime symmetries manifest becomes the kernel of our boundary/twistor transform, while the original Penrose transform is identified as a “square root” of CPT.'
author:
- Yasha Neiman
title: The holographic dual of the Penrose transform
---
Introduction
============
Quantum spacetime is infamously difficult to address directly. This leads one to search for alternative geometric frameworks, which may survive the breakdown of locality at the Planck scale. The most productive approach to date is AdS/CFT [@Maldacena:1997re; @Witten:1998qj; @Aharony:1999ti] – a retreat from the bulk spacetime onto its asymptotic boundary. There, one can operate with a fixed classical geometry, since the Planck length effectively vanishes due to an infinite warp factor. AdS/CFT relates two spacetime pictures, with two different notions of locality: an approximate locality in the higher-dimensional bulk, and a precise locality on the lower-dimensional boundary. The duality itself is of necessity non-local. Furthermore, the bulk and boundary pictures each contain a different set of gauge redundancies – the well-known price of locality – which are absent in the dual picture. A question then suggests itself: is there some third geometric framework, completely divorced from spacetime locality, underlying both the bulk and boundary descriptions? To find such a framework, one must focus on non-local, gauge-invariant objects in both bulk and boundary. Such is arguably the strategy behind the study of Ryu-Takayanagi surfaces [@Ryu:2006bv], kinematic space in the context of MERA [@Czech:2015qta], and other such relations between quantum-informational quantities and bulk geometry.
At the same time, there exists a much older proposal for a geometric framework to replace spacetime: Penrose’s twistor theory [@Penrose:1986ca; @Ward:1990vs]. There, we effectively trade locality for causality as the fundamental principle, replacing points with twistors – the “maximally lightlike” extended shapes in spacetime. Originally conceived as a framework for quantum General Relativity, twistor theory has now become a workhorse for scattering amplitude calculations in maximally supersymmetric Yang-Mills [@Adamo:2011pv] and supergravity [@Skinner:2013xp]. Might it be possible, then, to use twistor space as a basis for the non-local description underlying both bulk and boundary in AdS/CFT?
In the present paper, we answer this question in the affirmative, in the context of one simple model – the duality [@Klebanov:2002ja] between type-A higher-spin gravity in $AdS_4$ and a free $U(N)$ vector model on its 3d boundary. Higher-spin gravity [@Vasiliev:1995dn; @Vasiliev:1999ba] is an interacting theory of infinitely many massless fields, in this case one for each integer spin. On the boundary, these fields are dual to an infinite tower of conserved currents in the free CFT. The simplicity of this holographic model stems from its infinite-dimensional higher-spin symmetry – similar in some ways to supersymmetry, but stronger. We must note that, in this simple version, higher-spin gravity is highly unrealistic: while it does contain a massless spin-2 “graviton”, its interactions are nothing like those of GR, and in fact appear to be non-local at the cosmological scale. In this sense, we are dealing with a toy model. On the other hand, higher-spin gravity has the virtue of being formulated in four bulk dimensions, and is easily compatible with a positive cosmological constant.
A crucial simplifying feature of our higher-spin model is that it allows us to deal exclusively with free theories. In the bulk, we consider the linearized version of higher-spin gravity, i.e. free massless fields of all spins, which can be mapped into twistor space via the Penrose transform. On the boundary, we have the free CFT, which we map into twistor space using a novel “holographic dual” of the Penrose transform. This boundary version of the transform is more powerful than its bulk counterpart, since the correlators of the free CFT encode not only the linearized bulk theory, but also the bulk interactions. Thus, we’re essentially using the boundary CFT to solve the bulk theory, using twistor space as a common language between the two.
Note that twistor theory is a dimension-specific tool: it was originally constructed for massless 4d theories, subject either to conformal 4d symmetry (e.g. Yang-Mills) or to 4d isometries (e.g. GR or higher-spin gravity). On the other hand, AdS/CFT exploits the relation between conformal symmetry in $d$ dimensions and isometries in $d+1$ dimensions. Thus, the intersection between twistor theory and holography will naturally take place in either [AdS~5~/CFT~4~]{} or [AdS~4~/CFT~3~]{}. The [AdS~5~/CFT~4~]{} case was discussed in [@Adamo:2016rtr], and has the promise of general applicability: since the 4d boundary theory is conformal, one can always think of it as “massless”. In contrast, in the [AdS~4~/CFT~3~]{} case considered in this paper, we expect that twistor methods will be relevant only in the special setup of higher-spin theory, since it’s only there that the 4d bulk fields are all massless.
The rest of the paper is structured as follows. In section \[sec:summary\], we summarize the main results, with only a cursory explanation of the notations. Section \[sec:geometry\] is a geometric introduction to twistor space and its relation to bulk and boundary spinor spaces. Our geometry is carried out in 5d flat spacetime, within which both bulk and boundary are embedded. In section \[sec:algebra\], we introduce the higher-spin algebra, including structures that arise when focusing on a bulk or boundary point. In section \[sec:linear\_HS\], we formulate the linearized bulk theory and the Penrose transform. In section \[sec:spacetime\_subgroup\], we resume our discussion of higher-spin algebra, focusing on the representation of ordinary spacetime symmetries within the higher-spin adjoint and fundamental. This will lead us to a geometric viewpoint on the Penrose transform, which in turn will suggest its boundary dual. In section \[sec:CFT\], we discuss the boundary CFT in a bilocal language, and present the holographic dual of the Penrose transform. In section \[sec:holography\], we establish the holographic relationship between the bulk and boundary pictures, by calculating expectation values of local boundary currents. An analogous matching for the local field strengths of boundary sources is left for later work. Section \[sec:discuss\] is devoted to discussion and outlook.
Throughout the paper, we consider for simplicity Euclidean spacetime, i.e. the bulk is Euclidean Anti de Sitter space ($EAdS_4$). However, as discussed in section \[sec:discuss\], we envision an eventual application to Lorentzian de Sitter ($dS_4$).
Summary of results {#sec:summary}
==================
Penrose transform
-----------------
In some ways, higher-spin gravity is the most natural application of twistor theory, more so than Yang-Mills or General Relativity. In Yang-Mills and GR, twistors serve “merely” as the spinor representation of isometries or conformal transformations in 4d spacetime. In higher-spin theory, we utilize a greater power of these objects, using them to generate an infinite-dimensional extension of spacetime symmetries – the higher-spin (HS) group. The role of twistors in higher-spin algebra is identical to the role of vectors in Clifford algebra: $$\begin{aligned}
\text{Clifford algebra:}\quad \left\{\gamma_\mu,\gamma_\nu \right\} = -2\eta_{\mu\nu} \ ; \quad
\text{Higher-spin algebra:}\quad \left[Y_a,Y_b \right]_\star = 2iI_{ab} \ , \label{eq:Clifford_HS}\end{aligned}$$ where $Y_a$ are twistor coordinates, and $I_{ab}$ is the twistor metric. In both cases , the ordinary action of spacetime symmetries (realized as rotations in a higher-dimensional flat space) is implemented by the algebra’s adjoint representation, i.e. by multiplication on both sides. This should raise a curiosity about the *fundamental* representation: what if we multiply by the group element on one side only? In the case of Clifford algebra, this leads one to discover spinors. In the case of higher-spin algebra, it leads to the Penrose transform! Specifically, *the Penrose transform is a CPT reflection in the fundamental representation of the higher-spin group*. In other words, *the Penrose transform is a square root of CPT*: $$\begin{aligned}
\delta_x(Y)\star F(Y)\star\delta_x(Y) &= F(\text{CPT}\text{ of }Y\text{ around origin }x) \ ; \label{eq:CPT} \\
\pm F(Y)\star i\delta_x(Y) &\equiv C(x;Y) = \text{Penrose transform of }F(Y)\text{ at the point }x \ . \label{eq:sqrt_CPT}\end{aligned}$$ Here, $F(Y)$ is a spacetime-independent twistor function, $x$ is a bulk point, $C(x;Y)$ is a master field encoding a solution to the free massless field equations, and $\delta_x(Y)$ is a certain $x$-dependent delta function in twistor space. One may think of $\delta_x(Y)$ as a “twistor-bulk propagator”. The factor of $\pm i$ in is for later convenience.
We should point out that the statement is both old and new. On one hand, it was always clear that the twistor formalism of higher-spin theory is closely related to the Penrose transform (for a relatively recent treatment, see [@Gelfond:2008td]). Also, right-multiplication by a delta-function as in has long been recognized [@Didenko:2009td; @Didenko:2012tv; @Iazeolla:2011cb; @Iazeolla:2012nf; @Iazeolla:2017vng] as an important operation, relating the adjoint and “twisted adjoint” representations of higher-spin algebra, and allowing the construction of higher-spin invariants, as well as some explicit solutions to the Vasiliev equations. However, to our knowledge, it was never quite spelled out that this operation literally *is* the Penrose transform, i.e. that it relates free massless fields to *spacetime-independent* twistor functions. The reason for this is that the standard formulation of higher-spin theory works with “twistors” made up of spinors within a local orthonormal frame on a featureless base manifold. In such a framework, spacetime-independent twistor functions simply don’t arise as a natural object.
In contrast, in this paper, we work with global, spacetime-independent, Penrose-style twistors, associated with a background AdS~4~ spacetime. Specifically, our approach to higher-spin theory is a linearized version of the reformulation [@Neiman:2015wma] of the *full non-linear* Vasiliev equations on a fixed AdS~4~ background. At the linearized level, the existence and utility of such a formulation is not surprising. At the non-linear level, the reformulation [@Neiman:2015wma] is a less trivial matter, as it manages to avoid complicating the field equations, and retains the full higher-spin gauge symmetry. This is possible in higher-spin theory (as opposed to GR), because spacetime translations are contained in the local gauge group along with rotations, *independently from diffeomorphisms*. This in turn is related to the unfolded language of higher-spin theory, which bundles the fields’ spacetime derivatives together with the fields themselves.
Holographic dual of the Penrose transform
-----------------------------------------
Coming back to eqs. -, the next question is: can we find a context in which the $\sqrt{\text{CPT}}$ nature of the Penrose transform becomes manifest? It turns out that the answer is yes, and that it is intimately related to another “square root” relation – the fact that fundamental higher-spin fields in the bulk are dual to *quadratic* operators in the boundary CFT. In fact, the free vector model on the boundary is best expressed in a *bilocal* language, in which the relatively complicated local operators $\phi(\ell)\overset{\leftrightarrow}{\nabla}\dots\overset{\leftrightarrow}{\nabla}\bar\phi(\ell)$ are replaced by the simple product $\phi(\ell)\bar\phi(\ell')$, where $\ell,\ell'$ are boundary points. Consider, then, a boundary-bilocal object in the higher-spin algebra – a “twistor-boundary-boundary propagator”: $$\begin{aligned}
K(\ell,\ell';Y) = \frac{\sqrt{-2\ell\cdot\ell'}}{4\pi}\,\delta_{\ell}(Y)\star\delta_\ell'(Y) \ . \label{eq:K_summary}\end{aligned}$$ On this object, it turns out that the Penrose transform acts explicitly as a “square root” of CPT, by applying CPT to *one of the two* boundary points: $$\begin{aligned}
\begin{split}
i\delta_x(Y)\star K(\ell,\ell';Y) &= \pm K(\text{CPT}\text{ of }\ell\text{ around origin }x\ ,\ \ell'\ ;\ Y) \ ; \\
K(\ell,\ell';Y)\star i\delta_x(Y) &= \pm K(\ell\ ,\ \text{CPT}\text{ of }\ell'\text{ around origin }x\ ;\ Y) \ .
\end{split} \label{eq:K_sqrt_CPT}\end{aligned}$$ This property applies not only to CPT reflections, but to all of $SO(1,4)$, since the latter can be constructed (in (A)dS, but not in flat spacetime!) by combining CPT reflections around different origins. Thus, while $SO(1,4)$ is manifestly realized on arbitrary functions $f(Y)$ in the *adjoint* representation of the HS algebra, it is also manifestly realized in the *fundamental* representation when acting on $K(\ell,\ell';Y)$, by transforming one of the two boundary points $\ell,\ell'$. In particular, for the infinitesimal $SO(1,4)$ generators $M_{\mu\nu} = (-i/8)Y\gamma_{\mu\nu}Y$, we have: $$\begin{aligned}
\begin{split}
M_{\mu\nu}\star K(\ell,\ell';Y) &= \ell_\mu\frac{{\partial}K}{{\partial}\ell^\nu} - \ell_\nu\frac{{\partial}K}{{\partial}\ell^\mu} \ ; \\
-K(\ell,\ell';Y)\star M_{\mu\nu} &= \ell'_\mu\frac{{\partial}K}{{\partial}\ell'^\nu} - \ell'_\nu\frac{{\partial}K}{{\partial}\ell'^\mu} \ .
\end{split} \label{eq:K_sqrt_infinitesimal_summary}\end{aligned}$$ The $\sqrt{-\ell\cdot\ell'}$ prefactor in is necessary for eqs. - to hold, and it gives $K(\ell,\ell';Y)$ the appropriate conformal weight for a two-point function of massless scalars on the boundary. The numerical factor in is irrelevant to eqs. -, but is necessary for the CFT results below. As we will see, the sign ambiguities in are inherent to the HS algebra.
Moving on now from geometry to physics, our main result is that while $\delta_x(Y)$ solves the linearized bulk theory, $K(\ell,\ell';Y)$ solves the boundary CFT! Specifically, we begin with the CFT action with $U(N)$ singlet, single-trace sources, written in the spirit of [@Das:2003vw] in a bilocal form: $$\begin{aligned}
S_{\text{CFT}}[\Pi(\ell',\ell)] = -\int d^3\ell\,\bar\phi_I\Box\phi^I - \int d^3\ell' d^3\ell\,\bar\phi_I(\ell')\Pi(\ell',\ell)\phi^I(\ell) \ . \label{eq:S_summary}\end{aligned}$$ We then define a “holographic dual of the Penrose transform”, which packages the sources $\Pi(\ell,\ell')$ into a twistor function $F(Y)$: $$\begin{aligned}
F(Y) &= \int d^3\ell\,d^3\ell'K(\ell,\ell';Y)\,\Pi(\ell',\ell) \ . \label{eq:bdry_transform_summary}\end{aligned}$$ This allows us to write the partition function in the manifestly higher-spin-invariant form: $$\begin{aligned}
Z_{\text{CFT}}[F(Y)] \sim \exp\left(-\frac{N}{4}{\operatorname{tr}}_\star\ln_\star[1+F(Y)]\right) \equiv \left(\textstyle\det_\star[1+F(Y)]\right)^{-N/4} \ , \label{eq:Z_summary}\end{aligned}$$ where “${\operatorname{tr}}_\star$” stands for the HS-invariant trace operation ${\operatorname{tr}}_\star F(Y) = F(0)$. From , we can extract the expectation value of the bilocal operator $\phi^I(\ell)\bar\phi_I(\ell')$ in the presence of sources: $$\begin{aligned}
\left<\phi^I(\ell)\bar\phi_I(\ell')\right> = \frac{N}{4}{\operatorname{tr}}_\star\left(K(\ell',\ell;Y)\star[-1 + F(Y) + \dots] \right) \ , \label{eq:phi_phi_summary}\end{aligned}$$ where the dots indicate higher orders in the source $F(Y)$. This twistor formulation of the CFT makes global HS symmetry manifest, while doing away with the gauge redundancy of the sources $\Pi(\ell',\ell)$.
Crucially, we will see that, up to some subtleties involving discrete symmetries, the twistor functions $F(Y)$ in the bulk and boundary pictures can be identified with each other. Specifically, we will show that, away from sources, the asymptotic boundary data of the linearized bulk solution reproduces the linearized expectation values of the CFT operators, once the latter are translated into local currents. Thus, the 2-point correlators (more precisely, the 2-bilocal correlators) of the partition function are directly associated with the linearized bulk solution. The higher-point functions in can then be interpreted as encoding the effects of bulk interactions.
We note that the relation between conformal 3d fields $\phi^I(\ell),\bar\phi_I(\ell')$ and the fundamental HS representation was realized from different points of view in [@Shaynkman:2001ip; @Iazeolla:2008ix]. The partition function in a form similar to was obtained previously in [@Gelfond:2013xt]. On the CFT side, the main difference between our approach and that of [@Gelfond:2013xt] is that the latter operates directly with the current operators, while we are making contact with the fundamental fields $\phi^I(\ell),\bar\phi_I(\ell')$, i.e. with the underlying local structure of the boundary theory.
Avoiding contact terms {#sec:summary:contact}
----------------------
In the HS holography literature, when one calculates the correlation functions $\langle j(\ell_1)\dots j(\ell_n)\rangle$ of local CFT operators, the calculation is usually restricted to separated points, i.e. $\ell_1,\dots,\ell_n$ are all taken to be distinct. By themselves, these are not enough to capture the value of $Z_{CFT}$ for a general finite configuration of sources. Indeed, to calculate such values, we would need integrals of the form $d^3\ell_1\dots d^3\ell_n$, where some of the points $\ell_1,\dots,\ell_n$ may coincide, though only on lower-dimensional submanifolds of the integration domain. Thus, the full partition function at the single-trace level requires also some knowledge of the correlators’ behavior at coincident points. This extra requirement is similar to, but weaker than, a knowledge of the *multi-trace* correlators: the latter are equivalent to simply making single-trace insertion points coincide, as opposed to the coincidence appearing as a lower-dimensional possibility in a larger integral. This distinction is a consequence of the simplicity of our particular CFT: if the source-free action contained any multi-trace couplings, we would have no choice but to always take multi-trace insertions into account.
To be more specific, there are two kinds of problems that we can encounter on coincident-point submanifolds. First, the separated-point correlator may not be integrable through these submanifolds. Second, the answer may violate gauge invariance, or, equivalently, current conservation. Fixing such problems requires regularization, as well as adding contact terms both in the action and in the definition of the currents. For example, when a charge current $\mathbf{j} = i\phi\overset{\leftrightarrow}{\mathbf{\nabla}}\bar\phi$ (suppressing $U(N)$ indices) is coupled to a gauge potential $\mathbf{A}$, the current’s expectation value is divergent at points where $\mathbf{A}$ is nonzero. Specifically, the relevant 2-point function has a non-integrable $\sim 1/r^4$ short-distance singularity, which becomes $\sim 1/r^{2s+2}$ in the spin-$s$ case. This divergence is directly related to the fact that the true conserved current contains an extra contact term $\mathbf{A}\phi\bar\phi$; in other words, the derivative in the definition of $\mathbf{j}$ must be gauge-covariantized. The $\mathbf{A}\phi\bar\phi$ term has its own short-distance singularity, which cancels the previous one and leaves us with a finite & conserved current.
Most of these issues are resolved automatically by switching to the bilocal language . There, we only ever find-short distance singularities of the form $\sim 1/r$ (the fundamental propagator of the $\phi$ fields), which is integrable, and therefore doesn’t require regularization. From the local point of view, the bilocal language can be viewed as an extreme form of point-split regularizartion. Conversely, from the bilocal point of view, the local language corresponds to a singular choice of gauge, where the source $\Pi(\ell',\ell)$ is distributional with support on $\ell=\ell'$.
That being said, the bilocal language does not solve everything. In particular, given some values of the bilocal source, we may still wish to know the expectation value of a local current. It turns out that upon naive calculation, the resulting current isn’t locally conserved: the two points in the bilocal $\Pi(\ell',\ell)$ act as a source/sink pair (as one can see by examining eq. below). Thus, the need for contact terms arises whenever we’re interested in a local expectation value, even if the sources are bilocal.
Finally, we come to the fully nonlocal twistor formulation of the partition function. Here, we find that the need for contact terms seems to disappear entirely. This should not be too surprising, if we put two facts together:
1. The points at which the (local or bilocal) sources are non-vanishing are gauge-dependent. In particular, at any given point, one can gauge away the value of a spin-$s$ gauge potential and its first $2s-2$ derivatives.
2. The twistor language does away with both locality and gauge redundancy.
Specifically, as we’ll discuss in section \[sec:holography:general\_currents\], the currents that can be derived from a twistorial expression of the form are always automatically conserved.
Spacetime and twistor geometry {#sec:geometry}
==============================
In this section, we present some elements of geometry in the $EAdS_4$ bulk, its 3d boundary, and twistor space. Throughout, we view the bulk and boundary as embedded in a flat 5d spacetime. Similar embedding-space approaches to higher-spin theory and holography may be found e.g. in [@Bekaert:2012vt; @Didenko:2012vh], in the context of general dimensions. Those approaches employ a tensor formalism, while our emphasis will be on spinors and twistors. In particular, this section will focus on the embedding of bulk and boundary spinor spaces within the global twistor space.
Spacetime {#sec:geometry:spacetime}
---------
### Bulk and boundary
We define $EAdS_4$ as the hyperboloid of future-pointing unit timelike vectors in flat 5d Minkowski space ${\mathbb{R}}^{1,4}$: $$\begin{aligned}
EAdS_4 = \left\{x^\mu\in{\mathbb{R}}^{1,4}\, |\, x_\mu x^\mu = -1, \ x^0 > 0 \right\} \ . \label{eq:EAdS}\end{aligned}$$ The metric $\eta_{\mu\nu}$ of ${\mathbb{R}}^{1,4}$ has signature $(-,+,+,+,+)$. We use indices $(\mu,\nu,\dots)$ for ${\mathbb{R}}^{1,4}$ vectors, which we raise and lower using $\eta_{\mu\nu}$. The isometry group of $EAdS_4$ is just the rotation group $O(1,4)$ in the 5d spacetime (more precisely – the component $O^\uparrow(1,4)$ that preserves time orientation).
The tangent space at a point $x\in EAdS_4$ consists simply of the vectors $v^\mu$ that satisfy $x\cdot v = 0$. The $EAdS_4$ metric at $x$ can be identified with the projector onto this tangent space: $$\begin{aligned}
q_{\mu\nu}(x) = \eta_{\mu\nu} + x_\mu x_\nu \ .\end{aligned}$$ The covariant derivative of vectors in $EAdS_4$ can be defined as the flat ${\mathbb{R}}^{1,4}$ derivative, followed by a projection back onto the hyperboloid: $$\begin{aligned}
\nabla_\mu v_\nu = q_\mu^\rho(x)\, q_\nu^\sigma(x)\, {\partial}_\rho v_\sigma \ .\end{aligned}$$
In addition to the ambient ${\mathbb{R}}^{1,4}$ picture, it is sometimes useful to use an intrinsic coordinate system for $EAdS_4$. Of particular interest are the Poincare coordinates: $$\begin{aligned}
x^\mu(z,\mathbf{r}) = \frac{1}{z}\left(\frac{1+z^2+r^2}{2}, \mathbf{r}, \frac{1-z^2-r^2}{2} \right) \ , \label{eq:Poincare}\end{aligned}$$ where $\mathbf{r}$ is a flat 3d coordinate, and the metric reads: $$\begin{aligned}
dx_\mu dx^\mu = \frac{dz^2 + d\mathbf{r}\cdot d\mathbf{r}}{z^2} \ .\end{aligned}$$
The asymptotic boundary of $EAdS_4$ is the conformal 3-sphere of future-pointing null directions in ${\mathbb{R}}^{1,4}$. Thus, we represent boundary points by null vectors $\ell^\mu$, with the equivalence $\ell^\mu\cong\lambda\ell^\mu$. The $O(1,4)$ symmetry group then becomes the conformal symmetry of the boundary. The limit where a bulk point $x$ approaches the boundary can be represented as an extreme boost in ${\mathbb{R}}^{1,4}$, where the unit vector $x^\mu$ approaches a null direction $\ell^\mu$ as: $$\begin{aligned}
x^\mu\rightarrow \ell^\mu/z \ , \quad z\rightarrow 0 \ . \label{eq:limit}\end{aligned}$$
One can fix the conformal frame on the boundary by choosing a section of the ${\mathbb{R}}^{1,4}$ lightcone. Perhaps the most convenient is the flat section: $$\begin{aligned}
\ell^\mu(\mathbf{r}) = \left(\frac{1+r^2}{2}, \mathbf{r}, \frac{1-r^2}{2} \right) \ , \label{eq:flat}\end{aligned}$$ which can be viewed as the bulk-to-boundary limit of the Poincare coordinates . The section can be defined as the intersection of the lightcone $\ell\cdot\ell = 0$ with the null hyperplane: $$\begin{aligned}
\ell\cdot n = -\frac{1}{2} \ ; \quad n^\mu = \left(\frac{1}{2},\mathbf{0},-\frac{1}{2}\right) \label{eq:flat_hyperplane}\end{aligned}$$
The metric on the flat section is simply $d\ell_\mu d\ell^\mu = d\mathbf{r}\cdot d\mathbf{r}$. In particular, the ${\mathbb{R}}^{1,4}$ scalar product $\ell\cdot\ell'$ is directly related to the 3d Euclidean distance in the frame : $$\begin{aligned}
\ell\cdot\ell' = -\frac{1}{2}\left(\mathbf{r} - \mathbf{r'}\right)^2 \ . \label{eq:flat_distance}\end{aligned}$$
### Massless scalars and conserved currents on the boundary {#sec:geometry:spacetime:currents}
Boundary quantities with conformal weight $\Delta$ are represented by functions $f(\ell)$ on the lightcone, subject to the homogeneity condition $f(\lambda\ell) = \lambda^{-\Delta}f(\ell)$, or, equivalently: $$\begin{aligned}
\ell^\mu\frac{{\partial}}{{\partial}\ell^\mu}f(\ell) = -\Delta f(\ell) \ . \label{eq:homogeneity}\end{aligned}$$ In particular, a free massless scalar on the 3d boundary has conformal weight $\Delta=1/2$. Scalars with this weight admit a conformally covariant Laplacian $\Box$, which in the ${\mathbb{R}}^{1,4}$ language is given simply by [@Eastwood:2002su]: $$\begin{aligned}
\Box\phi(\ell) = \frac{{\partial}\phi(\ell)}{{\partial}\ell_\mu{\partial}\ell^\mu} \ . \label{eq:Laplacian}\end{aligned}$$ Here, it’s assumed that we’ve extended the function $\phi(\ell)$ into non-null values of $\ell^\mu$, where it remains subject to the homogeneity condition . The Laplacian does not otherwise depend on this artificial extension of $\phi(\ell)$ into $\ell\cdot\ell\neq 0$, as it vanishes for any function that is zero at $\ell\cdot\ell = 0$: $$\begin{aligned}
\Box\left((\ell\cdot\ell) f(\ell)\right) = 0 \ \ \text{at} \ \ \ell\cdot\ell = 0 \ , \ \ \text{for any } f(\ell) \text{ with weight } \Delta = 5/2 \ .\end{aligned}$$ One can verify explicitly that eq. defines the usual 3d Laplacian on the flat section .
Boundary currents of various spin and their conservation laws are also easy to describe in the $O(1,4)$-covariant framework. A spin-$s$ current is represented by a totally symmetric and traceless tensor $j^{\mu_1\dots\mu_s}$. To bring the tensor’s indices from ${\mathbb{R}}^{1,4}$ down to the boundary’s 3d tangent space, we impose a constraint and an equivalence relation: $$\begin{aligned}
\ell_{\mu_1}j^{\mu_1\mu_2\dots\mu_s} &= 0 \ ; \label{eq:current_tensor_constraint} \\
j^{\mu_1\mu_2\dots\mu_s} &\cong j^{\mu_1\mu_2\dots\mu_s} + \ell^{(\mu_1}\theta^{\mu_2\dots\mu_s)} \ , \label{eq:current_tensor_equivalence}\end{aligned}$$ where $\theta^{\mu_1\dots\mu_{s-1}}$ is a totally symmetric and traceless tensor satisfying $\ell_{\mu_1}\theta^{\mu_1\mu_2\dots\mu_{s-1}} = 0$. The presence of tensor indices makes the notion of conformal weight a bit subtle. In this paper, our tensor indices lie in the ${\mathbb{R}}^{1,4}$ ambient space, and we define the conformal weight $\Delta$ via $j^{\mu_1\dots\mu_s}(\lambda\ell) = \lambda^{-\Delta}j^{\mu_1\dots\mu_s}(\ell)$. For the corresponding tensor with indices in the boundary’s tangent or cotangent bundle, this implies a conformal weight of $\Delta+s$ or $\Delta-s$, respectively. For a spin-$s$ tensor $j^{\mu_1\dots\mu_s}$ with the particular weight $\Delta = s+1$, one can define a conformally covariant divergence: $$\begin{aligned}
(\operatorname{div} j)^{\mu_1\dots\mu_{s-1}} = \frac{{\partial}j^{\mu_1\dots\mu_{s-1}\mu_s}}{{\partial}\ell^{\mu_s}} \ , \label{eq:tensor_div}\end{aligned}$$ where we again extend $j^{\mu_1\dots\mu_s}(\ell)$ away from $\ell\cdot\ell = 0$, while maintaining the constraint and the homogeneity condition $j^{\mu_1\dots\mu_s}(\lambda\ell) = \lambda^{-s-1}j^{\mu_1\dots\mu_s}(\ell)$. To see that the result doesn’t otherwise depend on this artificial extension, we note that eq. can be rewritten in terms of the derivative $\ell\wedge({\partial}/{\partial}\ell)$, which only acts tangentially to the $\ell\cdot\ell = 0$ lightcone: $$\begin{aligned}
\left(\ell_\nu\frac{{\partial}}{{\partial}\ell^{\mu_s}} - \ell_{\mu_s}\frac{{\partial}}{{\partial}\ell^\nu} \right) j^{\mu_1\dots\mu_{s-1}\mu_s}
= (\operatorname{div} j)^{\mu_1\dots\mu_{s-1}}\ell_\nu + j^{\mu_1\dots\mu_{s-1}}{}_\nu \ .\end{aligned}$$ It remains to verify that the formula is consistent with the equivalence relation . It is here that the conformal weight $\Delta = s+1$ will be important. One must be careful to extend eq. away from $\ell\cdot\ell = 0$ in a way that doesn’t conflict with the constraint . To do this, we introduce a fixed null vector $n\neq\ell$, and replace $\ell^\mu$ in with: $$\begin{aligned}
\tilde\ell^\mu = \ell^\mu - \frac{\ell\cdot\ell}{\ell\cdot n}\,n^\mu \ .\end{aligned}$$ One then finds that the divergence is indeed consistent with , via: $$\begin{aligned}
\delta j^{\mu_1\dots\mu_s} = \tilde\ell^{(\mu_1}\theta^{\mu_2\dots\mu_s)} \quad \Longrightarrow \quad
\delta(\operatorname{div} j)^{\mu_1\dots\mu_{s-1}} = \frac{s-1}{s}\,\ell^{(\mu_1}\frac{{\partial}}{{\partial}\ell^\nu}\theta^{\mu_2\dots\mu_{s-1})\nu} \ .\end{aligned}$$
Twistors {#section:geometry:twistors}
--------
Here, we introduce spinors and twistors in $EAdS_4$ from the viewpoint described in [@Neiman:2013hca]. Our focus here is on algebraic properties; see [@Neiman:2013hca] for a more detailed geometric perspective.
The twistors of $EAdS_4$ are just the 4-component Dirac spinors of the isometry group $SO(1,4)$. We use indices $(a,b,\dots)$ for twistors. The twistor space is equipped with a symplectic metric $I_{ab}$, which is used to raise and lower indices via: $$\begin{aligned}
U_a = I_{ab}U^b \ ; \quad U^a = U_b I^{ba} \ ; \quad I_{ac}I^{bc} = \delta_a^b \ .\end{aligned}$$ Tensor and twistor indices are related through the gamma matrices $(\gamma_\mu)^a{}_b$, which satisfy the Clifford algebra $\{\gamma_\mu,\gamma_\nu\} = -2\eta_{\mu\nu}$. These 4+1d gamma matrices can be realized as the usual 3+1d ones, with the addition of $\gamma_5$ (in our notation, $\gamma_4$) for the fifth direction in ${\mathbb{R}}^{1,4}$. In $2\times 2$ block notation, the matrices $I_{ab}$ and $(\gamma_\mu)^a{}_b$ can be represented e.g. as: $$\begin{aligned}
\begin{split}
I_{ab} &= \begin{pmatrix} 0 & -i\sigma_2 \\ -i\sigma_2 & 0 \end{pmatrix} \ ; \\
(\gamma^0)^a{}_b &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \ ; \quad
(\gamma^4)^a{}_b = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \ ; \quad
(\gamma^k)^a{}_b = \begin{pmatrix} -i\sigma^k & 0 \\ 0 & i\sigma^k \end{pmatrix} \ , \label{eq:gamma_null}
\end{split}\end{aligned}$$ where $\sigma^k$ with $k=1,2,3$ are the Pauli matrices. The representation is geared to simplify the “null” matrices $\gamma_0\pm\gamma_4$. An alternative representation, which simplifies the “timelike” matrix $\gamma_0$, reads: $$\begin{aligned}
\begin{split}
I_{ab} &= \begin{pmatrix} -i\sigma_2 & 0 \\ 0 & i\sigma_2 \end{pmatrix} \ ; \\
(\gamma^0)^a{}_b &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \ ; \quad
(\gamma^4)^a{}_b = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \ ; \quad
(\gamma^k)^a{}_b = \begin{pmatrix} 0 & i\sigma^k \\ i\sigma^k & 0 \end{pmatrix} \ . \label{eq:gamma_timelike}
\end{split}\end{aligned}$$ The matrices $\gamma^\mu_{ab}$ are antisymmetric and traceless in their twistor indices. We define the antisymmetric product of gamma matrices as: $$\begin{aligned}
\gamma^{\mu\nu}_{ab} \equiv \gamma^{[\mu}_{ac}\gamma^{\nu]c}{}_b \ .\end{aligned}$$ The $\gamma^{\mu\nu}_{ab}$ are symmetric in their twistor indices. We use the matrices $\gamma_\mu^{ab}$ to convert between 4+1d vectors and traceless bitwistors as: $$\begin{aligned}
\xi^{ab} = \gamma_\mu^{ab}\xi^\mu \ ; \quad \xi^\mu = -\frac{1}{4}\gamma^\mu_{ab}\xi^{ab} \ . \label{eq:conversion_5d}\end{aligned}$$ Similarly, $\gamma_{\mu\nu}^{ab}$ can be used to convert between bivectors and symmetric twistor matrices: $$\begin{aligned}
f^{ab} = \frac{1}{2}\gamma_{\mu\nu}^{ab}f^{\mu\nu} \ ; \quad f^{\mu\nu} = \frac{1}{4}\gamma^{\mu\nu}_{ab} f^{ab} \ . \label{eq:conversion_bivectors}\end{aligned}$$ Useful identities include: $$\begin{aligned}
\begin{split}
&\gamma^\mu_{ab}\gamma_\nu^{ab} = -4\delta^\mu_\nu \ ; \quad \gamma^{\mu\nu}_{ab}\gamma_{\rho\sigma}^{ab} = 8\delta^{[\mu}_{[\rho}\delta^{\nu]}_{\sigma]} \ ; \quad
\gamma_\mu^{ab}\gamma^\mu_{cd} = I^{ab}I_{cd} - 4\delta^{[a}_{[c} \delta^{b]}_{d]} \ ; \quad \gamma_{\mu\nu}^{ab}\gamma^{\mu\nu}_{cd} = 8\delta^{(a}_{(c} \delta^{b)}_{d)} \ ; \\
&\epsilon^{abcd} = 3I^{[ab}I^{cd]} \ ; \quad \epsilon^{abcd}I_{cd} = 2I^{ab} \ ; \quad \epsilon^{abcd}\gamma^\mu_{cd} = -2\gamma^{\mu ab} \ ; \quad
\gamma_\mu^{[ab}\gamma_\nu^{cd]} = \frac{1}{3}\eta_{\mu\nu}\epsilon^{abcd} \ .
\end{split} \label{eq:twistor_identities}\end{aligned}$$ Here, $\epsilon^{abcd}$ is the totally antisymmetric symbol with inverse $\epsilon_{abcd} = 3I_{[ab}I_{cd]}$, such that $\epsilon_{abcd}\epsilon^{abcd} = 4!$. The metric $I_{ab}$ has unit determinant with respect to $\epsilon^{abcd}$. We use $\epsilon^{abcd}$ to define a measure on twistor space via: $$\begin{aligned}
d^4U \equiv \frac{\epsilon_{abcd}}{4!(2\pi)^2}\,dU^a dU^b dU^c dU^d \ . \label{eq:twistor_measure}\end{aligned}$$ Here and elsewhere, we include $2\pi$ factors in the measure, in such a way that they will not appear explicitly in our Fourier and Gaussian integrals. Note that our choice for the overall sign of $\epsilon^{abcd}$ is the opposite from that in [@Neiman:2013hca], and indeed, in the basis , we get $\epsilon^{1234} = -1$. This choice will end up being more convenient for relations such as .
### Index-free notation {#sec:geometry:twistors:index_free}
In order to streamline the formulas below, we now introduce some index-free notation for products in ${\mathbb{R}}^{1,4}$ and in twistor space. $x\cdot x$ will represent the scalar product $x_\mu x^\mu$ in ${\mathbb{R}}^{1,4}$. The twistor matrices $\delta_a^b$ and $(\gamma_\mu)^a{}_b$ will be written in index-free notation as $1$ and $\gamma_\mu$. Combined with the index conversion , this means that the matrix $(x^\mu\gamma_\mu)^a{}_b$ for a vector $x^\mu\in{\mathbb{R}}^{1,4}$ will be written simply as $x$ (this is just the Feynman slash convention, without the slash). Products in the index-free notation imply bottom-to-top index contractions. So, e.g. for two twistors $U^a,V^a$ and two vectors $\ell^\mu,x^\mu$, we have: $$\begin{aligned}
\begin{split}
&UV \equiv U_a V^a = -I_{ab}U^a V^b \ ; \quad \ell\cdot x \equiv \ell_\mu x^\mu = -\frac{1}{4}{\operatorname{tr}}(\ell x) \ ; \\
&(xU)^a \equiv x^a{}_b U^b \ ; \quad U\ell xU \equiv U_a\ell^a{}_b x^b{}_c U^c = -\ell_\mu x_\nu\gamma^{\mu\nu}_{ab} U^a U^b \ .
\end{split}\end{aligned}$$ A product $U\Gamma_1\dots\Gamma_n V$, where $U$ and $V$ are twistors and the matrices $\Gamma_1,\dots,\Gamma_n$ are either symmetric or antisymmetric, can be reversed as follows: $$\begin{aligned}
V\Gamma_n\dots\Gamma_1 U = (-1)^{n_\text{sym}+1}(U\Gamma_1\dots\Gamma_n V) \ , \label{eq:reverse}\end{aligned}$$ where $n_\text{sym}$ is the number of symmetric matrices among the $\Gamma_1,\dots,\Gamma_n$.
### Twistor integrals {#sec:geometry:twistors:integrals}
In calculations below, we will need to evaluate integrals over twistor space, as well as over various spinor subspaces. These integrals are somewhat delicate, because the relevant spaces are complex, and one has to worry about appropriate integration contours. To some extent, this is a result of our choice of signature: in Lorentzian $AdS_4$, the twistors and boundary spinors (but not the bulk spinors) have a natural real structure. However, this real structure doesn’t necessarily help, because the natural real contours may not be the ones along which the integrals converge. Luckily, the only integrals we will need explicitly are of delta functions and Gaussians. These can be defined by analytical continuation from appropriate real-line integrals.
The first integral formula that we’ll need is: $$\begin{aligned}
\int d^4U d^4V f(U)\, e^{iUV} = f(0) \ . \label{eq:delta_integral_raw}\end{aligned}$$ This can equivalently be written as: $$\begin{aligned}
\int d^4U\,\delta(U) f(U) = f(0) \ , \label{eq:delta_integral}\end{aligned}$$ where the twistor delta function is defined as: $$\begin{aligned}
\delta(U) = \int d^4V e^{iVU} \ . \label{eq:delta_U}\end{aligned}$$ The second twistor integral that we will use is the Gaussian: $$\begin{aligned}
\int d^4U\, e^{(UAU)/2} = \frac{\pm 1}{\sqrt{\det A}} \ ; \quad \det A = \frac{1}{8}\left({\operatorname{tr}}A^2\right)^2 - \frac{1}{4}{\operatorname{tr}}A^4 \ , \label{eq:Gaussian}\end{aligned}$$ where $A_{ab}$ is a symmetric twistor matrix, and we use its tracelessness for the last expression in . Note that the $2\pi$ factors are already taken care of by the definition of the measure.
The sign in is ambiguous due to the square root, and in general will depend on how exactly we analytically continue from the case of a real contour and real negative-definite $A_{ab}$. In fact, we’ll see that in the context of the HS symmetry group, this sign ambiguity is crucial, and cannot be globally fixed. Specifically, within the HS group, the subgroup $SO(1,4)$ of ordinary spacetime symmetries is represented by twistor Gaussians, and its topology is only consistent when the sign ambiguity is taken into account.
Finally, we note that the sign ambiguity in Gaussian integrals also reflects on the delta function . The integral in can be regularized and evaluated by inserting a broad Gaussian into the integrand. However, the result of this Gaussian integral is only defined up to sign. Therefore, while the integral involving $\delta(U)$ is well-defined, $\delta(U)$ itself is defined as a limit of ordinary functions only up to sign. An alternative way to see this is to define $\delta(U)$ as the limit of a series of ever-narrowing Gaussians, which are constrained to have a unit integral. Since these integrals are only defined up to sign, the same is true for the series that limits to $\delta(U)$.
Bulk spinors
------------
When we choose a point $x\in EAdS_4$, the Dirac representation of $SO(1,4)$ becomes identified with the Dirac representation of the rotation group $SO(4)$ at $x$. It then decomposes into left-handed and right-handed Weyl spinor representations, corresponding to $SO(4) = SO(3)_L\times SO(3)_R$. The decomposition is accomplished by a pair of projectors: $$\begin{aligned}
\begin{split}
P_L{}^a{}_b(x) &= \frac{1}{2}\left(\delta^a_b - x^\mu\gamma_\mu{}^a{}_b \right) = \frac{1}{2}\left(\delta^a_b - x^a{}_b \right) \ ; \\
P_R{}^a{}_b(x) &= \frac{1}{2}\left(\delta^a_b + x^\mu\gamma_\mu{}^a{}_b \right) = \frac{1}{2}\left(\delta^a_b + x^a{}_b \right) \ . \label{eq:projectors}
\end{split}\end{aligned}$$ These serve as an $x$-dependent version of the familiar chiral projectors in ${\mathbb{R}}^4$. We note that $P_L$ and $P_R$ get interchanged under the “antipodal map” $x^\mu\rightarrow-x^\mu$. In the Euclidean AdS context, this is a formal operation that takes us away from the hyperboloid and into its $x^0<0$ counterpart.
Given a twistor $U^a$, we denote its left-handed and right-handed components at $x$ as $u_{L/R}^a(x) = (P_{L/R}){}^a{}_b(x)U^b$. As in our treatment of tensors, it is possible to use the $(a,b,\dots)$ indices for both $SO(4,1)$ and $SO(4)$ Dirac spinors. The projectors $P^L_{ab}(x)$ and $P^R_{ab}(x)$ serve as the spinor metrics for the left-handed and right-handed Weyl spinor spaces. For a 2d spinor space, a symplectic metric also acts as a measure, i.e. we can define: $$\begin{aligned}
d^2u_L \equiv \frac{P^L_{ab}(x)}{2(2\pi)}\,dU^a dU^b \ ; \quad d^2u_R \equiv \frac{P^R_{ab}(x)}{2(2\pi)}\,dU^a dU^b \ . \label{eq:measure_chiral}\end{aligned}$$ Alternatively, the measures can be defined as the inverses of $P_L^{ab}$ and $P_R^{ab}$, as in: $$\begin{aligned}
\frac{du_L^a du_L^b}{2\pi} \equiv P_L^{ab}(x)\,d^2u_L \ ; \quad \frac{du_R^a du_R^b}{2\pi} \equiv P_R^{ab}(x)\,d^2u_R \ . \label{eq:measure_chiral_inverse}\end{aligned}$$ The two chiral spinor measures combine to form the twistor measure , via: $$\begin{aligned}
d^4U = d^2u_L d^2u_R \ . \label{eq:measure_product_bulk}\end{aligned}$$
The power of this formalism for describing spinors is that the twistors, i.e. the spinors of ${\mathbb{R}}^{1,4}$, are flat: we can transport them freely from one point in $EAdS_4$ to another. What changes from point to point is the twistor’s decomposition into left-handed and right-handed spinors. In particular, the covariant derivative for Weyl spinors in $EAdS_4$ can be constructed by embedding the spinor inside a twistor, taking the flat ${\mathbb{R}}^{1,4}$ derivative, and projecting back into the appropriate spinor space. For e.g. a left-handed spinor field $\psi_L^a(x)$, this can be written as: $$\begin{aligned}
\nabla_\mu\psi^a_L(x) = q_\mu^\nu(x) P_L{}^a{}_b(x)\,{\partial}_\nu\psi_L^b(x) \ . \label{eq:spinor_covariant}\end{aligned}$$ An important special case is the covariant derivative of the left-handed and right-handed components $y_L(x),y_R(x)$ of a *spacetime-independent* twistor $Y$: $$\begin{aligned}
\nabla_\mu\, y_L^a = -\frac{1}{2}(\gamma_\mu)^a{}_b\,y_R^b \ ; \quad \nabla_\mu\, y_R^a = \frac{1}{2}(\gamma_\mu)^a{}_b\,y_L^b \ .\end{aligned}$$ This is just Penrose’s twistor equation, in the presence of a cosmological constant.
A vector $\xi^\mu\in{\mathbb{R}}^{1,4}$, when evaluated at a point $x\in EAdS_4$, decomposes into an $O(4)$ scalar (the radial component, encoded by the scalar product $\xi\cdot x$) and an $O(4)$ vector (the tangential component, encoded by the vector $\xi^\mu + (\xi\cdot x)x^\mu$ or the bivector $\xi^{[\mu}x^{\nu]}$). For the twistor matrix $\xi = \xi^\mu\gamma_\mu$, this decomposition can be expressed in terms of chiral projections of the twistor indices: $$\begin{aligned}
O(4)\text{ scalar:} \quad &P_L\xi P_L = (\xi\cdot x)P_L \ ; \quad P_R\xi P_R = -(\xi\cdot x)P_R \ ; \\
O(4)\text{ vector:} \quad &P_L\xi P_R + P_R\xi P_L = \xi + (\xi\cdot x)x \ ; \quad P_L\xi P_R - P_R\xi P_L = \frac{1}{2}(\xi x - x\xi) \ . \label{eq:vector}\end{aligned}$$ In particlar, displacements $dx^\mu$ along the $EAdS_4$ hyperboloid have only mixed-chirality components, as in .
Boundary spinors {#sec:geometry:spinors_boundary}
----------------
At a boundary point $\ell^\mu$, the decomposition of twistor space is somewhat different. While the 4d bulk has two Weyl spinor spaces at each point, the 3d boundary has a single (2-component) Dirac spinor space. Let us now describe how this spinor space arises from the ${\mathbb{R}}^{1,4}$, twistorial perspective.
In the asymptotic limit , both the left-handed and right-handed projectors degenerate into multiples of $\ell^{ab}$: $$\begin{aligned}
P_L^{ab}(x)\rightarrow -\frac{1}{z}P^{ab}(\ell) \ ; \quad P_R^{ab}(x)\rightarrow \frac{1}{z}P^{ab}(\ell) \ , \label{eq:limit_spinor_spaces}\end{aligned}$$ where we’ve defined: $$\begin{aligned}
P^{ab}(\ell) \equiv \frac{1}{2}\ell^{ab} \ . \label{eq:P_ell}\end{aligned}$$ Thus, the two subspaces $P_L(x)$ and $P_R(x)$ degenerate into a single subspace $P(\ell)$, spanned by the bitwistor $P^{ab}(\ell) \sim \ell^{ab}$. Equivalently, $P(\ell)$ is the subspace annihilated by the matrix $\ell^a{}_b$: $$\begin{aligned}
u^a\in P(\ell) \quad \Longleftrightarrow \quad \ell^{[ab}u^{c]} = 0 \quad \Longleftrightarrow \quad \ell^a{}_b u^b = 0 \ .\end{aligned}$$ The subspace $P(\ell)$ can be identified as the spinor space on the 3d boundary. Though $P(\ell)$ is null under the twistor metric $I_{ab}$, one can use the inverse of the matrix to define a metric and a measure $d^2u$ on $P(\ell)$, in analogy with the bulk definition : $$\begin{aligned}
\frac{du^a du^b}{2\pi} \equiv P^{ab}(\ell)\, d^2u = \frac{1}{2}\ell^{ab} d^2u \ . \label{eq:measure}\end{aligned}$$ The measure $d^2u$ scales inversely with the null vector $\ell^\mu$, i.e. it has conformal weight $\Delta = 1$. We should therefore think of $P(\ell)$ as the space of boundary *co*spinors, i.e. the square roots of boundary covectors.
The space of *contravariant* boundary spinors, i.e. the square roots of boundary vectors, is the space $P^*(\ell)$ dual to $P(\ell)$ under the twistor metric. It is easy to see that this is the *quotient space* of twistors modulo terms in $P(\ell)$: $$\begin{aligned}
(u^*)^a \cong (u^*)^a + u^a \ , \quad u^a \in P(\ell) \ . \label{eq:dual_space}\end{aligned}$$ $P^*(\ell)$ can be equipped with a metric and measure inversely related to that of , i.e. given simply by the matrix : $$\begin{aligned}
d^2u^* \equiv \frac{P_{ab}(\ell)}{2(2\pi)}\,(du^*)^a (du^*)^b = \frac{\ell_{ab}}{8\pi}\,(du^*)^a (du^*)^b \ , \label{eq:dual_measure}\end{aligned}$$ with conformal weight $\Delta = -1$. Multiplication by the matrix defines a mapping between $P(\ell)$ and $P^*(\ell)$, via: $$\begin{aligned}
(u^*)^a \in P^*(\ell) \quad \longleftrightarrow \quad P^a{}_b(\ell) (u^*)^b = \frac{1}{2}\ell^a{}_b (u^*)^b \in P(\ell) \ . \label{eq:u_u*}\end{aligned}$$ This mapping is consistent with the measures ,. It can be viewed as the map between boundary spinors and cospinors via the spinor metric . It should be stressed that the spinor spaces $P(\ell)$ and $P^*(\ell)$ depend only on the *direction* of $\ell^\mu$, which corresponds to the choice of boundary point. However, the measures , and the mapping depend also on the *scaling* of $\ell^\mu$, which corresponds to a choice of conformal frame.
Note that for *bulk* spinors, there was no need for such subtleties. There, we have no arbitrary rescaling of the spinor metrics, and the chiral spinor spaces $P_L(x),P_R(x)$ are the same as their duals under the twistor metric $P_L^*(x),P_R^*(x)$. In particular, the measure can be viewed as the boundary limit of the bulk spinor measures .
At a bulk point $x$, an arbitrary twistor $U$ has a well-defined decomposition $U = u_L + u_R$. This is no longer true at a boundary point $\ell$: here, $U$ has a well-defined projection $u^*\in P^*(\ell)$, but its “$P(\ell) $ component” is ambiguous. However, one can *span* the twistor space by first choosing $u^*$, and then spanning the equivalence class by varying $u\in P(\ell)$. In this context, the two spinor measures , can be combined into the global twistor measure. From the identity $\epsilon_{abcd}\ell^{cd} = -2\ell_{ab}$, one can derive the explicit formula: $$\begin{aligned}
d^4U = -d^2u\,d^2u^* \ . \label{eq:measure_product_boundary}\end{aligned}$$
### Boundary currents in spinor form {#sec:geometry:spinors_boundary:currents}
The spinor language is especially well-suited for describing boundary currents of arbitrary spin and their conservation laws. A spin-$s$ boundary object (i.e. a rank-$s$ totally traceless and symmetric tensor) can be described by a totally symmetric rank-$2s$ spinor $j_\ell$ with indices in $P(\ell)$: $$\begin{aligned}
\ell^b{}_{a_1} j_\ell^{a_1 a_2\dots a_{2s}} = 0 \ ,\end{aligned}$$ or a totally symmetric rank-$2s$ spinor $j_*$ with indices in $P^*(\ell)$: $$\begin{aligned}
j_*^{a_1 a_2\dots a_{2s}} \cong j_*^{a_1 a_2\dots a_{2s}} + \ell^{(a_1}{}_c\,\lambda^{a_2\dots a_{2s})c}
\ \ \text{for any} \ \ \lambda^{a_2\dots a_{2s} c} \ . \label{eq:spinor_equivalence}\end{aligned}$$ These two representations are related by the mapping , i.e. by the spinor metric at $\ell$: $$\begin{aligned}
j_\ell^{a_1\dots a_{2s}} = \frac{1}{4^s}\,\ell^{a_1}{}_{b_1}\dots\ell^{a_{2s}}{}_{b_{2s}} j_*^{b_1\dots b_{2s}} \ . \label{eq:j_P_j*}\end{aligned}$$ Note that neither $j_\ell$ nor $j_*$ is the direct translation into twistor indices of the boundary tensor $j^{\mu_1\dots\mu_s}$ from section \[sec:geometry:spacetime:currents\]: $$\begin{aligned}
j^{a_1 b_1\dots a_s b_s} = \gamma_{\mu_1}^{a_1 b_1}\dots\gamma_{\mu_s}^{a_s b_s} j^{\mu_1\dots\mu_s} \ .\end{aligned}$$ As opposed to $j_\ell^{a_1\dots a_{2s}}$ and $j_*^{a_1\dots a_{2s}}$, the twistor indices on $j^{a_1 b_1\dots a_s b_s}$ are *not* totally symmetric. One can see from eqs. - that $j$ is a sort of intermediate between $j_\ell$ and $j_*$, with one index in every $a_k b_k$ pair lying in $P(\ell)$, and the other in $P^*(\ell)$. The dictionary between $j_\ell$, $j$ and $j_*$ can be viewed as two successive applications of the mapping : $$\begin{aligned}
j^{a_1 b_1\dots a_s b_s} &= \frac{1}{2^s}\delta^{[a_1}_{c_1} \ell^{b_1]}{}_{c_2}\dots\delta^{[a_s}_{c_{2s-1}} \ell^{b_s]}{}_{c_{2s}} j_*^{c_1 c_2\dots c_{2s-1}c_{2s}} \ ; \\
j_\ell^{a_1 a_2\dots a_{2s-1} a_{2s}} &= \ell^{a_1}{}_{c_1}\dots\ell^{a_{2s-1}}{}_{c_s} j^{c_1 a_2\dots c_s a_{2s}} \ ,\end{aligned}$$ or, restoring $j$ into tensor form: $$\begin{aligned}
j^{\mu_1\dots\mu_s}
&= \frac{1}{8^s}\ell_{\nu_1}\gamma^{\nu_1\mu_1}_{c_1 c_2}\dots\ell_{\nu_s}\gamma^{\nu_s\mu_s}_{c_{2s-1}c_{2s}} j_*^{c_1 c_2\dots c_{2s-1}c_{2s}} \ ;
\label{eq:j_j*} \\
j_\ell^{a_1 a_2\dots a_{2s-1} a_{2s}}
&= \ell^{\nu_1}\gamma_{\nu_1\mu_1}^{a_1 a_2}\dots\ell^{\nu_s}\gamma_{\nu_s\mu_s}^{a_{2s-1}a_{2s}} j^{\mu_1\dots\mu_s} \ .\end{aligned}$$ One can also translate $j_\ell$ rather than $j$ into tensor indices. This yields the tensor: $$\begin{aligned}
j_\ell^{\mu_1\nu_1\dots\mu_s\nu_s} &= \frac{1}{4^s}\gamma^{\mu_1\nu_1}_{a_1 a_2}\dots\gamma^{\mu_s\nu_s}_{a_{2s-1} a_{2s}} \, j_\ell^{a_1 a_2\dots a_{2s-1} a_{2s}}
= 2^s\ell^{[\mu_1}\delta^{\nu_1]}_{\rho_1}\dots\ell^{[\mu_s}\delta^{\nu_s]}_{\rho_s} j^{\rho_1\dots\rho_s} \ ,\end{aligned}$$ which is invariant under .
If $j$ has conformal weight $\Delta$, then $j_\ell$ and $j_*$ have weights $\Delta - s$ and $\Delta + s$, respectively. The conformally covariant divergence , which is well-defined for $\Delta = s+1$, is best expressed in spinor language in terms of $j_*$: $$\begin{aligned}
(\operatorname{div} j_*)^{a_1\dots a_{2s-2}} = \frac{1}{8}\gamma^{\mu\nu}_{bc}\ell_\mu\frac{{\partial}}{{\partial}\ell^\nu}\,j_*^{a_1 \dots a_{2s-2}bc} \ . \label{eq:spinor_div}\end{aligned}$$ When $j_*$ has the correct conformal weight $\Delta + s = 2s+1$, one can show that this operation is consistent with the equivalence relation . With the particular numerical factor in , $\operatorname{div} j_*$ is related to the tensorial expression via the spin-$(s-1)$ version of the map .
### More on the bulk-to-boundary limit {#sec:geometry:spinors_boundary:limit}
It is instructive to flesh out the limit in some more detail. For this purpose, we will need to know the *direction* from which the bulk point $x$ approaches the boundary point $\ell$. This direction can be encoded by a second boundary point $n$, where we normalize $\ell\cdot n = -1/2$ for convenience. We can then define the approach $x^\mu\rightarrow\ell^\mu/z$ as: $$\begin{aligned}
x^\mu = \frac{1}{z}\ell^\mu + zn^\mu \ , \label{eq:approach}\end{aligned}$$ such that $x\cdot x = -1$ is maintained throughout. The trajectory is just the geodesic from the boundary point $n$ to the boundary point $\ell$, which approaches $\ell$ as $z\rightarrow 0$. The spacelike unit tangent to the trajectory reads: $$\begin{aligned}
t^\mu = \frac{1}{z}\ell^\mu - zn^\mu \ . \label{eq:approach_t}\end{aligned}$$
For simplicity, let us choose a frame such that: $$\begin{aligned}
\ell^\mu = \left(\frac{1}{2}, \mathbf{0}, \frac{1}{2} \right) \ ; \quad n^\mu = \left(\frac{1}{2}, \mathbf{0}, -\frac{1}{2} \right) \ . \label{eq:frame}\end{aligned}$$ Then the trajectory is just the geodesic of changing $z$ at constant $\mathbf{r} = 0$ in the Poincare coordinates . In the frame , using the explicit gamma matrices , we can now observe the following. The spinor spaces $P(\ell)$ and $P_{L/R}(x)$ and are spanned by twistors of the form: $$\begin{aligned}
P(\ell): \ U_\ell = \begin{pmatrix} u \\ 0 \end{pmatrix} \ ; \quad P_L(x): \ U_L = \begin{pmatrix} u \\ zu \end{pmatrix} \ ; \quad P_R(x): \ U_R = \begin{pmatrix} u \\ -zu \end{pmatrix} \ , \label{eq:P_bulk_boundary}\end{aligned}$$ where $u$ is a 2-component spinor. This explicitly shows how $P_{L/R}(x)$ both converge towards $P(\ell)$.
It will be useful to identify the three twistors for given $u$ as representing “asymptotically the same” boundary spinor. They can be mapped explicitly onto each other using the following operators: $$\begin{aligned}
U_R &= -tU_L = txU_L = (\ell n - n\ell)U_L \ ; \\
U_L &= +tU_R = txU_R = (\ell n - n\ell)U_R \ ; \\
U_\ell &= \frac{1}{2}(1 + tx)U_{L/R} = \ell nU_{L/R} \ .\end{aligned}$$ Thus, the operator $tx = \ell n - n\ell$ asymptotically maps spinors in $P_L(x)$ to their “asymptotically equal” counterparts in $P_R(x)$ and vise versa, while the operator $(1+tx)/2 = \ell n$ maps them both to their “asymptotically equal” counterpart in $P(\ell)$. In other words, the projection $U\rightarrow \ell nU\in P(\ell)$ defines the “boundary limit” of a twistor $U$. In the language of section \[sec:spacetime\_subgroup\] below, the projector $\ell n$ can be interpreted as an infinite boost in the $\ell\wedge n$ plane.
Finally, let us work out the action $\Gamma\rightarrow \ell n\Gamma n\ell$ of the projector $\ell n$ on a complete basis of twistor matrices $\Gamma$: $$\begin{aligned}
\begin{split}
&1 \rightarrow 0 \ ; \quad \ell \rightarrow \ell \ ; \quad n \rightarrow 0 \ ; \quad \gamma_i \rightarrow 0 \ ; \\
&\ell n - n\ell \rightarrow 0 \ ; \quad \ell\gamma_i \rightarrow \ell\gamma_i \ ; \quad n\gamma_i \rightarrow 0 \ ; \quad \gamma_{ij} \rightarrow 0 \ .
\end{split} \label{eq:bilinear_limit_null}\end{aligned}$$ Here, we defined $\gamma_i = e_i^\mu\gamma_\mu$, where the indices $(i,j,\dots)$ run over the values $1,2,3$, and the basis $e_i^\mu$ spans the 3d subspace orthogonal to both $\ell$ and $n$. Using a basis with $\{x,t\}$ in place of $\{\ell,n\}$, eqs. become: $$\begin{aligned}
\begin{split}
&1 \rightarrow 0 \ ; \quad x \rightarrow \frac{1}{z}\ell \ ; \quad t \rightarrow \frac{1}{z}\ell \ ; \quad \gamma_i \rightarrow 0 \ ; \\
&tx \rightarrow 0 \ ; \quad x\gamma_i \rightarrow \frac{1}{z}\ell\gamma_i \ ; \quad t\gamma_i \rightarrow \frac{1}{z}\ell\gamma_i \ ; \quad \gamma_{ij} \rightarrow 0 \ .
\end{split} \label{eq:bilinear_limit}\end{aligned}$$
Bulk and boundary spinor spaces on an equal footing {#sec:geometry:spinors_equal}
---------------------------------------------------
For some purposes, in particular for the higher-spin two-point functions of section \[sec:algebra:two\_point\] below, one can avoid the distinction between bulk and boundary points. This feature is linked to covariance under the $O(1,5)$ group of bulk conformal transformations, though we will not pursue that angle explicitly.
Let us consider a 2-component spinor space, which may be either a boundary spinor space $P(\ell)$ or a bulk spinor space $P_L(x)$ or $P_R(x)$. This spinor space is spanned by a twistor matrix, which in index-free notation is again simply $P(\ell)$, $P_L(x)$ or $P_R(x)$. These can all be treated as special cases of: $$\begin{aligned}
P(\xi) = \frac{1}{2}\left(\sqrt{-\xi\cdot\xi} + \xi\right) \ , \label{eq:P_xi}\end{aligned}$$ where the matrix $P^{ab}(\xi)$ is determined by a timelike or null vector $\xi^\mu\in{\mathbb{R}}^{1,4}$. The special cases of bulk and boundary spinor spaces correspond to: $$\begin{aligned}
\xi^\mu = \ell^\mu \ \Rightarrow \ P(\xi) = P(\ell) \ ; \quad \xi^\mu = \pm x^\mu \ \Rightarrow \ P(\xi) = P_{R/L}(x) \ . \label{eq:spinor_spaces}\end{aligned}$$ Unifying eqs. and , we can define a metric and measure on the spinor space $P(\xi)$ via: $$\begin{aligned}
\frac{du^a du^b}{2\pi} \equiv P^{ab}(\xi)\, d^2u \ .\end{aligned}$$ We note the identity: $$\begin{aligned}
\epsilon_{abcd}P^{cd}(\xi) = 2P_{ab}(-\xi) \ ,\end{aligned}$$ which implies in particular that $P(-\xi)$ is the subspace orthogonal to $P(\xi)$. In other words, the space $P^*(\xi)$, i.e. the dual to $P(\xi)$ under the twistor metric, is just the quotient space of twistors modulo terms in $P(-\xi)$. For a boundary point, this reproduces the dual spinor space , since the spaces $P(\ell)$ and $P(-\ell)$ coincide (with a factor of $-1$ between the corresponding matrices). For a bulk point, this means that the space $P^*_{L/R}(x)$ dual to $P_{L/R}(x)$ is the space of twistors modulo terms in $P_{R/L}(x)$, which can be identified with $P_{L/R}(x)$ itself.
Consider now a pair of spinor spaces $P(\xi)$ and $P(\xi')$, associated with a pair of bulk or boundary points. The relationship between these spaces is governed by two invariants: $$\begin{aligned}
P_{ab}(\xi)P^{ab}(\xi') &= {\operatorname{tr}}\left(P(\xi)P(\xi')\right) = \sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi' \ ; \\
\frac{1}{2}\epsilon_{abcd}P^{ab}(\xi)P^{cd}(\xi') &= {\operatorname{tr}}\left(P(\xi)P(-\xi')\right) = \sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} + \xi\cdot\xi' \ . \end{aligned}$$ An arbitrary twistor $U$ can be decomposed along $P(\xi)$ and $P(\xi')$ as follows: $$\begin{aligned}
U = u + u' \ ; \quad
u = \frac{2P(\xi)P(-\xi')U}{{\operatorname{tr}}\left(P(\xi)P(-\xi')\right)} \in P(\xi) \ ; \quad u' = \frac{2P(\xi')P(-\xi)U}{{\operatorname{tr}}\left(P(\xi')P(-\xi)\right)} \in P(\xi') \ ,
\label{eq:U_u_u'}\end{aligned}$$ where the scalar product of the two components $u,u'$ reads: $$\begin{aligned}
uu' = \frac{U\xi\xi'U}{2\left(\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} + \xi\cdot\xi'\right)} \ . \label{eq:uu'}\end{aligned}$$ The twistor measure decomposes under as: $$\begin{aligned}
d^4U = \frac{1}{4}\epsilon_{abcd}P^{ab}(\xi)P^{cd}(\xi')\,d^2u\,d^2u' = \frac{1}{2}\left(\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} + \xi\cdot\xi'\right) d^2u\,d^2u' \ . \label{eq:U_u_u'_measure}\end{aligned}$$ The chiral decomposition $U = u_L + u_R$ at a single bulk point $x$ can be viewed as a special case of , with eqs. - reproducing the identities $u_L u_R = 0$ and $d^4U = d^2u_L d^2u_R$.
### Integrals over spinor spaces {#sec:geometry:spinors_equal:integrals}
In calculations below, we will need the 2-component spinor versions of the 4-component twistor integrals -. Consider a general (bulk or boundary) spinor space $P(\xi)$ as above. A Gaussian integral over $P(\xi)$ can be calculated as: $$\begin{aligned}
\int_{P(\xi)} d^2u\, e^{uAu/2} = \frac{\pm 1}{\sqrt{\det_{P(\xi)}(A)}} \ ; \quad \det\nolimits_{P(\xi)}(A) = -\frac{1}{2}{\operatorname{tr}}\left(P(\xi)A\right)^2 \ .
\label{eq:Gaussian_spinor}\end{aligned}$$ Here, $A$ is a symmetric twistor matrix, while $\det\nolimits_{P(\xi)}(A)$ is the determinant of $A$, viewed as a $2\times 2$ quadratic form over the spinor space $P(\xi)$.
The generic analog of the delta-function-type integral involves a pair of spinor spaces $P(\xi),P(\xi')$. The integral reads: $$\begin{aligned}
\int_{P(\xi)} d^2u \int_{P(\xi')} d^2u' f(u)\, e^{iuu'} = \frac{2}{P_{ab}(\xi)P^{ab}(\xi')}\,f(0)
= \frac{2}{\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi'}\,f(0) \ . \label{eq:delta_integral_spinor} \end{aligned}$$ In addition, at a single boundary point $\ell$, one can write the following delta-function-type integrals over the spinor space $P(\ell)$ and its dual space $P^*(\ell)$: $$\begin{aligned}
\int_{P(\ell)} d^2u \int_{P^*(\ell)} d^2u^* f(u)\, e^{iuu^*} = f(0) \ ; \quad \int_{P(\ell)} d^2u \int_{P^*(\ell)} d^2u^* f(u^*)\, e^{iuu^*} = f(0) \ .
\label{eq:delta_integral_boundary}\end{aligned}$$
The integral can be written explicitly in terms of a delta function as follows: $$\begin{aligned}
\int_{P(\xi)} d^2u\,\delta_{\xi'}(u) f(u) = \frac{2}{\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi'}\,f(0) \ ,\end{aligned}$$ where the spinor delta function is defined as: $$\begin{aligned}
\delta_\xi(U) = \int_{P(\xi)} d^2v\,e^{ivU} \ . \label{eq:delta_xi}\end{aligned}$$ In the particular cases of bulk and boundary spinor spaces, we denote these delta functions as: $$\begin{aligned}
\xi^\mu = \ell^\mu \ \Rightarrow \ \delta_\xi(U) = \delta_\ell(U) \ ; \quad \xi^\mu = \pm x^\mu \ \Rightarrow \ \delta_\xi(U) \equiv \delta_x^{R/L}(U) \ . \label{eq:spinor_deltas}\end{aligned}$$ The notation is meant to signify that $\delta_\xi(U)$ is a delta function with respect to $U$, with support on a 2d spinor space determined by $\xi$. Specifically, it has support on the subspace $P(-\xi)$ which is orthogonal to $P(\xi)$. For a boundary point, this means that $\delta_\ell(U)$ has support on $P(\ell)$, forcing the $P^*(\ell)$ component of $U$ to vanish. For a bulk point, it means that $\delta_x^{R/L}(U)$ has support on $P_{L/R}(x)$, forcing the $P_{R/L}(x)$ component to vanish. The boundary delta function $\delta_\ell(U)$ has conformal weight $\Delta = 1$, and can be used to rewrite the second integral in as: $$\begin{aligned}
\int_{P^*(\ell)} d^2u^*\,\delta_\ell(u^*)\, f(u^*) = f(0) \ .\end{aligned}$$
The comments from section \[sec:geometry:twistors:integrals\] regarding sign ambiguities in twistor integrals apply equally well to the spinor case. Gaussians are well-defined functions, but their integrals have a sign ambiguity that cannot be globally fixed. Conversely, delta functions have well-defined integrals, but they themselves are defined as limits of ordinary functions only up to sign. An additional subtlety arises when adding or comparing integrals over different spinor spaces, associated with different spacetime points. In that case, one must make a separate contour choice for every integral, and this choice may fail to be consistent across a large enough spacetime region.
Higher-spin algebra {#sec:algebra}
===================
Spacetime-independent structure
-------------------------------
In higher-spin theory, one introduces (spacetime-independent) twistor coordinates $Y^a$, which are acted on by the non-commutative star product: $$\begin{aligned}
Y^a\star Y^b = Y^a Y^b + iI^{ab} \ . \label{eq:star_basic}\end{aligned}$$ By associativity, this extends into a product on polynomials in $Y$: $$\begin{aligned}
f(Y)\star g(Y) = f \exp\left(iI^{ab}\overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^b}}\right) g \ . \label{eq:star_diff}\end{aligned}$$ In practical calculations, it is convenient to use the index-free notation of section \[sec:geometry:twistors:index\_free\], where some twistors are implicitly lower-index and some are upper-index. One can then use the formulas: $$\begin{aligned}
I^{ab}\overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^b}} = \overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y_a}}
= - \overleftarrow{\frac{{\partial}}{{\partial}Y_a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^a}} \ , \label{eq:ddY_ddY}\end{aligned}$$ where it is important that ${\partial}/{\partial}Y_a$ is *minus* the raised-index version of ${\partial}/{\partial}Y^a$. Together with rearrangements of the form , one can reduce calculations to convenient index-free expressions such as: $$\begin{aligned}
\begin{split}
U\Gamma_1\dots\Gamma_mY \left(I^{ab}\overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^b}}\right) Y\Gamma_{m+1}\dots\Gamma_nV
&= U\Gamma_1\dots\Gamma_n V \ ; \\
Y\Gamma_1\dots\Gamma_m \left(I^{ab}\overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^b}}\right) \Gamma_{m+1}\dots\Gamma_nY
&= -{\operatorname{tr}}(\Gamma_1\dots\Gamma_n) \ .
\end{split}\end{aligned}$$ The star product also extends to non-polynomial functions, where one must resort to an integral formula: $$\begin{aligned}
f(Y)\star g(Y) = f \exp\left(iI^{ab}\overleftarrow{\frac{{\partial}}{{\partial}Y^a}}\overrightarrow{\frac{{\partial}}{{\partial}Y^b}}\right) g
= \int d^4U d^4V f(Y+U)\, g(Y+V)\, e^{-iUV} \ . \label{eq:star_int}\end{aligned}$$
The higher-spin symmetry algebra is the infinite-dimensional Lie algebra of even (i.e. integer-spin) functions $f(Y)$ with the associative product . It contains as a subalgebra the generators of the $EAdS_4$ isometry group $O(1,4)$: $$\begin{aligned}
M_{\mu\nu} = -\frac{i}{8} Y\gamma_{\mu\nu}Y \quad ; \quad [M^{\mu\nu}, M_{\rho\sigma}]_\star = 4\delta^{[\mu}_{[\rho}\, M^{\nu]}{}_{\sigma]} \ . \label{eq:generators}\end{aligned}$$ The product respects a trace operation, defined simply by evaluating $f(Y)$ at $Y=0$: $$\begin{aligned}
{\operatorname{tr}}_\star f(Y) = f(0) \ ; \quad {\operatorname{tr}}_\star(f\star g) = {\operatorname{tr}}_\star(g\star f) = \int d^4U d^4V f(U)\,g(V)\,e^{-iUV} \ . \label{eq:str}\end{aligned}$$ Here, the equality ${\operatorname{tr}}_\star(f\star g) = {\operatorname{tr}}_\star(g\star f)$ relies on $f(Y),g(Y)$ being even functions. The ${\operatorname{tr}}_\star$ operation is usually denoted in the literature by “${\operatorname{str}}$”, since in certain generalizations of the algebra , the trace becomes a supertrace.
Another important object is the delta function : $$\begin{aligned}
\delta(Y) = \int d^4U\,e^{iUY} \ . \label{eq:delta}\end{aligned}$$ A star product with $\delta(Y)$ implements the Fourier transform: $$\begin{aligned}
f(Y)\star\delta(Y) = \int d^4U f(U)\,e^{iUY} \ ; \quad \delta(Y)\star f(Y) = \int d^4U f(U)\,e^{-iUY} \ . \label{eq:delta_Fourier}\end{aligned}$$ The following properties establish $\delta(Y)$ as a Klein operator of the algebra : $$\begin{aligned}
\delta(Y)\star\delta(Y) = 1 \ ; \quad \delta(Y)\star f(Y)\star\delta(Y) = f(-Y) \ , \label{eq:delta_Klein}\end{aligned}$$ i.e. $\delta(Y)$ (anti)commutes with even (odd) functions $f(Y)$. This implies that $\delta(Y)$ is invariant in the adjoint representation of the higher-spin symmetry group (recall that the symmetry includes only integer spins, i.e. only generators even in $Y$).
The star product $f\star g$, the trace ${\operatorname{tr}}_\star f$ and the invariant Klein operator $\delta(Y)$ are the only allowed ingredients in an expression that preserves (undeformed) higher-spin symmetry. The role of $\delta(Y)$ in this list is somewhat subtle. The issue is the contour ambiguity of the integral formula , which arises whenever we do higher-spin algebra with non-polynomial functions. As discussed in section \[sec:geometry:twistors:integrals\], even the simplest cases - delta functions and Gaussians - are associated with a sign ambiguity. In particular, one should be careful with assigning meaning to the sign of $\delta(Y)$ and its star products. While this sign ambiguity may not look like much, there is a sense in which it is the *only* information carried in star products with $\delta(Y)$. Indeed, since $\delta(Y)$ squares to unity, one may think of decomposing the space of functions $f(Y)$ into eigenspaces with eigenvalues $\pm 1$ under star-multiplication by $\delta(Y)$. Formally, this decomposition is accomplished by the pair of projectors: $$\begin{aligned}
{\mathcal{P}}_\pm(Y) = \frac{1 \pm \delta(Y)}{2} \ . \label{eq:projectors_delta}\end{aligned}$$ Conceptually, these projectors play an important role in the theory: as we will see, they are related to bulk antipodal symmetry, as well as to the two types of asymptotic boundary data (Neumann vs. Dirichlet or magnetic vs. electric). However, since the sign of $\delta(Y)$ is a priori ambiguous, one shouldn’t take the projectors too seriously. In particular, in section \[sec:linear\_HS:antipodal\], we will see in detail how they fail to be well-defined linear operators. In the present section, we will continue to list useful formal identities involving $\delta(Y)$. Later in the paper, we will make use of $\delta(Y)$ and the projectors ${\mathcal{P}}_\pm(Y)$, but with a dose of care and self-consciousness.
Structure at a bulk point
-------------------------
Choosing a bulk point $x\in EAdS_4$ picks out a preferred rotation group $SO(4) = SO(3)_L\times SO(3)_R$ out of the isometry group $SO(1,4)$. In the star-product language, the two chiral $SO(3)$’s are generated by bilinears $y^a_L y^b_L$ and $y^a_R y^b_R$, where we used the chiral decomposition $Y = y_L+y_R$ of the twistor $Y$ into Weyl spinors at $x$. Each of the chiral $SO(3)$’s can be extended into its own higher-spin subalgebra, given respectively by chiral functions $f(y_L)$ and $f(y_R)$. Since left-handed and right-handed spinors are orthogonal under the twistor metric, the two subalgebras commute. Explicitly, the chiral decomposition of the star product reads: $$\begin{aligned}
y_L^a\star y_L^b = y_L^a y_L^b + iP_L^{ab} \ ; \quad y_R^a\star y_R^b = y_R^a y_R^b + iP_R^{ab} \ ; \quad y_L^a\star y_R^b = y_R^b\star y_L^a = y_L^a y_R^b \ ,\end{aligned}$$ where we must keep in mind that the projectors $P_{L/R}$ and the Weyl spinors $y_{L/R}$ depend on the bulk point $x$.
Analogously to the role of $\delta(Y)$, delta functions with respect to $y_L$ and $y_R$ play the role of Klein operators for the left-handed and right-handed higher-spin subalgebras. We’ve already encountered these spinor delta functions in eqs. -: $$\begin{aligned}
\delta^L_x(Y) = \int_{P_L(x)} d^2u_L \,e^{iu_L Y} \ ; \quad \delta^R_x(Y) = \int_{P_R(x)} d^2u_R \,e^{iu_R Y} \ . \label{eq:delta_chiral}\end{aligned}$$ The delta function $\delta^{L/R}_x(Y)$ depends on the twistor $Y$ only through the spinor component $y_{L/R}$. These delta functions have star-product properties [@Didenko:2009td] analogous to eqs. -: $$\begin{aligned}
\begin{split}
f(y_L+y_R)\star\delta^L_x(y_L) &= \int d^2u_L\,f(u_L+y_R)\,e^{iu_L y_L} \ ; \\
\delta^L_x(y_L)\star f(y_L+y_R) &= \int d^2u_L\,f(u_L+y_R)\,e^{-iu_L y_L} \ ; \\
\delta^L_x(y_L)\star f(y_L+y_R)\star\delta^L_x(y_L) &= f(-y_L+y_R) \ ,
\end{split} \label{eq:delta_Fourier_bulk_raw}\end{aligned}$$ and similarly for $\delta^R_x(y_R)$. These can be written more covariantly as: $$\begin{aligned}
&f(Y)\star\delta^{L/R}_x(Y) = \int d^2u_{L/R}\,f(Y + u_{L/R})\,e^{iu_{L/R} Y} \ ; \label{eq:delta_Fourier_bulk_right} \\
&\delta^{L/R}_x(Y)\star f(Y) = \int d^2u_{L/R}\,f(Y + u_{L/R})\,e^{-iu_{L/R} Y} \ ; \label{eq:delta_Fourier_bulk_left} \\
&\delta^L_x(Y)\star f(Y)\star\delta^L_x(Y) = f(xY) \ ; \quad \delta^R_x(Y)\star f(Y)\star\delta^R_x(Y) = f(-xY) \label{eq:delta_Klein_bulk} \ .\end{aligned}$$ As a special case, we have: $$\begin{aligned}
\delta^{L/R}_x(Y)\star\delta(Y) = \delta(Y)\star\delta^{L/R}_x(Y) = \delta^{R/L}_x(Y) \ . \label{eq:bulk_delta_odd}\end{aligned}$$ The products of the chiral delta functions are $x$-independent: $$\begin{aligned}
\begin{split}
\delta^L_x(Y)\star\delta^L_x(Y) &= \delta^R_x(Y)\star\delta^R_x(Y) = 1 \ ; \\
\delta^L_x(Y)\star\delta^R_x(Y) &= \delta^R_x(Y)\star\delta^L_x(Y) = \delta^L_x(Y)\delta^R_x(Y) = \delta(Y) \ .
\end{split} \label{eq:bulk_delta_products}\end{aligned}$$ Finally, it will be helpful to explicitly express the $x$-dependence of $\delta^L_x(Y)$ and $\delta^R_x(Y)$. Taking $x$ derivatives of integrals such as is subtle, since the subspace over which we are integrating is itself a function of $x$. A useful workaround (e.g. for the $d^2u_L$ integral) is to perform a change of variables $u_L = P_L(x)u_L'$, where $u_L'$ can now be integrated over the left-handed spinor space $P_L(x')$ at an arbitrary *fixed* point $x'$. After taking the desired $x$ derivatives, we can replace $x'\rightarrow x$. By this method, we find: $$\begin{aligned}
\begin{split}
\nabla_\mu\delta^L_x &= -\frac{1}{2}(\gamma_\mu)^a{}_b\,y_R^b\,\frac{{\partial}\delta^L_x}{{\partial}y_L^a} = -\frac{i}{2}(y_L\gamma_\mu y_R)\star\delta^L_x
= \frac{i}{2}\,\delta^L_x\star(y_L\gamma_\mu y_R) \ ; \\
\nabla_\mu\delta^R_x &= \frac{1}{2}(\gamma_\mu)^a{}_b\,y_L^b\,\frac{{\partial}\delta^R_x}{{\partial}y_R^a} = -\frac{i}{2}(y_L\gamma_\mu y_R)\star\delta^R_x
= \frac{i}{2}\,\delta^R_x\star(y_L\gamma_\mu y_R) \ ,
\end{split}\end{aligned}$$ or, in more covariant notation: $$\begin{aligned}
\begin{split}
\nabla_\mu\delta^L_x &= -\frac{i}{4}(Y\gamma_\mu xY)\star\delta^L_x = \frac{i}{4}\,\delta^L_x\star(Y\gamma_\mu xY) \ ; \\
\nabla_\mu\delta^R_x &= -\frac{i}{4}(Y\gamma_\mu xY)\star\delta^R_x = \frac{i}{4}\,\delta^R_x\star(Y\gamma_\mu xY) \ . \label{eq:nabla_delta}
\end{split} \end{aligned}$$
Structure at a boundary point {#sec:algebra:boundary}
-----------------------------
Now, instead of a bulk point $x$, let us fix a boundary point $\ell$. The isometry group $SO(1,4)$, now viewed as the boundary conformal group, acquires three preferred subgroups, nested within each other:
1. Special conformal transformations around $\ell$ (or, equivalently, translations in a frame where $\ell$ is the point at infinity). These are generated by bilinears $YAY$, where the symmetric twistor matrix $A$ satisfies $\ell A = 0$.
2. Special conformal transformations and rotations around $\ell$. These are generated by bilinears $YAY$ where $\ell A - A\ell = 0$.
3. Special conformal transformations, rotations and dilatations around $\ell$. These are generated by bilinears $YAY$ where $\ell A - A\ell = \lambda\ell$ for some scalar $\lambda$.
Neither of these subgroups extends into an interesting higher-spin subalgebra. The only subgroup that extends at all is the first one. The corresponding higher-spin subalgebra ${\mathcal{A}}_0(\ell)$ consists of functions $f(Y)$ that satisfy $f(Y+u) = f(Y)$ for any $u\in P(\ell)$, i.e. of functions $f(y^*)$ over the boundary spinor space $P^*(\ell)$: $$\begin{aligned}
f(Y)\in{\mathcal{A}}_0(\ell) \quad \Longleftrightarrow \quad f(Y+u) = f(Y) \ \ \text{for} \ \ u\in P(\ell) \quad \Longleftrightarrow \quad f(Y) = f(y^*) \ .\end{aligned}$$ The special conformal transformations around $\ell$ are generated by the quadratic piece of this subalgebra. The entire subalgebra is commuting, and the star product is simply: $$\begin{aligned}
f,g\in{\mathcal{A}}_0(\ell) \quad \Longrightarrow \quad f(Y)\star g(Y) = f(Y)g(Y) \ .\end{aligned}$$
A special element of the subalgebra ${\mathcal{A}}_0(\ell)$ is the delta function with respect to $y^*$, which we’ve encountered in eqs. -: $$\begin{aligned}
\delta_\ell(Y) = \int_{P(\ell)} d^2u\, e^{iuY} \ . \label{eq:delta_ell}\end{aligned}$$ In the bulk-to-boundary limiting procedure , the delta function can be expressed as a rescaled limit of the bulk delta functions : $$\begin{aligned}
\delta_\ell(Y) = \lim_{x\rightarrow\ell/z} \frac{1}{z}\,\delta^R_x(Y) = -\lim_{x\rightarrow\ell/z} \frac{1}{z}\,\delta^L_x(Y) \ . \label{eq:delta_limit}\end{aligned}$$ However, unlike its bulk counterparts, $\delta_\ell(Y)$ is *not* a Klein operator. In particular, the star product $\delta_\ell(Y)\star\delta_\ell(Y)$ is divergent. The star product of $\delta_\ell(Y)$ with the global delta function $\delta(Y)$ reads: $$\begin{aligned}
\delta_\ell(Y)\star\delta(Y) = \delta(Y)\star\delta_\ell(Y) = -\delta_\ell(Y) \ . \label{eq:boundary_delta_odd}\end{aligned}$$ From the point of view of the bulk-to-boundary limit , these identities can be viewed as a limiting case of .
The delta function $\delta_\ell(Y)$ is a member not only of the subalgebra ${\mathcal{A}}_0(\ell)$, but of two additional (also degenerate) higher-spin subalgebras. These subalgebras, which we denote as ${\mathcal{A}}_\pm(\ell)$, consist of functions with the property: $$\begin{aligned}
f(Y)\in{\mathcal{A}}_\pm(\ell) \quad \Longleftrightarrow \quad f(Y+u) = e^{\pm iuY}f(Y) \ \ \text{for} \ \ u\in P(\ell) \ , \label{eq:A+-}\end{aligned}$$ Functions of the form can be thought of as “twisted” functions on $P^*(\ell)$; like the true functions on $P^*(\ell)$ that make up the subalgebra ${\mathcal{A}}_0(\ell)$, they depend freely only on a single two-component spinor. The star product in the subalgebras ${\mathcal{A}}_\pm(\ell)$ reads: $$\begin{aligned}
\begin{split}
f,g\in{\mathcal{A}}_-(\ell) \quad \Longrightarrow \quad f(Y)\star g(Y) = g(0)f(Y) = ({\operatorname{tr}}_\star g)f(Y) \ ; \\
f,g\in{\mathcal{A}}_+(\ell) \quad \Longrightarrow \quad f(Y)\star g(Y) = f(0)g(Y) = ({\operatorname{tr}}_\star f)g(Y) \ .
\end{split}\end{aligned}$$
The definition of the subalgebras ${\mathcal{A}}_\pm(\ell)$ can be expressed concisely in star-product form: $$\begin{aligned}
\begin{split}
f(Y)\in{\mathcal{A}}_-(\ell) \quad &\Longleftrightarrow \quad f(Y)\star \ell\,Y = 0 \ ; \\
f(Y)\in{\mathcal{A}}_+(\ell) \quad &\Longleftrightarrow \quad \ell\,Y\star f(Y) = 0 \ .
\end{split} \label{eq:A+-_star}\end{aligned}$$ From here, it follows that multiplication on the right (left) by a function in ${\mathcal{A}}_-(\ell)$ (${\mathcal{A}}_+(\ell)$) projects *any* function into the corresponding subalgebra: $$\begin{aligned}
f\in{\mathcal{A}}_-(\ell) \ \Rightarrow \ g\star f \in {\mathcal{A}}_-(\ell) \ ; \quad f\in{\mathcal{A}}_+(\ell) \ \Rightarrow \ f\star g \in {\mathcal{A}}_+(\ell) \ .\end{aligned}$$ In particular, since $\delta_\ell(Y)$ is an element of both ${\mathcal{A}}_-(\ell)$ and ${\mathcal{A}}_+(\ell)$, we have, for any $f(Y)$: $$\begin{aligned}
\begin{split}
f(Y)\star\delta_\ell(Y) &= \int_{P(\ell)} d^2u\,f(Y + u)\,e^{iuY} \in {\mathcal{A}}_-(\ell) \ ; \\
\delta_\ell(Y)\star f(Y) &= \int_{P(\ell)} d^2u\,f(Y + u)\,e^{-iuY} \in {\mathcal{A}}_+(\ell) \ ,
\end{split} \label{eq:delta_Fourier_boundary}\end{aligned}$$ while the product $\delta_\ell(Y)\star f(Y)\star\delta_\ell(Y)$ is divergent. The formulas can be recognized as boundary limits of -. There is no boundary analog of the Fourier-transform formulas , because at a boundary point $\ell$, twistor space does not decompose into an orthogonal pair of spinor spaces.
Structure at two and more points {#sec:algebra:two_point}
--------------------------------
A key object in our analysis will be the star product of two spinor delta functions $\delta^L_x(Y)$, $\delta^R_x(Y)$ or $\delta_\ell(Y)$ at a pair of bulk or boundary points. In this section, we will compute these products and discuss their properties. These two-point products are closely related to various propagators in the HS literature, such as the ${\mathcal{D}}$-functions of [@Gelfond:2008ur; @Gelfond:2008td] and the boundary-to-bulk propagators of [@Didenko:2012tv], and are quite similar in spirit to propagators in ordinary field theory. However, one should keep in mind an important detail: while the products $\delta\star\delta$ depend on two spacetime points, they depend on only *one* twistor variable $Y$, which is not associated with either point in particular. In section \[sec:spacetime\_subgroup:null\], we will discuss these two-point products from a different viewpoint, as group elements of the spacetime symmetry $SO(1,4)$.
### The general two-point product
The different kinds of two-point products can all be computed together, using the machinery of section \[sec:geometry:spinors\_equal\]. Recall from that the delta functions $\delta^L_x(Y),\delta^R_x(Y),\delta_\ell(Y)$ are all special cases of the general spinor delta function : $$\begin{aligned}
\delta_\xi(Y) = \int_{P(\xi)} d^2u\,e^{iuY} \ . \label{eq:delta_xi_again}\end{aligned}$$ The star-product formulas -,,, are all particular cases of the identities: $$\begin{aligned}
&f(Y)\star\delta_\xi(Y) = \int_{P(\xi)} d^2u\,f(Y+u)\,e^{iuY} \ ; \label{eq:delta_Fourier_xi_first} \\
&\delta_\xi(Y)\star f(Y) = \int_{P(\xi)} d^2u\,f(Y+u)\,e^{-iuY} \ ; \\
&\delta_\xi(Y)\star\delta(Y) = \delta(Y)\star\delta_\xi(Y) = \delta_{-\xi}(Y) \ .\end{aligned}$$ We can now compute the star product of two delta functions of the general type . First, using eq. , we get the integral expression: $$\begin{aligned}
\delta_\xi(Y)\star\delta_{\xi'}(Y) = \int_{P(\xi)} d^2u \int_{P(\xi')} d^2u'\,e^{i(uY + u'Y + uu')} \ .\end{aligned}$$ With some work, this integral can be brought to the form . To do this, we decompose the twistor $Y$ into a pair of spinors as in eq. : $$\begin{aligned}
Y = y + \bar y' \ ; \quad y \in P(\xi) \ ; \quad \bar y'\in P(-\xi') \ .\end{aligned}$$ The $\bar y'$ piece is identically orthogonal to $u'\in P(\xi')$, while the $y$ piece can be used to shift the integration variable $u\in P(\xi)$. This brings the integral into the form: $$\begin{aligned}
\begin{split}
\delta_\xi(Y)\star\delta_{\xi'}(Y) &= \int_{P(\xi)} d^2u \int_{P(\xi')} d^2u'\,e^{i(uY + y\bar y')} e^{iuu'}
= \frac{2e^{iy\bar y'}}{\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi'} \\
&= \frac{2}{\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi'}\,\exp\left(\frac{-iY\xi\xi'Y/2}{\sqrt{(\xi\cdot\xi)(\xi'\cdot\xi')} - \xi\cdot\xi'}\right) \ .
\end{split} \label{eq:general_two_point}\end{aligned}$$ where we used eq. in the first line and eq. in the second line. For particular cases of bulk/boundary points, the result reads: $$\begin{aligned}
\delta_\ell(Y)\star\delta_{\ell'}(Y) &= -\frac{2}{\ell\cdot\ell'}\exp\frac{iY\ell\ell' Y}{2\ell\cdot\ell'} \ ; \label{eq:delta_ell_ell} \\
\delta_\ell(Y)\star\delta^R_x(Y) &= -\frac{2}{\ell\cdot x}\exp\frac{iY\ell x Y}{2\ell\cdot x} \ ; \quad
\delta^R_x(Y)\star\delta_\ell(Y) = -\frac{2}{\ell\cdot x}\exp\frac{iYx\ell Y}{2\ell\cdot x} \ ; \label{eq:delta_ell_x} \\
\delta^R_x(Y)\star\delta^R_{x'}(Y) &= \frac{2}{1 - x\cdot x'}\exp\frac{iYxx'Y}{2(x\cdot x' - 1)} \ , \label{eq:delta_x_x}\end{aligned}$$ where one can substitute $\delta^R_x(Y)\rightarrow\delta^L_x(Y)$ via the antipodal map $x\rightarrow -x$, and likewise for $x'$.
### Properties of the boundary-boundary product
We now focus on the boundary-boundary two-point product , which possesses some remarkable properties. First, if we multiply by $\sqrt{-\ell\cdot\ell'}$, the result has the conformal weight $\Delta = 1/2$ of a 3d free massless scalar with respect to both boundary points. We can then evaluate the 3d conformal Laplacian , only to find that the massless wave equation is satisfied: $$\begin{aligned}
\sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y) &= \frac{2}{\sqrt{-\ell\cdot\ell'}}\exp\frac{iY\ell\ell' Y}{2\ell\cdot\ell'} \ ; \label{eq:K_raw} \\
\Box_\ell\left(\sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y)\right)
&= \Box_{\ell'}\left(\sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y)\right) = 0 \quad \forall \ell\neq\ell' \ . \label{eq:ell_ell_wave}\end{aligned}$$ At $\ell=\ell'$, the two-point product has a singularity, and the wave equation picks up a source term. Moreover, since this is an *essential* singularity, the source term will contain not just a delta distribution, but also an infinite tower of its derivatives. Thus, upon integration over $\ell$ or $\ell'$, the source will *not* appear localized at $\ell=\ell'$, as we will see explicitly in section \[sec:CFT:twistor:currents\].
Upon taking the higher-spin trace , the essential singularity in becomes a simple pole. In fact, up to a numerical factor, this trace is just the $\Delta = 1/2$ boundary-to-boundary propagator, which satisfies the wave equation with a point source: $$\begin{aligned}
{\operatorname{tr}}_\star\left(\sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y)\right) &= \frac{2}{\sqrt{-\ell\cdot\ell'}} \ ; \label{eq:str_ell_ell} \\
\Box_\ell{\operatorname{tr}}_\star\left(\sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y)\right) &= -8\pi\sqrt{2}\,\delta^{5/2,1/2}(\ell,\ell') \ . \label{eq:str_ell_ell_wave}\end{aligned}$$ Here, the superscripts on the boundary delta function $\delta(\ell,\ell')$ denote its conformal weight with respect to each argument. To derive the wave equation , we recall that in the flat frame , $\sqrt{-\ell\cdot\ell'}$ is just the 3d Euclidean distance $|\mathbf{r} - \mathbf{r'}|/\sqrt{2}$. The full Gaussian can now be understood as a Taylor series of derivatives of the propagator . This can be seen explicitly by converting the coefficients of different powers of $Y$ into boundary tensors, or more abstractly from eqs. , below.
Another feature of the boundary-boundary product is that it belongs simultaneously to the higher-spin subalgebras ${\mathcal{A}}_+(\ell)$ and ${\mathcal{A}}_-(\ell')$. As we can see from , the same is true for the more general product $\delta_\ell(Y)\star f(Y)\star\delta_{\ell'}(Y)$, where $f(Y)$ is an arbitrary function. On the other hand, it’s clear from the definition or that the intersection ${\mathcal{A}}_+(\ell)\cap{\mathcal{A}}_-(\ell')$ is one-dimensional. Therefore, for any function $f(Y)$, we must have: $$\begin{aligned}
\delta_\ell(Y)\star f(Y)\star\delta_{\ell'}(Y) = \lambda\,\delta_\ell(Y)\star\delta_{\ell'}(Y) \ , \label{eq:forgetful}\end{aligned}$$ for some $Y$-independent coefficient $\lambda$. Taking the higher-spin trace of both sides, we can express this coefficient as: $$\begin{aligned}
\lambda = -\frac{\ell\cdot\ell'}{2}{\operatorname{tr}}_\star\left(\delta_\ell(Y)\star f(Y)\star\delta_{\ell'}(Y)\right) \ . \label{eq:forgetful_coeff}\end{aligned}$$ Eq. is the underlying root of the “forgetful property” [@Didenko:2012tv] of higher-spin propagators.
As a special case of , we evaluate the three-point product: $$\begin{aligned}
\delta_\ell(Y)\star\delta_{\ell'}(Y)\star\delta_{\ell''}(Y) = \pm i\sqrt{-\frac{\ell\cdot\ell''}{2(\ell\cdot\ell')(\ell'\cdot\ell'')}}\,
\delta_\ell(Y)\star\delta_{\ell''}(Y) \ . \label{eq:delta_ell_ell_ell}\end{aligned}$$ The sign ambiguity is due to a Gaussian integration of the form . An efficient way to derive eq. is to use the result for the product of two delta functions, and then factor in the third delta function via ; thanks to eq. , it suffices to evaluate the result at $Y=0$.
Linearized higher-spin gravity {#sec:linear_HS}
==============================
In this section, we formulate linearized higher-spin gravity on $EAdS_4$, along with its solution via the Penrose transform. The formulas that appear here will receive a more geometric interpretation in section \[sec:spacetime\_subgroup\]. In section \[sec:linear\_HS:fields\], we describe free massless fields of arbitrary integer spin. In section \[sec:linear\_HS:Penrose\], we review the Penrose transform in (A)dS~4~. In section \[sec:linear\_HS:unfolded\], we introduce the unfolded formulation, which recasts both the field equations and the Penrose transform into HS-covariant star-product expressions. Finally, in section \[sec:linear\_HS:antipodal\], we discuss antipodal symmetry $x^\mu \leftrightarrow -x^\mu$ and its analogue in the twistor language.
Free massless fields in $EAdS_4$ {#sec:linear_HS:fields}
--------------------------------
Our starting point is a set of free massless fields, one for each integer spin. A field with spin $s>0$ is described by the self-dual and anti-self-dual parts of its field strength (i.e. the higher-spin generalization of the Maxwell tensor and the linearized Weyl tensor). These are encoded by purely left-handed and purely right-handed totally symmetric spinors with $2s$ indices. The field content is thus: $$\begin{aligned}
\text{Spin 0:} \quad C^{(0,0)} \ , \quad \text{Spin 1:} \quad C^{(2,0)}_{\alpha\beta}, C^{(0,2)}_{\dot\alpha\dot\beta} \ , \quad
\text{Spin 2:} \quad C^{(4,0)}_{\alpha\beta\gamma\delta}, C^{(0,4)}_{\dot\alpha\dot\beta\dot\gamma\dot\delta} \ , \quad \text{etc.} \ , \label{eq:HS_fields}\end{aligned}$$ where the numbers in parentheses signify the number of left-handed and right-handed spinor indices. We are temporarily introducing designated indices $(\alpha,\beta,\dots)$ and $(\dot\alpha,\dot\beta,\dots)$ respectively for left-handed and right-handed Weyl spinors at a bulk point $x$. These are the same as twistor indices $(a,b,\dots)$, but with $P_{L/R}(x)$ chiral projections implied. The spinor fields can also be expressed in tensor form, using the convention for converting a symmetric pair of twistor indices into an antisymmetric pair of tensor indices. The left-handed and right-handed parts of the spin-$s$ field strength combine into a single tensor, via: $$\begin{aligned}
\begin{split}
C_{\mu_1\nu_1\dots\mu_s\nu_s} &= C^L_{\mu_1\nu_1\dots\mu_s\nu_s} + C^R_{\mu_1\nu_1\dots\mu_s\nu_s} \ ; \\
C^L_{\mu_1\nu_1\dots\mu_s\nu_s} &= \frac{1}{4^s}\gamma_{\mu_1\nu_1}^{\alpha_1\beta_1}\dots \gamma_{\mu_s\nu_s}^{\alpha_s\beta_s} C^{(2s,0)}_{\alpha_1\beta_1\dots\alpha_s\beta_s} \ ; \\
C^R_{\mu_1\nu_1\dots\mu_s\nu_s} &= \frac{1}{4^s}\gamma_{\mu_1\nu_1}^{\dot\alpha_1\dot\beta_1}\dots \gamma_{\mu_s\nu_s}^{\dot\alpha_s\dot\beta_s} C^{(0,2s)}_{\dot\alpha_1\dot\beta_1\dots\dot\alpha_s\dot\beta_s} \ .
\end{split} \label{eq:C_tensor}\end{aligned}$$ The tensor field $C_{\mu_1\nu_1\dots\mu_s\nu_s}$ has the symmetries of a generalized Weyl tensor: it is totally traceless, antisymmetric within each $\mu_k\nu_k$ index pair, symmetric under the exchange of any two such pairs, and vanishes when antisymmetrized over any three indices. The right-handed and left-handed parts of $C_{\mu_1\nu_1\dots\mu_s\nu_s}$ are distinguished by their eigenvalues $\pm 1$ under a Hodge dualization of any $\mu_k\nu_k$ index pair: $$\begin{aligned}
-\frac{1}{2}\epsilon_{\mu_1\nu_1}{}^{\lambda\rho\sigma} x_\lambda C^{R/L}_{\rho\sigma\mu_2\nu_2\dots\mu_s\nu_s} = \pm C^{R/L}_{\rho\sigma\mu_2\nu_2\dots\mu_s\nu_s} \ , \label{eq:self_dual}\end{aligned}$$ where the minus sign on the LHS arises from the fact that the time component of $x_\lambda$ is negative.
Let us now write the field equations satisfied by the field strengths . The scalar field $C^{(0,0)}$ satisfies the wave equation for a conformally coupled massless scalar: $$\begin{aligned}
\nabla_\mu\nabla^\mu C^{(0,0)} = -2C^{(0,0)} \ , \label{eq:scalar_equation}\end{aligned}$$ while the fields with spin $s>0$ satisfy the free massless equations: $$\begin{aligned}
\nabla^{\alpha_1}{}_{\dot\beta}\,C^{(2s,0)}_{\alpha_1\alpha_2\dots\alpha_{2s}} = 0 \ ; \quad
\nabla_\beta{}^{\dot\alpha_1}\,C^{(0,2s)}_{\dot\alpha_1\dot\alpha_2\dots\dot\alpha_{2s}} = 0 \ . \label{eq:spinning_equation}\end{aligned}$$
The Penrose transform {#sec:linear_HS:Penrose}
---------------------
The Penrose transform [@Penrose:1986ca; @Ward:1990vs] is a closed-form general solution to the field equations - in terms of an even (but otherwise unconstrained) twistor function $F(Y)$. It is important that $F(Y)$ is a holomorphic function, i.e. without additional dependence on the complex-conjugate variable $\overline Y$; throughout this paper, we are taking this property of twistor functions for granted.
More specifically, each of the individual fields , i.e. each separate helicity, is captured by a twistor function $F(Y)$ of a particular *degree of homogeneity* $-2\pm 2s$. A general even function can be decomposed into eigenfunctions of the homogeneity operator $Y^a({\partial}/{\partial}Y^a)$, with even integer eigenvalues. In this way, a general even function $F(Y)$ contains a single free massless field of each helicity, i.e. precisely the higher-spin multiplet .
In the notations of this paper, the Penrose transform for each of the fields reads: $$\begin{aligned}
C^{(2s,0)}_{\alpha_1\dots\alpha_{2s}}(x) &= i\int_{P_R(x)} d^2u_R \left.\frac{{\partial}^s F_R(u_L+u_R)}{{\partial}u_L^{\alpha_1}\dots{\partial}u_L^{\alpha_{2s}}}\right|_{u_L = 0} \ ; \label{eq:Penrose_R_to_L} \\
C_{(0,2s)}^{\dot\alpha_1\dots\dot\alpha_{2s}}(x) &= i(-1)^s\int_{P_R(x)} d^2u_R\, u_R^{\dot\alpha_1}\dots u_R^{\dot\alpha_{2s}} F_R(u_R) \ , \label{eq:Penrose_R_to_R}\end{aligned}$$ where $F_R(Y)$ is an arbitrary even twistor function, and the factors of $i$ and $(-1)^s$ are for later convenience. The spin-0 field $C^{(0,0)}$ is contained in - a shared special case: $$\begin{aligned}
C^{(0,0)}(x) &= i\int_{P_R(x)} d^2u_R\, F_R(u_R) \ . \label{eq:Penrose_R_to_scalar}\end{aligned}$$ The $R$ subscript in $F_R(Y)$ is to indicate that the integrals in - are over the 2d spinor subspace $P_R(x)$. An alternative transform, using $P_L(x)$ instead, reads: $$\begin{aligned}
C_{(2s,0)}^{\alpha_1\dots\alpha_{2s}}(x) &= -i(-1)^s\int_{P_L(x)} d^2u_L\, u_L^{\alpha_1}\dots u_L^{\alpha_{2s}} F_L(u_L) \ ; \label{eq:Penrose_L_to_L} \\
C^{(0,2s)}_{\dot\alpha_1\dots\dot\alpha_{2s}}(x) &= -i\int_{P_L(x)} d^2u_L \left.\frac{{\partial}^s F_L(u_L+u_R)}{{\partial}u_R^{\dot\alpha_1}\dots{\partial}u_R^{\dot\alpha_{2s}}}\right|_{u_R = 0} \ , \label{eq:Penrose_L_to_R}\end{aligned}$$ where $F_L(Y)$ is again an arbitrary even twistor function, and we introduced an extra sign factor for later convenience. The transforms - can also be written in Dirac-spinor (i.e. twistor) indices, as: $$\begin{aligned}
\begin{split}
C^{(2s,0)}_{a_1\dots a_{2s}}(x) = i\int_{P_R(x)} d^2u \left.\frac{{\partial}^s F_R(U)}{{\partial}U^{a_1}\dots{\partial}U^{a_{2s}}}\right|_{U = u} = -i(-1)^s\int_{P_L(x)} d^2u\, u_{a_1}\dots u_{a_{2s}} F_L(u) \ ; \\
C^{(0,2s)}_{a_1\dots a_{2s}}(x) = i(-1)^s\int_{P_R(x)} d^2u\, u_{a_1}\dots u_{a_{2s}} F_R(u) = -i\int_{P_L(x)} d^2u \left.\frac{{\partial}^s F_L(U)}{{\partial}U^{a_1}\dots{\partial}U^{a_{2s}}}\right|_{U = u} \ .
\end{split}\end{aligned}$$ Here, the integrals automatically project the Dirac indices into the correct Weyl subspace in each case.
Proving that the fields - indeed satisfy the field equations - is rather straightforward. The main subtlety is the $x$-dependence of the spinor integration range, which must be taken into account when taking spacetime derivatives. This can be dealt with by same method as when deriving eq. , i.e. by performing a change of variables that shifts the $x$-dependence into the integrand. The details, in a slightly different language, can be found e.g. in [@Neiman:2013hca].
The above presentation of the Penrose transform differs somewhat from the one normally given in a twistor-theory textbook. The first difference is that that we’re starting in (A)dS spacetime, and treating twistors as the spinors of the isometry group $SO(1,4)$. Normally, one starts instead in flat spacetime, and treats twistors as the spinors of the conformal group $SO(2,4)$ (which, with our $EAdS_4$ signature, would actually be $SO(1,5)$). As far as the Penrose transform is concerned, this difference is merely superficial: both the transform and the free massless field equations are conformally covariant, so that Minkowski and (A)dS are equally good starting points. That being said, the unfolded, star-product-based formalism of the next subsection is *not* covariant under the 4d conformal group; there, the non-vanishing cosmological constant will be crucial.
Another difference between our presentation and the standard one is that the integrals in - are over ${\mathbb{C}}^2$ spinor subspaces (with measure $d^2u$), as opposed to their projective ${\mathbb{C}}{\mathbb{P}}^1$ versions (with measure $udu$). Thus, we are using the (well-known, but not as common) “non-projective” version of the transform. The projective vs. non-projective integrals are very closely related. In particular, the non-projective integrals - pick out the component of the twistor function $F_R(Y)$ with homogeneity $-2\pm 2s$ respectively, as one can show by rescaling the integration variable. This is the already-mentioned relation between helicity and the homogeneity of the twistor function. For a function $F_R(Y)$ of the “correct” homogeneity, the projective integral $u_R du_R$ will agree with the non-projective one up to numerical factors; essentially, the extra 1d integral in the non-projective case can be treated as $\int d\alpha/\alpha = \pm 2\pi i$. For a function $F_R(Y)$ of the “wrong” homogeneity, the projective integral is ill-defined, while the non-projective one evaluates to zero. Thus, the non-projective Penrose transform - is the same as the projective one, except that it allows us not to worry about mixing different spins/homogeneities in the integrand.
Finally, as we repeatedly discuss in this paper, integrals of the form - suffer from contour ambiguities. From the HS point of view, these are directly related to the analogous ambiguity in the integral definition of the star product. Due to these contour ambiguities, the Penrose transform is more properly defined in terms of sheaf cohomology. However, in keeping with the HS literature, we do not follow that more rigorous path, and instead continue working with ordinary functions, while keeping the ambiguity in mind. The advantage of this “naive” approach is that it allows us to treat $F_{L/R}(Y)$ and $C(x;Y)$ on an equal footing, as ordinary functions of the $Y$ variable.
Unfolded formulation and the higher-spin-covariant perspective {#sec:linear_HS:unfolded}
--------------------------------------------------------------
The next step is to rephrase the dynamics of our free massless fields in unfolded form. Let us introduce the full set of on-shell-inequivalent derivatives of the fields $C^{(2s,0)},C^{(0,2s)}$ for $s\geq 0$: $$\begin{aligned}
\begin{split}
\big(C^{(2s+k,k)}\big)_{\alpha_1\dots\alpha_{2s}\beta_1\dots\beta_k}{}^{\dot\beta_1\dots\dot\beta_k}
&= i^k\nabla_{(\beta_1}{}^{(\dot\beta_1}\dots\nabla_{\beta_k}{}^{\dot\beta_k)}\,C^{(2s,0)}_{\alpha_1\dots\alpha_{2s})} \ ; \\
\big(C^{(k,2s+k)}\big)^{\beta_1\dots\beta_k}{}_{\dot\beta_1\dots\dot\beta_k\dot\alpha_1\dots\dot\alpha_{2s}}
&= i^k\nabla^{(\beta_1}{}_{(\dot\beta_1}\dots\nabla^{\beta_k)}{}_{\dot\beta_k}\,C^{(0,2s)}_{\dot\alpha_1\dots\dot\alpha_{2s})} \ ,
\end{split} \label{eq:unfolding}\end{aligned}$$ where the factors of $i$ are for later convenience. We now have a field $C^{(m,n)}$ for every pair of integers $m,n$ such that $m+n$ is even, i.e. one field for every integer-spin representation of the bulk rotation group $SO(4)$. We can neatly package these into a single scalar master field $C(x;Y)$, which is an even function of the twistor coordinate $Y$: $$\begin{aligned}
\begin{split}
C(x;Y) &= \sum_{m,n} \frac{1}{m!n!}\, C^{(m,n)}_{\alpha_1\dots\alpha_m\dot\alpha_1\dots\dot\alpha_n}\,
y_L^{\alpha_1}\dots y_L^{\alpha_m}y_R^{\dot\alpha_1}\dots y_R^{\dot\alpha_n} \ ; \\
C^{(m,n)}_{\alpha_1\dots\alpha_m\dot\alpha_1\dots\dot\alpha_n}
&= (P_L)^{a_1}{}_{\alpha_1}\dots (P_L)^{a_m}{}_{\alpha_m}(P_R)^{a_{m+1}}{}_{\dot\alpha_1}\dots (P_R)^{a_{m+n}}{}_{\dot\alpha_n}\,
\left.\frac{{\partial}^{m+n} C}{{\partial}Y^{a_1}\dots{\partial}Y^{a_{m+n}}}\right|_{Y=0} \ ,
\end{split} \label{eq:master_field}\end{aligned}$$ where $y_{L/R} = P_{L/R}(x)Y$ are the chiral components of $Y$ at the point $x$. The field equations - and the definitions are all encapsulated in the following unfolded equation: $$\begin{aligned}
\nabla_\mu C = \frac{i}{2}\,C\star (y_L\gamma_\mu y_R) \ ,\end{aligned}$$ or, expressing $y_L$ and $y_R$ explicitly in terms of $Y$ and $x$: $$\begin{aligned}
\nabla_\mu C = \frac{i}{4}\,C\star (Y\gamma_\mu xY) \ . \label{eq:nabla_C}\end{aligned}$$ The star product in breaks down into three terms of the form $CYY$, $({\partial}C/{\partial}Y)Y$ and ${\partial}^2 C/{\partial}Y^2$. Among these, the $({\partial}C/{\partial}Y)Y$ piece accounts for the $x$ dependence of the chiral decomposition $Y = y_L+y_R$ in , while the other two account for the $x$ dependence of the component fields $C^{(m,n)}$ themselves. Specifically, the ${\partial}^2 C/{\partial}Y^2$ term encodes the flat-spacetime version of eqs. -, while the $CYY$ term corrects the second derivatives to account for the curvature of $EAdS_4$.
Having written the unfolded equation in the form , we recognize from that it is solved by the chiral delta functions $\delta^{L/R}_x(Y)$. Moreover, we see that the general solution can be expressed as: $$\begin{aligned}
C(x;Y) = F_R(Y)\star i\delta^R_x(Y) = i\int_{P_R(x)} d^2u_R\, F_R(Y + u_R)\,e^{iu_R Y} \ , \label{eq:C_solution_R}\end{aligned}$$ or, equivalently: $$\begin{aligned}
C(x;Y) = -F_L(Y)\star i\delta^L_x(Y) = -i\int_{P_L(x)} d^2u_L\, F_R(Y + u_L)\,e^{iu_L Y} \ , \label{eq:C_solution_L}\end{aligned}$$ where the $\pm i$ factors are chosen for later convenience, and we used eqs. - to obtain the explicit integral expressions. The spacetime-independent functions $F_{L/R}(Y)$ are Fourier transforms of each other: $$\begin{aligned}
F_R(Y) = -F_L(Y)\star\delta(Y) \ . \label{eq:F_LR}\end{aligned}$$ Moreover, using the decomposition , one can see that these functions are the same as the $F_{L/R}(Y)$ from section \[sec:linear\_HS:Penrose\]. Thus, we recognize eqs. - as the unfolded, HS-covariant formulation of the Penrose transform!
It may seem strange that the unfolded equation prefers $C\star(Y\gamma_\mu xY)$ over $(Y\gamma_\mu xY)\star C$. It turns out that the second possibility is in fact realized, if we replace $i\rightarrow -i$ in the definition of the unfolded fields. Equivalently, we can define an alternative master field $\tilde C$ as: $$\begin{aligned}
\tilde C(x;Y) &= \sum_{m,n} \frac{(-1)^m}{m!n!}\, C^{(m,n)}_{\alpha_1\dots\alpha_m\dot\alpha_1\dots\dot\alpha_n}\,
y_L^{\alpha_1}\dots y_L^{\alpha_m}y_R^{\dot\alpha_1}\dots y_R^{\dot\alpha_n} = C(x;y_R - y_L) = C(x;xY) \ ,\end{aligned}$$ for which the field equation and its solution read: $$\begin{aligned}
\nabla_\mu\tilde C &= -\frac{i}{4}(Y\gamma_\mu xY)\star\tilde C \ ; \label{eq:nabla_C_tilde} \\
\tilde C(x;Y) &= -i\delta^R_x(Y)\star F_R(Y) = i\delta^L_x(Y)\star F_L(Y) \ .\end{aligned}$$ In fact, it follows from that the Penrose transform $F_{L/R}(Y)$ is the same as in -.
Since $\delta^R_x(Y)$ and $\delta^L_x(Y)$ square to unity, we can explicitly solve for the master field at a point $x'$ in terms of the master field at a point $x$ via: $$\begin{aligned}
C(x';Y) = C(x;Y)\star\delta^R_x(Y)\star\delta^R_{x'}(Y) \ ,\end{aligned}$$ where the two-point product $\delta^R_x\star\delta^R_{x'} = \delta^L_x\star\delta^L_{x'}$ is given by the Gaussian : $$\begin{aligned}
\delta^R_x(Y)\star\delta^R_{x'}(Y) = \delta^L_x(Y)\star\delta^L_{x'}(Y) = \frac{2}{1 - x\cdot x'}\exp\frac{iYxx'Y}{2(x\cdot x' - 1)} \ . \label{eq:bulk_bulk_master}\end{aligned}$$ It is simultaneously a solution to the unfolded equation in $x'$ and to the “flipped” equation in $x$. The fact that the master field at $x'$ can be deduced from its value at a *single* point $x$ is a feature of the unfolded formalism.
Note that in all of the above, we did not require a higher-spin gauge connection. Instead, we directly wrote the linear field equations and their solutions in terms of gauge-invariant field strengths on the background $EAdS_4$ geometry. HS symmetry appears only as a *global* symmetry of the equations, parameterized by a spacetime-independent even function $\varepsilon(Y)$. The Penrose transform $F_R(Y)$ transforms under this symmetry in the adjoint: $$\begin{aligned}
\delta F_R = \varepsilon\star F_R - F_R\star\varepsilon \ ,\end{aligned}$$ and likewise for $F_L(Y)$. The master field $C(x;Y)$ transforms in the “twisted adjoint”: $$\begin{aligned}
\delta C = \varepsilon\star C - C\star\delta^R_x\star\varepsilon\star\delta^R_x \ ,\end{aligned}$$ where the product $\delta^R_x\star\varepsilon\star\delta^R_x = \delta^L_x\star\varepsilon\star\delta^L_x$ can be evaluated as in .
In section \[sec:CFT\], we will similarly describe the free $U(N)$ vector model (with external sources) in a language that renders global higher-spin symmetry manifest, while avoiding any gauge redundancy.
Antipodal symmetry {#sec:linear_HS:antipodal}
------------------
A special role is played by solutions with the antipodal symmetry: $$\begin{aligned}
C(-x;Y) = \pm C(x;Y) \ , \label{eq:C_antipodal}\end{aligned}$$ The antipodal map $x^\mu\rightarrow -x^\mu$ is the central element of the spacetime symmetry group $O(1,4)$. Under this map, the $EAdS_4$ hyperboloid is sent into its $x^0<0$ counterpart. Thus, strictly speaking, $C(-x;Y)$ is an analytic continuation of the solution $C(x,Y)$ into the antipodal $EAdS_4$. In the Poincare coordinates , the antipodal map corresponds to the operation $z\rightarrow -z$, which was invoked in the discussion [@Vasiliev:2012vf] of higher-spin holography. Indeed, as we will see in section \[sec:holography:asymptotics\], the two antipodal parities in directly correspond to the two types of asymptotic boundary data for each of the component fields in $C(x;Y)$ [@Vasiliev:2012vf]. A detailed analysis of this relation in the language of individual fields was carried out in [@Neiman:2014npa; @Halpern:2015zia] (see also [@Ng:2012xp]). The antipodal symmetry is also of significance in the de Sitter context [@Halpern:2015zia], as we will review in section \[sec:discuss\].
Let us now see how the symmetry is expressed at the level of spacetime-independent twistor functions. Plugging the identity $\delta^L_x(Y) = \delta^R_{-x}(Y)$ into the Penrose transform -, we obtain that is equivalent to any of the following: $$\begin{aligned}
F_L(Y) = \mp F_R(Y) \ \Longleftrightarrow \ F_R(Y)\star\delta(Y) = \pm F_R(Y) \ \Longleftrightarrow \ F_L(Y)\star\delta(Y) = \pm F_L(Y) \ . \label{eq:F_antipodal}\end{aligned}$$ Taking the Penrose transform of , we can also express as a star-product symmetry of $C(x;Y)$ at a single point $x$: $$\begin{aligned}
C(x;Y)\star\delta(Y) = \pm C(x;Y) \ . \label{eq:C_antipodal_delta}\end{aligned}$$ An arbitrary bulk solution $C(x;Y)$ can be decomposed into antipodally even and odd pieces in the sense of -. For the twistor functions $F_{L/R}(Y)$, as well as for $C(x;Y)$ viewed as a function of $Y$ at fixed $x$, the corresponding decomposition is accomplished by the projectors ${\mathcal{P}}_\pm(Y)$ from .
That being said, we must emphasize that conditions such as -, as well as the projectors ${\mathcal{P}}_\pm(Y)$, should be handled with caution, due to contour ambiguities in the star product, as well as in the delta functions $\delta(Y),\delta^{L/R}_x(Y)$ themselves. When in doubt, it is helpful to look back to the original condition in spacetime. We will now present a simple example that shows how - can fail to be well-defined linear properties, or, equivalently, how ${\mathcal{P}}_\pm(Y)$ can fail to be well-defined projectors.
Consider a conformally-coupled massless scalar field, with field equation . An important solution to this field equation is the boundary-to-bulk propagator $1/(\ell\cdot x)$. For $x^\mu$ timelike, i.e. on $EAdS_4$ and its antipodal image, this propagator is non-singular, and is odd under the antipodal map $x\rightarrow -x$. Now, consider a superposition of such propagators, obtained by integrating $\ell^\mu$ over an $S_3$ section of the ${\mathbb{R}}^{1,4}$ lightcone, i.e. over the 3-sphere $(\ell\cdot\ell = 0, \ \ell\cdot x_0 = -1)$, where $x_0$ is some future-pointing unit vector. This can be expressed as a conformally covariant $d^3\ell$ integral by inserting $1/(\ell\cdot x_0)^2$ into the integrand. The result reads: $$\begin{aligned}
\int \frac{d^3\ell}{(\ell\cdot x_0)^2}\,\frac{1}{(\ell\cdot x)} = 4\pi^2 \times \left\{
\begin{array}{ll}
\displaystyle 1/(x\cdot x_0 - 1) & \quad x\text{ future-pointing} \\
\displaystyle 1/(x\cdot x_0 + 1) & \quad x\text{ past-pointing}
\end{array} \right. \ , \label{eq:nonanalytic}\end{aligned}$$ where we recall that future-pointing vs. past-pointing $x^\mu$ correspond to points on the original $EAdS_4$ vs. the antipodal one. The key property of the bulk solution is that it’s still odd under $x\rightarrow -x$, but this is accomplished non-analytically: if we were to analytically continue the solution from future-pointing $x$ to past-pointing $x$, the result wouldn’t have a definite antipodal parity. As an aside, note that the RHS of is just a bulk-to-bulk propagator between $x_0$ and $x$. Therefore, is a simple example of the split representation [@Costa:2014kfa] of bulk-to-bulk propagators as boundary integrals.
Now, let us upgrade the statement to the master-field level. The master field for the boundary-to-bulk propagator $1/(\ell\cdot x)$ reads: $$\begin{aligned}
C_\ell(x;Y) = \frac{1}{2}\,\delta_\ell(Y)\star\delta_x^L(Y) = -\frac{1}{2}\,\delta_\ell(Y)\star\delta_x^R(Y) = \frac{1}{\ell\cdot x}\exp\frac{iY\ell xY}{2\ell\cdot x} \ .\end{aligned}$$ This is clearly antipodally odd, both in the spacetime sense of and in the star-product sense of : $$\begin{aligned}
C_\ell(x;Y)\star\delta(Y) = C_\ell(-x;Y) = -C_\ell(x;Y) \ . \end{aligned}$$ However, taking the linear superposition , we find: $$\begin{aligned}
\int \frac{d^3\ell}{(\ell\cdot x_0)^2}\,C_\ell(x;Y) = 4\pi^2 \times \left\{
\begin{array}{ll}
\displaystyle C^{(-)}_{x_0}(x;Y) & \quad x\text{ future-pointing} \\
\displaystyle C^{(+)}_{x_0}(x;Y) & \quad x\text{ past-pointing}
\end{array} \right. \ , \label{eq:nonanalytic_C}\end{aligned}$$ where $C^{(\pm)}_{x_0}(x;Y)$ is the master field corresponding to the bulk-to-bulk propagator $1/(x\cdot x_0 \pm 1)$, which we encountered in : $$\begin{aligned}
\begin{split}
C^{(-)}_{x_0}(x;Y) &= -\frac{1}{2}\,\delta^L_{x_0}(Y)\star\delta^L_x(Y) = -\frac{1}{2}\,\delta^R_{x_0}(Y)\star\delta^R_x(Y) = \frac{1}{x\cdot x_0 - 1}\exp\frac{iYx_0 xY}{2(x\cdot x_0 - 1)} \ ; \\
C^{(+)}_{x_0}(x;Y) &= +\frac{1}{2}\,\delta^R_{x_0}(Y)\star\delta^L_x(Y) = +\frac{1}{2}\,\delta^L_{x_0}(Y)\star\delta^R_x(Y) = \frac{1}{x\cdot x_0 + 1}\exp\frac{iYx_0 xY}{2(x\cdot x_0 + 1)} \ .
\end{split} \label{eq:nonanalytic_C_details}\end{aligned}$$ We now see that while the spacetime antipodal symmetry $C(-x;Y) = -C(x;Y)$ is preserved by the superposition , its star-product analogue $C(x;Y)\star\delta(Y) = -C(x;Y)$ is not. We also see exactly why this happens: the master field $C^{(\pm)}_{x_0}(x;Y)$ on each branch contains only the local Taylor series of the bulk solution. Therefore, the master field at each $x$ “sees” the analytic continuation of the bulk solution from the neighborhood of $x$, which does *not* have definite antipodal parity, instead of seeing the antipodally odd, but nonanalytic, global superposition -.
Higher-spin representation of spacetime symmetries and the Penrose transform {#sec:spacetime_subgroup}
============================================================================
As we’ve seen in eq. , the quadratic elements $Y_a Y_b$ of the higher-spin algebra generate the spacetime symmetry group $SO(1,4)$. In this section, we consider the finite group elements that arise by exponentiating these generators (the completion of $SO(1,4)$ into $O(1,4)$ will be addressed in section \[sec:spacetime\_subgroup:HS:antipodal\]). In the process, we will clarify the role of the delta functions $\delta(Y),\delta^{L/R}_x(Y),\delta_\ell(Y)$ with respect to spacetime symmetries. This in turn will lead us to the geometric interpretation - of the Penrose transform as a square root of CPT.
Clifford algebra {#sec:spacetime_subgroup:clifford}
----------------
As mentioned in eq. , HS algebra is just a simple variation on Clifford algebra, where the vector $\gamma_\mu$ subject to anticommutation relations is replaced with a twistor $Y_a$ subject to commutation relations. Correspondingly, our analysis below will closely mirror the well-known geometric properties of Clifford algebra (for a particularly spirited review of these, see [@GeometricAlgebra]). In Clifford algebra, commutation with the infinitesimal generators $\gamma_{[\mu}\gamma_{\nu]}/2$ realizes the standard action of the orthogonal group – in our case, $SO(1,4)$: $$\begin{aligned}
\left[\frac{1}{2}\gamma_{[\nu}\gamma_{\rho]}, \gamma^{\mu_1}\dots\gamma^{\mu_n} \right]
= 2\left(\delta^{\mu_1}_{[\nu}\gamma_{\rho]}\gamma^{\mu_2}\!\dots\gamma^{\mu_n} + \ldots + \gamma^{\mu_1}\!\dots\gamma^{\mu_{n-1}}\delta^{\mu_n}_{[\nu}\gamma_{\rho]}\right) \ . \label{eq:infinitesimal_Clifford}\end{aligned}$$ Alternatively, instead of starting with infinitesimal generators, one can construct $SO(1,4)$ out of some fundamental *finite* transformations. In particular, the adjoint action of $x = x^\mu\gamma_\mu$ is a reversal of the subspace orthogonal to a unit vector $x^\mu$. In our conventions, with $x^\mu$ timelike, this reads explicitly as: $$\begin{aligned}
x\,\gamma^{\mu_1}\!\dots\gamma^{\mu_n} x = \tilde\gamma^{\mu_1}\!\dots\tilde\gamma^{\mu_n} \ , \quad \text{where} \quad \tilde\gamma^\mu \equiv -(\delta^\mu_\nu + 2x^\mu x_\nu)\gamma^\nu \ . \label{eq:reflection_Clifford}\end{aligned}$$ By combining two such reversals with respect to a pair of axes $x,x'$, one obtains a finite rotation (or boost) by *twice the angle* between $x$ and $x'$: $$\begin{aligned}
xx'\gamma^{\mu_1}\!\dots\gamma^{\mu_n} x'x = \tilde\gamma^{\mu_1}\!\dots\tilde\gamma^{\mu_n} \ , \quad \text{where} \quad
\tilde\gamma^\mu \equiv \left(\delta^\mu_\nu + 2x^\mu x_\nu + 2x'^\mu x'_\nu + 4(x\cdot x')x^\mu x'_\nu \right)\gamma^\nu \ . \label{eq:finite_rotation_Clifford}\end{aligned}$$ In particular, a rotation by $\pi$ (in a spacelike plane) can be represented by $xx'$ with $x'$ perpendicular to $x$. A rotation by $2\pi$, represented by the algebra element $xx' = -1$, is obtained via $x' = -x$. The infinitesimal generators $\gamma_{[\mu}\gamma_{\nu]}/2$ can be obtained by expanding around $x=x'$.
In odd dimensions, such as our case with the embedding space ${\mathbb{R}}^{1,4}$, one can also go in the opposite direction, and derive the reflection by exponentiating the infinitesimal generators. The way to do this in ${\mathbb{R}}^{1,4}$ is to rotate by $\pi$ in a pair of planes orthogonal both to $x^\mu$ and to each other. If the rotation is performed in both planes at once, then, depending on the planes’ orientation, it will belong to either the left-handed or the right-handed subgroup of the 4d rotations $SO(4) = SO(3)_L\times SO(3)_R$ around $x^\mu$. We then obtain either $x$ or $-x$ as the reflection operator. When used in the adjoint, both $x$ and $-x$ produce the same reflection .
For comparison with the higher-spin case below, let us perform this calculation explicitly. We choose a frame such that $x^\mu = e_0^\mu$, and use the representation for the gamma matrices. Now, consider e.g. a right-handed rotation along the bivector $e_1\wedge e_2 + e_3\wedge e_4$. A rotation by an infinitesimal angle $\varepsilon$ in each of the two planes $e_1\wedge e_2$ and $e_3\wedge e_4$ is represented in Clifford algebra by: $$\begin{aligned}
1 + \frac{\varepsilon}{2}(\gamma_1\gamma_2 + \gamma_3\gamma_4) = 1 - i\varepsilon\begin{pmatrix} 0 & 0 \\ 0 & \sigma_3 \end{pmatrix} \ .\end{aligned}$$ Exponentiating, we obtain the operator for rotation by a finite angle $\theta$: $$\begin{aligned}
g_\theta = \exp\left(-i\theta\begin{pmatrix} 0 & 0 \\ 0 & \sigma_3 \end{pmatrix}\right) = \begin{pmatrix} 1 & 0 \\ 0 & \cos\theta - i\sin\theta\,\sigma_3 \end{pmatrix} \ . \label{eq:RH_Clifford}\end{aligned}$$ In particular, for $\theta = \pi$, we get the operator: $$\begin{aligned}
g_\pi = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = -\gamma_0 = -x \ .\end{aligned}$$ Similarly, a left-handed rotation along the bivector $e_1\wedge e_2 - e_3\wedge e_4$ will produce $+x$ as the reflection operator. Note that $g_{2\pi} = 1$, as it should be: while a single $2\pi$ rotation need not take us back to the identity, a combination of $2\pi$ rotations in a *pair* of planes must always do so.
So far, we considered the *adjoint* action $g\Gamma g^{-1}$ in Clifford algebra, where $g$ represents an $SO(1,4)$ group element. As we saw in or , this realizes the standard action of $SO(1,4)$ on the algebra element $\Gamma$, which consists of spin-0 and spin-1 pieces. The next natural question is what happens if one acts instead in the fundamental, i.e. simply via $g\Gamma$. The answer, of course, is that this transformation law describes spinors (or, in our case, twistors). In particular, the reflection is realized on twistors as $U\rightarrow \pm xU$.
When describing spinors from within the Clifford algebra itself (as Cartan had done originally), the geometric structure of $SO(1,4)$ becomes obscured. That is why it’s better to introduce separate indices for spinors, and to develop geometric intuition about spinor space in its own right. As we will see below, the situation in higher-spin algebra is different: there, the $SO(1,4)$ can be made manifest not only in the algebra’s adjoint representation, but also in the fundamental. In both the Clifford and HS cases, while the adjoint action of $SO(1,4)$ can be formulated on the individual vector $\gamma_\mu$ or twistor $Y_a$, the fundamental action mixes different powers of these objects.
Higher-spin algebra {#sec:spacetime_subgroup:HS}
-------------------
Let us now perform the analogous analysis for HS algebra in place of Clifford algebra. Since infinitesimal $SO(1,4)$ rotations are generated by $Y_a Y_b$, finite rotations will be generated, via exponentiation, by Gaussian functions. In addition, we will find that various reflections are represented by delta functions.
As a first step, let us identify the higher-spin analog of $x^\mu\gamma_\mu$ – the reflection that reverses the subspace orthogonal to $x^\mu$. As discussed above, we can construct this group element from the infinitesimal generators by performing a left-handed or right-handed rotation by $\pi$ in a pair of totally orthogonal planes. In the higher-spin algebra, such rotations are generated by the bilinears $y^a_L y^b_L$ or $y^a_R y^b_R$, respectively. As before, we fix $x^\mu = e_0^\mu$, and use the representation for the gamma matrices. A twistor $Y^a$ can now be decomposed as: $$\begin{aligned}
Y^a = \begin{pmatrix} y_L^0 \\ y_L^1 \\ y_R^0 \\ y_R^1 \end{pmatrix} \ ,\end{aligned}$$ where the top and bottom halves correspond to the left-handed and right-handed parts of $Y^a$ at $x^\mu$. Now, consider again a right-handed rotation along the bivector $e_1\wedge e_2 + e_3\wedge e_4$. In our chosen basis, we read off from eq. that a rotation by an infinitesimal angle $\varepsilon$ in each of the two planes $e_1\wedge e_2,e_3\wedge e_4$ is represented in HS algebra by $1 + (\varepsilon/2)y_R^0 y_R^1$. Exponentiating with the star product, we obtain the operator for rotation by a finite angle $\theta$: $$\begin{aligned}
R_\theta(Y) = \exp_\star\left(\frac{\theta}{2}\,y_R^0\, y_R^1 \right) = \frac{1}{\cos(\theta/2)}\exp\left(\tan\frac{\theta}{2}\,y_R^0\, y_R^1\right) \ . \label{eq:RH_Gaussian}\end{aligned}$$ One can verify this formula for the star-exponential by differentiating both sides with respect to $\theta$. Note the appearance of $\theta/2$ in eq. , as opposed to $\theta$ in its Clifford-algebra analog . In particular, for $\theta = 2\pi$, we get: $$\begin{aligned}
R_{2\pi}(Y) = -1 \ . \label{eq:sign_inconsistency}\end{aligned}$$ This signals a problem: even in a spinor representation, a $2\pi$ rotation in a *pair* of planes must return the identity. To resolve this contradiction, we must recall from section \[sec:geometry:twistors:integrals\] that the star product of Gaussians is only defined *up to sign*. Now that we understand Gaussians as $Spin(1,4)$ elements, eq. is teaching us that there is *no globally consistent way* to fix this sign ambiguity. For instance, one attempt to fix the sign ambiguity may be to define “the” Gaussian representing a $Spin(1,4)$ element as the one obtained by the shortest direct route from the identity, i.e. by exponentiating a generator through the smallest possible angle. However, this definition would break down for precisely the case we’re interested in: the reversal $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ of a 4d subspace in ${\mathbb{R}}^{1,4}$, which may be realized as a rotation with $\theta=\pi$.
Consider, then, the rotation with $\theta = \pi$. In this limit, the coefficients both outside and inside the exponent diverge, and the Gaussian becomes a delta function over the $y_R$ spinor space: $R_\pi(Y) \sim \delta^R_x(Y)$. To find the normalization, we integrate over $y_R$ using eq. : $$\begin{aligned}
\int R_\theta(y_R)\,d^2y_R = \frac{\pm i}{\sin(\theta/2)} \ .\end{aligned}$$ Thus, in the limit $\theta = \pi$, the integral is $\pm i$, and we identify the reflection operator as: $$\begin{aligned}
R_\pi(Y) = \pm i\delta^R_x(Y) \ . \label{eq:reflection_delta}\end{aligned}$$ As we will shortly see, the sign ambiguity here cannot be fixed. If we were to construct the reflection $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ via a left-handed rotation, we would instead get $\pm i\delta^L_x(Y)$ as the reflection operator. In other words, $\pm i\delta^L_x(Y)$ and $\pm i\delta^R_x(Y)$ in higher-spin algebra play the same geometric role as do $x$ and $-x$ in Clifford algebra. In fact, we’ve already seen in that the adjoint action of $\pm i\delta^{L/R}_x(Y)$ directly realizes the reflection $Y\rightarrow \pm xY$: $$\begin{aligned}
\begin{split}
\left(\pm i\delta^L_x(Y)\right)\star f(Y)\star \left(\pm i\delta^L_x(Y)\right)^{-1}_\star &= \delta^L_x(Y)\star f(Y)\star\delta^L_x(Y) = f(xY) \ ; \\
\left(\pm i\delta^R_x(Y)\right)\star f(Y)\star \left(\pm i\delta^R_x(Y)\right)^{-1}_\star &= \delta^R_x(Y)\star f(Y)\star\delta^R_x(Y) = f(-xY) \ .
\end{split} \label{eq:reflection_HS}\end{aligned}$$ As with eq. in Clifford algebra, we can now take the reflection as the fundamental geometric operation in place of the infinitesimal generators $Y_a Y_b$. In particular, the product of two reflections with respect to the unit vectors $x,x'$ gives a rotation in the corresponding plane by twice the angle between $x$ and $x'$: $$\begin{aligned}
\left(\pm i\delta^L_x(Y)\right)\star\left(\mp i\delta^L_{x'}(Y)\right) = \left(\pm i\delta^R_x(Y)\right)\star\left(\mp i\delta^R_{x'}(Y)\right) = \frac{2}{1 - x\cdot x'}\exp\frac{iYxx'Y}{2(x\cdot x' - 1)} \ .\end{aligned}$$ Note that in order to recover the identity in the limit $x=x'$, we must choose opposite signs for the two reflection operators. This demonstrates that the sign ambiguity cannot be fixed consistently.
As we can see, HS algebra is a kind of square root of Clifford algebra. In a sense, this is already clear from the definitions , since spinors are the “square roots” of vectors. However, the “square root” relationship between the algebras is more concrete than that. Indeed, we see e.g. in that the *adjoint* action of HS algebra realizes the *fundamental* action of Clifford algebra on the twistor $Y$. We also saw the angle $\theta/2$ appearing in as opposed to $\theta$ in , which led to a sign ambiguity *on top of* the ordinary double cover $SO(1,4)\rightarrow Spin(1,4)$.
We are now ready to apply this section’s geometric viewpoint to linearized higher-spin gravity. We recognize immediately that while the adjoint action of $\pm i\delta^{L/R}(Y)$ directly realizes the reflection $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ on $Y$, the fundamental action - realizes the Penrose transform. In this sense, the Penrose transform is the “square root” of a reflection. While the adjoint reflection acts on the argument $Y$, the Penrose transform acts on functions $f(Y)$ as a whole. This must be the case, since the $SO(1,4)$ transformation of an individual twistor does not have a square root: twistors are *already* a square root of ${\mathbb{R}}^{1,4}$ vectors.
For the final touch to our interpretation of the Penrose transform, we should spell out the spacetime significance of the reflection $-(\delta^\mu_\nu + 2x^\mu x_\nu)$. So far, the vector $x^\mu$ has been timelike, representing a radius vector on the $EAdS_4$ hyperboloid. However, eventually, the more physical case is Lorentzian $dS_4$ spacetime, given in ${\mathbb{R}}^{1,4}$ by *spacelike* unit vectors $x^\mu$. There, the subspace orthogonal to $x^\mu$ is the $dS_4$ tangent space at the point $x$, and the reflection $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ is the de Sitter analog of CPT, with $x$ as the origin. Our statement now follows: the Penrose transform is a square root of CPT.
### Rotations by $2\pi$ and the antipodal map {#sec:spacetime_subgroup:HS:antipodal}
So far in this section, we’ve been careful to distinguish $SO(1,4)$ from the full $O(1,4)$. The geometric transformations we’ve constructed up to now only cover $SO(1,4)$, i.e the even elements of $O(1,4)$. This includes the CPT reflections ,, since they reverse an even number of axes. To enlarge our scope to the full $O(1,4)$, we must add to our menu its central element: the antipodal map $x^\mu\rightarrow -x^\mu$, which reverses all 5 axes in ${\mathbb{R}}^{1,4}$. On a single twistor $Y^a$, the only way to represent this transformation non-trivially is by complex conjugation, which we will not consider here. With that option closed, we must resort, as with the Penrose transform, to acting on whole functions $f(Y)$. In fact, in section \[sec:linear\_HS:antipodal\], we’ve already seen how this happens – the antipodal map on bulk master fields $C(x;Y)$ is realized by multiplying either $C(x;Y)$ itself or its Penrose transform by $\delta(Y)$: $$\begin{aligned}
F_{L/R}(Y)\rightarrow F_{L/R}(Y)\star\delta(Y) \quad \Longleftrightarrow \quad C(x;Y) \rightarrow C(-x;Y) = C(x;Y)\star\delta(Y) \ .\end{aligned}$$ This follows from decomposing $\delta(Y) = \delta_x^L(Y)\star\delta_x^R(Y)$, which can be interpreted as the Penrose transform at $x$ followed by the inverse transform at $-x$. Thus, while $SO(1,4)$ is manifestly realized by the adjoint action of HS algebra, the antipodal map is realized by acting with $\delta(Y)$ in the fundamental.
To complete the picture, it remains to understand the geometric role of $\delta(Y)$ *when acting in the adjoint*. As we can see from , the answer is simply a $2\pi$ rotation: $$\begin{aligned}
\delta(Y)\star f(Y)\star\delta(Y) = f(-Y) \ .\end{aligned}$$ This can again be understood in terms of the decomposition $\delta(Y) = \delta_x^L(Y)\star\delta_x^R(Y)$: the product of two $\pi$ rotations along e.g. the bivectors $e_1\wedge e_2 + e_3\wedge e_4$ and $e_1\wedge e_2 - e_3\wedge e_4$ is simply a $2\pi$ rotation along $e_1\wedge e_2$.
In light of the above two roles of $\delta(Y)$, one can rephrase the $\sqrt{\text{CPT}}$ nature of the Penrose transform as follows: *the Penrose transform is to CPT as the antipodal map is to a $2\pi$ rotation*.
The null limit {#sec:spacetime_subgroup:null}
--------------
So far, we’ve considered spacetime symmetries through the lens of reflections around timelike (or spacelike) vectors $x^\mu$. As we’ve seen, this geometry relates naturally to the bulk higher-spin theory. To discuss the boundary theory, we must take the limit , where the reflection vector $x^\mu$ becomes null. The reflection matrix $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ then becomes: $$\begin{aligned}
-(\delta^\mu_\nu + 2x^\mu x_\nu) \ \longrightarrow \ \frac{2}{z^2}\left(-\ell^\mu\ell_\nu + O(z^2) \right) \ . \label{eq:null_reflection}\end{aligned}$$ The leading-order part of this matrix, renormalized so as to make it finite, is the degenerate “reflection matrix” $-\ell^\mu\ell_\nu$, which projects any vector onto $\ell^\mu$. Combining two such “reflections” with respect to a pair of null vectors $\ell,\ell'$, we get the matrix: $$\begin{aligned}
(-\ell^\mu\ell_\rho)(-\ell'^\rho\ell'_\nu) \sim -\ell^\mu\ell'_\nu \ . \label{eq:infinite_boost}\end{aligned}$$ Treating $\ell,\ell'$ as the null limits of highly boosted timelike vectors $x,x'$, we recognize the matrix as a boost by an infinite angle in the $\ell\wedge\ell'$ plane (again, renormalized for finiteness). This boost shrinks $\ell'^\mu$ to zero, stretches $\ell^\mu$ to infinity, and leaves untouched the subspace orthogonal to both. As a result, the renormalized matrix $-\ell^\mu\ell'_\nu$ leaves the $\ell^\mu$ component finite, while annihilating both the $\ell'^\mu$ and orthogonal components.
The degenerate “reflections” $-\ell^\mu\ell_\nu$ and “infinite boosts” $-\ell^\mu\ell'_\nu$ satisfy a “forgetful property”: any linear operation sandwiched between two reflections is reduced to the corresponding boost . Explicitly, for any matrix $M^\mu{}_\nu$, we trivially have: $$\begin{aligned}
(-\ell^\mu\ell_\rho)M^\rho{}_\sigma(-\ell'^\sigma\ell'_\nu) \sim -\ell^\mu\ell'_\nu \ . \label{eq:forgetful_matrix}\end{aligned}$$
In Clifford algebra, the analog of the matrix $-\ell^\mu\ell_\nu$ is the algebra element $\ell = \ell^\mu\gamma_\mu$; in higher-spin algebra, the corresponding element is the boundary spinor delta function $\delta_\ell(Y)$. These algebra elements and their products do not quite represent $SO(1,4)$ transformations, but renormalized limiting cases thereof. In fact, the renormalization is different in the different algebras: $-\ell^\mu\ell_\nu$, $\ell$ and $\delta_\ell(Y)$ all scale differently with $\ell^\mu$. Thus, it is tempting to apply the geometry of - in the context of Clifford or HS algebra, but one must be mindful that not every property might carry over.
It turns out that the HS algebra element $\delta_\ell(Y)$ closely resembles in its properties the “null reflection” matrix $-\ell^\mu\ell_\nu$, while the Clifford algebra element $\ell$ does not. In Clifford algebra, the analog of the “forgetful property” does not hold: the products $\ell\Gamma\ell'$ are not all proportional to each other, but span a 4d subspace, parameterized by varying $\Gamma$ over the Clifford algebra of the 3d hyperplane orthogonal to $\ell,\ell'$. In contrast, in HS algebra, the “forgetful property” *does* hold, as we’ve seen in eq. : all products of the form $\delta_\ell\star f\star\delta_{\ell'}$ are proportional to each other, since they must lie in the intersection of the subalgebras ${\mathcal{A}}_+(\ell)\cap{\mathcal{A}}_-(\ell')$. In particular, the product of three “null reflections” behaves similarly in spacetime and in HS algebra: $$\begin{aligned}
\begin{split}
(-\ell^\mu\ell_\rho)(-\ell'^\rho\ell'_\sigma)(-\ell''^\sigma\ell''_\nu) &= -(\ell\cdot\ell')(\ell'\cdot\ell'')\ell^\mu\ell''_\nu \\
&\text{vs.} \\
\delta_\ell(Y)\star\delta_{\ell'}(Y)\star\delta_{\ell''}(Y) &= \pm i\sqrt{-\frac{\ell\cdot\ell''}{2(\ell\cdot\ell')(\ell'\cdot\ell'')}}\,\delta_\ell(Y)\star\delta_{\ell''}(Y) \ ,
\end{split}\end{aligned}$$ where the different proportionality coefficients arise from the different scaling properties of $-\ell^\mu\ell_\nu$ and $\delta_\ell(Y)$.
Manifest $SO(1,4)$ in the higher-spin fundamental {#sec:spacetime_subgroup:manifest}
-------------------------------------------------
We are now ready to understand the geometric action of CPT – and thus of the entire $SO(1,4)$ – on the boundary two-point product in the higher-spin fundamental. In other words, we wish to calculate the action $\delta_\ell\star\delta_{\ell'}\star\delta^R_x$ of the CPT operator $\delta^R_x(Y)$ on the boundary two-point product $\delta_\ell(Y)\star\delta_{\ell'}(Y)$. The first step is to notice that the boundary-bulk product $\delta_{\ell'}\star\delta^R_x$ can be reduced to a boundary-boundary product. Specifically, we can read off from - the identity (switching temporarily from $\ell'$ to $\ell$ to simplify notations): $$\begin{aligned}
\delta_\ell(Y)\star\delta^R_x(Y) = -2(\ell\cdot x)\,\delta_\ell(Y)\star\delta_{\tilde\ell}(Y) \ , \label{eq:x_ell_to_ell_ell}\end{aligned}$$ where $\tilde\ell^\mu$ is the result of the CPT reflection $-(\delta^\mu_\nu + 2x^\mu x_\nu)$ acting on the null vector $\ell^\mu$: $$\begin{aligned}
\tilde\ell^\mu = -\ell^\mu - 2(\ell\cdot x)x^\mu \ . \label{eq:ell_tilde}\end{aligned}$$ In the context of $EAdS_4$ geometry, $\tilde\ell$ is the second boundary endpoint of the geodesic that begins at $\ell$ and passes through $x$.
Within the geometric framework of this section, the equality is not surprising. Up to renormalization, the product $\delta_\ell(Y)\star\delta^R_x(Y)$ represents an infinite boost in the timelike plane $\ell\wedge x$. The same boost can also be represented by $\delta_\ell(Y)\star\delta_{\tilde\ell}(Y)$, where $\tilde\ell$ is the second null vector in this plane. That is precisely the statement of eqs. -. Returning now to the task of calculating $\delta_\ell\star\delta_{\ell'}\star\delta^R_x$, we use and then to find: $$\begin{aligned}
\begin{split}
\delta_\ell(Y)\star\delta_{\ell'}(Y)\star\delta^R_x(Y) &= -2(\ell'\cdot x)\,\delta_\ell(Y)\star\delta_{\ell'}(Y)\star\delta_{\tilde\ell'}(Y)
= \pm i\sqrt{\frac{\ell\cdot\tilde\ell'}{\ell\cdot\ell'}}\,\delta_\ell(Y)\star\delta_{\tilde\ell'}(Y) \ .
\end{split}\end{aligned}$$ We have thus confirmed the first equation in : $$\begin{aligned}
K(\ell,\ell';Y)\star i\delta^R_x(Y) = \pm K(\ell,\tilde\ell';Y) \ , \label{eq:K_sqrt_CPT_1}\end{aligned}$$ where: $$\begin{aligned}
K(\ell,\ell';Y) \sim \sqrt{-\ell\cdot\ell'}\,\delta_\ell(Y)\star\delta_{\ell'}(Y) \ . \label{eq:K_raw2}\end{aligned}$$ For multiplication on the left, we similarly derive: $$\begin{aligned}
i\delta^R_x(Y)\star K(\ell,\ell';Y) = \pm K(\tilde\ell,\ell';Y) \ , \label{eq:K_sqrt_CPT_2}\end{aligned}$$ and identical formulas hold for $\delta^L_x(Y)$ in place of $\delta^R_x(Y)$. Note that the sign ambiguities in , cannot be consistently resolved. If we insisted on choosing a particular sign, then applying e.g. eq. twice, we would find a contradiction with the identity $\delta^R_x\star\delta^R_x = +1$.
As promised, we see that the CPT reflection operators $\pm i\delta^{R/L}_x(Y)$, acting on $K(\ell,\ell';Y)$ in the higher-spin fundamental, have the effect of applying CPT to *one* of the two boundary points $\ell,\ell'$. Thus, when acting on these bilocals, the Penrose transform is manifestly a square root of CPT. Furthermore, since all of $SO(1,4)$ can be constructed by combining reflections around different points $x$, we conclude that the same is true for a *general* $SO(1,4)$ operator $g$: acting with $g$ on $K(\ell,\ell';Y)$ in the higher-spin fundamental will result in the corresponding $SO(1,4)$ transformation of one of the two points $\ell,\ell'$. We can verify this directly by applying the $SO(1,4)$ generators to find the result quoted in : $$\begin{aligned}
\begin{split}
M_{\mu\nu}\star K(\ell,\ell';Y) &= \ell_\mu\frac{{\partial}K}{{\partial}\ell^\nu} - \ell_\nu\frac{{\partial}K}{{\partial}\ell^\mu} \ ; \\
-K(\ell,\ell';Y)\star M_{\mu\nu} &= \ell'_\mu\frac{{\partial}K}{{\partial}\ell'^\nu} - \ell'_\nu\frac{{\partial}K}{{\partial}\ell'^\mu} \ .
\end{split} \label{eq:K_sqrt_infinitesimal}\end{aligned}$$ Note that in this case, there are no sign ambiguities in the star products, since $M_{\mu\nu}\sim Y\gamma_{\mu\nu}Y$ is polynomial in $Y$.
The CFT in higher-spin-covariant twistor language {#sec:CFT}
=================================================
Overview
--------
In this section, we express the 3d free $U(N)$ vector model in twistor language, making its higher-spin conformal invariance manifest. We represent the conformal 3-sphere on which the CFT lives as the projective lightcone $(\ell_\mu\ell^\mu = 0,\ \ell^0 > 0,\ \ell^\mu\cong\lambda\ell^\mu)$ in ${\mathbb{R}}^{1,4}$. Thus, we are using the “embedding-space formalism” for CFT (see e.g. [@Weinberg:2010fx]). In section \[sec:CFT:local\], we express our CFT in the standard language of local operators and sources. In section \[sec:CFT:bilocal\], as a first step towards the twistor formalism, we express the theory and its correlators at separated points in a *bilocal* language. For the very special case of a free vector model, this language is more natural than the standard local one, because all single-trace operators are quadratic in the fundamental fields. Our bilocal formalism is inspired by the one in [@Das:2003vw]. However, unlike the authors of [@Das:2003vw], we do not treat the bilocal operators as a new “fundamental” field. Instead, we treat them straightforwardly as composite operators, coupled to bilocal sources. Of course, these quadratic CFT operators *do* become fundamental fields once we switch to the bulk description. In this sense, the CFT is a “square root” of the bulk theory. The results of the present section can be viewed as a consequence of this “square root” relation, combined with the “square root” relation - between the Penrose transform and CPT.
The local and bilocal languages for the CFT share some qualitative features. The bilocal sources, like the local gauge potentials, are gauge-redundant (in fact, their gauge redundancy is even larger). Conversely, the bilocal operators, like the local conserved currents, satisfy constraints. Finally, as discussed in section \[sec:summary:contact\], in a region with non-vanishing (either local or bilocal) sources, one would need contact terms to obtain finite and conserved expectation values for the local currents.
In section \[sec:CFT:twistor\], having established the bilocal language, we use it as a springboard towards a fully nonlocal, twistorial formulation of the CFT. In this formulation, the sources are no longer gauge-redundant, while the single-trace “currents” are constraint-free. At the same time, the theory’s global higher-spin symmetry becomes manifest. Our transform between the twistor and bilocal formulations is a boundary version of the bulk Penrose transform. Since the CFT is free even when the bulk is interacting, this boundary/twistor transform allows us to express the full partition function in the twistor language. Finally, as we discuss in section \[sec:holography:general\_currents\], the twistor language appears to automatically include all the necessary contact terms, so that we end up with conserved currents even at finite sources.
Local language {#sec:CFT:local}
--------------
We begin with the action of $N$ free massless scalars in the fundamental representation of an internal $U(N)$ symmetry: $$\begin{aligned}
S_{\text{CFT}} = -\int d^3\ell\,\bar\phi_I\Box\phi^I \ . \label{eq:S_free}\end{aligned}$$ Here, $I=1,\dots,N$ is an internal index; $\phi^I$ and their complex conjugates $\bar\phi_I$ are dynamical fields with conformal weight $\Delta=1/2$. We consider only $U(N)$ singlets to be observable (for example, one might imagine that the $U(N)$ is gauged with a very weak coupling). The single-trace primaries of the theory consist of an infinite tower of conserved currents $j^{(s)}$, one for each spin $s$. To write these out explicitly, we can use a flat 3d frame as in , with 3d spatial indices $(i,j,k,\dots)$. Then the spin-$s$ current $j^{(s)}$ reads [@Craigie:1983fb; @Anselmi:1999bb]: $$\begin{aligned}
j^{(s)}_{k_1\dots k_s} = \frac{1}{(2i)^s}\,\bar\phi_I
\left(\sum_{m=0}^s (-1)^m \binom{2s}{2m} \overset{\leftarrow}{{\partial}}_{(k_1}\dots\overset{\leftarrow}{{\partial}}_{k_m} \overset{\rightarrow}{{\partial}}_{k_{m+1}}\dots\overset{\rightarrow}{{\partial}}_{k_s)} - \text{traces}\right) \phi^I \ .
\label{eq:explicit_j}\end{aligned}$$ Here, the $1/i^s$ prefactor ensures that $j^{(s)}$ is real, while the $1/2^s$ prefactor is chosen for agreement with the bulk asymptotics in section \[sec:holography:example\_currents\]. We include in also the case $s=0$, i.e. the scalar “current” $j^{(0)} = \bar\phi_I\phi^I$. The spin-1 current $j^{(1)}_i$ is $1/2$ times the ordinary charge current for the $U(1)$ component of $U(N)$: $$\begin{aligned}
j^{(1)}_i = \frac{1}{2i}\,\bar\phi_I\overset{\leftrightarrow}{\partial}_i\phi^I \ ,\end{aligned}$$ while the spin-2 current $j^{(2)}_{ij}$ is $2$ times the theory’s stress-energy tensor: $$\begin{aligned}
j^{(2)}_{ij} = 2T_{ij} \ ; \quad T_{ij} = -\frac{1}{8}\left(\bar\phi_I{\partial}_i{\partial}_j\phi^I + \phi^I{\partial}_i{\partial}_j\bar\phi_I - 6{\partial}_{(i}\bar\phi_I{\partial}_{j)}\phi^I + 2g_{ij}{\partial}_k\bar\phi_I{\partial}^k\phi^I \right) \ . \label{eq:stress_tensor}\end{aligned}$$ For the stress tensor and the conserved currents of spin $s>2$, there are various related definitions that all satisfy a conservation law. The definition is the unique one that is totally symmetric and traceless. In particular, the stress tensor is the one derived by varying the metric in a theory of free massless scalars with conformal coupling.
When we’re not using the explicit formula with its flat 3d derivatives, we can use ${\mathbb{R}}^{1,4}$ indices $(\mu,\nu,\dots)$ for the currents $j^{(s)}$, as in section \[sec:geometry:spacetime:currents\]. Introducing sources $A^{(s)}_{\mu_1\dots\mu_s}$ for the single-trace operators , the free action becomes: $$\begin{aligned}
S_{\text{CFT}} = -\int d^3\ell\,\bar\phi_I\Box\phi^I - \int d^3\ell \sum_{s=0}^\infty A^{(s)}_{\mu_1\dots\mu_s}(\ell)\, j_{(s)}^{\mu_1\dots\mu_s}(\ell) \ . \label{eq:S_local}\end{aligned}$$ The sources $A^{(s)}_{\mu_1\dots \mu_s}$ are spin-$s$ gauge potentials. In particular, $A^{(1)}_\mu$ is an ordinary $U(1)$ gauge potential (times $2$), while $A^{(2)}_{\mu\nu}$ is a metric perturbation (times $1/2$).
Bilocal language {#sec:CFT:bilocal}
----------------
Having formulated our theory in the ordinary language of local operators and sources, let us now present its much simpler formulation in terms of bilocals. The idea is to notice that the local primaries $j^{(s)}_{\mu_1\dots\mu_s}(\ell)$ are just a complicated-looking Taylor expansion of the two-point inner product $\phi^I(\ell)\bar\phi_I(\ell')$. Note that before imposing the free field equations $\Box\phi^I = 0$, there is not enough information in the totally symmetric and traceless $j^{(s)}_{\mu_1\dots\mu_s}(\ell)$ to encode all possible configurations of $\phi^I(\ell)\bar\phi_I(\ell')$. However, *after* imposing the field equations, there is *too much* information. The currents $j^{(s)}_{\mu_1\dots\mu_s}(\ell)$ then satisfy constraints, i.e. conservation laws. In this situation, we might as well directly use the bilocal $\phi^I(\ell)\bar\phi_I(\ell')$ as our basic single-trace operator. The role of current conservation laws is then played by the field equations themselves. Coupling a bilocal source $\Pi(\ell',\ell)$ to the bilocal operator $\phi^I(\ell)\bar\phi_I(\ell')$, we write the CFT action in the form: $$\begin{aligned}
S_{\text{CFT}}[\Pi(\ell',\ell)] = -\int d^3\ell\,\bar\phi_I\Box\phi^I - \int d^3\ell' d^3\ell\,\bar\phi_I(\ell')\Pi(\ell',\ell)\phi^I(\ell) \ . \label{eq:S}\end{aligned}$$ The “conservation laws” (actually, just field equations) on $\phi^I(\ell)\bar\phi_I(\ell')$ induce a gauge redundancy on $\Pi(\ell',\ell)$: $$\begin{aligned}
\Pi(\ell',\ell) \rightarrow \Pi(\ell',\ell) + \Box_\ell f(\ell',\ell) + \Box_{\ell'} g(\ell',\ell) \ . \label{eq:gauge_Pi}\end{aligned}$$ In the large-$N$ limit, there are no constraints on $\phi^I(\ell)\bar\phi_I(\ell')$ other than the field equations, and thus captures the full gauge redundancy of $\Pi(\ell',\ell)$. For finite $N$, this is not the case: for example, for $N=1$, the product $\phi(\ell)\bar\phi(\ell')$ is determined by two functions $\phi(\ell),\bar\phi(\ell)$ of a single point $\ell$. Thus, for finite $N$, the redundancy in $\Pi(\ell',\ell)$ is greater. However, even then, this redundant parameterization of the single-trace sources remains legitimate.
The partition function of the theory is very easy to write down in the bilocal language. First, we write the action in a “matrix” notation: $$\begin{aligned}
S_{\text{CFT}}[\Pi(\ell',\ell)] = -\bar\phi_I(\Box + \Pi)\phi^I \ , \label{eq:S_matrix}\end{aligned}$$ where $\phi(\ell)$ is viewed as an infinite-dimensional vector, $\bar\phi(\ell)$ as a dual vector, and $\Box,\Pi$ as matrices/operators. The Gaussian path integral over $\phi$ and $\bar\phi$ immediately gives the partition function in the form: $$\begin{aligned}
Z_{\text{CFT}}[\Pi(\ell',\ell)] \sim \left(\det{(\Box + \Pi)}\right)^{-N} \sim \left(\det{(1 + G\Pi)}\right)^{-N} = \exp\left(-N{\operatorname{tr}}\ln(1+G\Pi)\right) \ , \label{eq:Z}\end{aligned}$$ where $G = \Box^{-1}$ is the boundary-to-boundary propagator: $$\begin{aligned}
G(\ell,\ell') = -\frac{1}{4\pi\sqrt{-2\ell\cdot\ell'}} \ , \label{eq:G}\end{aligned}$$ i.e. $G(\mathbf{r},\mathbf{r'}) = -1/(4\pi|\mathbf{r} - \mathbf{r'}|)$ in the flat frame .
The partition function is a combination of single-trace pieces of the form ${\operatorname{tr}}(G\Pi)^n$, which can be represented by “1-loop” Feynman diagrams as in figure \[fig:polygon\]. The $U(N)$ “color” factor is taken into account by the $N$ in the exponent in eq. .
![The Feynman diagram for a single-trace contribution ${\operatorname{tr}}(G\Pi)^n$ to the CFT partition function, drawn for $n=4$. Dashed lines represent the bilocal sources $\Pi(\ell',\ell)$. Solid lines represent propagators $G(\ell,\ell')$. Note that the diagram is in coordinate space rather than momentum space, and there is no loop integration involved.[]{data-label="fig:polygon"}](polygon.eps "fig:")\
Any UV divergences in the CFT’s Feynman diagrams (such as the diagram in figure \[fig:polygon\]) are associated with the short-distance divergence of the propagator , i.e. with the limit where some of the “external legs” of the $\Pi(\ell',\ell)$ factors coincide. As long as we are only interested in the bilocal source couplings and partition function , these short-distance singularities don’t seem to require any special treatment: the propagator behaves as $\sim 1/r$, which is integrable under the 3d volume measure $\sim r^2dr$. Thus, the partition function is well-defined without any contact-term corrections.
In contrast, the conserved local currents , the local source couplings and the bilocal gauge symmetry are all given up to contact terms, i.e. assuming separated points. We will not investigate these contact terms directly here. Instead, we will now switch to twistor language, where the need for contact terms, even for calculating local currents, seems to disappear entirely.
Twistor language {#sec:CFT:twistor}
----------------
### From bilocals to twistor functions {#sec:CFT:twistor:transform}
So far, we’ve made manifest the conformal $O(1,4)$ symmetry of the theory , but not its higher-spin extension. To this end, we will now employ the HS algebra of section \[sec:algebra\]. The boundary two-point products of section \[sec:algebra:two\_point\] will play a central role. First, let us package the bilocal source $\Pi(\ell',\ell)$ into a twistor function $\Pi(Y)$: $$\begin{aligned}
F(Y) &= \int d^3\ell\,d^3\ell'K(\ell,\ell';Y)\,\Pi(\ell',\ell) \ , \label{eq:Pi_transform}\end{aligned}$$ where the bilocal kernel $K(\ell,\ell';Y)$ is given by: $$\begin{aligned}
K(\ell,\ell';Y) = -\frac{\sqrt{-2\ell\cdot\ell'}}{4\pi}\,\delta_\ell(Y)\star\delta_{\ell'}(Y)
= -\frac{1}{\pi\sqrt{-2\ell\cdot\ell'}}\exp\frac{iY\ell\ell' Y}{2\ell\cdot\ell'} \ . \label{eq:K}\end{aligned}$$ The kernel $K(\ell,\ell';Y)$ is an even function of $Y$, and has conformal weight $\Delta = 1/2$ with respect to each of the boundary points $\ell,\ell'$.
The transform involves a loss of information: the original bilocal $\Pi(\ell',\ell)$ is a function of 6 coordinates, while $F(Y)$ only depends on 4. Nevertheless, we will see that $F(Y)$ is sufficient to express the partition function, i.e. it is a *complete* encoding of the “physically relevant” data in $\Pi(\ell',\ell)$. In fact, our transform can be viewed as stripping away the gauge redundancy in $\Pi(\ell',\ell)$. Indeed, we see from eq. that $K(\ell,\ell';Y)$ satisfies, up to contact terms, the same field equations $\Box_\ell K = \Box_{\ell'} K = 0$ as the bilocal operator $\phi^I(\ell)\bar\phi_I(\ell')$. Therefore, $F(Y)$ is invariant under the gauge symmetry . Thus, at large $N$, $F(Y)$ constitutes a *non-redundant* parameterization of the theory’s sources (recall that at finite $N$, there is additional redundancy in $\Pi(\ell',\ell)$, which is not captured by eq. ). What’s more, while the gauge redundancy is lost, the true global HS symmetry can now be made manifest. Indeed, we will see below that the partition function in terms of $F(Y)$ is manifestly HS-invariant, with $F(Y)$ transforming in the adjoint.
The remaining question is whether $F(Y)$ constructed through is an *arbitrary* even function of $Y$, i.e. whether the functions $K(\ell,\ell';Y)$ form a spanning set for the HS algebra. Skipping slightly ahead in the narrative, the answer is essentially yes. Specifically, there’s a one-to-one correspondence between $F(Y)$ and linearized bulk solutions (via the Penrose transform), and a one-to-one correspondence between linearized bulk solutions and allowed configurations of the linearized expectation values of the local currents $j^{(s)}_{\mu_1\dots\mu_s}(\ell)$. More precisely, the above statements are *almost* true, due to a pair of related subtleties. First, the Penrose transform involves contour ambiguities. Second, the one-to-one mapping between expectation values $\langle j^{(s)}_{\mu_1\dots\mu_s}(\ell)\rangle$ and bulk solutions involves a requirement of regularity on $EAdS_4$, without which one loses the relationship between the boundary data corresponding to $\langle j^{(s)}_{\mu_1\dots\mu_s}(\ell)\rangle$ and the boundary data corresponding to the sources $A^{(s)}_{\mu_1\dots\mu_s}(\ell)$. This regularity on $EAdS_4$ can be enforced by an $i\epsilon$ prescription on boundary-to-bulk propagators, i.e. it is yet another contour issue. Again related to the above is the question of how $F(Y)$ behaves under the “antipodal map” $F(Y)\rightarrow F(Y)\star\delta(Y)$. As we can see from , $K(\ell,\ell';Y)$ is odd under this map, at least for $\ell\neq\ell'$: $$\begin{aligned}
K(\ell,\ell';Y)\star\delta(Y) = -K(\ell,\ell';Y) \quad \forall \ell\neq\ell' \ . \label{eq:K_odd}\end{aligned}$$ However, as we’ve seen in section \[sec:linear\_HS:antipodal\], one cannot conclude the same for a linear superposition such as , and in fact $F(Y)\star\delta(Y)$ cannot be defined consistently for generic functions $F(Y)$.
To summarize, the encoding of the CFT sources into a twistor function $F(Y)$ is 1) complete, 2) free of the infinite-dimensional HS gauge redundancy, 3) capturing all the gauge-invariant information, 4) making global HS symmetry manifest, and 5) constraint-free, up to a set of closely related subtleties regarding contour choices, analiticity and discrete symmetries.
### The partition function in twistor language
In this section, we express the CFT partition function in terms of $F(Y)$. Remarkably, this can be done by rewriting each individual element in the CFT Feynman diagrams as an HS-covariant operation. The mechanism is captured by the following pair of identities: $$\begin{aligned}
{\operatorname{tr}}_\star K(\ell,\ell';Y) &= 4G(\ell,\ell') \ ; \label{eq:str_K} \\
K(\ell_1,\ell'_1;Y)\star K(\ell_2,\ell'_2;Y) &= G(\ell_2,\ell_1')\,K(\ell_1,\ell'_2;Y) \ . \label{eq:K_K}\end{aligned}$$ Here, the trace identity is just a restatement of eq. . The star-product identity follows from applying the three-point product formula twice: $$\begin{aligned}
\delta_{\ell_1}(Y)\star\delta_{\ell_1'}(Y)\star\delta_{\ell_2}(Y)\star\delta_{\ell_2'}(Y) = \frac{1}{2}\sqrt{\frac{(\ell_1\cdot\ell_2')}{(\ell_1\cdot\ell_1')(\ell_1'\cdot\ell_2)(\ell_2\cdot\ell_2')}}\,\delta_{\ell_1}(Y)\star\delta_{\ell_2'}(Y) \ .
\label{eq:delta_ell_ell_ell_ell}\end{aligned}$$ Here, we fix the sign ambiguity by considering the divergent limit where $\ell_1'=\ell_2$, which we regularize via $\ell_1'^\mu = \ell_2^\mu = zx^\mu$, with $x^\mu$ a highly boosted timelike unit vector, and $z\rightarrow 0$. The positive sign in then follows from the identity $\delta^R_x\star\delta^R_x = +1$. Note that this choice of sign in is incompatible with any global fixing of the original sign ambiguity in ; indeed, any such fixing would have given us a factor of $(\pm i)^2 = -1$. This is yet another example where certain sign ambiguities in the star product must be maintained for overall consistency.
Having established the identity , we note that the star product there radically alters the spatial dependence of the $K$’s. Indeed, the $K$ factors on the LHS have essential singularities at $\ell_1=\ell_1'$ and $\ell_2=\ell_2'$, while the RHS has an essential singularity at $\ell_1'=\ell_2$ and a simple pole at $\ell_1' = \ell_2$. This is an example of how the star product’s nonlocality in twistor space can get translated into nonlocality in spacetime.
We are now ready to employ eqs. - to compute the CFT partition function. The single-trace products that form the building blocks of $Z_{\text{CFT}}$, i.e. the one-loop Feynman diagrams from figure \[fig:polygon\], can be rewritten as: $$\begin{aligned}
{\operatorname{tr}}(G\Pi)^n = \frac{1}{4}{\operatorname{tr}}_\star\Big(\underbrace{F(Y)\star F(Y)\star\ldots\star F(Y)}_{n\text{ times}}\Big) \ . \label{eq:loop}\end{aligned}$$ Thus, the CFT Feynman diagrams make equally good sense in both the bilocal and higher-spin languages. In fact, the correspondence is at the level of individual diagram elements: via the identities -, every individual star product or higher-spin trace can be identified with a $G(\ell,\ell')$ propagator in the Feynman diagram.
The entire partition function can now be written in HS language, yielding the result : $$\begin{aligned}
Z_{\text{CFT}}[F(Y)] \sim \exp\left(-\frac{N}{4}{\operatorname{tr}}_\star\ln_\star[1+F(Y)]\right) = \left(\textstyle\det_\star[1+F(Y)]\right)^{-N/4} \ . \label{eq:Z_HS}\end{aligned}$$ Here, $\ln_\star[1+F(Y)]$ is defined by substituting star products in the Taylor expansion of $\ln(1+x)$, and we introduce the “star determinant” $\det_\star f \equiv \exp({\operatorname{tr}}_\star\ln_\star f)$.
The partition function is manifestly invariant under global HS symmetry, with the source $F(Y)$ transforming in the adjoint: $$\begin{aligned}
\delta F(Y) = \varepsilon(Y)\star F(Y) - F(Y)\star\varepsilon(Y) \ ; \quad \delta Z_{\text{CFT}} = 0 \ . \label{eq:Z_symmetry}\end{aligned}$$ Conversely, the symmetry completely fixes the invariant traces as the only possible ingredient in the partition function (up to a possible $\delta(Y)$ factor inside the trace; however, see eq. and the surrounding discussion). For this reason, the traces were introduced from a bulk perspective in [@Colombo:2012jx; @Didenko:2012tv] as the unique expressions for the $n$-point functions, with only their coefficients left undetermined. Here, we derived the traces directly from the boundary CFT, allowing us to fix their coefficients by writing down the full partition function .
### The single-trace currents in twistor space {#sec:CFT:twistor:currents}
We can construct a “current” operator conjugate to the source $F(Y)$ as an HS-covariant variational derivative: $$\begin{aligned}
\Phi(Y) = \frac{D}{DF(Y)} \ , \label{eq:Phi}\end{aligned}$$ where the derivative $D/Df(Y)$ a functional $\Theta[f(Y)]$ is defined via: $$\begin{aligned}
\delta\Theta = {\operatorname{tr}}_\star\left(\frac{D\Theta}{Df(Y)}\star\delta f(Y)\right) \ .\end{aligned}$$ As we can see from , this implies that $D/Df(Y)$ is actually a Fourier transform of the ordinary variational derivative: $$\begin{aligned}
\frac{D}{Df(Y)} = \int d^4U e^{iYU}\,\frac{\delta}{\delta f(U)} \ .\end{aligned}$$
The expectation value $\left<\Phi(Y)\right>$ reads: $$\begin{aligned}
\left<\Phi(Y)\right> = \frac{D\ln Z_{\text{CFT}}}{DF(Y)} = -\frac{N}{4}[1+F(Y)]_\star^{-1} = \frac{N}{4}\left(-1 + F(Y) + O(F^2) \right) \ . \label{eq:Phi_expectation}\end{aligned}$$ In particular, the linear piece of $\left<\Phi(Y)\right>$ is just a constant multiple of $F(Y)$. There is in fact no other possibility compatible with HS symmetry (again, with the subtle exception of multiplication by $\delta(Y)$, which we will touch on again in section \[sec:holography\]).
For comparison, in the original bilocal language of eqs. -, the current conjugate to $\Pi(\ell',\ell)$ is simply the bilocal operator $\phi^I(\ell)\bar\phi_I(\ell')$: $$\begin{aligned}
\phi^I(\ell)\bar\phi_I(\ell') &= \frac{\delta}{\delta\Pi(\ell',\ell)} \ ; \\
\begin{split}
\left<\phi^I(\ell)\bar\phi_I(\ell')\right> &= \frac{\delta\ln Z_{\text{CFT}}}{\delta\Pi(\ell',\ell)} = -N(1+G\Pi)^{-1}G \\
&= N\left(-G(\ell,\ell') + \int d^3\ell_1 d^3\ell_2\,G(\ell,\ell_1)\,\Pi(\ell_1,\ell_2)\,G(\ell_2,\ell') + O(\Pi^2) \right) \ .
\end{split} \label{eq:phi_phi_expectation}\end{aligned}$$ The currents $\Phi(Y)$ and $\phi^I(\ell)\bar\phi_I(\ell')$ are related via the chain rule for variational derivatives: $$\begin{aligned}
\begin{split}
\phi^I(\ell)\bar\phi_I(\ell') &= \int d^4Y \frac{\delta F(Y)}{\delta\Pi(\ell',\ell)}\frac{\delta}{\delta F(Y)}
= \int d^4Y K(\ell,\ell';Y) \int d^4U e^{-iYU} \Phi(U) \\
&= {\operatorname{tr}}_\star\left(\Phi(Y)\star K(\ell,\ell';Y)\right) \ .
\end{split} \label{eq:phi_phi}\end{aligned}$$ Equivalently, the perturbation of $F(Y)$ that couples to the operator $\phi^I(\ell)\bar\phi_I(\ell')$ is simply: $$\begin{aligned}
\frac{\delta F(Y)}{\delta\Pi(\ell',\ell)} = \frac{\delta{\left(\phi^I(\ell)\bar\phi_I(\ell')\right)}}{\delta\Phi(Y)} = K(\ell,\ell';Y) \ . \label{eq:phi_phi_source}\end{aligned}$$
Since the source $F(Y)$ is gauge-invariant (at large $N$) and constraint-free, we conclude that the operator $\Phi(Y)$ is constraint-free (at large $N$) and gauge-invariant. In contrast, the bilocal $\phi^I(\ell)\bar\phi_I(\ell')$, which depends on 6 rather than 4 coordinates, is constrained by field equations. In regions where the source $\Pi(\ell',\ell)$ vanishes, $\phi^I(\ell)\bar\phi_I(\ell')$ inherits from $K(\ell,\ell';Y)$ the source-free wave equation at $\ell\neq\ell'$. The same is *not* true in regions with non-vanishing $\Pi(\ell',\ell)$, even though this is not evident from eq. . The important subtlety here is that the star product has the power to reshuffle the singularity structure in the $\ell,\ell'$ dependence, as in . The true spatial dependence of $\phi^I(\ell)\bar\phi_I(\ell')$ in regions where $\Pi(\ell',\ell)$ is non-vanishing can be seen from the expansion .
Holography {#sec:holography}
==========
We are now ready to tie together our treatments of the bulk and boundary. Normally in AdS/CFT, one thinks in terms of local bulk fields, each of which is associated with asymptotic boundary data of two types, i.e. two complementary conformal weights. For massless fields in $AdS_4$, these conformal weights are integers, and the corresponding boundary data takes on an additional layer of meaning. For the conformally coupled massless scalar, the conformal weights are $\Delta = 1,2$; we refer to the corresponding boundary data as Dirichlet and Neumann, since that is their precise nature under a bulk conformal transformation that turns the asymptotic boundary into an ordinary hypersurface. For gauge fields with spin $s\geq 1$, the two different conformal weights arise for the gauge potential; in terms of the field strength, they correspond to its electric and magnetic parts, which have the same conformal weight but different parities.
In different setups, HS gravity is dual to a variety of vector models. The free vector model is the simplest case, which, from the bulk point of view, relies on two choices. First, one must choose the bulk interactions to be those of the type-A (i.e. parity-even) model; this will not make an explicit appearance in the present paper, since we only consider the bulk interactions indirectly, through the CFT. Second, one must choose Neumann boundary conditions for the scalar, and magnetic boundary conditions for the gauge fields of spin $s\geq 1$, i.e. treat the Neumann & magnetic boundary data as external sources, while the Dirichlet & electric boundary data will correspond to the CFT operators. As discussed in [@Vasiliev:2012vf], this is the choice of boundary data that preserves (global) HS symmetry, which is then reflected in the CFT.
In the present section, we will describe the HS/free-CFT holography from the twistorial perspective of sections \[sec:linear\_HS\],\[sec:CFT\]. We will then make contact with the standard language of bulk fields vs. local CFT operators by comparing the bulk fields’ Dirichlet/electric boundary data with the expectation values of the CFT currents.
The basic dictionary
--------------------
The fundamental entry in our holographic dictionary is to identify the twistor function $F(Y)$ that encodes the CFT sources in with either of the two functions $F_{R/L}(Y)$ that define the linearized bulk solution in -: $$\begin{aligned}
F(Y) = F_R(Y) \quad \text{or} \quad F(Y) = F_L(Y) \ . \label{eq:dictionary}\end{aligned}$$ In terms of the bulk master fields $C(x;Y)$, this implies: $$\begin{aligned}
F(Y) = -iC(x;Y)\star\delta^R_x(Y) \quad \text{or} \quad F(Y) = iC(x;Y)\star\delta^L_x(Y) \ . \label{eq:dictionary_C}\end{aligned}$$ With the substitution , the CFT partition function becomes a *nonlinear functional of the linearized bulk solution*, as envisaged in [@Didenko:2012tv; @Colombo:2012jx]: $$\begin{aligned}
Z \sim \left(\textstyle\det_\star[1-iC(x;Y)\star\delta^R_x(Y)]\right)^{-N/4} \ \text{or} \quad Z \sim \left(\textstyle\det_\star[1+iC(x;Y)\star\delta^L_x(Y)]\right)^{-N/4} \ . \label{eq:Z_bulk}\end{aligned}$$ As in [@Didenko:2012tv; @Colombo:2012jx], the partition function is given in terms of the master field at *any single bulk point* $x$, which, by virtue of the unfolded formulation, encodes the entire linearized bulk solution. In this way, having placed the bulk and boundary on a common footing via twistor functions, we are able to express the full partition function, including the effects of bulk interactions, in bulk terms.
To establish the relation , we will calculate on both sides the linearized expectation values of the local HS currents $j^{(s)}_{\mu_1\dots\mu_s}(\ell)$, as induced by the bilocal source $\Pi(\ell',\ell)$ at separated points. On the bulk side, this means calculating the electric field strengths at infinity. The distinction between $F_R(Y)$ and $F_L(Y)$ in does not affect these expectation values. This distinction is instead related to the value of the boundary gauge fields $A^{(s)}_{\mu_1\dots\mu_s}(\ell)$, as well as to the contour issues discussed in section \[sec:CFT:twistor:transform\]. This is because the two choices are related by $F(Y)\rightarrow -F(Y)\star\delta(Y)$, or, in terms of the bulk fields, by $C(x;Y)\rightarrow -C(-x;Y)$. The electric field strengths at infinity and the CFT currents are unaffected by this transformation, since they are associated [@Vasiliev:2012vf; @Neiman:2014npa] with the antipodally odd part of the bulk solution.
In a sense, an explicit calculation of $\left<j^{(s)}_{\mu_1\dots\mu_s}(\ell)\right>$ on both sides of the duality is actually unnecessary. The results are guaranteed to agree, simply because the higher-spin algebra contains every spin exactly once (actually twice for $s\geq 1$, corresponding to the two helicities; however, we can then use parity to distinguish their “electric” combination from the “magnetic” one). This is as it should be: since the higher-spin/free-CFT duality is in a sense the simplest of all holographic models, it *should* appear trivial – as trivial as eq. – once the correct language has been identified.
Thus, in practice, our calculation of the boundary currents will serve two aims: to provide a consistency check for the formalism, and to fix the proportionality coefficients between the boundary currents and the bulk electric fields at infinity.
Asymptotics of the bulk fields {#sec:holography:asymptotics}
------------------------------
In this section, we express the asymptotic boundary data of a bulk solution $C(x;Y)$ in terms of the twistor function $F_R(Y)$ (a similar analysis applies for $F_L(Y)$, with some sign changes). In accordance with the standard AdS/CFT prescription, we will focus on the asymptotics of the “fundamental” massless bulk fields , as opposed to the unfolded tower of derivatives . As a result, our expressions will generally not be HS-covariant, i.e. they will contain spinor integrals that cannot be reduced to star products.
### Spin 0 {#sec:holography:asymptotics:scalar}
The conformally-coupled massless scalar $C^{(0,0)}(x) = C(x;0)$ admits boundary data of two types: “Dirichlet data” $\varphi(\ell)$ with conformal weight $\Delta = 1$ and “Neumann data” $\pi(\ell)$ with weight $\Delta = 2$. At a bulk point $x$, the value of the scalar field can be found from the Penrose transform or as: $$\begin{aligned}
C(x;0) = i{\operatorname{tr}}_\star\left(F_R(Y)\star\delta^R_x(Y)\right) = i\int_{P_R(x)} d^2u_R\,F_R(u_R) \ . \label{eq:bulk_scalar}\end{aligned}$$ The Dirichlet boundary data can be read off directly from the bulk-to-boundary limit ,: $$\begin{aligned}
\varphi(\ell) = \lim_{x\rightarrow\ell/z} \frac{1}{z}\,C(x;0) = i{\operatorname{tr}}_\star\left(F_R(Y)\star\delta_\ell(Y)\right) = i\int_{P(\ell)} d^2u\,F_R(u) \ . \label{eq:Dirichlet}\end{aligned}$$
The Neumann boundary data will be given by the second term in the Taylor series in $z$: $$\begin{aligned}
C(x;0) = z\varphi(\ell) + z^2\pi(\ell) + O(z^3) \ .\end{aligned}$$ To extract it, we must take the bulk-to-boundary limit more carefully, as in , using a second boundary point $n$ to define the direction from which $x$ approaches $\ell$. Under , the chiral projector $P_R(x)$ takes the form: $$\begin{aligned}
P_R(x) = \frac{1}{2z}(\ell + z + z^2 n) = \frac{1}{z}P(\ell) + \frac{1}{2}(1 + zn) \ . \label{eq:approach_P}\end{aligned}$$ We can now rewrite the $P_R(x)$ integral in as an integral over $P(\ell)$, via the change of variables: $$\begin{aligned}
u_R = 2P_R(x)u = (1+zn)u \ .\end{aligned}$$ The measures $d^2u$ and $d^2u_R$ turn out to be related by a factor of $z$: $$\begin{aligned}
d^2u_R = \frac{P^R_{ab}(x)\,du_R^a du_R^b}{2(2\pi)} = \frac{2P^R_{ab}(x)\,du^a du^b}{2\pi} = \frac{zn_{ab}\,du^a du^b}{2\pi} = z d^2u \ .\end{aligned}$$ Here, in the third equality, we used $du_a du^a = 0$ for $u\in P(\ell)$, while the fourth equality follows from contracting eq. with $n_{ab}$. The bulk scalar thus becomes: $$\begin{aligned}
C(x;0) = iz\int_{P(\ell)} d^2u\,F_R\big((1+zn)u\big) = z\phi(\ell) + iz^2 n^{ab} \int_{P(\ell)}d^2u\,u_a\left.\frac{{\partial}F_R(U)}{{\partial}U^b}\right|_u + O(z^3) \ ,\end{aligned}$$ from which we extract: $$\begin{aligned}
\pi(\ell) = in^\mu\gamma_\mu^{ab} \int_{P(\ell)}d^2u\,u_a\left.\frac{{\partial}F_R(U)}{{\partial}U^b}\right|_u \ . \label{eq:Neumann_n}\end{aligned}$$ Now, recall that $n^\mu$ is an arbitrary null vector satisfying $\ell\cdot n = -1/2$. Since the result should not depend on the choice of $n$, we can rewrite it as: $$\begin{aligned}
\pi(\ell)\ell_\mu = -\frac{i}{2}\gamma_\mu^{ab} \int_{P(\ell)}d^2u\,u_a\left.\frac{{\partial}F_R(U)}{{\partial}U^b}\right|_u \ . \label{eq:Neumann}\end{aligned}$$ One can verify explicitly, using integration by parts, that the antisymmetric traceless part of the integral in is indeed proportional to $\ell_{ab}$ (or, equivalently, that it vanishes upon contraction with $\ell^b{}_c$).
Finally, let us point out the relation between the Dirichlet/Neumann boundary data and antipodal symmetry [@Vasiliev:2012vf; @Ng:2012xp; @Neiman:2014npa]. The Dirichlet data $\varphi(\ell)$ and the Neumann data $\pi(\ell)$ are associated with antipodally odd and even solutions respectively, in the sense that odd solutions have only $\varphi(\ell)$ non-vanishing, and even solutions have only $\pi(\ell)$ non-vanishing. As discussed in [@Halpern:2015zia], this property can be deduced from the fact that the conformal weight of $\varphi$ ($\pi$) is an odd (even) positive integer. In our present language, the antipodal symmetry of $\varphi$ and $\pi$ can be seen in two ways. First, one can read off from , the properties: $$\begin{aligned}
\varphi(-\ell) = -\varphi(\ell) \ ; \quad \pi(-\ell) = \pi(\ell) \ ,\end{aligned}$$ where we used the fact that the measure is odd under $\ell^\mu\rightarrow -\ell^\mu$. Second, we can apply the antipodal map to the bulk solution via $F_R(Y)\rightarrow F_R(Y)\star\delta(Y)$, which again results in: $$\begin{aligned}
\varphi(\ell) \rightarrow -\varphi(\ell) \ ; \quad \pi(\ell) \rightarrow \pi(\ell) \ .\end{aligned}$$ In deriving this result, it is crucial to keep track of the sign factor in .
### Spin $\geq 1$: chiral field strengths
The asymptotics for all the gauge fields with spin $s\geq 1$ can be described in a unified way using master fields. From the Penrose transform , we extract two generating functions for the field strengths at a bulk point $x$: $$\begin{aligned}
C(x;y_L) = i\int_{P_R(x)} d^2u_R\,F_R(y_L + u_R) \ ; \quad C(x;y_R) = i\int_{P_R(x)} d^2u_R\,F_R(u_R)\,e^{iu_R y_R} \ . \label{eq:bulk_chiral_fields_raw}\end{aligned}$$ Here, $y_L$ ($y_R$) is a left-handed (right-handed) spinor at $x$. The Taylor coefficients of $C(x;y_L)$ and $C(x;y_R)$ with respect to their spinor variables encode respectively the left-handed and right-handed field strengths $C^{(2s,0)}_{\alpha_1\dots\alpha_{2s}}(x),C^{(0,2s)}_{\dot\alpha_1\dots\dot\alpha_{2s}}(x)$ via eq. . The zeroth-order Taylor coefficient in both $C(x;y_L)$ and $C(x;y_R)$ is the spin-0 field $C(x;0)$.
As a step towards taking the boundary limit, let us note that the integrals in do not change if we add to $y_L$ or $y_R$ a spinor of the opposite chirality. In other words, the chiral master fields can be extended trivially into functions of an entire twistor $Y$: $$\begin{aligned}
\begin{split}
C_L(x;Y) &= C(x;P_L(x)Y) = i\int_{P_R(x)} d^2u_R\,F_R(Y + u_R) \ ; \\
C_R(x;Y) &= C(x;P_R(x)Y) = i\int_{P_R(x)} d^2u_R\,F_R(u_R)\,e^{iu_R Y} \ .
\end{split} \label{eq:bulk_chiral_fields}\end{aligned}$$ In the bulk-to-boundary limit , the left-handed and right-handed fields become: $$\begin{aligned}
C_L(x;Y) &= z\,{\mathcal{C}}_L(\ell;Y) + O(z^2) \ ; & {\mathcal{C}}_L(\ell;Y) &= i\int_{P(\ell)} d^2u\,F_R(Y + u) \ ; \label{eq:boundary_C_L} \\
C_R(x;Y) &= z\,{\mathcal{C}}_R(\ell;Y) + O(z^2) \ ; & {\mathcal{C}}_R(\ell;Y) &= i\int_{P(\ell)} d^2u\,F_R(u)\,e^{iuY} \ , \label{eq:boundary_C_R}\end{aligned}$$ where the factor of $z$ arises from the ratio of the measures $d^2u_R$ and $d^2u$. The boundary master fields ${\mathcal{C}}_{L/R}(\ell;Y)$ have conformal weight $\Delta=1$, and depend only on the $P^*(\ell)$ spinor component $y^*$ of the twistor $Y$: $$\begin{aligned}
{\mathcal{C}}_{L/R}(\ell,Y+u) = {\mathcal{C}}_{L/R}(\ell;Y) \qquad \forall u\in P(\ell) \ . \label{eq:C_boundary_projective}\end{aligned}$$ The Taylor expansion of ${\mathcal{C}}_{L/R}(\ell;y^*)$ in powers of $y^*$ generates the individual left-handed/right-handed field strengths of various spins. The fields defined in this way have spinor indices in $P(\ell)$, as in the $j_\ell$ representation of boundary currents from section \[sec:geometry:spinors\_boundary:currents\].
The asymptotic field strengths ${\mathcal{C}}_{L/R}(\ell;y^*)$ satisfy a Gauss law, i.e. each of the component fields with spin $s\geq 1$ has a vanishing divergence. To express and verify this fact explicitly, we must first use eq. or to convert the component fields into spinors with indices in $P^*(\ell)$. This is equivalent to converting ${\mathcal{C}}_{L/R}(\ell;y^*)$ into a function of $y\in P(\ell)$: $$\begin{aligned}
{\mathcal{C}}_{L/R}(\ell;Y) = {\mathcal{C}}_{L/R}^*(\ell;P(\ell)Y) \ , \end{aligned}$$ where ${\mathcal{C}}_{L/R}^*(\ell,y)$ can be given explicitly as: $$\begin{aligned}
\begin{split}
{\mathcal{C}}^*_L(\ell;y) &= -i\int d^4V\,F_R(V) \int_{P^*(\ell)}d^2u^*\,e^{iu^*(P(\ell)V - y)} \ ; \\
{\mathcal{C}}^*_R(\ell;y) &= i\int_{P^*(\ell)} d^2u^*\,F_R(P(\ell)u^*)\, e^{iu^*y} \ .
\end{split} \label{eq:C*}\end{aligned}$$ The vanishing of the divergence for each of the component field strengths can now be expressed as: $$\begin{aligned}
\ell_\mu\gamma_{ab}^{\mu\nu}\, \frac{{\partial}^3 {\mathcal{C}}_{L/R}^*(\ell;y)}{{\partial}\ell^\nu{\partial}y_a{\partial}y_b} = 0 \ , \label{eq:div_C}\end{aligned}$$ and one can easily check that this constraint is in fact satisfied by the expressions . Taking the ${\partial}/{\partial}\ell$ derivative of an integral over $P^*(\ell)$ requires some care, due to the $\ell$-dependence of the integration domain. The trick is to fix the integration range to some arbitrary 2d subspace of twistor space, which may then represent $P^*(\ell)$ for different values of $\ell$. One should keep track, however, of the $\ell$-dependence of the integration measure.
Finally, let us show how the boundary fields encoded in ${\mathcal{C}}_{L/R}(\ell;Y)$ can be arrived at through tensor language. We approach the boundary as in section \[sec:geometry:spinors\_boundary:limit\], moving the bulk point $x$ along towards the boundary point $\ell$ along the outwards-pointing tangent vector $t^\mu$. In the orthonormal tangent frame $t^\mu,e_i^\mu$, the components of the field strengths $C^{L/R}_{\mu_1\nu_1\dots\mu_s\nu_s}$ will scale as $z^{s+1}$ (this can be derived e.g. from the 4d conformal invariance of the free massless field equations). On the other hand, in a fixed frame in ${\mathbb{R}}^{1,4}$, the basis vector $t^\mu$ behaves asymptotically as $t^\mu\rightarrow\ell^\mu/z$, while the other basis vectors $e_i^\mu$ remain constant. Thus, in the fixed frame, $C^{L/R}_{\mu_1\nu_1\dots\mu_s\nu_s}$ will be dominated by components where the largest possible number of indices is pointing along $t^\mu$. This leaves us with the asymptotics: $$\begin{aligned}
\begin{split}
C^{L/R}_{\mu_1\nu_1\dots\mu_s\nu_s}(x) &= z^{s+1}\left(2^s t_{[\mu_1}\delta_{\nu_1]}^{\rho_1}\dots t_{[\mu_s}\delta_{\nu_s]}^{\rho_s}\,{\mathcal{C}}^{L/R}_{\rho_1\dots\rho_s}(\ell) + O(z) \right) \\
&= z\left(2^s \ell_{[\mu_1}\delta_{\nu_1]}^{\rho_1}\dots\ell_{[\mu_s}\delta_{\nu_s]}^{\rho_s}\,{\mathcal{C}}^{L/R}_{\rho_1\dots\rho_s}(\ell) + O(z) \right) \ .
\end{split} \label{eq:C_tensor_asymptotics}\end{aligned}$$ Here, in the first line, we insist on keeping the leading-order term within the tangent space of $EAdS_4$ at $x$; in the second line, we drop this requirement and substitute $t^\mu \rightarrow \ell^\mu/z$. The tensors ${\mathcal{C}}^{L/R}_{\mu_1\dots\mu_s}(\ell)$ are totally symmetric and traceless, with indices along $e_i^\mu$. More covariantly, these are boundary tensors in the sense of -, with conformal weight $\Delta = s+1$. The equivalence is associated with the different directions from which we could approach the boundary point $\ell$. Converting into spinor form as in , we get: $$\begin{aligned}
C^{(2s,0)}_{a_1\dots a_{2s}}(x) = z\,{\mathcal{C}}^L_{a_1\dots a_{2s}}(\ell) + O(z^2) \ ; \quad C^{(0,2s)}_{a_1\dots a_{2s}}(x) = z\,{\mathcal{C}}^R_{a_1\dots a_{2s}}(\ell) + O(z^2) \ , \label{eq:C_spinor_asymptotics}\end{aligned}$$ where ${\mathcal{C}}^L_{a_1\dots a_{2s}}(\ell)$ are totally symmetric boundary spinors with conformal weight $\Delta = 1$ and with indices in $P(\ell)$: $$\begin{aligned}
{\mathcal{C}}^{L/R}_{a_1\dots a_{2s}}(\ell) = \gamma_{a_1 a_2}^{\mu_1\nu_1}\dots\gamma_{a_{2s-1}a_{2s}}^{\mu_s\nu_s} \ell_{\mu_1}\dots\ell_{\mu_s} {\mathcal{C}}^{L/R}_{\nu_1\dots\nu_s}(\ell) \ . \label{eq:C_boundary_spinor_from_tensor}\end{aligned}$$ We can now pack these into master fields, in analogy with : $$\begin{aligned}
{\mathcal{C}}_{L/R}(\ell;Y) = \sum_{s=0}^\infty \frac{1}{(2s)!}\, Y^{a_1}\dots Y^{a_{2s}}\,{\mathcal{C}}^{L/R}_{a_1\dots a_{2s}}(\ell) \ . \label{eq:C_master_asymptotics}\end{aligned}$$ It is clear from eqs. and that the boundary master fields constructed in this way coincide with the ones in -.
### Spin $\geq 1$: electric and magnetic field strengths
A more standard decomposition of the asymptotic field strengths is into their electric and magnetic parts. These are given by the sum and difference of the chiral field strengths -: $$\begin{aligned}
{\mathcal{E}}(\ell;Y) = {\mathcal{C}}_R(\ell;Y) + {\mathcal{C}}_L(\ell;Y) \ ; \quad {\mathcal{B}}(\ell;Y) = {\mathcal{C}}_R(\ell;Y) - {\mathcal{C}}_L(\ell;Y) \ . \label{eq:E_B}\end{aligned}$$ ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$ again have conformal weight $\Delta = 1$, and depend only on the $P^*(\ell)$ spinor component of $Y$. Thus, their Taylor coefficients in $Y$ are totally symmetric spinors with indices in $P(\ell)$, which encode the electric and magnetic field tensors for the various spins. The (higher-spin) electric and magnetic Gauss laws follow directly from those for ${\mathcal{C}}_{L/R}(\ell;Y)$.
Let us unpack the definitions by working out their implications in tensor language. First, we identify the spin-0 components of ${\mathcal{E}}(x;Y)$ and ${\mathcal{B}}(x;Y)$: $$\begin{aligned}
{\mathcal{E}}(\ell;0) = 2\varphi(\ell) \ ; \quad {\mathcal{B}}(\ell;0) = 0 \ . \label{eq:E_B_spin_0}\end{aligned}$$ Thus, the spin-0 component of ${\mathcal{E}}$ is proportional to the Dirichlet data for the bulk scalar, while the spin-0 component of ${\mathcal{B}}$ vanishes. Next, we turn to the nonzero-spin components. Consider the bulk spin-$s$ field strength tensor . On a “time slice” (in quotes, since our bulk is Euclidean) with outward-pointing normal $t^\mu$, the field strength decomposes into electric and magnetic parts: $$\begin{aligned}
E_{\nu_1\nu_2\dots\nu_s}(x) &= t^{\mu_1}t^{\mu_2}\dots t^{\mu_s} C_{\mu_1\nu_1\mu_2\nu_2\dots\mu_s\nu_s}(x) \ ; \\
B_{\nu_1\nu_2\dots\nu_s}(x) &= t^{\mu_1}t^{\mu_2}\dots t^{\mu_s}\left(-\frac{1}{2}\epsilon_{\mu_1\nu_1}{}^{\lambda\rho\sigma}x_\lambda\right) C_{\rho\sigma\mu_2\nu_2\dots\mu_s\nu_s}(x) \ . \label{eq:magnetic_tensor}\end{aligned}$$ Thanks to the (anti)-self-duality of the field strength’s right-handed and left-handed components, this can be expressed equivalently as: $$\begin{aligned}
\begin{split}
E_{\nu_1\dots\nu_s}(x) &= t^{\mu_1}\dots t^{\mu_s} \left(C^R_{\mu_1\nu_1\dots\mu_s\nu_s}(x) + C^L_{\mu_1\nu_1\dots\mu_s\nu_s}(x) \right) \ ; \\
B_{\nu_1\dots\nu_s}(x) &= t^{\mu_1}\dots t^{\mu_s} \left(C^R_{\mu_1\nu_1\dots\mu_s\nu_s}(x) - C^L_{\mu_1\nu_1\dots\mu_s\nu_s}(x) \right) \ .
\end{split} \label{eq:E_B_bulk}\end{aligned}$$ Here, we can already see the origin of eqs. . To make the relation explicit, let us work out the asymptotics of $E_{\mu_1\dots\mu_s}(x)$ and $B_{\mu_1\dots\mu_s}(x)$ as our “time slice” approaches the boundary. From eq. , we can read off immediately: $$\begin{aligned}
E_{\mu_1\dots\mu_s}(x) = z^{s+1}{\mathcal{E}}_{\mu_1\dots\mu_s}(\ell) + O(z^2) \ ; \quad B_{\mu_1\dots\mu_s}(x) = z^{s+1}{\mathcal{B}}_{\mu_1\dots\mu_s}(\ell) + O(z^2) \ ,\end{aligned}$$ where: $$\begin{aligned}
{\mathcal{E}}_{\mu_1\dots\mu_s}(\ell) = {\mathcal{C}}^R_{\mu_1\dots\mu_s}(\ell) + {\mathcal{C}}^L_{\mu_1\dots\mu_s}(\ell) \ ; \quad {\mathcal{B}}_{\mu_1\dots\mu_s}(\ell) = {\mathcal{C}}^R_{\mu_1\dots\mu_s}(\ell) - {\mathcal{C}}^L_{\mu_1\dots\mu_s}(\ell) \ .\end{aligned}$$ To arrive at eqs. , all that remains is to convert the boundary tensors ${\mathcal{E}}_{\mu_1\dots\mu_s}(\ell)$ and ${\mathcal{B}}_{\mu_1\dots\mu_s}(\ell)$ into spinor form as in : $$\begin{aligned}
\begin{split}
{\mathcal{E}}_{a_1\dots a_{2s}}(\ell) &= \gamma_{a_1 a_2}^{\mu_1\nu_1}\dots\gamma_{a_{2s-1}a_{2s}}^{\mu_s\nu_s} \ell_{\mu_1}\dots\ell_{\mu_s} {\mathcal{E}}_{\nu_1\dots\nu_s}(\ell) \ ; \\
{\mathcal{B}}_{a_1\dots a_{2s}}(\ell) &= \gamma_{a_1 a_2}^{\mu_1\nu_1}\dots\gamma_{a_{2s-1}a_{2s}}^{\mu_s\nu_s} \ell_{\mu_1}\dots\ell_{\mu_s} {\mathcal{B}}_{\nu_1\dots\nu_s}(\ell) \ ,
\end{split} \label{eq:E_B_spinor_from_tensor}\end{aligned}$$ and then package them into master fields as in : $$\begin{aligned}
\begin{split}
{\mathcal{E}}(\ell;Y) &= \sum_{s=0}^\infty \frac{1}{(2s)!}\, Y^{a_1}\dots Y^{a_{2s}}\,{\mathcal{E}}_{a_1\dots a_{2s}}(\ell) \ ; \\
{\mathcal{B}}(\ell;Y) &= \sum_{s=0}^\infty \frac{1}{(2s)!}\, Y^{a_1}\dots Y^{a_{2s}}\,{\mathcal{B}}_{a_1\dots a_{2s}}(\ell) \ .
\end{split} \label{eq:E_B_master}\end{aligned}$$
Finally, we should address the antipodal symmetry of ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$. The antipodal map $F_R(Y)\rightarrow F_R(Y)\star\delta(Y)$ sends each of the integrals - into $-1$ times the other: $$\begin{aligned}
C_L(\ell;Y) \rightarrow -C_R(\ell;Y) \ ; \quad C_R(\ell;Y) \rightarrow -C_L(\ell;Y)\end{aligned}$$ We can therefore read off from that the electric fields ${\mathcal{E}}(\ell;Y)$ are antipodally odd, while the magnetic fields ${\mathcal{B}}(x;Y)$ are antipodally even [@Neiman:2014npa]: $$\begin{aligned}
{\mathcal{E}}(\ell;Y) \rightarrow -{\mathcal{E}}(\ell;Y) \ ; \quad {\mathcal{B}}(\ell;Y) \rightarrow {\mathcal{B}}(\ell;Y) \ .\end{aligned}$$ The same conclusion can be reached by the alternative methods that we’ve used for the spin-0 boundary data, i.e. by sending $\ell^\mu\rightarrow-\ell^\mu$ or examining the parity of the conformal weights of ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$. From this point of view, the different antipodal parities of ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$ arise from the antipodally odd $\epsilon_{\mu_1\nu_1}{}^{\lambda\rho\sigma}x_\lambda$ factor in the definition of the magnetic fields.
Electric fields at infinity from a bilocal boundary source {#sec:holography:example_asymptotics}
----------------------------------------------------------
Now that we’ve defined the electric fields at infinity, let us evaluate them for the particular case of a bilocal source concentrated at a pair of points $\ell_0,\ell'_0$. Thus, in the language of eq. , we choose the CFT sources as: $$\begin{aligned}
\Pi(\ell',\ell) = \delta^{5/2,1/2}(\ell,\ell_0)\,\delta^{5/2,1/2}(\ell',\ell'_0) \quad \Longrightarrow \quad F(Y) = K(\ell_0,\ell'_0;Y) \ , \label{eq:example_source}\end{aligned}$$ where the superscripts on the delta functions indicate their conformal weight with respect to each argument. Now, according to our holographic dictionary , we should construct the linearized bulk solution as the (right-handed or left-handed) Penrose transform of the twistor function $F(Y)$: $$\begin{aligned}
C(x;Y) = iK(\ell_0,\ell'_0;Y)\star\delta^R_x(Y) \quad \text{or} \quad C(x;Y) = -iK(\ell_0,\ell'_0;Y)\star\delta^L_x(Y) \ . \label{eq:ell_ell_x_raw}\end{aligned}$$ As we’ve seen in section \[sec:spacetime\_subgroup:manifest\], the result in both cases reads: $$\begin{aligned}
\begin{split}
C(x;Y) &= \pm K(\ell_0,\ -\ell'_0-2(\ell'_0\cdot x)x;\ Y) \\
&= \frac{\mp 1}{\pi\sqrt{2[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]}}\exp\frac{iY[\ell_0\ell'_0 + 2(\ell'_0\cdot x)\ell_0 x] Y}{2[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]} \ ,
\end{split} \label{eq:ell_ell_x}\end{aligned}$$ where the overall sign is ambiguous due to an intrinsic ambiguity in the star product. The bulk solution can be termed a “boundary-boundary-bulk” propagator. Note that one shouldn’t conclude from the expression that this propagator is even under the antipodal map $x\rightarrow -x$: the sign ambiguity in can be resolved in opposite ways for future-pointing vs. past-pointing $x^\mu$. In fact, we should conclude from that the propagator satisfies $C(x;Y)\star\delta(Y) = -C(x;Y)$, i.e. that it’s antipodally *odd*. This will be substantiated by our analysis of the solution’s asymptotic behavior.
The next step is to extract the left-handed and right-handed field strengths, as in eq. . Substituting $Y\rightarrow P_{L/R}(x)Y$ into the propagator , we get: $$\begin{aligned}
\begin{split}
C_L(x;Y) &= \frac{\mp 1}{\pi\sqrt{2[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]}}
\exp\frac{iY[\ell_0\ell'_0 + (\ell'_0\cdot x)\ell_0 x - (\ell_0\cdot x)\ell'_0 x - \ell_0\ell'_0 x] Y}{4[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]} \ ; \\
C_R(x;Y) &= \frac{\mp 1}{\pi\sqrt{2[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]}}
\exp\frac{iY[\ell_0\ell'_0 + (\ell'_0\cdot x)\ell_0 x - (\ell_0\cdot x)\ell'_0 x + \ell_0\ell'_0 x] Y}{4[\ell_0\cdot\ell'_0 + 2(\ell_0\cdot x)(\ell'_0\cdot x)]} \ ,
\end{split}\end{aligned}$$ where the only difference between the two expressions is in the sign of the last term in the exponent’s numerator. We can now take the bulk-to-boundary limit as in -, to get the asymptotic chiral field strengths: $$\begin{aligned}
{\mathcal{C}}_L(\ell;Y) = {\mathcal{C}}_R(\ell;Y) = \frac{\mp 1}{2\pi\sqrt{(\ell_0\cdot\ell)(\ell'_0\cdot\ell)}} \exp\frac{iY[(\ell'_0\cdot\ell)\ell_0\ell - (\ell_0\cdot\ell)\ell'_0\ell] Y}{8(\ell_0\cdot\ell)(\ell'_0\cdot\ell)} \ .\end{aligned}$$ From these, we find the electric and magnetic boundary data as in : $$\begin{aligned}
{\mathcal{E}}(\ell;Y) &= \frac{\mp 1}{\pi\sqrt{(\ell_0\cdot\ell)(\ell'_0\cdot\ell)}} \exp\frac{iY[(\ell'_0\cdot\ell)\ell_0\ell - (\ell_0\cdot\ell)\ell'_0\ell] Y}{8(\ell_0\cdot\ell)(\ell'_0\cdot\ell)} \ ; \label{eq:example_E_master} \\
{\mathcal{B}}(\ell;Y) &= 0 \quad \forall\ell\neq\ell_0,\ell'_0 \ ,\end{aligned}$$ where we’re careful to note that the magnetic field strengths vanish *away from the source points $\ell_0,\ell'_0$*. Our analysis here doesn’t capture the behavior at the source points themselves, and in fact we expect nonzero delta-function-like magnetic fields with support on $\ell_0,\ell'_0$. Since ${\mathcal{B}}(\ell;Y)$ is associated with antipodally even solutions, its vanishing substantiates our identification of the propagator - as antipodally odd. The possible *non*-vanishing of ${\mathcal{B}}(\ell;Y)$ at the source points themselves is related to the subtle interplay between antipodal symmetry and analyticity, which we discussed in section \[sec:linear\_HS:antipodal\].
Let us now extract the various tensor components of the electric master field . We begin with the spin-0 Dirichlet data : $$\begin{aligned}
\varphi(\ell) = \frac{1}{2}{\mathcal{E}}(\ell;0) = \frac{\mp 1}{2\pi\sqrt{(\ell_0\cdot\ell)(\ell'_0\cdot\ell)}} \ . \label{eq:example_Dirichlet_covariant}\end{aligned}$$ To extract the components with spin $s>0$, we expand into a Taylor series in $Y$ and compare with : $$\begin{aligned}
\begin{split}
{\mathcal{E}}^{a_1\dots a_{2s}}(\ell) ={}& \frac{\mp i^s (2s)!\,\gamma_{\mu_1\nu_1}^{(a_1 a_2}\dots\gamma_{\mu_s\nu_s}^{a_{2s-1}a_{2s})}}{8^s s!\,\pi \sqrt{(\ell_0\cdot\ell)(\ell'_0\cdot\ell)}}
\left(\frac{\ell^{\mu_1}\ell_0^{\nu_1}}{\ell_0\cdot\ell} - \frac{\ell^{\mu_1}\ell'^{\nu_1}_0}{\ell'_0\cdot\ell} \right) \ldots
\left(\frac{\ell^{\mu_s}\ell_0^{\nu_s}}{\ell_0\cdot\ell} - \frac{\ell^{\mu_s}\ell'^{\nu_s}_0}{\ell'_0\cdot\ell} \right)
\end{split}\end{aligned}$$ Next, we use to convert from spinors to tensors: $$\begin{aligned}
{\mathcal{E}}^{\mu_1\dots\mu_s}(\ell) = \frac{\mp i^s (2s)!}{8^s s!\,\pi \sqrt{(\ell_0\cdot\ell)(\ell'_0\cdot\ell)}}
\left(\frac{\ell_0^{\mu_1}}{\ell_0\cdot\ell} - \frac{\ell'^{\mu_1}_0}{\ell'_0\cdot\ell} \right) \ldots \left(\frac{\ell_0^{\mu_s}}{\ell_0\cdot\ell} - \frac{\ell'^{\mu_s}_0}{\ell'_0\cdot\ell} \right)
- \text{traces} \ . \label{eq:example_E_tensor_covariant}\end{aligned}$$ The trace pieces that are subtracted in can be represented using any 3d metric of the form $\eta_{\mu\nu} + 4\ell_{(\mu} n_{\nu)}$, where $n^\mu\in{\mathbb{R}}^{1,4}$ is a null vector satisfying $\ell\cdot n = -1/2$. Different choices of this 3d metric lead to tensors ${\mathcal{E}}^{\mu_1\dots\mu_s}(\ell)$ that are equivalent under .
Finally, let us make the boundary tensors more concrete by translating them into flat 3d boundary coordinates. To do this, we express $\ell_0$, $\ell'_0$ and $\ell$ in the flat conformal frame . As it stands, the tensor is not tangential to the flat section of the ${\mathbb{R}}^{1,4}$ lightcone. However, this can be fixed by adding a suitable multiple of $\ell^\mu$ to each tensor factor in . The $\ell_0^\mu/(\ell_0\cdot\ell)$ factors then become: $$\begin{aligned}
\frac{\ell_0^\mu}{\ell_0\cdot\ell} \cong \frac{\ell_0^\mu - \ell^\mu}{\ell_0\cdot\ell} = \frac{1}{|\mathbf{r} - \mathbf{r_0}|^2} \left(r^2 - r_0^2\, , \, 2(\mathbf{r} - \mathbf{r_0})\, , \, -(r^2 - r_0^2) \right) \ ,\end{aligned}$$ and likewise for the $\ell'^\mu_0/(\ell'_0\cdot\ell)$ factors. Plugging these back into and keeping only the values $\mu = 1,2,3$ for each index, we end up with the 3d tensor: $$\begin{aligned}
\begin{split}
{\mathcal{E}}_{k_1\dots k_s}(\mathbf{r}) ={}& \frac{\mp i^s (2s)!}{2^{2s-1} s! \pi |\mathbf{r} - \mathbf{r_0}| |\mathbf{r} - \mathbf{r'_0}|} \\
&\times \left(\frac{(\mathbf{r} - \mathbf{r_0})_{k_1}}{|\mathbf{r} - \mathbf{r_0}|^2} - \frac{(\mathbf{r} - \mathbf{r'_0})_{k_1}}{|\mathbf{r} - \mathbf{r'_0}|^2} \right) \ldots
\left(\frac{(\mathbf{r} - \mathbf{r_0})_{k_s}}{|\mathbf{r} - \mathbf{r_0}|^2} - \frac{(\mathbf{r} - \mathbf{r'_0})_{k_s}}{|\mathbf{r} - \mathbf{r'_0}|^2} \right) - \text{traces} \ .
\end{split} \label{eq:example_E_tensor}\end{aligned}$$ This time, the subtracted trace pieces can be written out unambiguously, using the flat 3d metric $\delta_{ij}$. For completeness, we translate into the flat frame also the scalar boundary data : $$\begin{aligned}
\varphi(\mathbf{r}) = \frac{\mp 1}{\pi |\mathbf{r} - \mathbf{r_0}| |\mathbf{r} - \mathbf{r'_0}|} \ . \label{eq:example_Dirichlet}\end{aligned}$$ Note that eqs. - have the same geometric structure as a 3-point function between two spin-0, $\Delta=1/2$ operators at $\mathbf{r_0},\mathbf{r'_0}$ and a spin-$s$, $\Delta=s+1$ operator at $\mathbf{r}$. This “coincidence” is of course predetermined by the boundary conformal symmetry.
Boundary currents from a bilocal source {#sec:holography:example_currents}
---------------------------------------
In this section, we calculate the linearized expectation values of the CFT currents induced by the bilocal source . First, we write the linearized expectation value of the bilocal operator : $$\begin{aligned}
\left<\phi^I(\ell)\bar\phi_I(\ell')\right>_{\text{linear}} = NG(\ell,\ell'_0)G(\ell_0,\ell') = \frac{N}{32\pi^2\sqrt{(\ell'_0\cdot\ell)(\ell_0\cdot\ell')}} \ , \label{eq:example_phi_phi_covariant}\end{aligned}$$ where we ignore both the zeroth order and all orders higher than 1 in the source dependence.
Translating into the flat boundary coordinates , we get: $$\begin{aligned}
\left<\phi^I(\mathbf{r})\bar\phi_I(\mathbf{r'})\right>_{\text{linear}} = \frac{N}{16\pi^2 |\mathbf{r} - \mathbf{r'_0}| |\mathbf{r'} - \mathbf{r_0}|} \ . \label{eq:example_phi_phi}\end{aligned}$$ The Taylor expansion of this around $\mathbf{r} = \mathbf{r'}$ reads: $$\begin{aligned}
\begin{split}
&\left<\phi^I(\mathbf{r})\overset{\leftarrow}{{\partial}}_{i_1}\dots\overset{\leftarrow}{{\partial}}_{i_m} \overset{\rightarrow}{{\partial}}_{j_1}\dots\overset{\rightarrow}{{\partial}}_{j_n} \bar\phi_I(\mathbf{r})\right>_{\text{linear}}
= \frac{(-1)^{m+n}(2m)!(2n)! N}{2^{m+n+4} \pi^2 m!n!} \\
&\qquad \times
\frac{(\mathbf{r} - \mathbf{r'_0})_{i_1}\dots(\mathbf{r} - \mathbf{r'_0})_{i_m}(\mathbf{r} - \mathbf{r_0})_{j_1}\dots(\mathbf{r} - \mathbf{r_0})_{j_n}}{|\mathbf{r} - \mathbf{r'_0}|^{2m+1} |\mathbf{r} - \mathbf{r_0}|^{2n+1}}
\ + \, \text{trace terms} \ ,
\end{split} \label{eq:example_phi_phi_taylor}\end{aligned}$$ where by “trace terms” we mean terms proportional to the flat 3d metric $\delta_{ij}$. We can now combine the derivatives to obtain the spin-$s$ currents : $$\begin{aligned}
\begin{split}
&\left<j^{(s)}_{k_1\dots k_s}(\mathbf{r})\right>_{\text{linear}} = \frac{(2s)!N}{4^{s+2}i^s\pi^2} \sum_{m=0}^s \frac{(-1)^{s-m}}{m!(s-m)!} \\
&\qquad \times \frac{(\mathbf{r} - \mathbf{r'_0})_{(k_1}\dots(\mathbf{r} - \mathbf{r'_0})_{k_m}(\mathbf{r} - \mathbf{r_0})_{k_{m+1}}\dots(\mathbf{r} - \mathbf{r_0})_{k_s)}}
{|\mathbf{r} - \mathbf{r'_0}|^{2m+1} |\mathbf{r} - \mathbf{r_0}|^{2(s-m)+1}} - \text{traces} \ .
\end{split}\end{aligned}$$ The above sum evaluates neatly into: $$\begin{aligned}
\begin{split}
&\left<j^{(s)}_{k_1\dots k_s}(\mathbf{r})\right>_{\text{linear}} = \frac{i^s(2s)! N}{4^{s+2}s!\pi^2 |\mathbf{r} - \mathbf{r_0}| |\mathbf{r} - \mathbf{r'_0}|} \\
&\qquad \times \left(\frac{(\mathbf{r} - \mathbf{r_0})_{k_1}}{|\mathbf{r} - \mathbf{r_0}|^2} - \frac{(\mathbf{r} - \mathbf{r'_0})_{k_1}}{|\mathbf{r} - \mathbf{r'_0}|^2} \right) \ldots
\left(\frac{(\mathbf{r} - \mathbf{r_0})_{k_s}}{|\mathbf{r} - \mathbf{r_0}|^2} - \frac{(\mathbf{r} - \mathbf{r'_0})_{k_s}}{|\mathbf{r} - \mathbf{r'_0}|^2} \right) - \text{traces} \ .
\end{split} \label{eq:example_j}\end{aligned}$$ If we had taken into account the source-independent term in the bilocal $\left<\phi^I(\ell)\bar\phi_I(\ell')\right>$, we would have gotten an additional divergent contribution to the spin-0 “current” $\langle j^{(0)}(\mathbf{r})\rangle = \langle\phi^I(\mathbf{r})\bar\phi_I(\mathbf{r})\rangle$, with no change to the currents of spin $s>0$.
Comparing now with the results - for the asymptotics of the linearized bulk fields, we find: $$\begin{aligned}
\left<\phi^I(\mathbf{r})\bar\phi_I(\mathbf{r})\right>_{\text{linear}} = \mp \frac{N}{16\pi}\,\varphi(\mathbf{r}) \ ; \quad
\left<j^{(s)}_{k_1\dots k_s}(\mathbf{r})\right>_{\text{linear}} = \mp \frac{N}{32\pi}\,{\mathcal{E}}_{k_1\dots k_s}(\mathbf{r}) \ . \label{eq:currents_holography}\end{aligned}$$ where, in the second equality, we take $s>0$. We’ve thus demonstrated the proportionality, and found the proportionality coefficients, between the Dirichlet/electric boundary data and the linearized expectations values of the corresponding single-trace operators. The $s$-independence of the coefficients in results from the particular normalization choice in our definition of the CFT currents.
The sign ambiguity in , which we’ve been carrying from eq. , can be fixed by hand by comparing with the standard dictionary in the spin-2 case, i.e. the correspondence between the bulk graviton and the CFT stress tensor. To do this, we’d have to fix a sign convention for the relation between the Weyl tensor $C_{\mu\nu\rho\sigma}$ and the corresponding bulk metric perturbation. In any case, this fixing of the signs doesn’t seem essential: in GR, the sign of the metric perturbation only becomes meaningful at the interacting level, where detailed analogies with HS gravity are not very useful.
General boundary currents and the extent of the holographic dictionary {#sec:holography:general_currents}
----------------------------------------------------------------------
Let us now extract the general lessons from our calculation in sections \[sec:holography:example\_asymptotics\]-\[sec:holography:example\_currents\]. In analogy with the bulk asymptotics ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$, let us define local master-field operators on the CFT side via: $$\begin{aligned}
J(\ell;Y) &= \frac{i}{8\pi}\int_{P(\ell)} d^2u \left(\Phi(u)\,e^{iuY} + \Phi(Y+u)\right) \ ; \label{eq:J} \\
H(\ell;Y) &= \frac{i}{2\pi N}\int_{P(\ell)} d^2u \left(\Phi(u)\,e^{iuY} - \Phi(Y+u)\right) \ . \label{eq:H}\end{aligned}$$ From eq. , we see that at first order in the sources, these operators have the expectation values: $$\begin{aligned}
\langle J(\ell;Y)\rangle_{\text{linear}} &= \frac{iN}{32\pi}\int_{P(\ell)} d^2u \left(F(u)\,e^{iuY} + F(Y+u)\right) \ ; \\
\langle H(\ell;Y)\rangle_{\text{linear}} &= \frac{i}{8\pi}\int_{P(\ell)} d^2u \left(F(u)\,e^{iuY} - F(Y+u)\right) \ .\end{aligned}$$ The $Y$ dependence of these master fields is only through the spinor component $y^*\in P^*(\ell)$.
The implication of the result is that, in regions where the source $\Pi(\ell',\ell)$ vanishes, $\langle J(\ell;Y)\rangle_{\text{linear}}$ encodes the linearized currents $\left<j^{(s)}_{\mu_1\dots \mu_s}(\ell)\right>_{\text{linear}}$ in the same sense that ${\mathcal{E}}(\ell;Y)$ encodes the electric boundary data $E_{\mu_1\dots\mu_s}(\ell)$, up to the sign ambiguity in the Penrose transform . At the same time, in complete analogy with ${\mathcal{E}}(\ell;Y)$ and ${\mathcal{B}}(\ell;Y)$, the tensor components of the operators - are automatically divergence-free. Putting everything together, we see that $J(\ell;Y)$ encodes a tower of spin-$s$ conformal primaries $j^{(s)}_{\mu_1\dots \mu_s}(\ell)$, conserved to all orders in the source $\Pi(\ell',\ell)$, which at linear order correctly reproduce the expectation values of the CFT currents in regions where $\Pi(\ell',\ell)$ vanishes. The most natural conclusion, then, is that $J(\ell;Y)$ encodes the conserved CFT currents to all orders in the source, *with all the necessary contact terms automatically included*.
As for the master field $H(\ell;Y)$, the result of section \[sec:holography:example\_asymptotics\] implies that its linearized expectation value vanishes in regions with $\Pi(\ell',\ell)=0$. By construction, $\langle H(\ell;Y)\rangle_{\text{linear}}$ is proportional to the magnetic boundary data ${\mathcal{B}}(\ell;Y)$. Thus, we expect that in regions with $\Pi(\ell',\ell)\neq 0$, it will encode the linearized magnetic field strengths associated with the source. This is the reason for our choice of coefficient in : we wanted to emphasize that $H(\ell;Y)$ is more closely related to the source $F(Y)$ than to the “current” $\Phi(Y)$. The interpretation of the full non-linear expectation value of $H(\ell;Y)$ is not entirely clear to us. Perhaps the most natural possibility is that it still encodes the source’s magnetic field strength, but with the non-abelian structure of higher-spin symmetry taken into account.
Discussion {#sec:discuss}
==========
In this paper, we’ve shown how a twistorial description underlies both bulk and boundary pictures in the higher-spin/free-CFT holography. In particular, our boundary/twistor transform , does the same for single-trace bilocals in the free $U(N)$ vector model as the Penrose transform has done for free massless fields in 4d.
Our main evidence that the bulk and boundary pictures as derived from twistor space are indeed holographically equivalent is the calculation of linearized boundary currents away from sources. Beyond this, much of the relationship between our twistor language and the standard local descriptions was left implicit. It should be worthwhile to explore this relationship further. In particular, one would like to express the boundary sources’ local field strengths in the bilocal language, and compare to the twistorial expression . One should also understand explicitly the local currents in regions where the source doesn’t vanish, and then check how (or whether) eq. contains the necessary information about contact terms.
On the bulk side, the main missing component in our approach, as in [@Didenko:2012tv], is the relation to the nonlinear Vasiliev equations. The unbroken global HS symmetry has allowed us to “cheat” by encoding the interactions as functionals of the linearized master fields. However, to make contact with the broader realm of higher-spin theory, one should understand how to go back and forth between this approach and Vasiliev’s picture of nonlinear bulk master fields.
From a fundamental perspective, the picture we laid out in this paper is very appealing: all the three geometric frameworks of bulk, boundary and twistor space are manifestly unified. Furthermore, the twistor function $F(Y)$ provides a clean diff-invariant & gauge-invariant encoding of the physical data on both bulk and boundary. Ideally, one would like to apply this kind of picture to more realistic holographic models, which contain General Relativity in the bulk. However, at the moment, it is unclear to us how that might happen. In our construction, we relied heavily on the fact that the boundary CFT is a free vector model – that is what allowed the bilocal formulation of the single-trace operators. Similarly, in the bulk, we made crucial use of the unfolded formulation of HS theory. It is what enabled us to cleanly encode a linearized bulk solution in terms of a master field at a single point $x$, which we could then use as an input for the “bulk” partition function .
While there are many reasons to study HS theory, the author’s personal motivation is that it provides the only known working model of [dS~4~/CFT~3~]{} [@Anninos:2011ui]. In that context, I am pursuing a program [@Halpern:2015zia] to extract the physics inside observers’ cosmological horizons. A key component in this program is the idea [@Parikh:2002py] to replace $dS_4$ with its “folded-in-half” version $dS_4/Z_2$, where the $Z_2$ refers to the antipodal map $x^\mu\rightarrow -x^\mu$. In [@Halpern:2015zia], we managed to derive the physics inside the horizon in this picture for the linearized limit of HS gravity, i.e. for free massless fields in the bulk. The motivation of the present work was to develop tools in order to translate those preliminary results into the twistor language of HS theory, and then extend them beyond the linearized limit. It is our hope that the language we developed here will prove powerful enough for the task.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Eugene Skvortsov, Slava Didenko, Mikhail Vasiliev, Antal Jevicki, Maxim Grigoriev, Vasudev Shyam and Adrian David for discussions. This work was supported by the Quantum Gravity Unit of the Okinawa Institute of Science and Technology Graduate University (OIST). During early stages of the work, YN was employed at Perimeter Institute, where he was supported by the Government of Canada through Industry Canada, by the Province of Ontario through the Ministry of Research & Innovation, as well as by NSERC Discovery grants. The initial calculations were performed while attending the MIAPP Programme “Higher Spin Theory and Duality”.
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---
abstract: 'For a class of one-dimensional autoregressive sequences $(X_n)$ we consider the tail behaviour of the stopping time $T_0=\min \lbrace n\geq 1: X_n\leq 0 \rbrace$. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of $T_0$ and on the analytical Fredholm alternative. Using this method, we show that $\mathbb{P}_x(T_0=n)\sim V(x)R_0^n$ for some $0<R_0<1$ and a positive $R_0$-harmonic function $V$. Further we prove that our conditions on the tail behaviour of the innovations are sharp in the sense that fatter tails produce non-exponential decay factors.'
address:
- 'Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany'
- 'Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany'
- 'Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany'
author:
- Günter Hinrichs
- Martin Kolb
- Vitali Wachtel
title: 'Persistence of one-dimensional AR(1)-sequences'
---
Introduction and setting
========================
The analysis of first hitting times of subsets of the state space by a Markov chain $(X_n)_{n\geq 0}$ is a subject with a long history, but still many recent contributions. For many applications it is important to gain precise control of the tail behaviour of hitting times. One of the aims of this work is to demonstrate the usefulness of the combination of analytic and probabilistic techniques using the example of autoregressive processes.
Let $\xi_k$ be independent, identically distributed random variables and let $a\in(0,1)$ be a fixed constant. An AR$(1)$-sequence is defined by $$\label{AR.def}
X_n=aX_{n-1}+\xi_n,\quad n\geq 1,$$ where the starting point $X_0$ of this process may be either deterministic or distributed according to any probabilistic measure $\nu$. If $X_0=x$ then we write $\mathbb{P}_x$ for the distribution of the process and if $X_0$ is distributed according to $\nu$ then we shall write $\mathbb{P}_\nu$ for the distribution of the process. The sequence $(X_n)_{n\in\mathbb{N}_0}$ defines a Markov chain with state space $\mathbb{R}$, whose properties have been analysed in a great number of papers and we only refer to some of the more recent contributions, such as [@AM17], [@AB11], [@Christensen12], [@Llaralde], [@Nov09], [@NK08]. Closest to our present contribution and the main stimulus for the present paper is the recent work [@AM17], where persistence probabilities for the process and its multidimensional versions have been studied. The focus of [@AM17] has been on deriving the existence and basic properties such as positivity and monotonicity of the persistence exponents $$\lambda_a:=\lim_{n \rightarrow \infty}\frac{1}{n}\log \mathbb{P}\bigl(X_0>0,X_1>0,\dots,X_n>0\bigr)$$ for rather general Markov chains (including multidimensional cases) and its calculation for some very specific chains. For our purposes, let us define $$T_0:=\min\{k \geq 1:\ X_k\leq 0\},$$ i.e. the first time at which the process becomes negative. Similar to [@AM17], we are going to study the tail behaviour of this stopping time, but, in contrast to [@AM17], we are aiming at precise instead of rough asymptotic results. This means we aim to find the precise, without logarithmic scaling, asymptotics of $$n \mapsto \mathbb{P}_x(T_0>n)$$ as $n\rightarrow \infty$. Under natural and in some sense minimal conditions we aim to show that the tail of the stopping time $T_0$ has an exactly exponential decay. We will focus on the one-dimensional situation.
Denoting by $P(x,dy)$ the transition probability of the Markov chain $(X_n)_{n\ge0}$ we observe that, for every $x>0$, $$\begin{split}
\mathbb{P}_x &\bigl(X_1>0,X_2>0,\ldots ,X_n>0\bigr)=\\
&\quad \quad\int_{(0,\infty)}P(x,dy_1)\int_{(0,\infty)}P(y_1,dy_2)\dots\int_{(0,\infty)}P(y_{n-1},(0,\infty))
\end{split}$$ and therefore the probability $\mathbb{P}_x \bigl(X_1>0,X_2>0,\ldots ,X_n>0\bigr)$ can be interpreted as the $n$-th power of the total mass of the substochastic transition kernel given by $$P_+(x,A):=\mathbf{1}_{(0,\infty)}(x)P(x,A \cap (0,\infty)).$$ From the Gelfand formula for the spectral radius it is tempting to connect $\lambda_a$ to the spectral radius of some operator induced by $P_+$. From an operator theoretic perspective the problem consists in the fact that, due to the unboundednes of the state space $(0,\infty)$, the substochastic kernel $P_+$ is usually not a (quasi-)compact operator on the standard Banach spaces of continuous or $p$-th power integrable functions. One way out is to find better adapted Banach spaces, which is often possible. A different strategy which we are going to present as well consists in analysing the behaviour of the Laplace transform $$\lambda \mapsto \mathbb{E}_x\bigl[e^{\lambda T_0}\bigr]$$ near the critical line and in this respect is classical. In fact, similar arguments appear in the investigation of other large time problems in the theory of stochastic processes such as [@L95] and [@SW75]. In order to deduce the required properties we will show that $\mathbb{E}_x\bigl[e^{\lambda T_0}\bigr]$ satisfies a suitable renewal equation and study some operator theoretic properties of the corresponding transition operator. This leads to a meromorphic representation for the function $\mathbb{E}_x\bigl[e^{z T_0}\bigr]$. The final step consists in showing that all singularities near the critical line $\{z:\, \Re z=\lambda_a\}$ are simple poles and in the subsequent application of the Wiener-Ikehara theorem. We want to emphasize that related results have been recently derived in [@CV17] using completely different methods and we will comment on connections below.
All our results will be valid for any stopping time $$T_r:=\min\{k \geq 1:\ X_k\leq r\},\quad r\in\mathbb{R}.$$ This is immediate from the observation that the sequence $X_n^{(r)}:=X_n-r$, $n\geq0$ satisfies with innovations $\xi_n^{(r)}:=\xi_n-(1-a)r$, $n\ge1$.
We want to stress at this point that even though we exclusively deal with the one-dimensional situation in this work, the approaches we present are more generally applicable, also in multidimensional situations and to processes of different type. As it seems that our analytic approaches are not well known in the probabilistic literature, they are at least rarely used in standard literature, we hope that the present contribution also serves as template on powerful analytic methods for persistence and quasistationary problems.
Several authors, see [@Christensen12; @Llaralde; @Nov09], have obtained exact expressions for the Laplace transform in the case when the distribution of the innovations is related to the exponential distribution. Unfortunately, the expressions are very complicated, and it is not clear how to invert them or how to use them for deriving the tail asymptotics for $T_0$. Moreover, it is not clear whether one can obtain an explicit expression for $\lambda_a$. If, for example, the innovations $\xi_n$ have density $e^{-\mu|x|}/(2\mu)$ then, as it has been shown in [@Llaralde], $$\mathbb{E}_0 s^{T_0}=\frac{s(as,a^2)_\infty}{(as,a^2)_\infty+(s,a^2)_\infty},$$ where $(u,q)_\infty=\prod_{k=0}^\infty(1-uq^k)$. Therefore, $e^{\lambda_a}$ is the minimal positive solution to the equation $(as,a^2)_\infty+(s,a^2)_\infty=0$. It is obvious that this solution lies between $1$ and $a^{-1}$, but an explicit expression is not accessible. In order to analyse the tail behaviour of $T_0$ we have to take into account all singularities of this function on the circle of radius $e^{\lambda_a}$, but this information is also rather hard to extract from the exact expression. Expressions in [@Christensen12] and in [@Nov09] are even more complicated. Summarising, all known explicit expressions for the Laplace transform of $T_0$ do not seem to provide any useful information on the asymptotic properties of $T_0$.
The structure of the paper is the following. In Section \[s:rough\] we prove under rather general assumptions the existence and positivity of the decay rate $$\lambda_a:=-\lim_{n\rightarrow \infty}\frac{1}{n}\log \mathbb{P}_x\bigl(T_0>n\bigr).$$ In Section \[s:boundedsupport\] we consider the situation where the distribution of the innovations has bounded support. In this case, one can use results in [@CV16] in order to show that the tails have a precise exponential decay. For unbounded innovations this approach is no longer possible.
In order to be able to cover also unbounded innovations we introduce in Section \[s:FuAn\] a different functional analytic approach relying on the concept of a quasicompact operator and the associated spectral theory.
Section \[s:renewal\] presents another way to deal with unbounded innovations, which is based on a renewal argument in combination with some basic operator theoretic arguments applied to the renewal operator.
Section \[s:reg\] deals with the situation of regularly varying case, where the foregoing theory is not applicable.
In Section \[s:dis\], the essential ingredients of our techniques and their applicability to different problems are discussed.
Rough asymptotics for persistence probabilities {#s:rough}
===============================================
In this section we establish a general result concerning the exponential decay of the tails of the hitting time $T_0$. Results of this type will be an essential ingredient for the further investigation of more detailed properties of the first hitting time $T_0$. More precisely, we shall derive asymptotics for the logarithmically scaled tail of $T_0$. In contrast to [@AM17] we study the one-dimensional situation, but there we are able to work under much weaker hypotheses.
Let us first recall some basic properties of the Markov process $(X_n)_{n\in\mathbb{N}_0}$. If $\mathbb{E}\log(1+|\xi_1|)<\infty$ then $X_n$ converges weakly to the distribution of the series $$X_\infty:=\sum_{k=1}^\infty a^{k-1}\xi_k.$$ This is immediate from the following expression for $X_n$: $$\label{eq.1}
X_n=a^nX_0+a^{n-1}\xi_1+a^{n-2}\xi_2+\ldots+\xi_n.$$ Let $\pi(dx)$ denote the distribution of $X_\infty$. This distribution is stationary for the Markov chain $X_n$, that is, $$\label{eq.2}
\mathbb{P}_\pi(X_n\in dx)=\pi(dx),\quad n\geq1.$$
\[thm.log\] Assume that the innovations $(\xi_n)_{n\in\mathbb{N}}$ satisfy $$\mathbb{E}\log(1+|\xi_1|)<\infty,\, \mathbb{E}(\xi_1^+)^\delta<\infty \text{ for some }\delta>0
\quad\text{and}\quad
\mathbb{P}(\xi_1>0)\mathbb{P}(\xi_1<0)>0\,.$$ Then, for every $a\in(0,1)$, $$\label{thm.log.1}
-\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}_x\bigl( T_0>n\bigr)
=\lambda_a\in (0,\infty),\quad x\in(0,\infty).$$ Furthermore, if the distribution of the innovations satisfies $$\label{thm.log.2}
\lim_{x\to\infty}\frac{\log\mathbb{P}(\xi_1>x)}{\log x}=0,$$ then $$\label{thm.log.3}
-\lim_{n\rightarrow\infty}\frac{1}{n}\log\mathbb{P}_x\bigl( T_0>n\bigr)
=0,\quad x\in(0,\infty).$$
If $\mathbb{P}(\xi_1<0)=0$ then $T_0=\infty$ almost surely. Furthermore, the assumption $\mathbb{P}(\xi_1>0)>0$ is imposed to avoid a trivial situation when the chain $X_n$ is monotone decreasing before hitting negative numbers. As it has been mentioned at the beginning of this section, the existence of the logarithmic moment yields the ergodicity of $X_n$. Finally, the finiteness of $\mathbb{E}(\xi_1^+)^\delta$ is needed for the positivity of $\lambda_a$ only.
Existence and finiteness of the limit $\lambda_a$ follow also from Theorem 2.3 in [@AM17] in a multidimensional situation at least under the condition that an exponential moment exists. We use a weaker moment assumption $\mathbb{E}(\xi_1^+)^\delta<\infty$. Since the finiteness of $\mathbb{E}(\xi_1^+)^\delta$ implies that $$\limsup_{x\to\infty}\frac{\log\mathbb{P}(\xi_1>x)}{\log x}\le-\delta,$$ we conclude that the moment assumption $\mathbb{E}(\xi_1^+)^\delta<\infty$ is optimal for the positivity of $\lambda_a$.
Relation is a simple consequence of relation (2.4) in [@AM17]. It should be noted that the proof of (2.4) does not use the assumption that the innovations are normally distributed and, consequently, we may use (2.4) in the proof of .
It follows from that, for $0\le x\le y$, $$\begin{aligned}
\label{eq.3}
\nonumber
\mathbb{P}_x(T_0>n)
&=\mathbb{P}\left(\min_{k\leq n}\left(xa^k+\sum_{j=1}^ka^{k-j}\xi_j\right)>0\right)\\
&\le \mathbb{P}\left(\min_{k\leq n}\left(ya^k+\sum_{j=1}^ka^{k-j}\xi_j\right)>0\right)
=\mathbb{P}_y(T_0>n).\end{aligned}$$ Besides $T_0$ we consider a slightly modified stopping time $$\widetilde{T}_0:=\min\{k\ge0:X_k\le0\}.$$
The monotonicity property implies that, for every $x>0$, $$\begin{aligned}
\mathbb{P}_\pi(\widetilde{T}_0>n)=\int_0^\infty\pi(dy)\mathbb{P}_y(T_0>n)
&\ge\int_x^\infty\pi(dy)\mathbb{P}_y(T_0>n)\\
&\ge \pi[x,\infty)\mathbb{P}_x(T_0>n).\end{aligned}$$ In other words, $$\label{eq.4}
\mathbb{P}_x(T_0>n)\le \frac{1}{\pi[x,\infty)}\mathbb{P}_\pi(T_0>n).$$
Next, multiplying by $a^{-n}$, we have $$a^{-n}X_n=X_0+\sum_{j=1}^n a^{-j}\xi_j=:X_0+S_n$$ and $$\left\{T_0>n\right\}=\left\{X_0+\min_{k\le n}S_k>0\right\}.$$ Since $X_0+S_k$ are sums of independent random variables, we may apply FKG inequality for product spaces, which gives the estimate $$\begin{aligned}
\label{eq.5}
\nonumber
\mathbb{P}(X_n>y|T_0>n)
&=\mathbb{P}\left(X_0+S_n>a^{-n}y\Big| X_0+\min_{k\le n}S_k>0\right)\\
&\ge \mathbb{P}\left(X_0+S_n>a^{-n}y\right)=\mathbb{P}(X_n>y),\end{aligned}$$ which holds for every distribution of $X_0$. Applying this inequality to the stationary process, we obtain $$\begin{aligned}
\mathbb{P}_\pi(\widetilde{T}_0>n+m)&=\int_0^\infty\mathbb{P}_\pi(X_n\in dy,\widetilde{T}_0>n)\mathbb{P}_y(T_0>m)\\
&=\mathbb{P}_\pi(\widetilde{T}_0>n)\int_0^\infty\mathbb{P}_\pi(X_n\in dy|\widetilde{T}_0>n)\mathbb{P}_y(T_0>m)\\
&\ge \mathbb{P}_\pi(\widetilde{T}_0>n)\int_0^\infty\mathbb{P}_\pi(X_n\in dy)\mathbb{P}_y(T_0>m)\\
&=\mathbb{P}_\pi(\widetilde{T}_0>n)\mathbb{P}_\pi(\widetilde{T}_0>m).\end{aligned}$$ Then, by the Fekete lemma, $$\label{eq.6}
-\lim_{n\to\infty} \frac{1}{n}\log\mathbb{P}_\pi(\widetilde{T}_0>n)=:\lambda_a(\pi)\in[0,\infty).$$ By the same argument, for every fixed $x$ we have $$\begin{aligned}
\mathbb{P}_x(T_0>n+m)
&\ge\mathbb{P}_x(T_0>n)\int_0^\infty\mathbb{P}_x(X_n\in dy)\mathbb{P}_y(T_0>m)\\
&\ge\mathbb{P}_x(T_0>n)\int_x^\infty\mathbb{P}_x(X_n\in dy)\mathbb{P}_y(T_0>m).\end{aligned}$$ Using now the monotonicity property , we get $$\label{eq.6a}
\mathbb{P}_x(T_0>n+m)\ge \mathbb{P}_x(T_0>n)\mathbb{P}_x(X_n\ge x)\mathbb{P}_x(T_0>m).$$ If $x$ is such that $\pi[x,\infty)>0$ then $\mathbb{P}_x(X_n\ge x)\to\pi[x,\infty)$ and we may apply again the Fekete lemma: $$\label{eq.7}
-\lim_{n\to\infty} \frac{1}{n}\log\mathbb{P}_x(T_0>n)=:\lambda_a(x)\in[0,\infty).$$ By , $\lambda_a(x)$ is decreasing in $x$. Moreover, from , and , we infer that $$\lambda_a(x)\ge\lambda_a(\pi)\quad\text{for all }x\text{ such that }\pi[x,\infty)>0.$$ If $\pi[x,\infty)>0$, then there exist $\varepsilon>0$ and $m_0$ such that $\mathbb{P}_x(X_{m_0}>x+\varepsilon)>0$. Then, using the Markov property at time $m_0$ and the monotonicity of $P_y(T_0>n)$, we conclude that $\lambda_a(x)=\lambda_a(y)$ for all $y\in[x,x+\varepsilon]$. As a result, we have $$\lambda_a(x)=\lambda_a(\pi)=:\lambda_a \quad
\text{for all }x\text{ such that }\pi[x,\infty)>0.$$ According to Theorem 1 in [@NK08], the assumption $\mathbf{E}(\xi_1^+)^\delta<\infty$ for some $\delta>0$ implies that $\lambda_a>0$. Thus, we have the same exponential rate for all starting points in the support of the measure $\pi$.
If $\mathbb{P}(\xi_1>x)>0$ for all $x>0$, then we have the logarithmic asymptotic behaviour for all positive starting points.
Consider now the case when innovations are bounded from above. Let $R$ denote the essential supremum of $\xi_1$, that is, $\mathbb{P}(\xi_1\le R)=1$ and $\mathbb{P}(\xi_1>R-\varepsilon)>0$ for every $\varepsilon>0$. It is easy to see that $\pi([x,\infty))>0$ for every $x\in(0,R/(1-a))$ and $\pi([x,\infty))=0$ for each $x>R/(1-a)$. If the starting point $x$ is greater than $R_*:=R/(1-a)$ then the sequence $X_n$ decreases before it hits $(0,R_*)$ and there exists $m=m(x,\varepsilon)$ such that $X_m\le R_*+\varepsilon$. Further, for every starting point in $[R_*,R_*+\varepsilon)$ the hitting time of $(-\infty,R_*)$ is stochastically bounded by a geometric random variable with parameter $1-p_\varepsilon$, where $p_\varepsilon:=\mathbb{P}(\xi_1>R-\varepsilon)>0$. Since we know the exponent for all starting points in $(0,R_*)$, we conclude that $$-\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}_{x}(T_0>n)
=\min\{\lambda_a,\log(1/p_\varepsilon)\},\quad x\ge R_*.$$ Letting now $\varepsilon\to0$, we obtain $$-\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}_{x}(T_0>n)
=\min\{\lambda_a,\log(1/p)\},\quad x\ge R_*,$$ where $p=\mathbb{P}(\xi_1=R)$. Now, noting that $$\mathbb{P}_x(T_0>n)\ge \mathbb{P}(\xi_1=\xi_2=\ldots=\xi_n=R)=p^n,\quad x>0,$$ we infer that $\lambda_a\le \log(1/p)$. Consequently, $$-\lim_{n\to\infty}\frac{1}{n}\log\mathbb{P}_{x}(T_0>n)
=\lambda_a,\quad x\ge R_*.$$ This completes the proof of .
It is clear that the distribution of $\xi_1$ is a uniform minorant for distributions of $X_1$, that is, $$\mathbb{P}_x(X_1>y)\ge \mathbb{P}(\xi_1>y),\quad\text{for all }x,y>0.$$ Therefore, $$\mathbb{P}_x(T_0>n)\ge \mathbb{P}_\mu(T_0>n-1),$$ where $\mu$ denotes the distribution of $\xi_1$. follows now from (2.4) in [@AM17].
We conclude this section with the following open problem: For which starting distributions $\mu(dx)=\mathbb{P}(X_0\in dx)$ the statement of Theorem \[thm.log\] remains valid? A general observation that the persistence exponent may depend on the initial distribution of an AR(1)-sequence has been made in Proposition 2.2 in [@AM17]. The reader can find conditions on the initial distribution in [@AM17], in [@CV17] and also in Section 4.2 below ensuring the validity of Theorem 1. Until now a complete characterization does not seem to exist. At this point we want to emphasize that this phenomenon is well known in the theory of quasistationary distributions and is strongly related to the fact that quasistationary distributions are not unique in general (see e.g. [@CMSM13]). Conditions ensuring uniqueness of quasistationary distributions are given in [@CV16].
Innovations with bounded to the right support:\
approach via quasistationarity {#s:boundedsupport}
===============================================
Exponential decay of $\mathbb{P}(T_0>n)$ is often related to a quasi-stationary behaviour of $(X_n)$ conditioned on the event $\{T_0>n\}$. Recall that the quasistationarity implies that $$\mathbb{P}(T_0>n+1|T_0>n)\to e^{-\lambda_a}.$$ So, the logarithmic asymptotics in Theorem \[thm.log\] should also follow from the quasistionarity of $(X_n)$. Furthermore, it is quite natural to expect that the knowledge on the rate of convergence towards a quasi-stationary distribution will imply preciser statements on the tail behaviour of $T_0$. This has been recently confirmed in [@CV16], where necessary and sufficient conditions for an exponentail speed of convergence in the total variation norm to a quasistationary distribution have been provided. There, it has also been shown that such fast convergence yields a purely exponential decay of $\mathbb{P}(T_0>n)$.
Let us formulate conditions from [@CV16] in terms of the AR(1)-sequence $(X_n)_{n\in\mathbb N_0}$.\
[**First condition**]{}: there exist a probability measure $\nu$ and constants $n_0\ge1$, $c_1>0$ such that $$\label{A1-cond}
\mathbb{P}_x(X_{n_0}\in\cdot|T_0>n_0)\ge c_1\nu(\cdot)\quad\text{for all }x>0.$$ [**Second condition**]{}: there exists a constant $c_2$ such that $$\label{A2-cond}
\mathbb{P}_\nu(T_0>n)\ge c_2\mathbb{P}_x(T_0>n)\quad\text{for all }n\ge1\text{ and }x>0.$$ It is rather obvious that can not be valid for all $x>0$. This observation implies that the results in [@CV16] are not applicable to AR(1)-sequences with unbounded to the right innovations. So, we shall assume that innovations $\xi_k$ are bounded. Let $R$ denote, as in the previous section, the essential supremum of $\xi_1$. Then the invariant measure lives on the set $(-\infty,R_*]$, where $R_*=R/(1-a)$. If the starting point lies in $(0,R_*]$ then the chain $X_n$ does not exceed $R_*$ at all times. Consequently, we have to find restrictions on the distribution of innovations which will ensure the validity of and for $x\le R_*$ only.
We begin by showing that holds for a quite wide class of innovations.
\[lem:A2-cond\] Assume that the distribution of innovations satisfy $$\label{lem.A2.assump}
\mathbb{P}(\xi_1\le -aR_*)+\mathbb{P}(\xi_1=R)<1.$$ Then there exist $\delta>0$ and a constant $c$ such that, for every $x\in[R_*-\delta,R_*]$, $$\label{lem.A2.0}
\mathbb{P}_{R_*}(T_0>n)\le c\mathbb{P}_x(T_0>n),\quad n\ge1.$$ In particular, the condition is valid for any $\nu$ with $\nu[R_*-\delta,R_*]>0$.
If the assumption does not hold, i.e., $$\mathbb{P}(\xi_1\le -aR_*)+\mathbb{P}(\xi_1=R)=1,$$ then one can easily see that, for all $x\in(0,R_*]$ and $n\ge 1$, $$\mathbb{P}_x(T_0>n)=\left(\mathbb{P}(\xi_1=R)\right)^n.$$ Therefore, does not restrict the generality. $\diamond$
Clearly, yields the existence of $\gamma>0$ such that $$\mathbb{P}(\xi_1>-aR_*+\gamma)>\mathbb{P}(\xi_1=R).$$ Further, there exists $m=m(\gamma)$ such that, uniformly in starting points $x\in(0,R_*]$, $$\mathbb{P}_x(X_{m-1}>R_*-\gamma/a,T_0>m-1)\ge \left(\mathbb{P}(\xi_1=R)\right)^{m-1}.$$ Consequently, $$\begin{aligned}
\mathbb{P}_x(T_0>m)
&\ge\mathbb{P}_x(X_{m-1}>R_*-\gamma/a,T_0>m-1)\mathbb{P}(\xi_m>-aR_*+\gamma)\\
&>\left(\mathbb{P}(\xi_1=R)+\varepsilon\right)^m\end{aligned}$$ for all $x\in(0,R_*]$ and some $\varepsilon>0$. Since $$\begin{aligned}
\mathbb P_x(T_0>nm) \ge \left( \min_{x>0}\mathbb P_x(T_0>m) \right)^{\left\lfloor\frac nm\right\rfloor} = \left( \mathbb P(\xi_1=R)+\varepsilon \right)^{\left\lfloor\frac nm\right\rfloor m}\,,\end{aligned}$$ we infer that $$\mathbb{P}(\xi_1=R)<e^{-\lambda_a}.$$ Therefore, there exists $\delta>0$ such that $$\mathbb{P}(\xi_1>R-\delta)<e^{-\lambda_a},$$ which is equivalent to $$\label{lem.A2.1}
\varepsilon(\delta):=\mathbb{P}_{R_*}(X_1>R_*-\delta)<e^{-\lambda_a}.$$
Taking into account the monotonicity property , we get $$\begin{aligned}
\mathbb{P}_{R_*}(T_0>n)
&\le \mathbb{P}_{R_*}(X_1\le R_*-\delta)\mathbb{P}_{R_*-\delta}(T_0>n-1)\\
&\hspace{1cm}+\mathbb{P}_{R_*}(X_1> R_*-\delta)\mathbb{P}_{R_*}(T_0>n-1)\\
&=(1-\varepsilon(\delta))\mathbb{P}_{R_*-\delta}(T_0>n-1)
+\varepsilon(\delta)\mathbb{P}_{R_*}(T_0>n-1).\end{aligned}$$ Iterating this estimate, we obtain $$\begin{aligned}
\label{lem.A2.2}
\mathbb{P}_{R_*}(T_0>n)\le\frac{1-\varepsilon(\delta)}{\varepsilon(\delta)}
\sum_{k=1}^n\varepsilon^k(\delta)\mathbb{P}_{R_*-\delta}(T_0>n-k).\end{aligned}$$ It follows from that $$\mathbb{P}_{R_*-\delta}(T_0>n-k)\le
\frac{\mathbb{P}_{R_*-\delta}(T_0>n)}
{\mathbb{P}_{R_*-\delta}(T_0>k)\mathbb{P}_{R_*-\delta}(X_k>R_*-\delta)}.$$ Plugging this into , we have $$\begin{aligned}
\mathbb{P}_{R_*}(T_0>n)\le\frac{1-\varepsilon(\delta)}{\varepsilon(\delta)}
\frac{\mathbb{P}_{R_*-\delta}(T_0>n)}{\inf_k \mathbb{P}_{R_*-\delta}(X_k>R_*-\delta)}
\sum_{k=1}^\infty\frac{\varepsilon^k(\delta)}{\mathbb{P}_{R_*-\delta}(T_0>k)}.\end{aligned}$$ The summability of the series on the right hand side follows from Theorem \[thm.log\] and from estimate . Furthermore, the convergence of $X_n$ towards the stationary distribution $\pi$ implies that $\inf_k \mathbb{P}_x(X_k>x)$ is positive. Thus, there exists a constant $c$ such that $$\mathbb{P}_{R_*}(T_0>n)\le c \mathbb{P}_{R_*-\delta}(T_0>n),\quad n\ge 1.$$ The monotonicity of $\mathbb{P}_x(T_0>n)$ completes the proof of the first claim.
Using the monotonicity property once again and applying , we obtain $$\begin{aligned}
\mathbb{P}_{\nu}(T_0>n)&\ge \frac{\nu[R_*-\delta,R_*]}{c}\mathbb{P}_{R_*}(T_0>n)\\
&\ge \frac{\nu[R_*-\delta,R_*]}{c}\mathbb{P}_{x}(T_0>n),\quad x\in(0,R_*].\end{aligned}$$ This completes the proof of the lemma.
We now turn to the condition .
\[lem:A1-cond\] Assume that the distribution of $\xi_1$ has an absolutely continuous component with the density function $\varphi(x)$ satisfying $$\label{lem.A1.1}
\varphi(y)\ge \varkappa>0\quad\text{for all }y\in[R-y_0,R],\ y_0>0.$$ Then, for every measurable $A\subseteq[R_*-y_0,R_*-ay_0]$, $$\liminf_{n\to\infty}\inf_{x\in[0,R_*]}\mathbb{P}_x(X_n\in A|T_0>n)\ge
\varkappa\pi[R_*-y_0,R_*){\rm Leb}(A)$$ ($\rm Leb$ denoting the Lebesgue measure).
By the Markov property, $$\begin{aligned}
\mathbb{P}_x(X_{n+1}\in A|T_0>n+1)
&=\frac{\mathbb{P}_x(X_{n+1}\in A,T_0>n+1)}{\mathbb{P}_x(T_0>n+1)}\\
&\ge\frac{\int_A\left(\int_0^{R_*}\varphi(z-ay)\mathbb{P}_x(X_n\in dy, T_0>n)\right)dz}{\mathbb{P}_x(T_0>n)}.\end{aligned}$$ Applying , we get $$\begin{aligned}
\mathbb{P}_x(X_{n+1}\in A|T_0>n+1)
&\ge \varkappa\int_A\mathbb{P}_x\left(X_n\in\left[\frac{z-R}{a},\frac{z-R+y_0}{a}\right]\Big|T_0>n\right)dz.\end{aligned}$$ For every $z\ge R_*-y_0$ we have $\frac{z-R+y_0}{a}\ge R_*$. Therefore, $$\begin{aligned}
\mathbb{P}_x(X_{n+1}\in A|T_0>n+1)
&\ge \varkappa\int_A\mathbb{P}_x\left(X_n\ge\frac{z-R}{a}\Big|T_0>n\right)dz.\end{aligned}$$ Furthermore, for every $z\le R_*-ay_0$ one has $\frac{z-R}{a}\le R_*-y_0$. Thus, using now and recalling that $X_n$ is increasing in the starting point, we conclude that $$\begin{aligned}
\inf_{x\in[0,R_*)}\mathbb{P}_x(X_{n+1}\in A|T_0>n+1)
\ge \varkappa \mathbb{P}_0(X_n\ge R_*-y_0){\rm Leb}(A).\end{aligned}$$ Letting here $n\to\infty$, we get the desired estimate.
Combining these two lemmata with Proposition 1.2 in [@CV16], we get
\[prop.bounded\] Assume that the innovations $\xi_i$ are a.s. bounded and that their distribution possesses an absolutely continuous component satisfying . Then there exists a positive function $V(x)$ such that, for each $x\in(0,R_*]$, $$\mathbb{P}_x(T_0>n)\sim V(x)e^{-\lambda_a n}\quad\text{as }n\to\infty.$$
Functional analytic approaches {#s:FuAn}
==============================
In this section we combine probabilistic insights with some basic functional analytic observations in order to derive the precise tail behaviour of $T_0$. We want to stress that even though we call this approach functional analytic we will only make use of rather fundamental properties of compact operators combined with assertions of Perron-Frobenius type. The functional analytic ingredients can be found in standard references such as [@AB06], [@D07], [@MN91] and [@S71].
Quasi-compactness approach for bounded innovations. {#sec:quasi-bounded}
---------------------------------------------------
The initial idea from the introduction can be most straightforwardly carried through for bounded innovations $\xi_i$. We assume that they have a density $\varphi$ which is strictly positive on all of its support $[-A,B]$ ($A,B>0$) and consider $P_+f(x):=\mathbb Ef(ax+\xi_1)$ (where $f(y):=0$ for $y<0$) as an operator on $C\left(\left[0,\frac B{1-a}\right]\right)$ with the supremum norm. In this case, we are going to show that $P_+$ is compact with a simple largest eigenvalue $e^{-\lambda_a}$ strictly between 0 and 1 and can then conclude $$\label{expAbfb}
\mathbb P_x(T_0>n)=P_+^n\mathbf 1_{[0,\infty)}(x) = V(x)e^{-\lambda_an} + O(e^{-(\lambda_a+\varepsilon)n})$$ for some nonnegative $V$ and $\varepsilon>0$. Apart from condition , which is not needed in this approach, the result is contained in Theorem \[prop.bounded\]. Our main purpose here is to finally lead over to those cases of unbounded innovations in which conditions from [@CV16] are not valid.
For any continuous $f$ and $x,y\in\left[0,\frac B{1-a}\right]$, $|P_+f(x)|\le\|f\|$ and $$\begin{aligned}
\label{gleichgr13}
|P_+f(x)-P_+f(y)| =& \Big|\int[\varphi(z-ax)-\varphi(z-ay)]f(z)\text dz\Big| \\
\le& \|f\|\int|\varphi(z-ax)-\varphi(z-ay)|\text dz\,.\end{aligned}$$ This goes to zero for $y\to x$, i.e. $P_+$ maps bounded families to equicontinuous ones and, therefore, is compact. $(X_n)$ clearly reaches zero from any starting point in $\left[0,\frac B{1-a}\right]$ if $\xi_1,\dots,\xi_{\left\lceil\frac{B}{1-a}\frac2A\right\rceil}<-\frac A2$, which happens with positive probability, so $$\begin{aligned}
\|P_+^{\left\lceil\frac{B}{1-a}\frac2A\right\rceil}\| = \|P_+^{\left\lceil\frac{B}{1-a}\frac2A\right\rceil}1_{[0,\infty)}\| = \sup_x\mathbb P_x\left(T_0>\left\lceil\frac{B}{1-a}\frac2A\right\rceil\right)<1\,.\end{aligned}$$ Consequently, all eigenvalues of $P_+$ must have modulus less than 1. We now invoke the following generalization of the Perron-Frobenius theorem (see Theorems 6 and 7 in [@S64]):\
**Theorem A** (see [@S64]).
“Quasicompact” should now be thought of as “compact”, the general version will be needed and explained in the proof of Theorem \[th:quasicompact\].
In our case, $B=C\left(\left[0,\frac B{1-a}\right]\right)$ and for $K$ we take the cone of nonnegative functions. It is closed, proper (i.e. $K\cap -K=\{0\}$) and fundamental (this means that $K$ spans a dense subset of $B$, which is clear since $K-K=B$). $P_+$, taking the role of $T$, is positive. The density $\varphi$ was assumed to be strictly positive, so positivity holds true even in the stronger sense that $P_+$ maps nonnegative functions which are somewhere strictly positive to functions which are everywhere strictly positive. For our choice of the space $B$ its dual space $B^*$ consists of all functionals $f\mapsto\int f\text d\mu$ with finite signed Borel measures $\mu$, positive functionals correspond to positive measures. Therefore, this strong positivity implies that $f^*(Tf)>0$ for all $f>0$ and all positive $f^*\in B^*$.
Theorem A is therefore applicable. It allows for the general conclusion that, for some $\varepsilon >0$ and any $f>0$, $$\label{Thochnallg}
T^n f = r(T)^n\,u^*(f)\,u + \mathcal{O}\bigl((r(T)-\varepsilon)^n\bigr)$$ in the space $B$, which, applied to our setting with $f=\mathbf1_{\left[0,\frac{B}{1-a}\right]}$, yields .
In fact, the computation also works for bounded measurable $f$ and shows that $P_+$ is compact on the corresponding space $B\left(\left[0,\frac B{1-a}\right]\right)$. It should be noted that the corresponding dual space consists of signed finitely additive and absolutely continuous measures. Consequently, is also applicable for indicator functions $f=\mathbf1_A$ with measurable $A\subset \left[0,\frac{B}{1-a}\right]$ and yields $$\mathbb P_x(X_n\in A,T_0>n) = \nu(A)V(x)e^{-\lambda_an}+\mathcal O(e^{-(\lambda_a+\varepsilon)n})$$ with same factor $\nu(A)$ which is strictly positive unless $A$ has measure zero and, together with , $$\mathbb P_x(X_n\in A\mid T_0>n) = \nu(A) + \mathcal O(e^{-\varepsilon n}).$$ In other words, we have an exponentially fast convergence to a quasistationary distribution $\nu$.
Quasi-compactness approach for unbounded innovations
----------------------------------------------------
If the innovations $\xi_i$ are absolutely continuous, but are unbounded, $P_+$ still maps bounded families to equicountinous ones and has the same positivity properties, but is in general not compact. A way out is to choose a more suitable Banach space. This can be, for example, explicitely carried through in the case of innovations with standard normal distributions. In this case, $X$ is a discretized Ornstein-Uhlenbeck process: The latter is given, e.g., by the stochastic differential equation $$dZ_t = \theta Z_t\,dt+\sigma dB_t,\quad Z_0=x,$$ where $\theta,\sigma>0$ are constants and $(B_t)_{t\geq 0}$ denotes a Brownian motion. The random variable $Z_t$ is normal distributed with $$\mathbb{E}[Z_t] = x e^{-\theta t}\quad \text{and}\quad \text{Var}[Z_t]=\frac{\sigma^2}{2\theta}(1-e^{-2\theta t}).$$ For the chain $X_n$ with standard normal distributed innovations we have $$\mathbb{P}_x\bigl(X_1 \in B) = \int_B\frac{1}{\sqrt{2\pi}}e^{-\frac{(ax-y)^2}{2}}\,dy.$$ Taking $$\theta=\log(a^{-1})\quad \text{and}\quad \sigma^2=\frac{2\log(a^{-1})}{1-a^2}\,,$$ we see that $Z_1$ and $X_1$ are identically distributed.
$Z$ has $\mathcal N(0,\frac{\sigma^2}{2\theta})$ as the stationary distribution and $Pf(x):=\mathbb Ef(ax+\xi_1)$ is a self-adjoint compact operator with norm 1 on $B:=L^2\left(\mathbb R,\mathcal N\left(0,\frac{\sigma^2}{2\theta}\right)\right)$. This is a well-known result and can be seen e.g. from its diagonal representation in the Hermite polynomials, which form a complete set of eigenfunctions. Consequently, also $$P_+:=1_{[0,\infty)}P1_{[0,\infty)}$$ is compact and self-adjoint. Its norm is strictly below 1. (Otherwise, one could find a normalised sequence $(f_n)$ with $\|P_+f_n\|\ge1-\frac1n$ for all $n\in\mathbb N$. A subsequence $(f_{n_k})$ would have a weak limit $f$ with $\|f\|\le1$ and, by compactness, $P_+f_{n_k}\to P_+f$ in norm, in particular $\|P_+f\|=1$. Then, $P1_{[0,\infty)}f$ would have norm 1, but the spectral representation of $P$ shows that $g\equiv1$ is the only function with $\|g\|\le1$ and $\|Pg\|=1$. Therefore, one can apply Theorem A in the same way as for bounded innovations. A result of a similar type has been shown in [@AB11], but the conclusion drawn has been somewhat weaker.
With somewhat more effort, one could also work on the space of continuous functions with the norm $\|f\|:=\sup_x|f(x)\sqrt{\pi(x)}|$, where $\pi$ is the density of the $\mathcal N(0,\frac{\sigma^2}{2\theta})$ distribution, so the crucial step was to introduce a suitable weight function, whereas $L^2$ instead of $C$ brought only computational benefits in this particular case.
Let us now consider general AR(1)-processes with possibly unbounded innovations. Assume that for some $M>0$, some $\varepsilon >0$ and $\tilde{\lambda}=\lambda_a+\varepsilon$ we have $$\label{e:quasicomp}
\Lambda(x):=\mathbb{E}_x\bigl[ e^{\tilde{\lambda} {T}_M}\bigr] < \infty\,,$$ where $${T}_M:=\min\{k\ge1:X_k\le M\}\,.$$ This property will be later shown to hold, whenever the innovations have moments of all orders, i.e. we do not require existence of an exponential moment. We introduce the Banach space $B(\mathbb R_0^+)$ of measurable functions on $[0,\infty)$ equipped with the norm $$\| f \|_\Lambda := \|\Lambda^{-1}f\|_{\infty} < \infty\,.$$ $\Lambda^{-1}$ is decreasing in $x\in\mathbb R_0^+$ and clearly takes values in $(0,1]$. In contrast to other weight functions with these properties, it is computationally well tractable by the aid of the Markov property.
Actually the approach we now present is an adaption of a standard approach to the ergodicity of Markov chains via quasicompactness (see e.g. Chapter 6 in [@R84] as well as [@HH00]). In the literature on quasistationary distributions an approach of this type has been presented in [@G01]. Our main goal is to indicate via the example of autoregressive processes that the results of [@CV17] can be to a large extent reproved via the concept of quasicompact operators. We hope that our outline of standard analytic ideas in a probabilistic context iluminates further the interrelationship between probabilistic and analytic concepts related to persistence exponents/quasistationarity on the one hand and spectral theoretic ideas on the other.
\[prop:trop\] The transition operator $P$ $$Pf(x):=\mathbb{E}_x\bigl[f(X_1),{T}_0>1\bigr]$$ defines a bounded operator on $(B(\mathbb{R}_0^+),\|\cdot\|_\Lambda)$. Moreover, the spectral radius $r(P)$ is lower bounded by $e^{-\lambda_a}$.
For the boundedness, we observe that for $f \in B(\mathbb{R_+})$ with $\|f\|_\Lambda\leq 1$ $$\begin{split}
\|Pf\|_\Lambda\leq \sup_{x >0}\bigl|\Lambda(x)^{-1}(P\Lambda)(x)\bigr| \,.
\end{split}$$ By the Markov property, $$\label{Markov7}
\mathbb E_x\left[e^{\tilde\lambda T_M}\mid X_1\right]
= e^{\tilde\lambda}\mathbf1_{0<X_1\le M} + e^{\tilde\lambda} \Lambda(X_1)\mathbf1_{X_1>M} \,.$$ Therefore, for every $x\ge0$ we have $$\begin{split}
(P\Lambda)(x)=& \mathbb E_x[\Lambda(X_1),0<X_1\le M] + \mathbb E_x[\Lambda(X_1),X_1>M] \\
\le&\Lambda(M) + e^{-\tilde\lambda}\mathbb E_x \left[e^{\tilde\lambda T_M},X_1>M\right] \\
\le& \Lambda(M)+e^{-\tilde\lambda}\Lambda(x)\,,
\end{split}$$ proving that $P$ is bounded. In order to prove the assertion concerning the spectral radius we observe that, for $x>0$, $$\|P^n\|\geq \frac{1}{\Lambda(x)}(P^n\mathbf{1})(x)$$ and that therefore $$r(P):=\lim_{n\rightarrow \infty}\|P^n\|^{1/n}\geq e^{-\lambda_a}$$ by the Gelfand formula.
\[th:quasicompact\] Assume that condition is satisfied and that the innovations are distributed according to a density $\varphi$. Then the operator $P$ is quasicompact on $(B(\mathbb{R}_0^+),\|\cdot\|_\Lambda)$. If, in addition, $\varphi>0$ a.e., the spectral radius $r(P)$ is an isolated eigenvalue with algebraic multiplicity $1$ and all other spectral values have absolut value strictly smaller than $r(P)$.
Let us decompose $P$ into the sum of the following operators: $$\begin{split}
U_1:= P\mathbf{1}_{[0,M]}\,,\quad U_2:= \mathbf{1}_{[0,M]}P\mathbf{1}_{(M,\infty)}\,,\quad
U_3:= \mathbf{1}_{(M,\infty)}P\mathbf{1}_{(M,\infty)}\,.
\end{split}$$ The operators $U_1$ and $U_2$ are easily seen to be compact. For this, we first note that $$\lbrace U_1f ; \|f\|_\Lambda\leq 1\rbrace = \lbrace \mathbb{E}_x[f(X_1)\Lambda(X_1),0<X_1\leq M] ;\|f\|\leq 1\rbrace$$ is bounded and equicontinuous. This follows immediately from the fact that $\Lambda(z)\le \Lambda(M)$ for $0\le z\le M$ and $$\begin{aligned}
\mathbb{E}_x[f(X_1)\Lambda(X_1),0<X_1\leq M] = \int_0^M\Lambda(z)f(z)\varphi(z-ax)\text dz\,.\end{aligned}$$ The boundedness also implies $$\lim_{x\to\infty}\sup_{f:\|f\|_\Lambda\le1}\Lambda^{-1}(x)\mathbb E_x[f(X_1),0<X_1\le M)]=0\,,$$ so $\lbrace U_1f ; \|f\|_\Lambda\leq 1\rbrace$ is precompact and $U_1$ is compact.
By , for $x\le M$, $$\begin{aligned}
U_2(\Lambda f)(x) = \mathbb E_x\left[f(X_1)\Lambda(X_1),X_1>M\right]
= e^{-\tilde\lambda} \mathbb E_x\left[f(X_1)e^{\tilde\lambda T_M},X_1>M\right]\,,\end{aligned}$$ so for all $f$ with $\|f\|\le1$ we have $$|U_2(\Lambda f)(x)|\le e^{-\tilde\lambda}\Lambda(M)\,,$$ i.e. $\{U_2f ; \|f\|_\Lambda\le1\}$ is bounded w.r.t. $\|\cdot\|_\Lambda$, and $$\begin{aligned}
|U_2(\Lambda f)(y)-U_2(\Lambda f)(x)| \le e^{-\tilde\lambda}\sum_{k=2}^N |I_{f,k}(y)-I_{f,k}(x)| + 2e^{-\tilde\lambda}\sup_{0\le x\le M}|R_{1,N}(x)| \end{aligned}$$ with $$\begin{aligned}
I_{f,k}(x) :=& e^{\tilde\lambda k}\mathbb E_x\left[f(X_1)\mathbf1_{X_1>M,\,T_M=k}\right] \\
=& e^{\tilde\lambda k}\int_M^\infty\dots\int_M^\infty\int_{-\infty}^Mf(x_1)\Phi_{x,k}(x_1,\dots,x_k)\text d(x_k,\dots,x_1)\,,\end{aligned}$$ where $$\Phi_{x,k}(x_1,\dots,x_k):=\varphi(x_1-ax)\varphi(x_2-ax_1)\dots\varphi(x_k-ax_{k-1})$$ stands for the common density of $X_1,\dots,X_k$ given $X_0=x$, and $$\begin{aligned}
R_{f,N}(x):= \mathbb E_x\left[f(X_1)e^{\tilde\lambda T_M},T_M>N\right]\,.\end{aligned}$$ Clearly, $$\sup_{0\le x\le M}R_{1,N}(x)=R_{1,N}(M)\to0\quad\text{as }N\to\infty.$$ Moreover, $$\Phi_{y,k}(x_1,\dots,x_{k})=\Phi_{x,k}(x_1-a(y-x),\dots,x_{k}-a^{k}(y-x))\,,$$ so $$\begin{aligned}
|I_{f,k}(y)-I_{f,k}(x)|\le e^{\tilde\lambda k}\int\dots\int|\Phi_{y,k}-\Phi_{x,k}|\text d(x_1,\dots,x_k)\xrightarrow{y\to x}0\,.\end{aligned}$$ (This is clear for continuous $\Phi$ with compact support and follows easily for general $\Phi$ if one approximates them in $L^1$ by such ones.) Consequently, $\{\left.(U_2f)\right|_{[0,M]}; \|f\|_\Lambda\le1\}$ is equicontinuous, $U_2$ is compact, too, and $$P=L+U_3,$$ where $L$ is a compact operator.
We now estimate the operator norm of $U_3$: For $x>M$, yields $$(U_3\Lambda)(x)=\mathbb{E}_x\bigl[\Lambda(X_1),X_1>M\bigr] \leq e^{-\tilde{\lambda}}\Lambda(x)$$ and therefore we have for the operator norm of $U_3$ $$\|U_3\| \leq e^{-\tilde{\lambda}},$$ which, again by the Gelfand formula, tells us that $$r(U_3)=\lim_{n\rightarrow\infty}\| U_3^{n}\|^{1/n} \leq e^{-\tilde{\lambda}}<e^{-\lambda_a}\le r(P)\,,$$ the latter by Proposition \[prop:trop\].
According to the definition in [@S64], which calls an operator P quasicompact if $P^n = L + U$ for some $n\in\mathbb N$, compact operator $L$ and bounded operator $U$ with $\rho(U)<\rho(P)^n$, $P$ is quasicompact. Applying Theorem A as in Section \[sec:quasi-bounded\] allows to deduce the remaining statements of the theorem.
Observe that the main ingredient in the above proof is only the finitenes of $\mathbb{E}_x\bigl[e^{\tilde{\lambda}T_M}]$ for some $M>0$, which together with some ’local’ compact perturbation argument ensures that there is a gap seperating the largest eigenvalue from the remaining parts of the spectrum. Therefore, we expect that the method will work in other settings, too.
\[cor:qc\] Assume that condition is satisfied and let us assume that the innovations have a strictly positive continuous density. Then there exists $\delta>0$ such that for every $x>0$ and every measurable set $A\subset (0,\infty)$ $$\mathbb{P}_x\bigl(X_n \in A;T_0>n)= V(x)e^{-\lambda_a n}+O\Bigl((e^{-\lambda_a}-\delta)^n\Bigr).$$
This result is partly contained in the recent work [@CV17] by Champagnat and Villemonais. If $\mathbb{E}e^{\theta(\xi_1)\log\xi_1}<\infty$ for some function $\theta(x)$ such that $\theta(x)\uparrow \infty$ and $\ell(x):=\theta(x)\log x$ is concave, then $$\frac{\mathbb{E}e^{\ell(ax+\xi_1)}}{e^{\ell(x)}}\le\frac{e^{\ell(ax)}\mathbb{E}e^{\ell(\xi_1)}}{e^{\ell(x)}}
\le e^{-\theta(ax)\log(a)}\mathbb{E}e^{\ell(\xi_1)}\to0\quad\text{as }x\to\infty$$ and, consequently, in this case their Proposition 7.2 can be applied to AR$(1)$-sequences and gives the same type of convergence towards a quasi-stationary distribution. It is obvious that $\theta(x)=\log^b(x)$ with $b>0$ satisfies the conditions mentioned above. We shall see later that holds for innovations having all power moments. Therefore, the condition $\mathbb{E}e^{\log^{1+b}\xi_1}<\infty$ is slightly stronger than the existence of all power moments.
Alternative approach for unbounded innovations {#s:renewal}
==============================================
In this section we present another approach to investigate the tails of the hitting times, which, in contrast to Perron-Frobenius-type methods, has the potential to deal with situations where the transition operator is not quasicompact and to identify additional polynomial decay factors. Apart from birth/death processes and one-dimensional diffusions quasistationary convergence in cases with no spectral gap has not yet been established. Moreover, only basic properties of compact operators play a role and therefore the functional analytic machinery will be more straightforward.
As an alternative to the search for a weight function, we now start with the following observation: The larger $X_n$ is, the more it is diminished by the prefactor $a$ in the recursion $X_{n+1}=aX_n+\xi_{n+1}$. In contrast, adding $\xi_{n+1}$ has always the same absolute effect, no matter how large $X_n$ is. Therefore, in some sense, it is easier for the process to reach average positive values from extremely large values than to reach zero from average values, so, also in the case of unbounded innovations, the main part in estimating $\mathbb P_x(T_0>n)$ should still be to analyse what happens for not too large $x$.
The transition operator will in general not be (quasi-)compact on the usual space $C(\mathbb R_0^+)$, but, as shown on the next pages, a modified functional analytic approach, in which one basically works on the continuous functions on some interval and keeps under control what happens outside, is possible. In contrast, the “coming down from infinity” which we just described has not the uniform character that would be needed to apply the results in [@CV16].
Here is the main result of this section:
\[thm.analytic\] Assume that the distribution of innovations has a density $\varphi(x)$ which is positive a.e. on $\mathbb R$. Assume also that $\mathbb{E}(\xi_1^+)^t<\infty$ for all $t>0$ and $\mathbb{E}(\xi_1^-)^\delta<\infty$ for some $\delta>0$. Then there exist $\gamma>0$ and a positive function $V$ such that $$\label{thm.an.1}
\mathbb{P}_x(T_0=n)=e^{-\lambda_a (n+1)}
V(x)
+O\left(e^{-(\lambda_a+\gamma)n}\right).$$ The function $V$ is $e^{\lambda_a}$-harmonic for the transition kernel $P_+$, that is, $$V(x)=e^{\lambda_a}\int_0^\infty P_+(x,dy)V(y)=e^{\lambda_a}\mathbb{E}[V(X_1);T_0>1],\quad x\ge0.$$
Again, this result describes not only the exact asymptotic behaviour of $\mathbb{P}_x(T_0=n)$ but states also that the remainder term decays exponentially faster than the main term. It is worth mentioning that the existence of all power moments required in Theorem \[thm.analytic\] is the minimal moment condition. More precisely, we shall show in Proposition \[prop:reg.tails\] that if the tail of $\xi_1$ is regularly varying then it may happen that $e^{\lambda_a n}\mathbb{P}(T_0>n)\to0$.
The starting point of the approach, which we are going to use in this section, is based on the following renewal-type decomposition for the moment generating function of $T_0$. First define $$\sigma_r:=\inf\{n\ge1: X_n> r\},\quad r>0.$$ Fix $\lambda<\lambda_a$. Then, for $x\le r$ we have $$\begin{aligned}
\mathbb{E}_x\left[e^{\lambda T_0}\right]
&=\mathbb{E}_x\left[e^{\lambda T_0}; T_0<\sigma_r\right]
+\mathbb{E}_x\left[e^{\lambda T_0}; T_0>\sigma_r\right]\\
&=\mathbb{E}_x\left[e^{\lambda T_0}; T_0<\sigma_r\right]
+\mathbb{E}_x\left[e^{\lambda \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\lambda T_0}\right]\right]\\
&=\mathbb{E}_x\left[e^{\lambda T_0}; T_0<\sigma_r\right]
+\mathbb{E}_x\left[e^{\lambda \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\lambda T_0};T_r=T_0\right]\right]\\
&\hspace{1cm}+\mathbb{E}_x\left[e^{\lambda \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\lambda T_0};T_r<T_0\right]\right].\end{aligned}$$ Using now the Markov property at time $T_r$, we obtain the equation $$\label{main_decomp}
\mathbb{E}_x\left[e^{\lambda T_0}\right]=F_\lambda(x)+
\int_0^r K_\lambda(x,dy)\mathbb{E}_y\left[e^{\lambda T_0}\right],$$ where $$\label{def_F}
F_\lambda(x)=\mathbb{E}_x\left[e^{\lambda T_0}; T_0<\sigma_r\right]
+\mathbb{E}_x\left[e^{\lambda \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\lambda T_0};T_r=T_0\right]\right]$$ and $$\label{def_K}
K_\lambda(x,dy)=
\mathbb{E}_x\left[e^{\lambda \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\lambda T_r}\mathbf{1}_{\{T_r<T_0\}}\mathbf 1_{\text dy}(X_{T_r})\right]\right]\,.$$
To analyse the renewal equation , we first have to derive some properties of the functions $F_\lambda$ and the operators $K_\lambda$. More precisely, we first show that there exists $r>0$ such that $F_\lambda(x)$ and $K_\lambda(x,dy)$ can be extended analytically for $\Re\lambda<\lambda_a+\varepsilon$.
Estimates for stopping times $T_0\wedge\sigma_r$ and $T_r$.
-----------------------------------------------------------
The main purpose of this paragraph is to show that, under the conditions of Theorem \[thm.analytic\], $T_0\wedge\sigma_r$ and $T_r$ have lighter tails than $T_0$. This fact will play a crucial role in the study of properties of $F_\lambda$ and $K_\lambda$.
\[lem:two-sided\] Assume that $\mathbb{E}|\xi_1|^\delta$ is finite for some $\delta>0$. Then for every $r$ such that $\pi[r,\infty)>0$ there exists $\varepsilon_r>0$ such that $$\label{two-sided.1}
\sup_{x\in(0,r)}\mathbb{P}_x(T_0\wedge\sigma_r>n)\le C_re^{-(\lambda_a+\varepsilon_r)n},
\quad n\ge0.$$
Clearly, $$\mathbb{P}_x(T_0\wedge\sigma_r>n)=\mathbb{P}_x\left(\max_{k\le n}X_k<r, T_0>n\right),\quad n\ge1.$$ Fix some $n_0\ge1$ and consider the sequence $\mathbb{P}_x\left(\max_{k\le \ell n_0}X_k<r, T_0>\ell n_0\right)$ in $\ell\in\mathbb N$. By the Markov property, $$\begin{aligned}
&\mathbb{P}_x\left(\max_{k\le \ell n_0}X_k<r, T_0>\ell n_0\right)\\
&=\int_0^r\mathbb{P}_x\left(X_{(\ell-1)n_0}\in dy;\max_{k\le (\ell-1) n_0}X_k<r, T_0>(\ell-1) n_0\right)
\mathbb{P}_y\left(\max_{k\le n_0}X_{k}<r,T_0>n_0\right).\end{aligned}$$ It is easy to see that functions $\mathbf{1}_ {\{\max_{k\le n_0}X_k\ge r\}}$ and $\mathbf{1}_{\{T_0>n_0\}}$ are increasing functions in every innovation $\xi_k$, $k\le n_0$. Thus, by the FKG-inequality for product spaces, $$\mathbb{P}_y\left(\max_{k\le n_0}X_{k}\ge r,T_0>n_0\right)\ge
\mathbb{P}_y\left(\max_{k\le n_0}X_{k}\ge r\right)\mathbb{P}_y\left(T_0>n_0\right).$$ In other words, $$\begin{aligned}
\mathbb{P}_y\left(\max_{k\le n_0}X_{k}<r,T_0>n_0\right)
&\le \mathbb{P}_y\left(\max_{k\le n_0}X_{k}<r\right)\mathbb{P}_y\left(T_0>n_0\right)\\
&\le \mathbb{P}_0\left(\max_{k\le n_0}X_{k}<r\right)\mathbb{P}_r\left(T_0>n_0\right).\end{aligned}$$ Consequently, $$\begin{aligned}
\label{two-sides.3}
\nonumber
&\mathbb{P}_x\left(\max_{k\le \ell n_0}X_k<r, T_0>\ell n_0\right)\\
\nonumber
&\hspace{0.5cm}\le\mathbb{P}_0\left(\max_{k\le n_0}X_k<r\right)\mathbb{P}_r(T_0>n_0)
\mathbb{P}_x\left(\max_{k\le (\ell-1) n_0}X_k<r, T_0>(\ell-1) n_0\right)\\
&\hspace{0.5cm}\le\ldots\le
\left(\mathbb{P}_0\left(\max_{k\le n_0}X_k<r\right)\mathbb{P}_r(T_0>n_0)\right)^\ell,
\quad x\in(0,r).\end{aligned}$$ By Theorem \[thm.log\], $\mathbb{P}_r(T_0>n_0)=e^{-\lambda_a n_0+o(n_0)}$ as $n_0\to\infty$. Since $r-X_n$ is an AR($1$)-sequence, we may apply Theorem \[thm.log\] to this sequence: $$\mathbb{P}_0\left(\max_{k\le n_0}X_k<r\right)=e^{-\widetilde{\lambda}_an_0+o(n_0)},
\quad n_0\to\infty$$ for some $\widetilde{\lambda}_a>0$. Therefore, there exists $n_0$ such that $$\frac{1}{n_0}\log\mathbb{P}_0\left(\max_{k\le n_0}X_k<r\right)\mathbb{P}_r(T_0>n_0)<-\lambda_a-\frac{\widetilde{\lambda}_a}2\,.$$ Combining this estimate with , we obtain .
We next show that a similar estimate holds for $T_r$.
\[prop:all.mom\] Assume that $\mathbb{E}(\xi_1^+)^t<\infty$ for all $t>0$. Then for all $A\in(1,1/a)$ and all $\lambda>0$ there exists $r_0=r_0(A,\lambda)$ such that, for all $r\ge r_0$, $$\mathbb{E}\bigl[ e^{\lambda T_r}\bigr]\le 2e^\lambda\mathbb{E}\left(\frac{X_0}{r}\right)^{\lambda/\log A}$$ for any distribution of $X_0$ with support in $(r,\infty)$.
Define $$y_j=rA^j,\quad j\ge0.$$ Let us consider an ’aggregated’ chain $(Y_n)_{n\ge 0}$ defined by the transition kernel $$\mathbb{P}(Y_1=j|Y_0=k)=\mathbb{P}_{y_k}(X_1\in(y_{j-1},y_j]),\ k,j\ge1$$ and $$\mathbb{P}(Y_1=0|Y_0=0)=1,\quad
\mathbb{P}(Y_1=0|Y_0=k)=\mathbb{P}_{y_k}(X_1\le y_0)\,,\, k\ge1.$$ Similarly we define the initial distribution: $$\mathbb{P}(Y_0=j)=\mathbb{P}(X_0\in(y_{j-1},y_j])\,,\,j\ge1\,.$$ Define also the stopping times $$\tau:=\inf\{n: Y_n=0\}
\quad\text{and}\quad
\theta:=\inf\{n:Y_n\ge Y_{n-1}\}.$$ We first estimate the distribution of $Y_\theta$. Noting that $Y_\theta=j\ge1$ implies that $Y_{\theta-1}\le j$, we have $$\max_{k\ge1}\mathbb{P}(Y_\theta=j|Y_0=k)=\max_{k\le j}\mathbb{P}(Y_\theta=j|Y_0=k).$$ Furthermore, using the fact that $\theta\le j$ for $Y_0\le j$ and $Y_\theta=j$, we infer that $$\max_{k\ge1}\mathbb{P}(Y_\theta=j|Y_0=k)\le j\max_{k\le j}\mathbb{P}(Y_1=j|Y_0=k).$$ By the definition of $Y_n$, $$\begin{aligned}
\mathbb{P}(Y_1=j|Y_0=k)
&=\mathbb{P}_{y_k}(X_1\in(y_{j-1},y_j])\\
&\le \mathbb{P}_{y_k}(X_1>y_{j-1})\le\mathbb{P}(\xi_1>(1-aA)u_{j-1}),\quad k\le j.\end{aligned}$$ As a result, $$\label{all.mom.1}
\max_{k\ge1}\mathbb{P}(Y_\theta=j|Y_0=k)\le j\mathbb{P}(\xi_1>(1-aA)u_{j-1})=:q_j(r),
\quad j\ge1.$$ Set $q_0(r):=1-\sum_{j=1}^\infty q_j(r)$. It is easy to see that $\lim_{r\to\infty}\sum_{j=1}^\infty q_j(r)=0$. Therefore, $\{q_j(r)\}$ is a probability distribution for all $r$ large enough.
Noting that the chain $(Y_n)$ is stochastically monotone and using , we have, for arbitrary initial $Y_0$, $$\tau_0\le Y_0+\mathbf{1}_{\lbrace Y_\theta \ge 1\rbrace}\tau^{(q)}_0\quad\text{in distribution},$$ where $\tau^{(q)}_0$ is independent of $Y_0,\ldots,Y_\theta$ and is distributed as $\tau_0$ corresponding to the initial distribution $q_j(r)/(1-q_0(r)),\ j\ge1$. Combining this inequality with the bound $\mathbb{P}(Y_\theta\ge1)\le 1-q_0(r)$, we obtain $$\label{all.mom.2}
\mathbb{E} e^{\lambda\tau_0}
\le\mathbb{E}e^{\lambda Y_0}\left(1+(1-q_0(r))\mathbb{E}e^{\lambda\tau^{(q)}_0}\right).$$
If $Y_0$ is distributed according to $q_j(r)/(1-q_0(r))$, then we conclude from that $$\label{all.mom.3}
\mathbb{E}e^{\lambda\tau^{(q)}_0}
\le \frac{\mathbb{E}e^{\lambda Y_0}}{1-(1-q_0(r))\mathbb{E}e^{\lambda Y_0}}.$$ It follows from the Chebyshev inequality that $$q_j(r)\le j\frac{\mathbb{E}(\xi_1^+)^t}{(1-aA)^tu_{j-1}^t}
=\frac{\mathbb{E}(\xi_1^+)^tA^t}{(1-aA)^tr^t}jA^{-jt},\quad j\ge1.$$ Consequently, $$\begin{aligned}
(1-q_0(r))\mathbb{E}e^{\lambda Y_0}&=\sum_{j=1}^\infty q_j(r)e^{\lambda j}\\
&\le \frac{\mathbb{E}(\xi_1^+)^te^\lambda}{(1-aA)^tr^t}\sum_{j=1}^\infty j(e^\lambda A^{-t})^{j-1}
=\frac{\mathbb{E}(\xi_1^+)^te^\lambda}{(1-aA)^tr^t}\left(1-e^\lambda A^{-t}\right)^{-2}.\end{aligned}$$ For all $t$ such that $A^t\ge 2e^{\lambda}$ we have $$(1-q_0(r))\mathbb{E}e^{\lambda Y_0}\le 4\frac{\mathbb{E}(\xi_1^+)^te^\lambda}{(1-aA)^tr^t}.$$ As a result, $$(1-q_0(r))\mathbb{E}e^{\lambda Y_0}\le\frac{1}{2}$$ for all $r$ large enough. Combining this estimate with , we obtain $$(1-q_0(r))\mathbb{E}e^{\lambda \tau^{(q)}_0}\le 1.$$ Plugging this into leads to $$\mathbb{E} e^{\lambda\tau_0}\le2\mathbb{E}e^{\lambda Y_0}.$$ The stocahstic monotonicity of $X_n$ implies that $T_r\le\tau_0$ in distribution. Combining this with the fact that $Y_0\le \frac{\log(X_0/r)}{\log A}+1$, we obtain the desired inequality for the chain $X_n$.
\[cor:all.mom\] Under the assumptions of Lemma \[prop:all.mom\], there exist $r>0$ and $\varepsilon>0$ such that, for all $\lambda\le\lambda_a+2\varepsilon$, $$\sup_{x\in[0,r]}
\mathbb{E}_x\left[e^{\lambda\sigma_r}\mathbf{1}\{T_0>\sigma_r\}\mathbb{E}_{X_{\sigma_r}}[e^{\lambda T_r}]\right]
<\infty\,.$$
By the Markov property at time $\sigma_r$ and by Lemma \[prop:all.mom\], $$\begin{aligned}
\label{revis.1}
\nonumber
&\mathbb{E}_x\left[e^{\lambda\sigma_r}\mathbf{1}\{T_0>\sigma_r\}\mathbb{E}_{X_{\sigma_r}}[e^{\lambda T_r}]\right]\\
\nonumber
&\hspace{1cm}=\sum_{k=1}^\infty e^{\lambda k}\int_r^\infty\mathbb{P}_x(T_o>\sigma_r=k,X_{\sigma_r}\in dy)\mathbb{E}_y[e^{\lambda T_r}]\\
&\hspace{1cm}\le\frac{2e^\lambda}{r^{\lambda/\log A}}\sum_{k=1}^\infty e^{\lambda k}\int_r^\infty\mathbb{P}_x(T_o>\sigma_r=k,X_{\sigma_r}\in dy)y^{\lambda/\log A}.\end{aligned}$$ Noting that $$\begin{aligned}
\label{revis.2}
\mathbb{P}_x(T_0>\sigma_r=k,X_{\sigma_r}>z)\le \mathbb{P}_x(T_0>\sigma_r>k-1)\mathbb{P}(\xi_1>(1-a)z),\quad z>r,\end{aligned}$$ we obtain $$\begin{aligned}
\mathbb{E}_x\left[e^{\lambda\sigma_r}\mathbf{1}\{T_0>\sigma_r\}\mathbb{E}_{X_{\sigma_r}}[e^{\lambda T_r}]\right]
\le C(\lambda,A,r)\sum_{k=1}^\infty e^{\lambda k} \mathbb{P}_x(T_0>\sigma_r>k-1).\end{aligned}$$ The desired bound follows now from Lemma \[lem:two-sided\].
Properties of $F_\lambda$ and $K_\lambda$.
------------------------------------------
We start this subsection by stating properties of the function $F_\lambda$ that are important for our approach.
\[l:pos-function\] For every complex $z$ with real part $\Re (z)\le \lambda_a+\varepsilon$ the function $F_{z}$ defines a continuous function on $[0,r]$, which is strictly positive if additionally $z\in \mathbb{R}$.
We first show that $F_z$ is well-defined for all $z$ with $\Re (z)<\lambda_a+\varepsilon$. Indeed, it follows from Lemma \[lem:two-sided\] that $$\Big|\mathbb{E}_x\left[e^{z T_0};T_0<\sigma_r\right]\Big|
\le \mathbb{E}_x\left[e^{\Re (z) T_0\wedge\sigma_r}\right]\le C$$ uniformly in $x\in[0,r]$ and in $z$ with $\Re (z)<\lambda_a+\varepsilon$ for every $\varepsilon<\varepsilon_r$. Applying Corollary \[cor:all.mom\], we also conclude that $$\begin{aligned}
&\Big|\mathbb{E}_x\left[e^{z \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{z T_0};T_r=T_0\right]\right]\Big|\\
&\hspace{2cm}\le \mathbb{E}_x\left[e^{\Re (z) \sigma_r}\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb{E}_{X_{\sigma_r}}\left[e^{\Re (z) T_r}\right]\right]\le C\end{aligned}$$ uniformly in $x\in[0,r]$ and in $z$ with $\Re (z)<\lambda_a+\varepsilon$. Therefore, $F_z(x)$ is bounded for all $x\in[0,r]$ and all $z$ with $\Re (z)<\lambda_a+\varepsilon$.
It is also clear that, for all $x\in[0,r]$, $$F_\lambda(x)\ge \mathbb{E}_x\left[ e^{\lambda(T_0\wedge\sigma_r)}\right]\ge 1,
\quad \lambda\in[0,\lambda_a+\varepsilon).$$ Thus, it remains to show the continuity of $F_z$. Fix some $N\ge1$. Since the innovations have an absolutely continuous distribution, the probabilities $\mathbb{P}_x(\sigma_r>T_0=k)$ are continuous in $x$. As a result, $\sum_{k=1}^N e^{zk}\mathbb{P}_x(\sigma_r>T_0=k)$ is continuous in $x$. Noting that, according to Lemma \[lem:two-sided\], $$\begin{aligned}
\max_{x\in[0,r]}\Big|\mathbb{E}_x\left[e^{z T_0}\mathbf{1}_{\{\sigma_r>T_0>N\}}\right]\Big|\to0
\quad\text{as }N\to\infty,\end{aligned}$$ we infer that $\mathbb{E}_x\left[e^{z T_0};T_0<\sigma_r\right]$ is continuous on $[0,r]$.
Using the continuity of the distribution of innovations once again, we conclude that $$\mathbb{E}_x\left[e^{z\sigma_r}\mathbf{1}_{\{\sigma_r<T_0\wedge N\}}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}\mathbf{1}_{\{T_r=T_0<N\}}]\right]$$ is continuous in $x\in[0,r]$. Therefore, it remains to show that, as $N\to\infty$, $$\label{pos.f.1}
\sup_{x\in[0,r]}\Big|\mathbb{E}_x\left[e^{z\sigma_r}\mathbf{1}_{\{T_0>\sigma_r\ge N\}}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}]\right]\Big|\to0$$ and $$\label{pos.f.2}
\Big|\mathbb{E}_x\left[e^{z\sigma_r}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}\mathbf{1}_{\{T_r\ge N\}}]\right]\Big|\to0.$$
Similar to the derivation of , $$\begin{aligned}
&\left|\mathbb{E}_x\left[e^{z\sigma_r}\mathbf{1}_{\{T_0>\sigma_r\ge N\}}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}]\right]\right|\\
&\hspace{1cm} \le \mathbb{E}_x\left[e^{\Re(z)\sigma_r}\mathbf{1}_{\{T_0>\sigma_r\ge N\}}\mathbb{E}_{X_{\sigma_r}}[e^{\Re(z) T_r}]\right]\\
&\hspace{1cm} \le \frac{2e^{\Re(z)}}{r^{\Re(z)/\log A}}\sum_{k=N}^\infty e^{\Re(z) k}\int_r^\infty\mathbb{P}_x(T_o>\sigma_r=k,X_{\sigma_r}\in dy)y^{\Re(z)/\log A}.\end{aligned}$$ Using now , one gets $$\left|\mathbb{E}_x\left[e^{z\sigma_r}\mathbf{1}_{\{T_0>\sigma_r\ge N\}}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}]\right]\right|
\le C(\Re(z),A,r)\sum_{k=N}^\infty e^{\Re(z) k}\mathbb{P}(T_0>\sigma_r>k-1).$$ From this bound and Lemma \[lem:two-sided\] we infer that is valid for all $z$ with $\Re (z)<\lambda_a+2\varepsilon$.
Applying the Cauchy-Schwarz inequality, we obtain $$\left|\mathbb{E}_y\left[e^{zT_r}\mathbf{1}_{\{T_r\ge N\}}\right]\right|
\le \left(\mathbb{E}_y\left[e^{2\Re(z)T_r}\right]\right)^{1/2}\mathbb{P}_y^{1/2}(T_r\ge N).$$ From this bound and Lemma \[prop:all.mom\], we get $$\begin{aligned}
\Big|\mathbb{E}_x\left[e^{z\sigma_r}\mathbb{E}_{X_{\sigma_r}}[e^{z T_r}\mathbf{1}_{\{T_r\ge N\}}]\right]\Big|
\le C\mathbb{E}_x\left[e^{\Re(z)\sigma_r}X_{\sigma_r}^{\Re(z)/\log A}\mathbb{P}^{1/2}_{X_{\sigma_r}}(T_r\ge N)\right]\end{aligned}$$ Since $\lim_{N\to\infty}\mathbb{P}^{1/2}_{X_{\sigma_r}}(T_r\ge N)$ almost surely and the family $e^{\Re(z)\sigma_r}X_{\sigma_r}^{\Re(z)/\log A}$, $\Re(z)\le \lambda_a+\varepsilon$ is uniformly integrable, we conclude that is valid.
We now establish an essential further property of the family of kernels $K_{\lambda}$, which exactly is the reason for introducing them.
\[lem:K\_lambda\] Consider $K_\lambda f:=\int_0^rK_\lambda(\cdot,dy)f(y)$, where $r$ is as in Lemma \[prop:all.mom\]. Then, under the assumptions of Theorem \[thm.analytic\], for all $\lambda$ with $\Re(\lambda)\le\lambda_a+\varepsilon$, $K_\lambda$ is a compact operator on the Banach space $X=C([0,r])$ equipped with the supremum norm.
Furthermore, if $\lambda\in[0,\lambda_a+\varepsilon)$ then for every $X \ni f > 0$ (i.e. everywhere nonnegative and somewhere strictly positive) and all $x\in[0,r]$, $K_{\lambda}f(x)>0$.
We have to show that, for fixed $\lambda$, $\{K_\lambda f: f\in C([0,r]),\|f\|\le1\}$ is equicontinuous. This will imply that $K_\lambda$ maps to $C([0,r])$, in particular $\|K_\lambda 1\|<\infty$, and the proof can then be concluded by noting that $\sup\{\|K_\lambda f\| :f\in C([0,r]),\|f\|\le1\}=\|K_\lambda1\|<\infty$, which, together with equicontinuity, yields compactness.
For equicontinuity, one has to bring the smoothing effect of the density into play. We write $$\Phi_{x,k,l}(x_1,\dots,x_{k+l}):=\varphi(x_1-ax)\cdots\varphi(x_{k+l}-ax_{k+l-1})$$ for the common density of $X_1,\dots,X_{k+l}$ and write $$\begin{aligned}
&K_\lambda f(x) = \sum_{k,l=1}^NI_{f,k,l}(x) + R_{f,N,1}(x)+R_{f,N,2}(x)\end{aligned}$$ and $$\label{comp1}
\begin{split}
&|K_\lambda f(y)-K_\lambda f(x)| \\
\le& \sum_{k,l=1}^N|I_{f,k,l}(y)-I_{f,k,l}(x)| + 2\sup_{x\in[0,r]}\left(|R_{1,N,1}(x)|+ |R_{1,N,2}(x)|\right)
\end{split}$$ with $$\begin{aligned}
I_{f,k,l}(x) =& e^{\lambda(k+l)}\mathbb{E}_x\bigl[\mathbf{1}_{X_1.\dots,X_{k-1}\in]0,r[,\, X_k\ge r,\, X_{k+1},\dots,X_{k+l-1}>r,\, X_{k+l}\in]0,r]} f(X_{k+l})\bigr] \\
=& e^{\lambda(k+l)}\int_0^r\dots\int_0^r\int_r^\infty\dots\int_r^\infty\int_0^r\Phi_{x,k,l}(x_1,\dots,x_{k+l})f(x_{k+l})\text dx_{k+l}
\cdots\text dx_1\,,\end{aligned}$$ $$\begin{aligned}
R_{f,N,1}(x)=\mathbb{E}_x\bigl[e^{\lambda\sigma_r}\mathbf{1}_{N<\sigma_r<T_0}\mathbb{E}_{X_{\sigma_r}}\bigl[e^{\lambda T_r}\mathbf{1}_{T_r<T_0}f(X_{T_r})\bigr]\bigr]\end{aligned}$$ and $$\begin{aligned}
R_{f,N,2}(x) = \mathbb{E}_x\bigl[e^{\lambda\sigma_r}\mathbf{1}_{\lbrace \sigma_r<T_0, \sigma_r\le N\rbrace}\mathbb{E}_{X_{\sigma_r}}\bigl[e^{\lambda T_r}\mathbf{1}_{\lbrace N<T_r<T_0\rbrace}f(X_{T_r})\bigr]\bigr] \,.\end{aligned}$$ Since $$\Phi_{y,k,l}(x_1,\dots,x_{k+l})=\Phi_{x,k,l}(x_1-a(y-x),\dots,x_{k+l}-a^{k+l}(y-x))\,,$$ we can conclude $$\begin{aligned}
|I_{f,k,l}(y)-I_{f,k,l}(x)|
\le \int_0^\infty\dots\int_0^\infty|\Phi_{y,k,l}-\Phi_{x,k,l}|\text d(x_1,\dots,x_{k+l})\xrightarrow{y\to x}0\,,\end{aligned}$$ independently of $f$. (This is clear for continuous $\Phi$ with compact support and follows easily for general $\Phi$ if one approximates them in $L^1$ by such ones.)
It follows from and that $$\begin{aligned}
\lim_{N\to\infty}\sup_{x\in[0,r]}|R_{1,N,1}(x)|=0\end{aligned}$$ and $$\begin{aligned}
\lim_{N\to\infty}\sup_{x\in[0,r]}|R_{1,N,2}(x)|=0.\end{aligned}$$ So, if we go back to equation , choose first $N$ large and then $y$ sufficiently close to $x$, equicontinuity follows.
We now prove the positivity of $K_\lambda$ for real values of $\lambda$. If $f(z)\ge\varepsilon>0$ for $z\in I:=[z_0-\delta,z_0+\delta]$, then $$K_\lambda f(x)\ge \varepsilon \mathbb{E}_x\left[\mathbf{1}_{\{T_0>\sigma_r\}}
\mathbb P_{X_{\sigma_r}}\left(T_r<T_0,X_{T_r}\in I\right)\right]\,.$$ The expression is bounded from below by $\varepsilon\mathbb P_x(X_1\in(r,r+\delta],X_2\in I)$, which is positive if the density $\varphi$ is positive on all of $\mathbb R$.
Fredholm alternative and the proof of Theorem \[thm.analytic\]
--------------------------------------------------------------
We will now make use of the so-called analytic Fredholm alternative. For the convenience of the reader we formulate a suitable version of this result.\
**Theorem B** (Theorem 1 in [@S68]).
We are now going to apply this result in order to deduce the subsequent proposition:
For $z \in \mathbb{C}$ with $\Re z <\lambda_a + \varepsilon$, the operator valued function $$\lbrace z \in \mathbb{C} \mid \Re z <\lambda_a + \varepsilon\rbrace \ni z\mapsto R_z:= (I-K_z)^{-1}$$ is meromorphic.
It suffices to show that for some $z \in \mathbb{C}$ with $\Re z <\lambda_a+\varepsilon$ the inverse of $I-K_z$ exists and defines a bounded operator. This follows from the fact that for $\lambda$ sufficiently small the operator norm of $K_{\lambda}$ can be easily seen to be strictly smaller than one and the inverse thus can be shown to exist by the Neumann series. An application of Theorem B therefore shows that $(I-K_z)^{-1}$ is meromorphic on the required domain.
Now we are able to draw a conclusion analogously to Corollary 1 of [@SW75]:
If for $z \in \mathbb{C}$ with $\Re z <\lambda_a + \varepsilon$ the function $u=u_z$ is a solution of $$u=F_z+K_zu$$ then the $X$-valued function $u$ is meromorphic in $z$. Therfore, we conclude that the function $$z\mapsto L(x,z):=\mathbb{E}_x\bigl[ e^{zT_0}\bigr]$$ has a meromorphic continuation to $\lbrace z \in \mathbb{C}\mid \Re z< \lambda_a+ \varepsilon \rbrace$.
We now aim to study the dominant poles.
\[prop:gf\] The function $L(z)$ which meromorphically extends the Laplace transform of $T_0$ has a simple pole at $z=\lambda_a$ and no other poles on the interval $\{z=\lambda_a+i\psi,\ \psi\in(-\pi,\pi)\}$.
We first observe that $$\mathbb{E}_x\bigl[z^{T_0}\bigr]=\sum_{n=0}^{\infty}\mathbb{P}_x(T=n)z^{n}$$ is a power series with non-negative coefficients and therefore by Pringsheim’ s theorem (see e.g. Theorem 4.1.2 in [@MN91]) the radius of convergence $r_a=e^{\lambda_a}$ is a singularity. Therefore $\lambda_a$ is a singularity of $L$ and as $L$ is meromorphic we conclude that $\lambda_a$ is a pole. Observe that this implies that the operator valued function $(I-K_z)^{-1}$ has a pole at $z=\lambda_a$. Therefore, we conclude that $K_{\lambda_a}$ has the eigenvalue $1$.
Observe next that for $\lambda<\lambda_a+\varepsilon$ the operator $K_{\lambda}$ is positive and compact on the Banach lattice $X$. Using the second part of Lemma \[lem:K\_lambda\] we conclude by the classical Krein-Rutman theorem as given in Theorem 4 of [@S64] that the spectral radius $r(K_{\lambda})>0$ is in fact an eigenvalue with algebraic multiplicity one and the associated eigenfunction can be chosen to be non-negative. Furthermore, by Theorem 2.1 in [@D08] the spectral radius is continuous in $\lambda$.
We now claim that $1$ coincides with the spectral radius of $K_{\lambda_a}$. Assume contrary that the spectral radius $r(K_{\lambda_a})$ of $K_{\lambda_a}$ is strictly bigger then $1$. Under this assumption and the continuity of the spectral radius we conclude that for $\lambda<\lambda_a$ we still have $\rho(K_{\lambda})>1$. We now claim, that this contradicts that the equation $$\label{e:renewaleq}
L_\lambda=F_{\lambda}+K_{\lambda}L_\lambda$$ holds. Observe that by Lemma \[l:pos-function\] the function $F_{\lambda}$ is lower bounded by $1$ and iterating equation we conclude that for $n\geq 1$ $$L_{\lambda} \geq K_{\lambda}^n\mathbf{1}.$$ This contradicts the fact $\|K_{\lambda}\mathbf{1}\|_{\infty} \rightarrow\infty$ if $n\rightarrow \infty$ as a consequence of the spectral radius formula $\rho(K_{\lambda})=\lim_{n\rightarrow\infty}\|K_{\lambda}^n\|^{1/n}$.
We now aim to show that the function $L$ has a simple pole at $\lambda_a$. Here, we can observe that the derivative $\frac{d}{d\lambda}K_{\lambda}$ for real $\lambda < \lambda_a+\varepsilon$ defines a positive operator on the Banach space $X$. Therefore, we have shown, that the properties $P1$, $P2$ and $P3$ in [@SW75] are satisfied and, as shown on page 233 of [@SW75], we can then conclude via Corollary 1 in [@SW75] that the pole at $\lambda_a$ is in fact simple.
Now assume that $L(z)$ has another pole at $z=\lambda_a+i\psi$, i.e. $K_z$ has an eigenvalue 1 with some eigenfunction $g$ there. Applying the triangle inequality to the definition of $K_zg$, one obtains $|g(x)|=\left|K_{z}g(x)\right|\le K_{\lambda_a}|g(x)|$ for all $x\in[0,r]$. The positivity of the density $\varphi$ implies that, for any $m,n\in\mathbb N$, it happens with positive probability that $\sigma_r=m,T_r=n$ and the whole expression inside the expectation value defining $K_z$ is nonzero. Consequently, its phase is not a.s. constant unless $\psi$ is a multiple of $2\pi$, the triangle inequality is even strict and, by compactness of $[0,r]$, $$\inf_{x\in[0,r]}\frac{K_{\lambda_a}|g(x)|}{|g(x)|}>1\,.$$ Now we look at $K_{\lambda_a}$ as an operator on the real-valued continuous functions $C([0,r])$. Since $r(K_{\lambda_a})=1$, what we just found would mean that the min-max principle in the form $$r(K_{\lambda_a}) = \max_{h\in\mathcal K}\inf_{\substack{h'\in \mathcal H' \\ \langle h,h'\rangle>0}} \frac{\langle K_{\lambda_a}h,h'\rangle}{\langle h,h'\rangle}$$ with $\mathcal K$ denoting the nonnegative elements of $C([0,r])$ and $\mathcal H':=\{h\mapsto h(x)\mid x\in[0,r]\}$ is violated. However, all conditions from [@Marek66] ensuring its validity are satisfied:
- $C([0,r])=\mathcal K-\mathcal K$ and the cone $\mathcal K$ is closed and normal ($\|g+h\|\ge\|g\|$ for all normalised $g,h\in\mathcal K$).
- $\mathcal H'$ is total, i.e. if $\langle h,h'\rangle\ge0$ for all $h'\in\mathcal H'$, then $h\in\mathcal K$.
- $K_{\lambda_a}$ is semi non-supporting: For all nonzero $h\in\mathcal K$, $K_{\lambda_a}h(x)>0$ for all $x$ as in Lemma \[lem:K\_lambda\] and therefore one can conclude for all nonzero $h'\in\mathcal K'$ (the dual cone, i.e. positive measures) $\langle K_{\lambda_a}h,h'\rangle>0$.
- The resolvent $\lambda\mapsto(\lambda-K_{\lambda_a})^{-1}$ has only finitely many singularities with $|\lambda|=r(K_{\lambda_a})$, all of them poles, because $K_{\lambda_a}$ is compact.
It follows from the Proposition \[prop:gf\] that the generating function $\mathbb{E}_x[z^{T_0}]$ has the unique simple pole at $e^{\lambda_a}$ and that there are no further poles on the disc or radius $e^{\lambda_a+\gamma}$ with some $\gamma>0$. Then we have the following representation $$\mathbb{E}_x[z^{T_0}]=\frac{V(x)}{e^{\lambda_a}-z}+g_x(z),$$ where $g_x$ is analytical on the disc with radius $e^{\lambda_a+\gamma}$. Therefore, $$\begin{aligned}
\label{thm.an.1wdh}
\mathbb{P}_x(T_0=n)=V(x)e^{-\lambda_a(n+1)}+O(e^{-(\lambda_a+\gamma)n}). \end{aligned}$$ Since the left hand side is positive, we infer that the function $V$ is positive as well. Therefore, it remains to show that $v$ is $e^{\lambda_a}$-harmonic for $P_+$. Let $r$ be so large that $\mathbb{E}_y[e^{\lambda_a T_r}]$ is finite. For every $x\ge r$ one has the inequality $$\mathbb{P}_x(T_0\ge n)\le\sum_{m=1}^n\mathbb{P}_x(T_r=m)\mathbb{P}_r(T_0\ge n-m).$$ It follows from that $\mathbb{P}_r(T_0\ge k)\le C(r)e^{-\lambda_a(k+1)}$, $k\ge0$. Thus, $$\label{thm.an.2}
\mathbb{P}_x(T_0\ge n)\le C(r)e^{-\lambda_a n}\sum_{m=1}^n\mathbb{P}_x(T_r=m)
\le C(r)\mathbb{E}_x[e^{\lambda_a T_r}]e^{-\lambda_a(n+1)}.$$ It is immediate from that $$\label{thm.an.3}
\mathbb{P}_x(T_0\ge n)= \frac{V(x)}{1-e^{-\lambda_a}}e^{-\lambda_a(n+1)}
+O\left(e^{-(\lambda_a+\gamma)n}\right).$$ From this equality and from we infer that $$\label{thm.an.4}
V(x)\le \frac{C(r)}{1-e^{-\lambda_a}}\mathbb{E}_x[e^{\lambda_a T_r}],\quad x\ge r.$$ Since $$\mathbb{E}_x[e^{\lambda_a T_r}]=\int_0^r P(x,dy)e^{\lambda_a}+
\int_r^\infty P(x,dy)e^{\lambda_a}\mathbb{E}_y[e^{\lambda_a T_r}],$$ we infer that $$\label{thm.an.5}
\int_r^\infty P(x,dy)\mathbb{E}_y[e^{\lambda_a T_r}]<\infty$$ and, in view of , $$\label{thm.an.6}
\int_0^\infty P(x,dy)V(y)<\infty.$$
Fix some $A>r$ and consider the equality $$\begin{aligned}
\label{thm.an.7}
\mathbb{P}_x(T_0\ge n+1)&=\int_0^\infty P(x,dy)\mathbb{P}_y(T_0\ge n)\\
\nonumber
&=\int_0^A P(x,dy)\mathbb{P}_y(T_0\ge n)+\int_A^\infty P(x,dy)\mathbb{P}_y(T_0\ge n).\end{aligned}$$ Combining and , we obtain $$\label{thm.an.8}
\lim_{A\to\infty}\limsup_{n\to\infty}
e^{\lambda_a(n+1)}\int_A^\infty P(x,dy)\mathbb{P}_y(T_0\ge n)=0.$$ Furthermore, by and , $$\label{thm.an.9}
\lim_{A\to\infty}\lim_{n\to\infty}e^{\lambda_a(n+1)}\int_0^A P(x,dy)\mathbb{P}_y(T_0\ge n)
=\int_0^\infty P(x,dy)V(y).$$ Plugging and into , we obtain $$\lim_{n\to\infty} e^{\lambda_a(n+1)}\mathbb{P}_x(T_0\ge n+1)=\int_0^\infty P(x,dy)V(y).$$ According to , the limit on the left hand side equals to $e^{-\lambda_a}V(x)$. As a result we have the equality $$e^{-\lambda_a}V(x)=\int_0^\infty P(x,dy)V(y).$$ Therefore, the proof is complete.
Innovations with regularly varying tails {#s:reg}
========================================
The main purpose of this section is to show that the finiteness of all moments of $\xi_1^+$ is necessary for getting purely exponential decay for the tail of $T_0$. More precisely, we are going to show that if the right tail is regulaly varying then, independent of the index of regular variation, the asymptotic behaviour of $\mathbb{P}_x(T_0>n)$ depends on the slowly varying component of $\mathbb{P}(\xi_1>x)$. In particular, it may happen that $e^{\lambda_a n}\mathbb{P}_x(T_0>n)\to0$ as $n\to\infty$.
\[prop:reg.tails\] Assume that $$\label{reg.tails.0}
\mathbb{P}(\xi_1>x)=x^{-r}L(x)$$ for some $r>0$ and some slowly varying function $L$. Then, for all $M\ge0$, $$\label{reg.tails.1}
\liminf_{n\to\infty}\frac{1}{n}\log\mathbb{P}(T_M>n)\ge r\log a.$$ If, in addition, $$\label{reg.tails.1a}
L(x)=O(\log^{-2r-2} x),$$ then, for all $M$ sufficiently large, $$\label{reg.tails.2}
\lim_{n\to\infty} a^{-rn}\mathbb{P}(T_M>n)=0.$$ In fact, one even has $$\sum_{n=0}^{\infty}a^{-rn}\mathbb{P}(T_M>n)<\infty.$$
It is immediate from and that $$\lambda_a=-r\log a$$ for innovations satisfying and . We conjecture that the same holds under the assumption and that the asymptotic behavior of the slowly varying function $L$ affects lower order corrections only. $\diamond$
Fix some $A>a^{-1}$ and define $u_k=MA^k$, $k\ge0$. Then $$\begin{aligned}
\mathbb{P}_{u_k}(X_1\ge u_{k-1})=\mathbb{P}(au_k+\xi_1>u_{k-1})=\mathbb{P}(\xi_1>-(aA-1)u_{k-1}). \end{aligned}$$ Therefore, using the Markov property and the stochastic monotonicity of $X_n$, we get $$\begin{aligned}
\mathbb{P}_{u_k}(T_M>k-1)
&\ge\mathbb{P}_{u_k}(X_1\ge u_{k-1},X_2>u_{k-2},\ldots,X_{k-1}\ge u_1)\\
&=\prod_{j=2}^{k}\mathbb{P}_{u_j}(X_1\ge u_{j-1})=\prod_{j=1}^{k-1}\mathbb{P}(\xi_1>-(aA-1)u_{j}).\end{aligned}$$ It follows now from the assumption $\mathbb{E}\log(1+|\xi_1|)<\infty$ that $$\inf_{k\ge2}\prod_{j=1}^{k-1}\mathbb{P}(\xi_1>-(aA-1)u_{j})=:p(M,A)>0.$$ This implies that $$\mathbb{P}_M(T_M>k)\ge p(M,A)\mathbb{P}(\xi_1>u_k).$$ Since the tail of $\xi_1$ is regularly varying of index $-r$, we obtain $$\liminf_{k\to\infty}\frac{1}{k}\log\mathbb{P}_M(T_M>k)\ge -r\lim_{k\to\infty}\log u_k
=-r\log A.$$ Letting now $A\downarrow a^{-1}$, we arrive at .
In order to prove the second statement we consider events $$B_n:=\left\{\xi_k\le h\frac{a^{-(n-k+1)}}{(n-k+1)^2}\text{ for all }k<T_M\wedge n\right\}.$$ On the event $\{T_M>n\}\cap B_n$ one has $$\begin{aligned}
X_n=a^nM+\sum_{k=1}^na^{n-k}\xi_k\le a^n M+\frac{h}{a}\sum_{k=1}^n\frac{1}{(n-k+1)^2}
\le aM+\frac{2h}{a}.\end{aligned}$$ Thus, for every $M\ge \frac{2}{a(1-a)}h$ and all $n\ge1$ one has $X_n\le M$ and, consequently, $\{T_M>n\}\cap B_n=\emptyset$. This implies that $$\mathbb{P}_M(T_M>n)=\mathbb{P}_M(T_M>n;B_n^c)
\le\sum_{k=1}^n\mathbb{P}_M(T_M>k-1)\mathbb{P}\left(\xi_k>h\frac{a^{-(n-k+1)}}{(n-k+1)^2}\right).$$ From the assumption \[reg.tails.1a\] on $L(x)$ we get $$\mathbb{P}\left(\xi_k>h\frac{a^{-(n-k+1)}}{(n-k+1)^2}\right)
\le\frac{c(a)}{h^r}\frac{a^{r(n-k+1)}}{(n-k+1)^2}.$$ Therefore, $$\begin{aligned}
\mathbb{P}_M(T_M>n)\le\frac{c(a)}{h^r}\sum_{j=0}^{n-1}\mathbb{P}_M(T_M>j)\frac{a^{r(n-j)}}{(n-j)^2},
\quad n\ge1.\end{aligned}$$ Multiplying both sides with $s^n$ and summing over all $n$, we obtain $$\begin{aligned}
\sum_{n=0}^\infty s^n\mathbb{P}_M(T_M>n)
&\le 1+\sum_{n=1}^\infty s^n \frac{c(a)}{h^r}\sum_{j=0}^{n-1}\mathbb{P}_M(T_M>j)\frac{a^{r(n-j)}}{(n-j)^2}\\
&=1+\frac{c(a)}{h^r}\sum_{j=0}^\infty s^j\mathbb{P}_M(T_M>j)\sum_{n=j+1}^\infty\frac{(sa^r)^{n-j}}{(n-j)^2}\\
&=1+\frac{c(a)}{h^r}\sum_{j=0}^\infty s^j\mathbb{P}_M(T_M>j)\sum_{n=1}^\infty \frac{(sa^r)^{n}}{n^2}.\end{aligned}$$ In other words, $$\begin{aligned}
\sum_{n=0}^\infty s^n\mathbb{P}_M(T_M>n)\le
\left(1-\frac{c(a)}{h^r}\sum_{n=1}^\infty \frac{(sa^r)^{n}}{n^2}\right)^{-1}.\end{aligned}$$ If $h$ is so large that $h^r>4c(a)$ then we have $$\begin{aligned}
\sum_{n=0}^\infty a^{-rn}\mathbb{P}_M(T_M>n)\le 2.\end{aligned}$$ This yields .
In order to see, that spectral properties remain relevant in the situation at hand we observe the following assertion formulated in terms of Tweedie’s **R**-theory (see [@T74a] and [@T74b]).
Assume that the conditions and are satisfied. Then for $r$ large enough the operator $K_{\lambda_a}$ is well defined and the spectral radius $r(K_{\lambda_a})$ belongs to $[0,1]$. Under the assumption $r(K_{\lambda_a})<1$ the Laplace transform of $T_0$ remains bounded up to the critical line. In particular, the Submarkovian transition operator $P$ is $R$-transient in the sense of Tweedie.
Until now a complete analysis of persistence probabilities including the effects of polynomial decay factors as well as thorough investigation the quasistationary behaviour of AR(1) sequences with heavy tailed innovations does not seem to exist and constitutes an interesting open problem.
Discussion {#s:dis}
==========
In this section we summarise the general ideas of the two main analytic approaches used in this work and comment on their possible applicability to other models. As a matter of fact both approaches use different tools but also share some structural similarites. For a Markov process $(X_n)_{n \in \mathbb{N}_0}$ we want to find the precise tail behaviour of the first hitting time $T_B$ of a measurable subset $B$ of the state space.
- The first approach is in essence spectral theoretic. We first analyse the decay rate $$\theta_B=-\lim_{n\rightarrow\infty}\frac{1}{n}\log \mathbb{P}_x\bigl(T_B> n\bigr)$$ and make sure that $\theta_B$ does not depend on the starting point. The second step consists in showing that for a strictly larger measurable set $B'\supset B$ one has $$\label{e:discussion}
\theta_B < \theta_{B'}\,.$$ This allows to introduce a suitable weighted Banach space and under appropriate conditions on the distribution of the innovations to prove the quasicompactness of the killed transition kernel. Application of a suitable result of Perron-Frobenius type allows to establish precise exponential decay of the tails of $T_B$.
- The second approach consists in analysing the Laplace transform $$\lbrace z \in \mathbb{C} \mid \Re z<\theta_B\rbrace \ni \lambda \mapsto F_\lambda(x):= \mathbb{E}_x\bigl[e^{\lambda T_B}\bigr]$$ near the abscissa of convergence $\theta_B$. We prove that $F_{\lambda}(x)$ has a meromorphic extension to $\lbrace z \in \mathbb{C} \mid \Re z<\theta_B+\varepsilon\rbrace $ for some $\varepsilon>0$ with $\lambda_B$ being a pole. This allows to deduce the precise exponential decay using a suitable Tauberian theorem. In order to prove the existence of a meromorphic extension we derive a renewal type equation in terms of the transition kernel $K_\lambda$. The existence of a meromorphic extension is shown using the analytic Fredholm alternative and suitable properties of $K_{\lambda}$. In particular, we need to show that the operators $K_{\lambda}$ are compact for all $\lambda$ with $\Re \lambda < \lambda_a+\varepsilon$ and satisfy the conditions of a suitable Perron-Frobenius theorem. Here results of the type given in are again essential as well as absolute continuity and strict positivity of the transition kernel.
We believe that these methods can be applied without big changes to some other Markov chain models. For example, to max-autoregressive processes or to random exchange processes, which are quite closely related to AR($1$)-sequences, see [@Zerner18].
In both approaches, we have assumed that the underlying distributions are absolutely continuous with almost everywhere positive density and have sufficiently light tails. The positivity on the whole axis has been used only to give very simple proofs of the posivity of compact operators. We believe that this positivity can be shown also in the case when the density is zero on significant parts of the axis. Probably one will have the positivity of an appropriate power of the operator, but this does not restrict the applicability of Perron-Frobenius type results. The absolute continuity assumption seems to be really crucial for both approaches, and it is not clear whether one can replace it by the existence of an absolute continuous component. A very challenging problem is to study the case of discrete innovations.
The second approach might also be applicable to chains with a certain periodicity, presumably resulting in poles of higher order. A different behaviour of the pole(s) also can be expected in the presence of distributions with regularly varying tails. Getting completely rid of absolute continuity seems so be much more tricky. Moreover, the stochastic monotonicity of AR(1)-sequences was extensively used, a lack of this property is expected to result in largely technical complications.
The authors of [@CV17] are able to prove similar results using a different approach. Their conditions are of the following types (see page 8 in [@CV17]):
- Local minorization of Doeblin-type
- Two global Lyapunov criteria
- Local Harnack inequality
- Aperiodicity
The approach used by Champagnat and Villemonais is more related to coupling ideas of Doeblin type. The global Lypunov criteria are related to our condition whereas the local minorization, the local Harnack inequality as well as the aperiodicity are closer to the condition used in our approaches to prove quasicompactness with a leading eigenvalue of mulitiplicity one in approach one and compactness of the renewal operator with a leading eigenvalue of multiplicity one in approach two. All three approaches have their merits. As far as possible extensions to AR(1)-processes with fat tailed innovations are concerned, the approach in Section \[s:renewal\] seems to be most promising.
**Acknowledgement.** The authors would like to thank a referee for his useful report and for suggestion to add the final section. This project has been initiated during the subbatical stay of VW at the Technion, Israel. He thanks the Humboldt foundation and the Technion for the financial support. MK would like to thank his student P. Trykacz for critical reading of the final draft.
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---
abstract: 'In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we prove that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n$ grows significantly faster than $\sqrt[k]{h/(ek)}$ ; and tends to 0, if $n$ grows significantly slower than $\sqrt[k]{h/(ek)}$ .'
address:
- 'Instituto de Matemáticas UNAM and Instituto de Física UASLP, Mexico'
- 'Instituto de Matemáticas, Universidad de Antioquia, Colombia'
author:
- Frank Duque
- 'Alejandro Roldán-Correa'
- 'Leon A. Valencia'
bibliography:
- 'kpercolation.bib'
title: 'Accessiblility Percolation with Crossing Valleys on $n$-ary Trees'
---
[^1]
[^2]
Introduction {#S: Introduction}
============
Let $T$ be a rooted tree (whose edges are directed from fathers to children). Assume that the vertices of $T$ are labeled with independent and identically distributed continuous random variables; given a vertex $v\in T$, we denote by $w(v)$ its label. Let $$P=v_0\rightarrow v_1\rightarrow \cdots \rightarrow v_h$$ be a path in $T$ from the root to a leaf. We say that $P$ is a $k$*-accessible* if there is a subsequence $S:=v_{r(0)},v_{r(1)},v_{r(2)},\ldots,v_{r(t)}$ of the vertices in $P$ such that: $$w({v_{r(0)}})<w({v_{r(1)}})<w({v_{r(2)}})<\cdots<w({v_{r(t)}}),$$ each $k$ consecutive vertices of $P$ contains at least one vertex in $S$, $v_0=v_{r(0)}$ and $v_h=v_{r(t)}$. In Figure \[fig:kAccessiblilityInT\] we illustrate a labelled tree and its $2$-accessible paths. We denote by $\theta_k(T)$ the probability of having a $k$-accessible path in $T$.
![An example of a labelled tree without $1$-accessible paths, and for which the only $2$-accessible paths are labeled as: $53$,$99$,$68$,$4$,$71$; and $53$,$65$,$13$,$78$,$26$,$91$.[]{data-label="fig:kAccessiblilityInT"}](kAccessiblilityInT){width=".6\linewidth"}
Motivation.
-----------
Fitness landscapes are by construction a static concept that assigns fitness values to the points of an underlying configuration space [@DefLandscapes2003]. This concept was first proposed in 1932 by Sewall Wright [@LandscapeDef1932] (as a mapping from a set of genotypes to fitness); since then, it has attracted particular interest in evolutionary biology and in evolutionary computation, because it offers an approach to conceptualize and visualize how an evolving population may change over time, in various population genetics models [@FitnessLandscape2014Book].
For the case of an haploid asexual population on a given fitness landscape, framing by the ‘strong selection, weak mutation’ (SSWM) regime, a mutations path is considered selectively accessible if the fitness values encountered along it are monotonically increasing [@JKAJ2011; @WDDH2006; @WWC2005]. Motivated by the concept of selective accessibility, a new kind of percolation was introduced by Nowak and Krug in [@NK2013], in which they studied the probability of having a monotonically increasing path in an graph whose vertices has been labelled with random numbers [@NK2013; @RZ2013; @ApHc2016; @BBS2017; @Li2017].
The model of accessibility percolation was introduced by Nowak and Krug [@NK2013] as follows. Imagine a population of some life form endowed with the same genetic type (genotype). If a mutation occurs, a new genotype is created which can die out or replace the old one. Provided that natural selection is sufficiently strong, the latter only happens if the new genotype has larger fitness. As a consequence, on longer timescales the genotype of the population takes a path through the space of genotypes along which the fitness is monotonically increasing [@G1984].
The hypercube is a graph whose vertices are all possible $N$-tuples in $\lbrace 0,1 \rbrace^N$ (for some positive integer $N$), where two vertices are connected by an undirected edge, if the number of coordinates at which they differ is one. In many basic mathematical models of genetic mutations, the genotype sequence space is represented by the hypercube: each genome is represented as a node of the hypercube and each mutation involves the flipping of a single bit from 0 (the “wild” state) to 1 (the “mutant” state) [@JKAJ2011; @WWC2005; @Martinsson2014]. An $f$-ary (complete $f$-ary) tree with height $h$, is a rooted tree (whose edges are directed from fathers to children) whose vertices at distance at most $h-1$ to the root, have $f$ children. Being a simpler case and motivated by the hypercube case, Nowak and Krug in [@NK2013] study the problem of determining the growth of $f$ as a function of $h$ for having accessibility percolation in $f$-ary trees.
According to the ruggedness or smoothness of the landscape, there are some topological features (as peaks, valleys, ridges etc.) that block the selectively accessible mutations paths towards the highest peak in the landscape. The evolutionary process that may allow escaping those topological features is known as valley crossing, which is not allowed in the selectively accessible mutation paths. However, in natural populations of sufficient size, a number of double mutants is present at all times, and the crossing valleys can be relatively facile [@JKAJ2011; @WC2005; @WDFF2009]; the SSWM assumption may therefore seem very restrictive.
A simple way of allowing peaks and valleys on the accessibility percolation model, consists in “ allowing holes” in the monotonicity of the path that the genotype population takes. In this paper, as a variation of the concept of accessibility percolation introduced by Nowak and Krug in [@NK2013], we introduce the concept of $k$-accessibility percolation in which crossing small valleys is allowed. Imagine a population of some organisms endowed with the same genotype. If a mutation occurs, a new genotype is created. The genotype die, when in the path $P$ of consecutive mutations, there is not a subsequence $S$ with increasing fitness and small holes.
Previous results
----------------
Given a function $f:\mathbb{Z}_+\rightarrow \mathbb{Z}_+$, we denote by $T_h(f)$ the $f(h)$-ary tree with height $h$ and we denote by $\mathcal{T}_f$ the sequence of trees $\lbrace T_h(f)\rbrace_{h\in \mathbb{Z}_+}$. We denote by $\theta_k(\mathcal{T}_f):=\lim_{h\rightarrow \infty} \theta_k (T_{h}(f))$ provided the limit exists, when $\theta_k(\mathcal{T}_f)>0$ we say that there is $k$-percolation in $\mathcal{T}_f$.
In the first work on accessibility percolation, Nowak and Krug studied the problem of determining the growth of $f$, with respect to $h$, such that there is $1$-accessibility percolation on $\mathcal{T}_f$ [@NK2013]. Nowak and Krug prove that: if $f$ is a super-linear function, then there is $1$-accessibility percolation in $\mathcal{T}_f$; and if $f$ is a sub-linear function, then there is not $1$-accessibility percolation in $\mathcal{T}_f$. They also studied the linear case, $f(h)=\alpha h$, establishing the existence of a threshold between $e^{-1}$ and $1$ on the scaling constant factor; there is $1$-accessibility percolation in $\mathcal{T}_f$ when $\alpha>1$, and there is not $1$-accessibility percolation in $\mathcal{T}_f$ when $\alpha<e^{-1}$. See Theorem \[thm:NK\]
\[thm:NK\] $\theta_1 (\mathcal{T}_f)>0$, if $f(h)\geq \left\lfloor \alpha h \right\rfloor $ for $h$ large enough and $\alpha>1$. $\theta_1 (\mathcal{T}_f)=0$, if $f(h)\leq \left\lfloor \alpha h \right\rfloor$ for $h$ large enough and $\alpha \leq e^{-1}$.
Continuing with the work of Nowak and Krug, Roberts and Zhao [@RZ2013] determined that $\alpha=e^{-1}$ is a threshold on the scaling constant factor for $1$-accessibility percolation on $\mathcal{T}_f$, see Theorem \[thm:NK1p\] and Theorem \[thm:NK1perc\].
\[thm:NK1p\] $$\theta_1 (\mathcal{T}_f)=
\begin{cases}
1, & \quad \text{if } f(h)\geq \left\lfloor \alpha h \right\rfloor \text{ for $h$ large enough and } \alpha>e^{-1},\\
0, & \quad \text{if } f(h)\leq \left\lfloor \alpha h \right\rfloor \text{ for $h$ large enough and } \alpha \leq e^{-1}.\\
\end{cases}$$
\[Roberts-Zhao [@RZ2013]\]\[thm:NK1perc\] If $f(h)=\left( \frac{1+\beta_h}{e} \right)h$ where $\beta_h \rightarrow 0$ as $h \rightarrow \infty$, then $$\theta_1 (\mathcal{T}_f)=
\begin{cases}
1, & \quad \text{if } h\beta_h/\log h \rightarrow \infty. \\
0, & \quad \text{if } \log h -2h\beta_h \rightarrow \infty,\\
\end{cases}$$
Accessibility percolation has been also studied, recently, for the case when the underlying graph is the hypercube; see [@Martinsson2014; @martinsson2015; @ApHc2016; @BBS2017; @Li2017; @BBS2017].
Our results.
------------
We denote by $\Omega(g)$ the functions $t$ such that $\lim_{h\rightarrow\infty}(t(h)/g(h))\geq k>0$. We denote by $\omega(g)$ the functions $t$ such that $\lim_{h\rightarrow\infty}(t(h)/g(h)) = \infty$ (for a more formal definition of $\Omega$ and $\omega$ the reader may change $\lim_{h\rightarrow\infty}$ by $\liminf_{h\rightarrow\infty}$). Therefore $\omega\left(\sqrt[k]{\log(h)}\right)$ denotes a function $t$ such that, for any constant $C>0$, $t(h)\geq C\sqrt[k]{\log(h)}$ for $h$ large enough; similarly $\Omega(h^c)$ denotes a function that, for some constants $c,C>0$, it is above $Ch^c$ for $h$ large enough.
In this paper we determine the growth of $f$, as a function of $h$, for which there is $k$-accessibility percolation on $\mathcal{T}_f$. Our main result is the following.
\[thm:main\] Let $\mathcal{T}_f$ be the sequence of $n$-ary trees $\lbrace T_h(f)\rbrace_{h\in \mathbb{Z}_+}$. Then $$\label{eq:mainthm}
\theta_k(\mathcal{T}_f)=
\begin{cases}
1, & \quad \text{if } f(h)\geq \sqrt[k]{h/(ek)}+\omega\left(\sqrt[k]{\log(h)}\right), \\
0, & \quad \text{if } f(h)\leq \sqrt[k]{h/(ek)}- \Omega(h^c) \text{ and } c>0.\\
\end{cases}$$
The reader may note that, from Theorem \[thm:main\] it follows that
$$\theta_k(\mathcal{T}_f)=
\begin{cases}
1, & \quad \text{if } f(h)\geq \sqrt[k]{h/(ek)}c \text{ for $h$ large enough and } c>1, \\
0, & \quad \text{if } f(h)\leq \sqrt[k]{h/(ek)}c \text{ for $h$ is large enough and } c<1.
\end{cases}$$
Preliminaries
=============
Before proceeding with the proof of Theorem \[thm:main\], we introduce the concept of $k$-transitive closure any we state an equivalent version of Theorem \[thm:NK1perc\].
We define the $k$-transitive closure of $G$, $G^k$, as the graph obtained from it, by adding new edges from each vertex $u$ to each vertex $v$, with the property that $G$ does not already contain the directed edge from $u$ to $v$ but does contain a directed path from $u$ to $v$ with length at most $k$.
Although the concept of $k$-accessibility percolation was only introduced for trees, it can be easily extended to directed graphs where: there is a single vertex distinguished as the source, some vertices distinguished as sinks, and all of the maximal paths are directed paths from the source to some sink; we name the graphs that satisfy the previous statements as monotone graphs. For the case of monotone graphs, we say that a path in it is $1$-accessible if: it starts in the source, ends at some sink and its vertices have increasing labels. Similar to the case of trees, we denote by $\theta_1(G)$ the probability of having a $1$-accessible path in $G$.
In Figure \[fig:1AccessiblilityInT3\] we illustrate the $2$-transitive closure of the graph depicted in Figure \[fig:kAccessiblilityInT\] and its $1$-accessible paths. Note that the $1$-accessible paths in the graph depicted in Figure \[fig:1AccessiblilityInT3\] correspond to the $2$-accessible paths in the graph depicted in Figure \[fig:kAccessiblilityInT\].
\[def1\] Let $T$ be a rooted tree with height $h$ and $T^k$ be its $k$-transitive closure. Then $\theta_1(T^k)=\theta_k(T)$.
Let $T$ be a rooted tree with height $h$ and $T^k$ be its $k$-transitive closure. If $P$ is a $k$-accessible path in $T$, there is a subsequence $S$ of the vertices in $P$ with increasing labels, that contains at least one vertex in each $k$ consecutive vertices in $P$ and that contains the root and a leaf of $T$; therefore $S$ is a $1$-accessible path in $T^k$. On the other direction, if there is a $1$-accessible path in $T^k$, the vertices in such path define a subsequence $S$ of the vertices in some path $P$ in $T$ that make $P$ a $k$-accessible path.
![An illustration of the $2$-transitive closure of the graph depicted in Figure \[fig:kAccessiblilityInT\], the $1$-accessible paths in such graph are labeled as: $53$,$68$,$71$; and $53$,$65$,$78$,$91$.[]{data-label="fig:1AccessiblilityInT3"}](1AccessiblilityInT2){width=".6\linewidth"}
For proving Theorem \[thm:main\] we use the following proposition, that it is an equivalent version of Theorem \[thm:NK1perc\].
\[prop:nk\] $$\theta_1 (\mathcal{T}_f) =
\begin{cases}
1, & \quad \text{if } f(h)= \frac{h}{e}+ \omega\left(\log(h) \right), \\
0, & \quad \text{if } f(h)= \frac{h}{e}+ \frac{\log(h)}{2e} - \omega(1).\\
\end{cases}$$
Let $f(h)$, $g(h)$ and $\beta_h$ be such that $$f(h)= \frac{h}{e}+ g(h)\log(h)= \frac{1+\beta_h}{e}h.$$ As $\beta_h=g(h)\frac{h}{\log(h)}$, it follows that, $h\beta_h/\log h$ goes to infinity when $h$ goes to infinity, if and only if $g(h)=\omega(1)$. Therefore, by Theorem \[thm:NK1perc\], if $f(h)= \frac{h}{e}+\omega(\log(h))$ then $f(h)= \frac{h}{e}+\omega(1)\log(h)$ and $\theta_1 (\mathcal{T}_f)=1$.
Let $f(h)$, $g(h)$ and $\beta_h$ be such that $$f(h)= \frac{h}{e}+\frac{\log(h)}{2e}- g(h)= \frac{1+\beta_h}{e}h.$$ As $\beta_h=\frac{\log(h)- 2e\cdot g(h)}{2h}$, it follows that, $\log h -2h\beta_h$ goes to infinity when $h$ goes to infinity, if and only if $g(h)=\omega(1)$. Therefore, by Theorem \[thm:NK1perc\], if $f(h)= \frac{h}{e}+\frac{\log(h)}{2e}-\omega(1)$ then $\theta_1 (\mathcal{T}_f)=0$.
Proof of Theorem \[thm:main\]
==============================
Now we proceed to prove Theorem \[thm:main\]. The proof is divided two steps according with the two cases in Equation \[eq:mainthm\].
Step 1 {#step-1 .unnumbered}
------
Let $\mathcal{T}_f$ be the sequence of $n$-ary trees $\lbrace T_h(f)\rbrace_{h\in \mathbb{Z}_+}$. In this section we assume $$f(h)\geq \sqrt[k]{h/(ek)}+\omega \left( \sqrt[k]{\log(h)} \right)$$ and we prove that $\theta_k(\mathcal{T}_f)=1$.
Let $\mathcal{T}^k_f$ be the sequence of graphs $\lbrace T_h^k(f)\rbrace_{h\in \mathbb{Z}_+}$ where $T_h^k(f)$ is the $k$-transitive closure of $T_h(f)$. Let $g(h)= h/e + \omega \left(\log(h) \right)$, $T^k=T_h^k(f)$ and $T'=T_{\left\lfloor h/k\right\rfloor }(g)$. By Remark \[def1\], provided the limit exists, $$\theta_k(\mathcal{T}_f)=\lim_{h\rightarrow \infty}\theta_k(T_h(f))=
\lim_{h\rightarrow\infty}\theta_1(T_h^k(f)).$$ By Proposition \[prop:nk\] $$\lim_{h\rightarrow\infty}\theta_1(T_h(g))=\theta_1( \mathcal{T}_g)=1.$$ To prove that $\theta_k(\mathcal{T}_f)=1$, it is enough to show that $\theta_1(T^k)\geq \theta_1(T')$.
Let $G$ be the subgraph of $T^k$, obtained from removing the vertices in $T$, whose distance from the root is not a multiple of $k$. Note that $G$ is a tree contained in $T^k$ with height $\left\lfloor h/k\right\rfloor$. Also note that, for $h$ large enough, $$\text{deg}_G(v)\geq \left( \sqrt[k]{h/(ek)}+\omega \left(\sqrt[k]{\log(h)} \right) \right) ^k \geq {h/(ek)}+\omega \left({\log(h)} \right) \geq g(\left\lfloor h/k\right\rfloor);$$ thus the non leaves vertices of $G$ have degree at least $g(\left\lfloor h/k\right\rfloor)$. Therefore, for $h$ large enough $T'\subset G \subset T^k.$ Therefore, the probability of having at least one $1$-accessible path in $T'$, is a lower bound of the probability of having at least an $1$-accessible path in $T^k$.
Step 2 {#step-2 .unnumbered}
------
Let $\mathcal{T}_f$ be the sequence of $n$-ary trees $\lbrace T_h(f)\rbrace_{h\in \mathbb{Z}_+}$. In this section we assume $f(h)\leq \sqrt[k]{h/(ek)}- \Omega(h^c)$ for some $0<c<1$, and we prove that $\theta_k(\mathcal{T}_f)=0$.
Let $\mathcal{T}^k_f$ be the sequence of graphs $\lbrace T_h^k(f)\rbrace_{h\in \mathbb{Z}_+}$ where $T_h^k(f)$ is the $k$-transitive closure of $T_h(f)$. Let $g(h)=\frac{h}{e}$, $T=T_h(f)$, $T^k=T_h^k(f)$ and $T'=T_{ h/k}(g)$. By Remark \[def1\], provided the limit exists, $$\theta_k(\mathcal{T}_f)=\lim_{h\rightarrow \infty}\theta_k(T_h(f))=
\lim_{h\rightarrow\infty}\theta_1(T_h^k(f)).$$ By Proposition \[prop:nk\] $$\lim_{h\rightarrow\infty}\theta_1(T_h(g))=\theta_1( \mathcal{T}_g)=0.$$ To prove that $\theta_k(\mathcal{T}_f)=0$, it is enough to show that $\theta_1(T^k)\leq \theta_1(T')$. For this we define a tree $H^k$ such that $$\theta_1(T^k)\leq \theta_1(H^k) \leq \theta_1(T').$$
$H^k$ will denote a tree, whose vertices are in correspondence with the paths in $T^k$ that start in the root. As an outline of the construction of $H^k$ note that, any path in $T^k$ is characterized by the following two issues. The first issue is the vertex $v$ in which the path ends. Suppose that $v$ is at distance $l$ to the root. The second issue is the information about the levels in which it has no vertices; it can be represented by a subset $s$ of $\lbrace 1,2,\ldots,l-1 \rbrace$ that does not contain $k$ consecutive numbers. Therefore the vertices in $H^k$ will be determined by a vertex $v$ in $T^k$ and a $s$ subset of $\lbrace 1,2,\ldots,l-1 \rbrace$ that does not contain $k$ consecutive numbers. About the edges in $H^k$, two vertices will be adjacent in $H^k$ if: one of the corresponding paths in $T^k$ contains the other corresponding path, and those corresponding only differ in one edge. As an example of this outline, consider the paths $v_0,v_1,v_3$ and $v_0,v_1,v_3,v_5$ in the graph $T^2$ illustrated in Figure \[fig:ex-H:Tg2\]; in $H^2$, illustrated in Figure \[fig:ex-H:H\], those paths are represented by $v_3^{ \{2\}}$ and $v_5^{ \{2,4\}}$, respectively, and they are adjacent.
We define $H^k$ formally as follows.
Let $\mathcal{A}_l$ be the set whose elements are the subsets $s$ of $ \lbrace 1,2,\ldots ,l-1 \rbrace$ that do not contain $k$ consecutive numbers. Let $H^k=(V,E)$ be the graph whose vertices and edges are defined as follows. See Figure \[fig:ex-H\].
\[htb\]
[.2]{} ![An example of the subgraphs of $T$ (\[fig:ex-H:Tg\]), $T^2$ (\[fig:ex-H:Tg2\]) and $H^2$ (\[fig:ex-H:H\]), respectively, corresponding to six consecutive vertices in the path $v_0, v_1, v_2, v_3, v_4, v_5, v_5, v_5$ depicted Figure \[fig:kAccessiblilityInT\]. []{data-label="fig:ex-H"}](Tg "fig:"){width=".26\linewidth"}
[.2]{} ![An example of the subgraphs of $T$ (\[fig:ex-H:Tg\]), $T^2$ (\[fig:ex-H:Tg2\]) and $H^2$ (\[fig:ex-H:H\]), respectively, corresponding to six consecutive vertices in the path $v_0, v_1, v_2, v_3, v_4, v_5, v_5, v_5$ depicted Figure \[fig:kAccessiblilityInT\]. []{data-label="fig:ex-H"}](Tg2 "fig:"){width=".5\linewidth"}
[.6]{} ![An example of the subgraphs of $T$ (\[fig:ex-H:Tg\]), $T^2$ (\[fig:ex-H:Tg2\]) and $H^2$ (\[fig:ex-H:H\]), respectively, corresponding to six consecutive vertices in the path $v_0, v_1, v_2, v_3, v_4, v_5, v_5, v_5$ depicted Figure \[fig:kAccessiblilityInT\]. []{data-label="fig:ex-H"}](H-2 "fig:"){width="0.9\linewidth"}
The set of vertices in $H^k$ is defined as $$V:=\lbrace v^s:v\in T \textnormal{ and } s\in \mathcal{A}_l \textnormal{, where $l$ is the distance of $v$ to the root} \rbrace.$$ The edges in $H^k$ are defined recursively as follows. Let $v$ be a vertex of $T$ at distance $l$ to the root and let $s\in \mathcal{A}_l$. We say that $w^{s'}$ is a son of $v^s$ if:
- $w$ is a son of $v$ in $T$ and $s=s'$
- $w$ is a descendant of $v$ at distance $1<j\leq k$ in $T$ and $s'=s\cup\lbrace l+1,l+2,\ldots l+j-1 \rbrace$
Given a vertex $v$ in $T^k$, we claim that: the paths in $T^k$ from the root to $v$, the elements in $\mathcal{A}_l$, and the paths in $H^k$ from the root to $v^s$ (for some $s\in\mathcal{A}_l$), are in one-to-one correspondence. For each path $P$ in $T^k$ from the root to $v$, there exist one and only one $s$ in $\mathcal{A}_l$ whose elements are the indices $i$ such that $v_i$ is not in $P$. For each $s$ in $\mathcal{A}_l$, there is one and only one path in $P$ that starts in the root and ends in $v$; such path contains the vertices $v_i$ such that $i$ is not in $s$. For each path in $H^k$ from the root to $v^s$, obviously there is one and only one $s\in \mathcal{A}_l$. Now suppose that $s$ is in $\mathcal{A}_l$, there is one and only one path $P$ in $H^k$ from the root to $v^s$; such path contains the vertices $v_i^{s(i)}$ where $i$ is not in $s$ and $s(i)=s\cap \{1,2,\dots, i-1\}$. As an example consider: In the graph $G^2$ for the graph $G$ in Figure \[fig:kAccessiblilityInT\], the path $v_0,v_1,v_3,v_5$; in $\mathcal{A}_5$, the set $\{2,4\}$; and in $H^2$, the path $v_0^{\phi},v_1^{\phi},v_3^{\{2\}},v_5^{\{2,4\}}$ (See Figure \[fig:ex-H\]).
Assume that the vertices in $H^2$ are labelled with independent and identically distributed random variables with the same distribution as the labels in $T$.
\[htb\]
[.3]{} ![An example of a sequence of graphs in which the graph depicted in Figure \[fig:ex-H:H\] is obtained from the graph in Figure \[fig:ex-H:Tg2\]. It illustrates how to obtain the sequence of graphs used in Lemma \[lem:Tg2-H-2\].[]{data-label="fig:ex-H-etapas"}](H0 "fig:"){width=".34\linewidth"}
[.33]{} ![An example of a sequence of graphs in which the graph depicted in Figure \[fig:ex-H:H\] is obtained from the graph in Figure \[fig:ex-H:Tg2\]. It illustrates how to obtain the sequence of graphs used in Lemma \[lem:Tg2-H-2\].[]{data-label="fig:ex-H-etapas"}](H1 "fig:"){width=".36\linewidth"}
[.36]{} ![An example of a sequence of graphs in which the graph depicted in Figure \[fig:ex-H:H\] is obtained from the graph in Figure \[fig:ex-H:Tg2\]. It illustrates how to obtain the sequence of graphs used in Lemma \[lem:Tg2-H-2\].[]{data-label="fig:ex-H-etapas"}](H2 "fig:"){width="0.50\linewidth"}
[.4]{} ![An example of a sequence of graphs in which the graph depicted in Figure \[fig:ex-H:H\] is obtained from the graph in Figure \[fig:ex-H:Tg2\]. It illustrates how to obtain the sequence of graphs used in Lemma \[lem:Tg2-H-2\].[]{data-label="fig:ex-H-etapas"}](H3 "fig:"){width=".73\linewidth"}
[.6]{} ![An example of a sequence of graphs in which the graph depicted in Figure \[fig:ex-H:H\] is obtained from the graph in Figure \[fig:ex-H:Tg2\]. It illustrates how to obtain the sequence of graphs used in Lemma \[lem:Tg2-H-2\].[]{data-label="fig:ex-H-etapas"}](H4 "fig:"){width=".85\linewidth"}
\[lem:Tg2-H-2\] $\theta_1(H^k)\geq \theta_1(T^k).$
In what follows we define a sequence of graphs $H_0, H_1, \ldots, H_t$, such that $H_0=T^k$, $H_t=H^k$ and $\theta_1(H_i)\leq \theta_1(H_{i+1})$, for $i=0,1,\ldots, t-1$. The graph $H_0$ is obtained from $T^k$ changing each vertex $v$ for a vertex $v^\phi$, see Figure \[fig:ex-H:H0\]. Given some $H_i$, we say that a vertex $v\in H_i$ is divisible if: (see vertex $v_j^s$ in Figure \[fig:divisiones1\])
- The subgraph of $H_i$, induced by $v$ and its descendants vertices, is a directed tree. We denote such tree as $T(v)$.
- There are at least two edges that starts in an ancestor of $v$ and ends at $v$. From such edges we denote by $e(v)$ the edge that starts in the ancestor $x=u^s$ of $v$ such that $u$ is the nearest to the root in $T$.
\[htb\]
[.5]{} ![ Illustration of the proof of $\theta_1(H_i)\leq \theta_1(H_{i+1})$. []{data-label="fig:divisiones"}](divisiones1 "fig:"){width=".9\linewidth"}
[.5]{} ![ Illustration of the proof of $\theta_1(H_i)\leq \theta_1(H_{i+1})$. []{data-label="fig:divisiones"}](divisiones2 "fig:"){width=".9\linewidth"}
Whenever $H_i$ has at least one divisible vertex, we define $H_{i+1}$ as follows. Let $v^s$ be a divisible vertex in $H_i$. Let $v_0,v_1, \ldots, v_j$ be a path in $T$ from the root to $v$, then $v_0^{s(0)},v_1^{s(1)}, \ldots, v_j^{s(j)}$ is a path in $H_i$ from the root to $v^s$. Let $x=v_{j'}^{s(j')}$ be the vertex where $e(v)$ starts. Let $Q$ be the indices of the vertices that $e(v)$ jumps, $i.e.$ $Q:=\lbrace \{ j'+1, j'+2, \ldots j-1\} \rbrace$. Let $T^Q$ be a copy of $T(v)$ where each vertex $u^{s'}$ is changed by $u^{s'\cup Q }$. $H_{i+1}$ is defined as the graph obtained from $H_i$ by: removing $e(v)$, adding $T^Q$, and adding an edge $e'(v)$ that starts in $x$ and ends at the root of $T^Q$. See Figure \[fig:ex-H-etapas\].
The reader may notice that, if $H_i$ does not have divisible vertices then $H_i=H^k$. It remains to prove that $\theta_1(H_i)\leq \theta_1(H_{i+1})$.
Given a vertex $y$, in a labeled directed graph $G$, we denote by $[y\leadsto G]$ the event of having a path in $G$ that starts in $y$, ends at a sink of $G$ and has increasing labels; we denote by $[y\not\leadsto G]$ its complement. For the case when $y$ is the root it is denoted by $0$.
Consider in the graph in Figure \[fig:divisiones1\], the event of having an $1$-accessible path that contains the edge $e(v_j^s)$. Note that such event occurs, if and only if, there is an increasing path from $0$ to $x$, $w(x)<w(v^s_j)$ and $[v^s_j \leadsto T(v_j^s)]$. Similarly, consider the event in Figure \[fig:divisiones2\], the event of having an $1$-accessible path that contains the edge $e'(v_j^s)$. Note that such event occurs, if and only if, there is an increasing path from $0$ to $x$, $w(x)<w(v_j^{s\cup Q})$ and $[v_j^{s\cup Q} \leadsto T^Q]$. With the notation introduced in the construction of $H_{i+1}$, define $\Gamma_1$ has the event of having an $1$-accessible path in $H_i$ that contains the edge $e(v_j^s)$, and define $\Gamma_2$ has the event of having an $1$-accessible path in $H_{i+1}$ that contains the edge $e'(v_j^s)$.
This lemma follows from the following facts:
- Note that $\Gamma_1$ is the event of having an increasing path from $0$ to $x$, $w(x)<w(v_j^s)$ and $[v_j^s \leadsto T(v_j^s)]$; and $\Gamma_2$ is the event of having an increasing path from $0$ to $x$, $w(x)<w(v^{s\cup Q}_j)$ and $[v^{s\cup Q}_j \leadsto T^Q]$. Therefore, as $$\begin{aligned}
\mathbf{P}\left( w(x)<w(v^{s}_j) \right)&=\mathbf{P}\left( w(x)<w(v^{s\cup Q}_j) \right) \text{ and }\\
\mathbf{P}\left( [v_j^s \leadsto T(v_j^s)] \right)&=\mathbf{P}\left( [v^{s\cup Q}_j \leadsto T^Q] \right) \text{ then }\\
\mathbf{P}\left( \Gamma_1 \right)&=\mathbf{P}\left( \Gamma_2 \right).
\end{aligned}$$
- As the only edge that joins $T^Q$ with the vertices in $H_i$ is $e'(v_j^s)$, and $H_i$ and $H_{i+1}$ only differ in $e(v_j^s)$, $e'(v_j^s)$ and $T^Q$ then $H_{i} -e(v_j^s)=H_{i+1} -\lbrace e'(v_j^s),T^Q\rbrace$. Therefore, as there are no paths in $H_{i+1} -e'(v_j^s)$ from the root to a leaf in $T^Q$ then $$\begin{aligned}
\mathbf{P} \left( [0\leadsto H_{i+1} -e'(v_j^s)] \right)
&= \mathbf{P} \left( \left[0\leadsto H_{i+1} -\lbrace e'(v_j^s),T^Q\rbrace \right] \right)\\
&=\mathbf{P} \left( [0\leadsto H_{i} -e(v_j^s)] \right)
\end{aligned}$$
- Let $[0\leadsto x]$ the event of having an increasing path from $0$ to $x$. Then $$\begin{aligned}
\mathbf{P}\left([0\leadsto H_i-e(v)] \big| \Gamma_1\right)
&\geq \mathbf{P}\left([0\leadsto H_i-e(v)] \big| [0\leadsto x],w(x)<w(v^s_j) \right)\\
& = \mathbf{P}\left([0\leadsto H_{i+1} -\lbrace e'(v_j^s),T^Q\rbrace] \big| [0\leadsto x],w(x)<w(v^{s\cup Q}_j) \right)\\
& =\mathbf{P}\left([0\leadsto H_{i+1} -\lbrace e'(v_j^s),T^Q\rbrace] \big| \Gamma_2 \right)\\
& =\mathbf{P}\left([0\leadsto H_{i+1} - e'(v_j^s)] \big| \Gamma_2 \right)
\end{aligned}$$
- $\theta_1(H_{i+1})-\theta_1(H_{i})\geq 0$. $$\begin{aligned}
&\theta_1(H_{i+1})-\theta_1(H_{i})= \mathbf{P} \left( [0\leadsto H_{i+1}] \right) - \mathbf{P} \left( [0\leadsto H_{i}] \right) \\
&=\mathbf{P} \left( [0\leadsto H_{i+1}]\cap [0\leadsto H_{i+1} -e'(v)] \right) -\mathbf{P} \left( [0\leadsto H_{i}]\cap [0\leadsto H_{i} -e(v)] \right)\\
&+\mathbf{P} \left( [0\leadsto H_{i+1}]\cap [0\not\leadsto H_{i+1} -e'(v)] \right)
- \mathbf{P} \left( [0\leadsto H_{i}]\cap [0\not\leadsto H_{i} -e(v)] \right)\\
&=\mathbf{P} \left( [0\leadsto H_{i+1} -e'(v)] \right)
-\mathbf{P} \left( [0\leadsto H_{i} -e(v)] \right)\\
&+\mathbf{P} \left( \Gamma_2\cap [0\not\leadsto H_{i+1} -e'(v)] \right)
- \mathbf{P} \left( \Gamma_1\cap [0\not\leadsto H_{i} -e(v)] \right)\\
&=\mathbf{P} \left( \Gamma_2 \right) -\mathbf{P} \left( \Gamma_2\cap [0\leadsto H_{i+1} -e'(v)] \right) \\
&- \mathbf{P} \left( \Gamma_1 \right) +\mathbf{P} \left( \Gamma_1\cap [0\leadsto H_{i} -e(v)] \right)\\
&=\mathbf{P} \left( \Gamma_1 \right)\left[
\mathbf{P} \left( [0\leadsto H_{i} -e(v)] \big| \Gamma_1 \right)
-\mathbf{P} \left( [0\leadsto H_{i+1}-e'(v) ] \big| \Gamma_2 \right)
\right] \geq 0
\end{aligned}$$
$\theta_1(T')\geq \theta_1(H^k)$
By definition we require to prove that: the probability of having a $1$-accessible path in $T'$, is an upper bound for the probability of having a $1$-accessible path in $H^k$.
Let $H'\subset H^k$ be the tree induced by the vertices of $H^k$ at distance at most $h/k$ to the root. Notice that $\theta_1(H') \geq \theta_1(H^k)$. Also notice that, if $v$ is a vertex of $H'$ at distance at most $h/k-1$ to the root then (for $h$ large) $$\text{deg}_{H'}(v)=\sum_{j=1}^k \left\lfloor f(h) \right\rfloor ^j \leq \sum_{j=1}^k f(h) ^j = \frac{f(h)^{k+1}-1}{f(h)-1}$$ We claim that $\text{deg}_{H'}(v)\leq g(\lfloor h/k \rfloor)$ (recall, $g$ was defined as $g(h)=\frac{h}{e}$), from which $H'\subset T'$ and, as both trees have the same height, then $\theta_1(T') \geq \theta_1(H')$.
Now, to finish this proof, we prove that the claim holds. Let $A=\sqrt[k]{h/(ek)}$. As $f(h)\leq \sqrt[k]{h/(ek)}- \Omega(h^c)$ then for $c<x<1$ and $h$ large enough, $ f(h)\leq A-A^{x}$. It is enough to prove that $$\frac{\left(A-A^{x} \right)^{k+1} -1}{\left(A-A^{x} \right)-1}\leq A^k$$ but it follows from that $$\left(A-A^{x} \right)^{k+1} -1= A^{k+1}-(k+1)A^{k+x}+o(A^{k+x})$$ and $$A^k\left(A-A^{x} -1 \right)= A^{k+1}-A^{k+x} +A^k.$$
**Acknowledgments:** The authors are thankful to Ricardo Restrepo for helpful discussions.
[^1]: Research Partially supported by FORDECYT 265667 (Mexico)
[^2]: Research Partially supported by Universidad de Antioquia (Colombia).
|
---
abstract: 'Infrared-Faint Radio Sources (IFRS) are objects which are strong at radio wavelengths but undetected in sensitive Spitzer observations at infrared wavelengths. Their nature is uncertain and most have not yet been associated with any known astrophysical object. One possibility is that they are radio pulsars. To test this hypothesis we undertook observations of 16 of these sources with the Parkes Radio Telescope. Our results limit the radio emission to a pulsed flux density of less than 0.21 mJy (assuming a 50% duty cycle). This is well below the flux density of the IFRS. We therefore conclude that these IFRS are not radio pulsars.'
author:
- |
A. D. Cameron,$^{1,2}$ M. Keith,$^2$ G. Hobbs,$^2$ R. P. Norris,$^2$ M. Y. Mao$^{2,3,4}$ & E. Middelberg$^5$\
$^1$ School of Physics, University of New South Wales, Sydney, NSW 2052, Australia\
$^2$ CSIRO Astronomy & Space Science, PO Box 76, Epping NSW 1710, Australia\
$^3$ School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, 7001, Australia\
$^4$ Anglo-Australian Observatory, PO Box 296, Epping, NSW, 1710, Australia\
$^5$ Astronomisches Institut, Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany\
bibliography:
- 'journals.bib'
- 'modrefs.bib'
- 'psrrefs.bib'
- 'crossrefs.bib'
nocite:
- '[@naa+06; @mnc+08]'
- '[@mnt+08]'
- '[@norris2010]'
- '[@lacy2010]'
- '[@middelberg2010]'
- '[@norris2010]'
- '[@mnt+08]'
- '[@middelberg2010]'
- '[@mnc+08]'
- '[@middelberg2010]'
title: 'Are the infrared-faint radio sources pulsars?'
---
[ ]{}
surveys, pulsars: general
Introduction {#sec:Intro}
============
Infrared-Faint Radio Sources (IFRS) are objects which typically have flux densities of several mJy at 1.4 GHz, but are undetected at 3.6 $\mu$m using sensitive *Spitzer Space Telescope* observations with $\mu$Jy sensitivities [@naa+06]. They were an unexpected discovery in the Australia Telescope Large Area Survey (ATLAS), which, covering seven square degrees at 1400 MHz to $\sim$30 $\mu$Jy, is the widest deep field radio survey attempted thus far (Norris et al. 2006; Middelberg et al. 2008a). There are $\sim 50$ IFRS in ATLAS, accounting for 2.5% of the current ATLAS source catalogue. Most of the IFRS are unresolved at 1.4GHz and have flux densities of up to tens of mJy.
The ATLAS project surveys two regions of sky. The first region includes the Chandra Deep Field South (CDFS) [@grt+01] which encompasses the Great Observatories Origins Deep Survey (GOODS) field [@gfk+04]. The second region coincides with the European Large Area ISO Survey-South1 (ELAIS-S1). Both regions have also been covered by the *Spitzer* Wide-Area Infrared Extragalactic (SWIRE) survey program [@lsrr+03]. The second region is close to (but does not overlap) the region covered by the recent Parkes High Latitude Pulsar Survey [@bjd+06].
The primary goals of ATLAS are to trace the cosmic evolution of active galactic nuclei (AGN) and star-forming galaxies. The detected sources are expected to follow relatively well-known spectral energy distributions (SED) that predict infrared emission detectable by *Spitzer* from most strong radio sources. Since the discovery of IFRS in 2006, several publications have attempted to understand their nature and emission mechanisms. Very long baseline interferometry (VLBI) observations of a total of six IFRS by @Norris07b and Middelberg et al. (2008b) have resulted in the detection of high-brightness temperature cores in two IFRS. Recently, @hms+10 investigated the four IFRS in the GOODS field, for which ultra-deep Spitzer imaging had recently become available, and attempted to fit template SEDs corresponding to known classes of galaxies and quasars. Norris et al. (2010) have extended this work using deep Spitzer imaging data from the Spitzer Extragalactic Representative Volume Survey project (Lacy et al. 2010), and have shown that most IFRS sources have extreme values of S20/S3.6, which can be fitted by a high-redshift radio-loud galaxy or quasar. They also stacked the deep Spitzer data to show that a typical IFRS must have a median 3.6 $\mu$m flux density of no more than 0.2 $\mu$Jy, giving extreme values of the radio-infrared flux density ratio. Middelberg et al. (2011) have found the spectra of IFRS are remarkably steep, with a median spectral index of $-$1.4 and a prominent lack of spectral indices flatter than $-$0.7. We note that the systematic steepening of spectral indices can not be the result of variability.
While unidentified radio sources have been well-known for many decades, they are usually identified once sufficiently sensitive optical/IR observations become available. The IFRS differ from previous classes of unidentified sources in their extreme ratio of radio/IR flux density. For example, the ratio of 20 cm to 3.6 $\mu$m flux density of the IFRS studied here is typically 1000, and is 8000 in the case of ES247 (Norris et al. 2010). This may be compared with the ratio of $~$10 expected for starburst galaxies and $~100$ for a typical radio-loud quasar template [@ewmd+94]. They bear some similarities to the high-redshift radio galaxies studied by @ssd+07 and @jts+09 but appear in many cases to be even more extreme. Hence, the nature of IFRS remains unclear, largely because they are invisible to even the most sensitive optical/infrared observations. While at least some are likely to be high-redshift radio galaxies or quasars, it is also likely that the class may encompass several types of object. Other candidates for IFRS include heavily obscured AGN, stray radio lobes, radio relics and radio pulsars ([@naa+06; @Norris07b; @ga08]; Middelberg et al. 2008b). @Norris07b propose that IFRS may even represent a phase of AGN evolution.
For 16 of these sources the measured flux densities and spectral indices are consistent with those measured for pulsars in our own galaxy [@mhth05][^1] . As the ATLAS survey regions are well away from the Galactic plane, it is unlikely that all 16 of these sources are pulsars, but the possibility remains that some are. If so, then it is important to identify them so that (a) they can be removed from the extragalactic source statistics, and (b) so they can be accounted for in models of pulsar populations.
However, it is difficult to estimate how many pulsars would be expected in any given region of the sky. The most recent pulsar surveys have covered regions close to the Galactic plane (e.g., [@mlc+01]). These pulsar surveys have discovered a large number of new pulsars, but relatively few millisecond pulsars. The Galaxy is likely to contain a similar number of normal and millisecond pulsars [@lml+98], but the millisecond pulsars are likely to have travelled significant distances from the Galactic plane and are difficult to find due to their relatively low luminosities, their fast spin-rates and because they are commonly in binary systems, which disrupts the regular periodicity of their emitted pulses. The only deep 20-cm pulsar survey covering a large area that is far from the Galactic plane is the Parkes High Latitude Pulsar Survey, which covers Galactic longitudes $220^\circ < l < 260^\circ$ and latitudes $|b|<60^\circ$ [@bjd+06], with 6456 pointings of 265 seconds each. This survey detected 42 pulsars with a survey sensitivity of pulsed flux density $\sim0.5$ mJy. Even though this survey made relatively few discoveries, the pulsars detected away from the Galactic plane were of great astrophysical interest. They included the first double pulsar system PSR J0737$-$3039 [@lbk+04] and three other millisecond pulsars. @bjd+06 discussed the Galactic latitude distribution of their detections in detail, and suggested a roughly uniform distribution. However, this survey was relatively insensitive to millisecond pulsars, because of the restricted sampling time and number of frequency channels, and so the total number of millisecond pulsars detected was small. Hence, it is currently impossible to make a reasonable estimate of the number of pulsars likely to be detected per square degree at high Galactic latitude. With a survey sensitive to millisecond pulsars it seems reasonable that at least one detectable pulsar exists in the ATLAS survey region of roughly seven square degrees, but because this number is so poorly known, it adds a further motivation for this study.
It should be noted that detections and discoveries of pulsars in radio continuum data are not without precedent. Examples include PSR B1937$+$21, the first millisecond pulsar [@bkh+82], PSR B1821$-$24, found during a VLA imaging search of globular clusters [@hhb+85], PSR B1951$+$32, initially identified as a steep-spectrum polarised point source central to the supernova remnant CTB 80 [@Strom+87], and PSR J0218$+$4232, originally uncovered by the Westerbork Synthesis Radio Telescope (WSRT) [@ndf+95].
Observations
============
[lllll]{}Source & Right ascension & Declination & 20cm Flux density & Spectral index\
& (hms) & (dms) & (mJy) &\
ES011 & 00:32:7.444 & $-$44:39:57.8 & 9.5 & $-$1.44\
ES318 & 00:37:05.54 & $-$44:07:33.7 & 2.0 & $-$0.78\
ES419 & 00:33:22.80 & $-$43:59:15.4 & 4.4 & $-$1.35\
ES427 & 00:34:11.59 & $-$43:58:17.0 & 22.3 & $-$1.08\
ES509 & 00:31:38.63 & $-$43:52:20.8 & 22.7 & $-$1.02\
ES749 & 00:29:05.23 & $-$43:34:03.9 & 10.3 & $-$1.08\
ES798 & 00:39:07.93 & $-$43:32:05.8 & 11.7 & $-$0.81\
ES973 & 00:38:44.14 & $-$43:19:20.4 & 11.6 & $-$1.15\
ES1259 & 00:38:27.171 & $-$42:51:33.8 &4.5\
\
CS114 & 03:27:59.89 & $-$27:55:54.7 & 7.7 & $-$1.34\
CS164 & 03:29:00.20 & $-$27:37:54.8 & 1.6 & $-$0.92\
CS215 & 03:29:50.02 & $-$27:31:52.6 &1.6 & $-$0.76\
CS241 & 03:30:10.22 & $-$28:26:53.0 & 1.0 & $-$1.96\
CS255 & 03:30:24.08 & $-$27:56:58.7 & 0.5\
CS415 & 03:32:13.97 & $-$27:43:51.1 & 2.6 & $-$2.38\
CS538 & 03:33:30.20 & $-$28:35:11.2 & 2.0 & $-$0.88\
We selected 16 of the IFRS based on their flux density and spectral index. The sources range in flux density from 0.5mJy to 23mJy and possess spectral indices from $-$2.4 to $-$0.2. An explanation of the calculation and derivation of these values, including the problems associated with resolution and scintillation, may be found in Middelberg et al. (2011), from which the IFRS values were taken. These parameters are certainly similar to those of the known pulsar population, which have flux densities that range from 0.01 to 1100mJy at an observing frequency close to 1400MHz and have known spectral indices ranging between $-3.5$ and $+0.9$ with a median of $-1.7$ [@mhth05]. It should be noted that the lower value of the pulsar flux density range is not due to any physical characteristic of pulsars, but is instead due to the physical limitations in the sensitivity of our instrumentation.
In Table \[tb:sources\] we tabulate the source names, positions, flux density and spectral index of the IFRS. For each source we carried out a 35 minute observation using the Parkes 64-m radio telescope equipped with the 20-cm 13-multibeam receiver (although only the central beam was utilised for our analysis), with 340MHz of bandwidth centred on 1352MHz. We used the Berkeley Parkes Swinburne Recorder (BPSR), a high resolution digital filterbank, providing 870 frequency channels, 2-bit samples every 64 $\mu$s. This observing setup is almost identical to the High Time Resolution Universe Pulsar Survey project [@kjv+10], using a similar bandwidth, observing frequency backend system and sampling rate, with high sensitivity to millisecond pulsars. The data were recorded to magnetic tape for off-line processing. The data were processed using the [Hitrun]{} pipeline (see [@kjv+10] for details) to search for periodic signals in dedispersed time series with trial dispersion measures (DMs) in the range of 0 to 1000cm$^{-3}$pc.
We determine the theoretical sensitivity of our observations using the radiometer equation, which gives the fundamental limiting flux density of the central beam, $$S_{min}=\frac{{\sigma}(T_{sys}+T_{sky})}{G\sqrt{2B{\tau}_{obs}}}\sqrt{\frac{W}{1-W}}.$$ In this equation, $\sigma$ is the cutoff S/N for a positive detection, $T_{sys}$ is the system noise temperature, $T_{sky}$ is the sky noise temperature, $G$ is the system gain, $B$ is the observing bandwidth, $\tau_{obs}$ is the integration time of the observations and $W$ is the fractional pulse width. The typical value of $W$ for pulsars is approximately 10 percent of the pulse period, although it can be as high as 50 percent. Scaling our parameters from those of @kjv+10, setting $\sigma=8$, $T_{sys}=23$ K, $T_sky=0.7$ K and $G=0.735$, we arrive at a sensitivity to pulsed emission of $S_{min}=0.07$ mJy for a 10 percent pulse duty cycle, or $S_{min}=0.21$ mJy for a 50 percent pulse duty cycle. As all of our chosen IFRS candidates possess continuum flux densities above the higher of the two flux density limits, if any of them are indeed pulsars they should be easily detectable by our observations at the $8\sigma$ confidence level.
Results and discussion
======================
The folded pulse profiles (intensity as a function of pulse phase) were viewed by eye for all candidates with a signal-to-noise ratio above $8\sigma$. The observed pulse profiles for each candidate and their pulse periods were clearly caused by radio frequency interference or processing artefacts. For example, one candidate with a signal-to-noise ratio of $13 \sigma$ was detected with a frequency of 50.01 Hz and a sinusoidal pulse profile, indicating RFI originating with the mains power supply. We note that the minimum flux density of the chosen IFRS is 0.5 mJy, and the median flux density of the IFRS is approximately 4.4 mJy. If these flux densities are due to pulsars, then with a duty cycle of $W=0.5$ and an $8\sigma$ detection level of 0.21 mJy, the faintest of the IFRS should have been detected at a confidence level of $19\sigma$ with most of the sources ruled out by more than $100\sigma$. We conclude that none of the IFRS observed in this project show emission consistent with being a radio pulsar.
The absence of pulsar detections in this project emphasises the difficulty of identifying pulsar candidates through continuum surveys. Despite the four detections described in Section \[sec:Intro\], the vast majority of continuum (i.e., low time resolution) pulsar surveys have found no positive results (e.g. [@kcc+00] and [@kcb+00]). Also, while our analysis attempted to take advantage of the typically steep values of pulsar spectral indices in selecting our targets, the principle means of searching for pulsar candidates in continuum surveys has been to identify sources with significant linear and/or circular polarisation [@hmxq98; @hdv+09]. Further studies using these selection criteria may yield better results.
Finally, the Evolutionary Map of the Universe (EMU) survey[^2], to be carried out using the Australian Square Kilometre Array Pathfinder (ASKAP), is likely to detect 70 million radio sources with a sensitivity limit of $\sim 10 \mu$Jy. Over a million of these sources are likely to be currently undetected IFRS. However, at this point in time, we cannot make a prediction as to the number of these IFRS which may be pulsars, or indeed the fraction of pulsars expected in the total EMU source count. These values, as well as the techniques which may be used to more accurately isolate candidate pulsars from large-scale continuum surveys such as EMU, are the subject of ongoing research and are to be published in later work.
Conclusions
===========
We have searched for short-term radio pulsations originating from 16 of the ATLAS infrared faint sources and find that pulsed emission cannot account for the observed radio flux density. Since the sensitivity limit of our observations is well below the observed flux densities of the chosen IFRS, it is unlikely that any of these enigmatic sources are simply close-by pulsars in our own Galaxy. Hence, the nature of these sources is yet to be determined.
Acknowledgements {#acknowledgements .unnumbered}
================
GH is the recipient of an Australian Research Council QEII Fellowship (project \#DP0878388). The Parkes telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
[^1]: http://www.atnf.csiro.au/research/pulsar/psrcat/
[^2]: http://www.atnf.csiro.au/people/rnorris/emu/
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